Jagiellonian University, Krakw Faculty of Physics, Astronomy and Applied Informatics PhD dissertation Physical sciences Physics Symmetry and Classifcation of Multipartite Entangled States Adam Burchardt Supervisor Prof. dr hab. Karol łyczkowski Krakw, September 2021 Research supported by Narodowe Centrum Nauki Grant number DEC-2015/18/A ST2/00274 cioci Basi Separable states are all alike; every entangled state is entangled in its own way. — LewTolstoy, Anna Karenina(Paraphrasis) Iam immensely gratefultoallpeoplewhohavebeeninmylifeforthelast threeyears. Firstofall,tomysupervisor, Karolłyczkowski,whointroduced metotheworldofquantuminformation,whichIlikedsomuch. Thankyou for the scientifc problemsyou presented to me, for the adviceyou gave me, and for the timeyou devoted to me. Thankyouto all colleagues and collaborators with whomI have had the opportunity towork, talk, andchatover the pastyears: Zahra Raissi, Gonçalo Quinta, Rui André, Jak Czartowski, Marcin Rudzi«ski, Grzegorz Rajchel-Mieldzio¢, Wojtek Bruzda, Suhail Ahmad Rather, Arul Lakshminarayan, BaichuYu,Pooja Jayachandran,Valerio Scarani,Yu Cai, Nicolas Brunner, Paweł Mazurek, Máté Farkas, Jerzy Paczos, Marcin Wierzbi«ski, Waldemar Kłobus, MahaswetaPandit,Tamás Vértesi, Wiesław Laskowski, Adrian Kołodziejski, Felix Huber, Markus Grassl, Dardo Goyeneche, M¯aris Ozolos, Timo Simnacher, AntonioAcín, Albert Rico Andrés, Gerard Angles Munne, KamilKorzekwa, Oliver Reardon-Smith, Roberto Salazar, Alexssandre de Oliveira, Michał Eckstein, Simon Burton, John Martin, DavidLyons, and Fereshte Shahbeigi, among many others. I express my special thanks to two of mycolleagues: Konrad Szyma«ski and Stanisław Czachski, who greatly helped me in the last days of preparing this dissertation. I would like to thank Agata Hadasz, Agnieszka Hakenszmidt and Agnieszka Golakfor all their help duringmy stayin Krakow. Allofthiswouldnothavebeenpossiblewithoutthehelpofmyfamily, friends, andbeloved. Abstract One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantifcation of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classifcation of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation aswell as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states andinvestigate their entanglementproperties. Thirdly, we studythe well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new statesbelonging to this family, for instance an AME state of4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approachfor the general problem of verifcation of the equivalencebetween anypair of arbitrary quantum states based ona singlepolynomial entanglement measure. Streszczenie Zjawisko spl¡tania kwantowego jest jednym z kluczowych przejaww mechaniki kwantowej. Jakkolwiek spl¡tanie kwantowe dwucz¡stkowychsystemw jest dobrzepoznanym zagadnieniem, nasze zrozumienie spl¡tania w systemach wielocz¦±ciowych jest nadal niezadowalaj¡ce. Niniejsza rozprawa dotyczy rnych aspektw kwantyfkacji spl¡tania w stanach wielocz¡stkowych oraz roli symetrii w takichukładach. Po pierwsze, prezentujemyzwi¡zek mi¦dzy klasyfkacj¡ wielocz¡stkowego spl¡taniaa teori¡w¦złwibadamy rodzin¦ stanw,w ktrych splatanie jestodporne na utrat¦poszczegnych cz¡stek. Konstruujemy takie stany wykorzystuj¡c reprezentacj¦ Majorany, a tak»e niekte metody kombinatoryczne. Po drugie, wprowadzamy klasy stanw wysoce symetrycznych, ale niewpełni symetrycznychibadamywłasno±ci ich spl¡tania. Po trzecie, badamy szeroko dyskutowan¡ klas¦ stanw AME (ang. Absolutely Maximally Entangled). Z jednej stronypokazujemy konstrukcj¦ nowych stanw nale»¡cych do tej rodziny, ze stanem AME dla czterechpodukładow,ka»dyz sze±ciomapoziomami,z drugiej za± zajmujemy si¦ problemem rwnowa»no±ci takich stanw. Na koniec przedstawiamy nowatorskiepodej±ciedo ognegoproblemuweryfkacjirwnowa»no±ci dwchdowolnychstanwkwantowych w oparciuopojedyncz¡ miar¦ spl¡tania. This PhD Dissertation is based on the following publications and preprints available online [A] G. M. Quinta, R. André, A. Burchardt, and K. łyczkowski Cutresistant links and multipartite entanglement resistant toparticle loss, Phys. Rev. A 100, 062329 (2019). [B] A. Burchardt, Z. Raissi, Stochastic local operations with classical communication of absolutely maximally entangled states, Phys. Rev. A 102,022413 (2020). [C] A. Burchardt, J. Czartowski, and K. łyczkowski, Entanglement in highly symmetric multipartite quantum states, Phys. Rev. A 104, 022426 (2021). [D] S. Rather∗ , A. Burchardt∗, W. Bruzda, G. Rajchel-Mieldzio¢, A. Lakshminarayan, K. łyczkowski, Thirty-six entangledoÿcers of Euler: Quantum solution to a classically impossible problem, ArXiv: 2104.05122 (2021). ∗Contributed equally [E] A. Burchardt, G. M. Quinta, R. André, Entanglement Classifcation via Single Entanglement Measure, ArXiv: 2106.00850 (2021). Although publications and preprints mentioned above do notbelong to the mainbodyofthis thesis,we attach themas appendices for the convenience of the reader. List of Abbreviations AME Absolutely MaximallyEntangled LM Local Monomial LU Local Unitary MOLS Mutually Orthogonal Latin Square OA Orthogonal Array OLS Orthogonal Latin Square QECC Quantum Error Correction Code SL Special linear SLIP SL-symmetric InvariantPolynomial SLOCC Stochastic Local Operations with Classical Communication Contents 1 Introduction 7 2 Quantum states and topological links 18 2.1 Motivation............................. 18 2.2 Linksandquantum states .................... 19 2.3 In search for m-resistant states of N-qubit system . . . . . . . 20 2.4 In search for m-resistant states of N-qudit system . . . . . . . 25 2.5 Asymptotic case ......................... 27 2.6 Conclusions............................ 29 3 Highly symmetric states and groups 30 3.1 Motivation............................. 31 3.2 Groupof symmetryofa quantum state . . . . . . . . . . . . . 32 3.3 Symmetric states related to(hyper)graphs . . . . . . . . . . . 35 3.4 Entanglementproperties ..................... 36 3.5 ExamplesofDicke-like states .................. 39 3.6 Quantum circuits ......................... 44 3.7 Hamiltonians ........................... 47 3.8 Conclusions............................ 51 4 Absolutely maximally entangled states 53 4.1 AMEand k-uniform states .................... 53 4.2 Orthogonal arrays ........................ 55 4.3 Di˙erent linear structures .................... 58 4.4 Conclusions............................ 59 5 Thirty-six entangled oÿcers of Euler 61 5.1 Thirty-sixoÿcersofEuler.................... 61 5.2 AME stateoffourquhex..................... 63 5.3 Structureofthe AME(4,6) state ................ 64 5.4Verifcationof 2-unitarity .................... 67 5.5 Conclusions............................ 71 6 Classifcation of absolutely maximally entangled states 72 6.1 Local equivalence of AME and k-uniform states . . . . . . . . 72 6.2 Localequivalences, case2kmentbecamea milestone forthe developmentof quantum mechanics and our understandingoftheworld. Moreover,quantumentanglementhasbecome the heart of a completely new and dynamically developing feld of science lyingintheintersectionofQuantumPhysicsandInformation Theory:Quantum Information Theory. Qubits. A central notion in Quantum Information Theory is qubit, an abstract term, distilledfromvarious concretephysical realizations. Arguably, the simplest quantum-mechanical systemisatwo-level(ortwo-state) system. Such a systemmightbephysically realized asthespinofthe electron, which inagiven reference frameis eitherupordown,orasthepolarizationofa single photon, with distinguishedvertical and horizontalpolarization. In an abstract way, a pure qubit state is a coherent superposition of the aforementioned distinguished basis states |0i , |1i, i.e. a linear combination |ψi = α |0i + β |1i , |α|2 + |β|2 =1. (1.1) Such an object is nowerdays refered as qubit -quantum binary unit[Sch95], aquantum analogue of a classical bit. In a classical system, a bit is always in precisely one of two states, either 0 or 1. However, quantum mechanics allowsthequbittobeina coherentsuperpositionofboth statessimultaneously, a property that is fundamental to quantum mechanics and quantum computing. Two-qubit system. The most common variant of the EPR paradox (formulatedby Bohm[Boh51, BA57, RDB+09]) was expressed in terms of the quantum mechanical formulation of spin. In the more recentnotations, it mightbe explained asa particular stateofatwo-qubit system,i.e.a linear combination of four vectors |00i , |01i , |10i , |11i representing directions of both spins. So-called EPR state mightbe thus written as 1 |EPRi = √ (|00i + |11i). (1.2) 2 Suchstateis strongly correlatedinanyreference frame,inawaythat classical correlations do not allow. While studying quantum entanglement, it is useful to abstract from certain local properties of a state which do not a˙ect global entanglement and quantum correlations. An example of this type of abstraction is a Local Unitary (LU) equivalence of two given states. Having saythat, it is known that any two-qubit state is LU-equivalent to the following system √ |ψi = p |00i + p1 − p |11i , (1.3) 1 for some parameter p ∈ [0, ]. In particular, for p =0 the state is separable, 2 1 while for p = the state is maximally entangled, and coincides with the 2 famous EPR state. In such a way, we obtained a satisfactory quantifcation of entanglement in two-qubit systems. Indeed, the exact amount of entanglement is measuredby two relatedvalues (p, 1 − p), which are also known asSchmidtcoeÿcients[HW08]. The closerthevalues p and 1−p are to each other, the more entangled the state is. Furthermore, the number of non-vanishing values amongp, 1−p is known as Schmidt rank, which is equal to 1 for separable states and equal to 2 for entangled states. This results in the coarse-grained classifcation of entanglement intwo-qubit systems. Indeed, with respect to theSchmidt ranktwo classes of states are distinguished: separable and entangled. This straightforward division into two classes of bipartite states coincides with the division under another class of local operations: Stochastic Local Operations with Classical Communications (SLOCC)[DVC00]. SLOCC operations include not only local unitary rotations but also additions of ancillas (i.e. enlarging the Hilbert space), measurements, and throwing away parts of the system, eachperformed locally ona given subsystem[DVC00, BPR+00]. As it was shown, mathematically SLOCC operations are represented via invertiable operators [DVC00]. SLOCC operators cannot generate entanglementbetween subsystems, however, they might enhance or strengthen the existing entanglement with some non-vanishing probability of success. This is refected in the fact that there are only two states which are not equivalent with respect to SLOCCtransformations: separable |00i state and entangled |EPRi state. Animportant featureofboth typesoflocaloperations:LUand SLOCC are that each of them provides an equivalence relation on the state-space, and hence dividesitinto equivalence classes[Kra10a]. Any two statesfrom one class areinterconvertibleby anadequatelocaloperator,while such a transformation cannotbe provided for states from di˙erent classes. In that way,quantum states whichexhibit the same(forLU) or similar(for SLOCC) entanglement properties are grouped together, which gives the solution for the problem of entanglement classifcation of two-qubit systems. As we already discussed, the problem of entanglement classifcation of two-qubit states is well-understood. Indeed, there are infnitely many LU 1 equivalence classes, indexed by one real parameter p ∈ [0, ], see Eq.(1.3). 2 On the other hand, there is only one SLOCC-equivalence class of genuinely entangled states, which mightbe representedbytheEPR state. In 1997 Scott Hill and William K. Wootters introduced the notion of Concurrence C, an entanglement measure, which for any two-qubit pure state state |ψi = c00 |00i + c01 |01i + c10 |10i + c11 |11i reads C(|ψi) := 2|c00c11 − c01c10|∈ [0, 1] (1.4) as anabsolute valueof the degreetwopolynomialin the state-coeÿcients [HW97]. As it was observed, the value of concurrenceis not only independentofLUoperationperformed on anyqubit,but also on anylocalSpecial Linear operation (SL), i.e. local invertible matrices with determinant one. Anyinvertibleoperationmightbe presentedasSLoperationuptotheglobal constant. In that way, local SL operations are relevant to SLOCC operations on qubits. The notion of concurrencewas an important step towards a modern approach to the problem of entanglement quantifcation via SLpolynomial invariant (SLIP), measures. SLIP measures are at the same time polynomials in the state coeÿcients and invariantunder SLOCC operations [ES14], which signifcantly facilitates the problem of quantifyingentanglement resources. Three-qubit system. In the late 1980s Daniel Greenberger, Michael Horne and Anton Zeilinger studied three-qubit entangled state the tripartite generalisationofthe EPR pair[GHZ07]: 1 |GHZi = √ (|000i + |111i). 2 Bouwmeester et al. frstly presented the experimental observation of the stateaboveasapolarizationentanglementfor three spatially separatedpho-tons[BPD+99]. Since then, many other experimental realisations of GHZ statewere performed. From the theoreticalpointof view, there aretwo important features of GHZ state. On one hand it is maximally entangled, i.e each of its reductions to the one-particle subsystem is maximally mixed, whichis often a desiredproperty. On the other hand, however, state GHZ is not robust against particleloss. Indeed, all its reductions to thetwo-particle subsystem form an unentangled mixed state, in other words its two-particle correlations are of a classical nature. There isyet another particular three-partite entangled state 1 |Wi = √ (|001i + |010i + |100i), 3 introduced by in 2000 by Wolfgang D, Guifre Vidal, and Ignacio Cirac in their seminal paper “Three qubits can be entangled in two inequivalent ways”[DVC00]. Contrary to the GHZ state, theW state is robust against the particle loss, indeed, each of its two-particle subsystems exhibits quantum correlations. On the other hand, the one-particle subsystems are not maximally-mixed, as it was for GHZ state. As it was shown, the aforementioned GHZ and W states are the only two distinct SLOCC-non-equivalentstates of genuinely entangled three-qubit states [DVC00]. Furthermore, there are infnitely many LU-equivalence classesofgenuinelyentangledthree qubit states, parametrizedbythree real parameters[DVC00, Sud01]. In 2000ValerieCo˙man, Joydip Kundu, and William K.Wootters introduced the frst entanglement measure related to the 3-body quantum τ(3) correlations in the system: the three-tangle [CKW00]. Three-tangle distinguishes between GHZ and W states and achieves two extreme values τ(3)(GHZ)=1 and τ(3)(W)=0 on both states. While the GHZ state exhibits maximal 3-body and vanishing 2-body quantum correlations, the quantum correlations in the W state are reversed. As it was shown, the three-tangleisa SLIP measure, henceinvariantunder SLOCCoperations. In that way, the three-tangle provides a satisfactory method for distinguishing between di˙erent SLOCC classes of genuinely entangled three-qubit states. Figure 1.1:Two distinct three-partite qubit states: GHZ andWrepresented in the form of links. On the left Borromean rings, characterized by the fact thatif oneringis cut, the remainingtwobecome disconnected. This resembles entanglementin GHZ state, which is not robust against particleloss.Ontherightanotherlinkof three knots,forwhichcutofanycomponent does not disconects the link itself. This is relevant to the robustness of the W state. Four-qubit system. If the discussion of entangled in three-qubit systems did not convince the reader that entanglement in the multipartite system has a rich and complex form, we shall proceed to the four qubit states. Asweshall see,a systemof four qubitsis alreadylarge enoughto revealtwo substantial problems concerning multipartite-entanglement: the existence of maximally entangled states and the classifcation of entanglement. In the following,I discuss briefyboth problems. Atsushi Higuchi and Anthony Sudbery in their seminal paper “How entangled can two couples get?” investigate entanglement in system of four qubits concerning bipartitions of the system [HS00]. In particular they showed that there is no four-partite qubit system, which is maximally entangled with respect to any of bipartitions for two-partite systems: 12|34, 13|24, and 14|23. Furthermore, they showed that the following state 1 2πi 4πi |M4i = √ |0011i+ |1100i+ e 3 (|1010i+ |0101i)+ e 3 (|1001i+ |0110i) , 6 (1.5) maximize the average entropy of entanglement for such bipartitions. As it was later observed, state |M4i, as well its conjugate transpose exhibits intriguing symmetric properties[LW11, CLSW10]. Contrary to tripartite states |GHZi and |Wi, state |M4i is not fullypermutation-invariant. Nevertheless state |M4i isinvariant underany evenpermutationof qubits. Such states for which anyelement of an alternating subgroup of thepermutation group, A4 lem of classifcation of entanglement in four-qubit states arouses great interest[VDDMV01, CD07, CW07, LLHL09, VES11, GW13, SdVK16, GGM18]. The four-qubit states were successfully divided into nine families, most of which contain an infnite number of SLOCC-classes [VDDMV01, CD07, SdVK16]. In particular, the so called Gabcd family with the most degrees of freedom is representedbythe following highly-symmetric states a + da − d |Gabcdi = �|0000i + |1111i + �|0011i + |1100i  22 b + cb − c + �|0101i + |1010i + �|0110i + |1001i . 22 Furthermore, non-SLOCC-equivalent four qubit states canbe e˙ectively distinguishedbythree independent SLIP measures [VES11,LT03]. General N-qubit system. Quantum states with non-equivalent entanglementrepresent distinct resources and hence maybe useful for di˙erent protocols. The idea of clustering states into classes exhibiting di˙erent qualities underquantum information processing tasks resulted in their classifcation under SLOCC. Such a classifcation was successfully presented for two, three and four qubits[DVC00, VDDMV01, GW13, LLHL09]. However, the full classifcation of larger systems is completely unkown. Even the much simpler problem of detecting if two N-qubit states (N> 4)are SLOCCequivalent is, in general, quite demanding[VES11, ZZH16, GW11, BR20, SHKS20]. It was shown that for multipartite N ≥ 4 systems there exist infnitely manySLOCC classes[DVC00]. In particular, the set of equivalence classes under SLOCC operations depends at least on 2(2N − 1) − 6N parameters, whichgrowsexponentiallywiththenumberof qubits[DVC00].Evenlarger number of LU-equivalence classes was also investigated[Kra10b, GRB97, dVCSK12]. So far, great e˙ort has been made to reduce this verifcation problem to the problem of computing entanglement measures, which by defnition take the same value for the equivalent states[LT03, GT09, DO09, VES11, LL13, Sza12, JYKH19, HLT13, HLT16]. Nevertheless, starting from four
partite states, one needs to compute values of at least three independent measures, to decide with certainty on local equivalence. Furthermore, the number of independent measures grows exponentially with the number of qubits[DVC00], making it intractable to use thisapproach to discriminate locally equivalent states[LBS+07]. In fact, this rapid increasing diÿcultyto verify the entanglement equivalence between two states is common to any procedure[LBS+07]. Qutrits andbeyond. So-far, in our study of entanglement we focused onthemultipartitequbit systems,consistingoftwo-level subsystems. These theoretical considerations go hand in hand with experiments, since manipulationandrelative controlover several qubitshave alreadybecomea standard task[MSB+11, RGF+17]. Even though two-level subsystems are the most common multipartite states, they are certainly not the most general from botha theoreticalandexperimentalpointof view. Recently,muchattention hasbeen paid to the qutrites. Qutrits are realizedby a 3-level quantum system, that maybeina superpositionof threemutually orthogonal quantum states[NJDH+13]. SimilarlytoEq.(1.1),a qutrit statemightbe writtenin the form |ψi = α |0i + β |1i + γ |2i , |α|2 + |β|2 + |γ|2 =1. Although precise manipulationof such states turnedouttobea demanding task, there are some successful indirect methodsfor it[LWL+08]. Further
more, there is an ongoingdevelopment of quantum computers using qutrits and qubits withmultiple states[NJDH+13]. Overall, there is no reason to restrict the number of levels of each subsysteminthe generalconsiderationsofmultipartitequantum states[WHSK20]. There are several advantages of using larger dimensional subsystems in quantum computations, including the increasing the variety of quantum gates available, applications to adiabatic quantum computing devices[ADS13, ZE12, PLX+08], or to topological quantum systems[CHW15, BCRS16]. In this thesis, beside of multipartite qubit systems, I consider homogeneous multipartite qudit systems, i.e. systems of N particles in which each particle has the same number of levels d (d =2 for qubits, d =3 for qutrits, etc.). The heterogenous multipartite systems are also being investigated[HESG18], and some e˙ort hasbeen made to classify them [Miy03, MV04, MW03]. Absolutely Maximally Entangled states. Besides the diÿcult problem of the entanglement classifcation in the N qudit system, we prompt a simpler question about which states represent maximum entanglement in such a system. This question is ambiguous, there is no unique way of generalizing the maximally entangled state of two-qubit into anyother system. In particular, according to di˙erent entanglement measures for multipartite states(like the tangle, the Schmidt measure, the localizable entanglement, or geometric measure of entanglement), the states with the largest entanglementdo notoverlapin general[Kra10a]. Oneof thepossibleinterpretations of states with largest entanglement, however, was signalized by the work of Higuchi and Sudbery[HS00]. It resulted in the well-established notion of Absolutely Maximally Entangled (AME) states[HCR+12]. AME states are those multipartite quantum states that carry absolute maximum entanglement for allpossible partitions. AMEstates arebeing appliedin several branches of quantum information theory: in quantum secret sharing protocols[HCR+12], in parallel open-destination teleportation [HC13], in holographic quantum error correcting codes[PYHP15], among manyothers. Di˙erent families of AME states havebeen introduced[Rai99, Hel13] and the problemof their existenceisbeinginvestigated[Sco04, HS00, HGS17]. It hasbeen demonstrated that the simplest class of AME states, namely AME states with the minimal support, is in one-to-one correspondence with the classical error correction codes[RGRA17]and combinatorial designs known as Orthogonal Arrays [GRDMZ18]. Henceforward, the two-wayinteraction with combinatorialdesigns and quantum error correction codes is observed [Sco04, GZ14]. AME statesare special cases of k-uniform states characterizedbythe propertythatallof their reductionstok parties aremaximally mixed[AC13]. Structure and aims of the thesis. As we have demonstrated so far, the quantifcation and classifcation of entanglement for multipartite states is an involving long-distance project. Characterization of di˙erent classes of entanglement in multipartite quantum systems remains a major issue relevant for various quantum information tasks and is interesting from the point of view of foundations of quantum theory. In this dissertation, we will show the progress in several directions of these complex issues. The main goals and results of the thesis are presented in the six next chapters of the dissertation. We briefy present the scope of the proceeding chapters and indicate the main resultsbelow. If notspecifed di˙erently, the author’s contributions to thework coveredbythischapterwere signifcant. Chapter2 constitutesa linkbetween the classifcationofmultipartite entanglementand knot theory. We introduce the notion of m-resistant states, states whichremains entangled after losing an arbitrary subset of m particles, butbecomes separable afterlosingany numberofparticleslargerthan m.I construct several families of N-particle states with the desired property of entanglement resistance to particle loss. Chapter3 discusses highly-symmetric states and their properties. Inspiredby remarkable entanglement propertiesofa state |M4i, I introduce the notion of G-symmetric states of N-qubits, for anysubgroup of thepermutation group symmetric G 4)systems, for which the classifcation is completely unknown. In addition, I have developed a novel method to obtain the normal form of a 4-qubit state which bypasses the possibly infnite iterative procedure. Chapter8 summarizes resultsobtained in the dissertation and outlines the open problems andperspectives for further research. Appendices contain original research papers, that do not belong to the dissertation, but havebeen included for the convenience of the reader. Chapter 2 Quantum states and topological links In thischapter,Iintroduce the notionof m-resistant states. We saythat an entangled quantum state of N subsystems is m-resistant if and only if it remains entangled after losing an arbitrary subset of m particles, andbecomes separable after losing any number of particles larger than m. Furthermore,I establishan analogytothe problemofdesigningatopologicallink consisting of N rings characterized by the fact that by cutting any (m + 1) of them, the remaining linkbecomes unknotted. This analogy allows us to exhibit several N-particle states with the desired property of entanglement resistance to particle loss. An extension of some parts of this chapter, in which the author’s contributionwasnot substantial canbe foundinthejoint work [QABZ19], attachedto the dissertation in AppendixA. 2.1 Motivation An e˙ort to study multipartite entanglement usingthe knot theory was proposed by Aravind, who related some N-partite quantum states with a link composed of N closed rings[Ara97]. The |GHZi state,whichbecomes separable after any measurement in the computational basisperformed on its subsystem, was an initial inspiration. In such a way |GHZi state can be associated with a particular confguration of three rings, called Borromean rings, see Figure 1.1. Indeed, Borromean link is characterized by the fact that if one ring is cut, the remaining two become disconnected. Usage of othertoolsborrowedfrom knot theoryto analyzemultipartiteentanglement was further advocated in[KL04, KM19]. The aforementioned analogy is, however, not basis independent. On the one hand, measuring the |GHZi state in the computational basis collapses always in a separable state, on the other hand, measurement in the rotated basis |+i , |−i basis results into an entangled state. This motivated Sugita[Sug07] to revise the above analogyby proposing the partial trace of a subsystem as an alternative interpretation for cutting related ring. In thatway, thephysical process corresponding to cutting the ringbecomes basis independent[MSS13]. This analogybetween quantum entanglement and linkedrings,was later explored[QA18], and willbe used here. Partial traceovergiven subsystemscanbephysicallyinterpretedasnotregistering themby a measurement device or asthe lossof related particles. Studying quantum entanglement resistant to a particle loss thus corresponds to the question, whether a given reduced density matrix represents an entangled state[NMB18, OB18, KNM02, KZM02]. 2.2 Links and quantum states Recently, Neven et al. introduced the notion of m-resistant quantum Nqubit states which entanglement of the reduced state of N − m subsystems isfragilewithrespecttolossofanyadditional subsystem[NMB18]. Defnition 1. An entangled state |ψi of N parties is called m-resistant if: • |ψi remains entangled as any m of its N subsystems are traced away; • |ψi becomes separableifa partial traceisperformedoveran arbitrary set of m +1 subsystems. The frst non-trivial examples of m-resistance arise for N =3 qubits, whichis simple enough to study on intuitive grounds. The m =0 resistance isexemplifedbythe GHZ state 1 |GHZi = √ (|000i + |111i) , (2.1) 2 since tracing out any party of GHZ state results in a separable state. The correspondinglink has thepropertiesofthewell-know Borromean link, depicted in Figure 1.1. In particular, if any ring in Borromean link is cut, the remainingtwo ringsbecome separated. Such a connectionbetweenBorromean link and the GHZstatewas already noticedbyAravind[Ara97].For arbitrary number N of rings, the natural generalization of Borromean link is called Brunnian link[Bru92]. Brunnian linkischaracterizedbythe same propertythat cut of anyofits rings disconnect the link, see Figure 2.3. On the other hand, the three-qubit state[DVC00], 1 |Wi = √ (|100i + |010i + |001i) (2.2) 3 possesses propertyofbeingm =1 resistant. Indeed, a partial trace over any of its subsystems will result in an entangled mixed state. A corresponding link which mimics this property is represented on Figure 1.1. It was frst considered asa representationof theW statein[Ara97]. 2.3 In search for m-resistant states of N-qubit system In[QABZ19], we presented the general form of the symmetric mixed state of N qubits with the propertyofbeing m resistant. This form was obtained by relating to a given m-resistant link to the associated quantum state. Unlike the case of mixed states, similar identifcationbetween links and pure states is more intricate, if at all possible. It seems that the problem of fnding pure m-resistant states for a given m and the arbitrary number N of qubits is not trivial. Obviously, the desired propertyis fully symmetric with respect to the exchange of subsystems. Therefore a search for such states among symmetric states seems to be a reasonable approach. Below, we presentthepartialsolutionto this problem, whichis basedontheMajorana representation[AMM10, MGB+10]. The stellar representation of Majorana[MM10, AMM10, BZ17] provides an alternative intuition on the geometry of symmetric states. Any permutation-invariant state of N qubits might be wtitten in the following form 1 |ψi = √ X |η1ii1 ... |ηN iiN (2.3) K σ∈SN where K isa suitable normalization constant, the sum runsover allpermutations σ ∈ SN of particles indices i1,i2,...,iN , and θj θj  |ηji= cos |0iX + e iφj sin |1iX . (2.4) X 22 The pairs of angles (θj,φj) represent apoint on the sphere, and are called Majorana stars. Thus, one may defne a fully symmetric N-qubit state by fxing N points on the sphere. For this reason, the name stellar representationis alsocommon,as eachpoint representsa starinthesky,whilea group of stars forms a constellation. Among manyadvantages, the Majorana representation ascribes some geometricalintuitionof entanglementofa symmetric state. If onechooses all of the stars at a singlepoint the corresponding state is separable. For example, N degenerated stars on the Northpole represent the separable state |0 ··· 0i. Nevertheless, as the degeneracy is lifted, most of the non-trivial constellation of stars corresponds to an entangled state of N qubits. Furthermore,the distancebetweenthe starsisrelatedtothe degreeofentanglement [ZS01a, GKZ12], although the precise criterium to quantify entanglement is not uniquely defned. We introduce the following family of states: 1 sN ! |ψN i = |0i⊗N − (−1)N+m |DN i (2.5) mm m q1+ �N  m where |DN i are so-called Dicke states [Dic54]: m  |Dm X σ|1 ··· 10 ··· 0 i , (2.6) N i∝ σ∈SN |{z} |{z} k times N−m times with the summation goingover allpermutations σ. This family of states was motivatedbyinvatigation of N =3, 4 qubit states. Indeed, for N =3, 4 and any non-trivial value of m, the state |ψN i provides example of m-resistant m state. In particular in the case of N =3 qubits presented constellations give us m =0, and m =1 resistant states respectively: 1 |ψ03 i = √ (|000i + |111i) , (2.7) 2 1 |ψ13 i = √ (3 |000i + |011i + |101i + |110i) . (2.8) 12 It is straightforward to show that the constellations presented on Figure 2.1 give the desired answer for both non-trivial values of m. Similarly, states |ψ4 i are m =0, 1, 2 resistant states of four qubits for respective values of m m. Related constellations are presented on Figure 2.2. Note that |ψN i may m be directly related to the distribution ofm stars on the Northpole, with all others evenly distributed along the equator. (a) 0-resistant state (b) 1-resistant state Figure 2.1: Constellations defning m-resistantstatesof3qubits. The middle arrow servesasa referencepointingtothe Northpole. The constellationin (a) corresponds to the state |GHZi and is illustratedbythe Borromean link, see Figure 1.1, while (b) presents an exemplary 1-resistant state of three qubits. (a) 0-resistant state (b) 1-resistant state (c) 2-resistant state Figure 2.2: Constellations defning m-resistant states of 4 qubits for m = 0, 1, 2. The middle arrow serves asa referencepointing totheNorthpole. Presented constellations are related to three links presented on Figure 2.3. The agreement between the above construction and m-resistant states breaks down for N =5 parties. Indeed, the state |ψ5 i = is in fact 2 1 resistant. In all other cases,i.e. m =0, 2 and 3, related pure states are 0, 2, and 3-resistant respectively. One may naturally inquire whether another symmetric state of fve qubits might be in fact 1-resistant. We have tested all combinations wheretwo stars are liftedby ageneral latitude, aswell as four stars, and it ispossible to show that no 1-resistant state exists for these families of states. Obviously, this does not prove that a pure 1-resistant state of 5 qubits does not exist, although it is tantalizing to conjecture so. Ingeneral,wehaveverifed the following. Figure 2.3: On the left, an example of a 0-resistant link of 4 rings, which is a 4-component Brunnian link. Notice that after cutting a single ring all three remaining ringsbecome disconnected. In the center, an exampleof a 1-resistant link of 4 rings. After cutting any single ring one arrives at the Borromean link. On the right, an example of a 2-resistant link of 4 rings. After cutting any two rings the remaining two are still connected. Proposition 1. For any number of particles N ≥ 3, the family of states Eq. (2.5): 1 sN ! |ψN i = |0i⊗N − (−1)N+m |DN i (2.9) mm m q1+ �N  m provides examples of m =0, (N −3), and (N −2)-resistant states respectively. Proof. Observe that |ψN i is zero resistant. Indeed,by(2.5),wehave 0 1 |ψ0 N i = |0i⊗N − (−1)N |1i⊗N  . (2.10) 2 This state is a generalization of the GHZ state for N qubits, which, after anypartial trace, returns a densitymatrix of the form 1 trk ρˆN = |0ih0|⊗(N−k) + |1ih1|⊗(N−k) , (2.11) 0 2 where trk denotes the trace over anyset consisting of k qubits. This density matrixis separable for any k> 0, and thus the state |ψN i is 0-resistant for 0 any N. Regarding the state |ψN i, after simple algebraic operations, one fnds N−2 that any N − 2 partial traces will result in the densitymatrix proportional to ⎛ α22 +(α02)2 00 −α02 ⎞ 0 α21 α21 0 trN−2 ρˆN ∝ , (2.12) N−2⎜ 0 α21 α21 0 ⎟ ⎝⎠ −α02 00 1 where we defne αij ≡ �N−i . The partially transposed matrix has one of its j eigenvalues equal to λ1 = −N2 +3N − 4 (2.13) which is negative for any N. Thus,bythepositive partial transpose (PPT) test, we confrm that each 2-qubit reduced density matrix is always entangled, and the state |ψN i is (N − 2)-resistant for any N. N−2 Finally, focusing on the state |ψN i,weperform thepartial traceover N−3 any set of N − 2 qubits and obtain a two-qubit state ⎛ ⎜⎜ ⎞ ⎟⎟ α23 +(α03)2 000 0 α22 α22 0 0 α22 α22 0 0 0 0 α21 trN−2 ρˆN ∝ N−3 . (2.14) ⎝ ⎠ Thepartially transposed matrix of size four has allpositive eigenvalues for N> 1. In this case the PPT test guarantees that the resulting state is always separable. We must then look into the reduced densitymatrix with one less partial trace, in order to check if it is entangled. Such a 3-qubit densitymatrix is proportional to ⎛ ⎜ ⎜⎜ ⎜ ⎜ ⎞ ⎟ ⎟⎟ ⎟ ⎟ α33 + (α03)2 0 0 0 0 0 0 α03 0 α32 α32 0 α32 0 0 0 0 α32 α32 0 α32 0 0 0 trN−3 ρˆN ∝ N−3 0 00 α31 0 α31 α31 0 . 0 α32 α32 0 α32 0 0 0 0 0 0 α31 0 α31 α31 0 0 0 0 α31 0 α31 α31 0 α03 0 0 0 0 0 0 1 ⎜⎝ ⎟⎠ (2.15) By partially transposing anyqubit,we will obtaina matrix whichpossesses an eigenvalue equal to λ1 = 13 − 7N + N2 − p193 − 202N + 79N2 − 14N3 + N4 (2.16) which is negative for all N. PPT criterion implies that all the subsystems are entangled. This proves that the state |ψN i is (N − 3)-resistant. N−3 As a fnal remark to the section, we note that the constellations for all 0-resitant states follow the same rule, which is a regular N-sided poligon. This result hasbeen previously found in[NMB18]. 2.4 In search for m-resistant states of N-qudit system As we mentioned in Section 2.3, we did not fnd a general construction of an m-resistant pure qubit state of N qubit system. Hence, we have expanded our searchfor non-symmetric states with subsystems of a larger local dimension d ≥ 3. We present the general formula for a m-resistant N-qudit pure state for any N ≥ 2m. Our construction is based on a particular family of combinatorial designs and their connection to quantum states. In particular, we use the notion of orthogonal arrays (OA), and the established link[GZ14]betweenmultipartite quantum states andOA. This connection willbe expanded and further usein Chapter4 fordesigninganother family of quantum states. Orthogonal arrays[ASH99] are combinatorial arrangements, tables with entries satisfyinggiven orthogonal properties. A close connectionbetween OA and codes, entangled states, error-correcting codes, uniform states has been established[ASH99]. Therefore,investigation of the connectionsbetween OA and resistant states seems to be a natural approach. Firstly, we briefy present the concept of OA, secondly, we demonstrate relations betweenOA and resistant states. An orthogonal array OA(r, N, d, k) is a table composed by r rows, N columns with entries taken from 0,...,d − 1 in such a waythat eachsubset of k columns contains allpossible combination of symbols with the same amountofrepetitions. Thenumberof suchrepetitionsis called the index of theOAand denotedby λ. One mayobserve, that theindex ofOAis related to the other parameters: r λ = . (2.17) dk Figure2.4presents anexampleof anOA.Apure quantum state consistingof r terms mightbe associated withOA (r, N, d, k), simplybyreading all rows ofOA[GZ14, ASH99]. The state of N qudits associated with the orthogonal arrayOA (r, N, d, k) willbe denoted asby |φ(N,k)i . d The crucial quantity for our purpose, related to OA, is its index. It preserves the following information: how many repetitions of any sequence i1,...,ik there are for eachsubsystem of k rows. Forλ =1, anysequence appears only once, and suchan arrayis called index unity array. We emphasize their remarkable role in the search for resistantquantum states. Proposition 2. For any orthogonal array of index unity OA �dk, N, d, k, where N ≥ 2k, the relevant quantum state |(N, m)di is k − 1-resistant. 30 + |32130i 3 3 021 3 21 + |33021i 03 + |31203i 32130 3 12 + |30312i 31203 3 02 + |23102i 30312 3 13 + |22013i 23102 2 20 + |21320i 22013 2 31 + |20231i 21320 2 10 + |13210i 20231 2 01 + |12301i 13210 1 32 + |11032i 12301 1 23 + |10123i 11032 1 33 + |03333i 10123 1 22 + |02222i 03333 0 11 + |01111i 02222 0 00 |(5, 1)4i∝|00000i 01111 0 0 0000 0 Figure 2.4: Orthogonal array of unity index OA �42 , 5, 4, 2 obtained from the Reed-Solomon code of length 5 over Galois feld GF(4). Each subset consistingoftwo columns containsallpossible combinationofsymbols. Here, twosuchsubsetsarehighlighted.Therelevantquantumstateisobtainedby forming a superposition of states corresponding to consecutive rows of the array – see expression on the right-hand side. Fortheproofofabove statement,we referto[QABZ19].FromtheOA presented in Figure 2.4, we obtain, for example, the following fve-ququart, 1-resistant state: |(5, 1)4i ∝|00000i + |01111i + |02222i + |03333i + |10123i + |11032i + |12301i + |13210i + |20231i + |21320i + |22013i + |23102i + |30312i + |31203i + |32130i + |33021i . (2.18) Bushprovided the general method for constructingOAs of index unity [Bus52]. n Theorem 1 (Bush, 53’). If d is a prime power, i.e d = pfor some prime number p and natural number n, then we can construct the array OA �dk,d +1, d, k. Note that combining Bush’s result with Proposition2 provides the exis
tence of m-resistant N-qudit states for N ≥ 2(m + 1). Proposition 3. For any N ≥ 2(m + 1) there exists the N-qudit state which is m-resistant. The local dimension d is the smallest primepower larger than N − 1. Eq.(2.18)presents an example of a m-resistant qudit state obtained by reading consequtive roves of OA. A more interested reader might easily reproduce other resistant states withthehelpofavailableOAtables[Slo].We organize all obtained results in Table 2.1. They encourage us to pose the following conjecture. Conjecture 1. For anyN and m, there exists an m-resistant N-qudit state in some local dimension d. Furthermore, we leave as a list of related open problems. 1. Investigate whether there exist pure states of N qubits with the mresistance property, for any m =0, 1,...N − 2. 2. If such a state do not exists, for each N fnd the minimal local dimension d such that there exist m-resistant states of N qudits. 3. For anyclass of m-resistantstates of N qubits fnd a state for whichits average entanglementafter partial trace over anyset of m parties is the largest, if measured with respect to a given measure of entanglement. 2.5 Asymptotic case Consider a system consisting of N subsystems with d levels eachand assume that N is large. It is well known that a generic pure state of such a system istypically strongly entangled[ZS01b,HLW06]. Therefore the partial trace of the corresponding N-partyprojectoroverany choiceof subsystemstraced awaybecomes mixed.Typically, the more parties k are traced out, the more mixed is the state describing the remaining N − k subsystems. If the number k of subsystems removed is equal to N/2 the reduced state typically has a full rank so it can belong to the separable ball around the maximally mixed state[ZHSL98, GB02]. It was indeed observed[KNM02, KZM02,HLW06]that for k ≈ N/2 atransition from entangled reduced states tothe reductionswithpositivepartialtransposetakesplace.Furthermore,if m N 4 5 6 7 8 0 |ψ4 0i |ψ5 0i |ψ6 0i |ψ7 0i |ψ8 0i 1 |ψ4 1i |φ1,5i 4 |φ1,6i 5 |φ1,7i 7 |φ1,8i 7 2 |ψ4 2i |ψ5 2i AME |φ2,7i 7 |φ2,8i 7 3 |ψ5 3i |ψ6 3i |φ3,8i 7 4 |ψ6 4i |ψ7 4i 5 |ψ7 5i |ψ8 5i 6 |ψ8 6i Table 2.1: Collection of m-resistant states of N parties obtained so far. Three families: |ψ0 N i, |ψN i and |ψN i of m-resistant N-qubit states dis N−3N−2 cussed in Section 2.3 are presented on the di˙erently shaded blue background. Furthermore, m-resistant qudit states |φm,N i constructed by virtue of or d thogonal arrays are demonstrated on the green background. The number in subscript is relevant to the local dimension d of a state. Observe that the localdimension d of qudit states is usually(but not always) equal to N − 1. Finally, a six-qubit state AME(6,2) provides an example of a 2-resistant state. k ≈ 3N/5 the remaining subsystem consisting of 2N/5 particles is typically separable[ASY12, ASY14] and thistransitionbecomes sharp if N →∞. Thus one can expect that a generic pure state of a N–system state is mresistant with m typically of the order of 3N/5 independently of the local dimension d. 2.6 Conclusions Inthischapter,Ipresentedan analogybetweenquantum statesandtopologicallinks,inwhichtheoperationoflosinga subsystemis relatedtoneglecting an associated ring.Furthermore,Iintroduced the notionof m-resistant states, which entanglement is resistant forloss of any m particles but fragile for loss of any larger subsystem (Defnition 1). I presented two methods of constructing m-resistant states: Section 2.3 is based on the Majorana repre
sentation of symmetric states, while Section 2.4 on the combinatorial notion of Orthogonal Arrays. Related results are summarized in Proposition1and Propositions 2 and 3 respectively. Table 2.1 presents both aforementioned families of states at glance. Finally, Section 2.5 discusses the typical resistance propertyof large systems. Chapter 3 Highly symmetric states and groups In this chapter, I present an approach for constructing highly-entangled quantum states which is based on the group theory. Introduced states resembletheDicke states,whereastheinteractionsoccuronlybetweenspecifc subsystems relatedbythe actionof thechosen group. The states constructed by this technique exhibit desired symmetry properties and form a natural resource for less-symmetric quantum informational tasks.Furthermore,Iintroduce a larger class of genuinely entangled states based on an arbitrary network structure, graphically represented as a (hyper)graph, with excitations appearing only in particular subsystems represented by (hyper)edges. Iinvestigate the entanglement properties ofboth families of states and show an interesting phenomenon: most of the entanglement is concentrated between nodesofdistancetwo andis absentbetween immediate neighbours. In addition I present two di˙erent methods of constructing introduced states: frstly,Ipropose quantum circuits whose complexityis comparable with the complexityof quantum circuits proposed for Dicke states, secondly,Ipresent such states as a ground states of Hamiltonians with 3-body interactions. Idemonstrate the viabilityof the provided constructionsbysimulating one of the considered states on available quantum computers: IBM -Santiago andAthens.An extensionof some parts,inwhichthe author’spartwasnot substantial, aswell as proofsof presented results, canbe foundin the joint paper[BCZ21], attachedtothe dissertationinAppendixC. 3.1 Motivation Among di˙erenttypesofentangled states,permutationinvariant statesattracteda lot of attention inboth continuous [SAI05, AI07] and discrete [DPR07, BG13]variable systems. A remarkable example of such states is dueto Dicke[Dic54], Dicke state with k excitations in a system of N-qubits isdefned as[SGDM03], |DkN i∝ X σ|1 ··· 10 ··· 0 i  , (3.1) σ∈SN |{z} |{z} k times N−k times where the summation runs over all permutations in the symmetric group SN . On one hand, symmetry of the Dicke states simplifes their theoretical [LYZZ19]and experimental[LPV+14, WKK+09]detection,furthermore facilitates the tasks of quantum tomography[MAT+15]. On the other hand, the entanglement of Dicke states turnedout tobe maximallypersistent and robust for the particle loss[Dic54,BR01], such states provide inherent resources in numerous quantum information contexts, including quantum secret sharing protocols[GKBW07],opendestination teleportation[MJPV99], quantum metrology[PSO+18], and decoherence free quantum communica-tion[BEG+04]. So far most of the scientifc attention was focused on fully symmetric tasks, for example, parallel teleportation[HC13], or symmetric quantum secret sharing protocols[HCR+12]. In various realistic situations, however, such a full symmetrybetween collaborating systemsis notpossible, required or even desirable. As an example, it was shown that there is no four-qubit state which is maximally entangled with respect to all possible symmetric partitions[HS00]. Such a state would allow for the parallel teleportation protocol oftwo qubitsbetween anytwotwo-qubit subsystems. Nevertheless, the following state: 1  |1100i + |0110i + |0011i + |1001i 2 allows for the teleportationofa single qubitto an arbitrary subsystem, and additionally for the parallel teleportation across the partition 13|24. As we already discussed in Introduction, the following state 1 2πi 4πi  |M4i = √ |0011i+ |1100i+ e 3 (|1010i+ |0101i)+ e 3 (|1001i+ |0110i), 6 (3.2) turned out to maximize the average entropyof entanglement over all bipartitions of four-qubit state. As it was later observed, state |M4i is not fully permutational invariant, but invariant under anyevenpermutation of qubits [LW11, CLSW10]. On one hand, it is especially reasonable to share resources in a not fully symmetric way in variants of quantum secrets sharing schemes, allowing only some parties for cooperation. On the other hand, various molecules in Nature (like benzene) stand out with remarkable symmetries, the general investigations of entanglement in highly symmetric systems may shed some light on the nature of correlation in relevant chemical molecules. In recentyears, correlations and the entanglement containedinchemicalbounds was investigated[SBS+17, SPM+15, DMD+21], and a special attention was dedicatedto highly symmetrical molecules[SPM+15]. Although for most molecules the total correlation between orbitals seems to be classical, the general signifcance of entanglement in chemical bonds seems to be present [DMD+21]. 3.2 Group of symmetry of a quantum state Apure state|ψi is called symmetric ifitisinvariantunderanypermutationof its subsystems, i.e. σ |ψi = |ψi for anyelement σ from thepermutation group SN , where N denotes number of subsystems. This might be generalized to a very natural defnition of the group of symmetry of a quantum state. Defnition 2. A state |ψi of N subsystems is called H-symmetric, where H is a subgroup of the permutation group, H< SN , i˙ it is permutation invariant for any σ ∈ H, and only for suchpermutations. We begin with several examples of states with restricted group of symmetry. Firstly, consider a system consisting of three qubits. The celebrated states[DVC00]: |GHZi∝|000i + |111i , |Wi∝|001i + |010i + |100i . are fully symmetric, see Figure 3.1. We also may construct three–qubit quantum states with other types of symmetry. For example |χ1i∝|001i + |010i is S2-symmetric state. This state is, however, bi-separable with respect to the partition A|BC. Nevertheless, a similar example |χi canbe found among genuinely entangled states: |χ2i∝|001i + |010i +2 |100i +2 |111i . Furthermore the following state 2πi 4πi |χ3i∝|001i + e 3 |010i + e 3 |100i exhibits the alternating A3-symmetry, symmetry of all evenpermutations of three qubits. Note that the above state is entangled and locally equivalent to the celebrated |W i state. In that way, we constructed states with all possible types of discrete symmetries in a three-qubit setting. Observe that there exists an easy recipe for the construction of an Hsymmetric state, where Hsition fora H-symmetric stateisalsopossible.Weproceedwiththefollowing defnition. Defnition 3. For a given subgroupH< SN , we defne the N-qubit Dickelike state with k excitations is the following way:  |Dk ∝ X σ|1 ··· 10 ··· 0 i (3.4) N iH σ∈H |{z} |{z} k times N−k times where the summation runsover allpermutations from the group H< SN . Aswe shall see, the normalization constantin Eq.(3.4)highly depends on the structure of the subgroup H and the number of excitations, and is diÿcult to present in a consistent way. Note that the group symmetry of the Dicke-like state |DkN iH is not necessarily given by H. The general analysis of the group of symmetries is tightly connected with the partially ordered set (poset) of all subgroups of SN , which has rather complicated structure [Bra01]. Anysymmetric state |ψi canbe written in the computational basis as: |ψi∝ X |φσ(1)i···|φσ(N)i , (3.5) σ∈SN where the sumis takenover allpermutations σ ∈SN in the symmetric group. In such a way, symmetric states have an e˙ective representation, called Stel-lar representation [MM10], asN points (stars) on the Bloch sphere, corresponding to vectors |φii ,..., |φN i, as we already discussed in Chapter 2. Stellarrepresentation turnedouttoplay arolebyclassifcationofentanglement in symmetric quantum states[MM10, RM11, GKZ12]. Furthermore, special symmetry conditions imposed on the stars, are related with highly entanglement properties of resulted states[MGB+10,BDGM15, DGM17]. I introduce a natural generalization of this approach – the generalized stellar representation, whichis suitable for quantum states exhibiting modes symmetries, i.e. for which summation in Eq.(3.5) runs over a subgroup H of the symmetric group SN . This might possibly restrict the group of symmetries in the resulting state. As for symmetric states, consider N points on the Bloch sphere: |φ1i ,..., |φN i , and the followingproduct |ψi∝ X |φσ(1)i···|φσ(N)i , (3.6) σ∈H where the sum runs over allpermutations σ from the group H. We might representthe state |ψi asacollectionof N points on the Blochsphere relevant tovectors |φii ,..., |φN i with indicated action of the group H, see Figure 3.1. Notice that a given constellation of ‘stars’ at the Bloch sphere with the selected symmetry group H do not represent uniquely the quantum state. The important information is carried in how the group H is contained in SN , which mathematically might be expressed by immersion H,→SN of the group H into the symmetric group SN . Figure 3.1: On the left, stellar representation of three qubits |Wi state, by two stars at the Northpole anda single star at the Southpole. Stellar representation of a Dicke state |Dk i consists of k stars at the SouthPole and N N − k stars at the NorthPole in general[GKZ12]. Similar holds true for the Dicke-like states. On the right, |D42 iC4 defned in Example 1 is represented byfour stars on the Blochsphere,with indicated actionof theCyclic group C4. 3.3 Symmetric states related to (hyper)graphs In this section, I propose a scheme to associate to a given graph with N vertices a single pure quantum state of an N-party system. Such a representation refects not only the symmetry, but also the structure of a quantum circuit under which presented families of quantum states can be constructed. Let us emphasize that states constructed in that way are completely di˙erent form, the so-called graph states, known also as cluster states [BR01, HEB04, AB06, HC13]. A graph G is a pair (V, E) where V is a fnite set, and E is a collection of two-element subsets of V . We refer to elements of V as vertices, and elements of E as edges respectively. A hypergraph is a natural generalisation of a graph, in which edges are arbitrary (not necessarily 2-element) subsets of V . A hypergraph is called uniform if its all edges consist of the same number of elements equal to k, we refer to such an object as k-hypergraph. In particular, a 2-hypergraph is simply a graph. We shall denote the number of verticesby N. Moreover,we assume thatverticesofthe(hyper)graph are labeledby numbers 1,...,N. Defnition 4. Witha given(hyper)graph G =(V, E), we associate a quantum state of N qubits in the following way: 1 |Gi := X |ψei (3.7) p|E| e∈E where |ψei is a tensor product of |1i onpositions labelledby indices which form the (hyper)edge e and |0i on otherpositions. We shall refer to such states as excitation–states. Figure 3.2 illustrates this defnition. Figure 3.2: So-called telescope state |Ψteli [EOS09] represented as an excitation-state. Relatedhypergraphis not uniform.Telescope stateis simply obtained by reading over all edges. Note that as the empty set cannot form thehyperedge in thehypergraph, theket |0000i cannot appear in the superposition which forms the excitation state. There is the following relation between the automorphisms group of a complete k-hypergraph and the group of symmetry of related state. Observation 1. Consider an excitation-state |Gi related to the hypergraph G. The group of symmetries of a state |Gi is an automorphisms group of the related hypergraph G. 3.4 Entanglement properties In this section,I present results concerning entanglement properties of general excitation states and Dike-like states. In particular, Proposition6 and Corollary1provides separabilitycriterion, while Propositions5,7and8 provides formulasfor exactvalueofa concurrencefor regular graphs.Aswe shall see, entanglement properties of excitation states refect the structure of the (hyper)graph. Webeginbyrecalling the notion of Concurrence, denotedby C [HW97]. Concurrence is an entanglement measure, for any two-qubit mixed state ρ its concurrence reads, C(ρ) := max{λ1 − λ2 − λ3 − λ4, 0}, (3.8) where λ1,...,λ4 denote squarerootsoftheeigenvaluesofa Hermitian matrix √ ρρ ˜√ ρ ordered decreasingly, where ρ ˜=(σy ⊗ σy)ρ ∗ (σy ⊗ σy) is, the so-called, spin-fipped form of ρ. Note that the above defnicion coincides withEq.(1.4)for all puretwo-qubit states. The generalized concurrence Cv|rest measures the entanglementbetween the subsystem v and the rest of the system. For pure states, this quantityis determinedbythe relevant reduced densitymatrix[HW97], Cv|rest = √ 2 detρv,hence itsvaluesbelong to the range [0, 1]. The distribution of bipartite entanglement, measuredbythe concurrence satisfes so-called monogamyinequality[CKW00,OV06]: C2 ≥ C2 + ...C2 , (3.9) v1|v2...vN v1v2 v1vN where v2 ...vN are vertices relevant to subsystems. In further considerations, we restrict our analysis to the k-uniform hypergraphs, and assume our graphs tobe connected. Disconnected graphs are relevantforthetensorproductoftwo excitation-states,andhencemightbe analyzed separately. We use the followingnotation. By distance d between vertices v0 and w, we understand the minimal number of vertices v1,...vd, such that there exist (hyper)edges e1,...,ed: vi−1,vi ∈ ei and vd = w. The degree dv of the vertex v is the number of edges on which v is incident. The joint neighborhood nvw of two vertices is the number of sets W suchthatboth: W ∪ v and W ∪ w constitute an edge. Finally, the section svw is the number of edges on which v and w are incident. Notice that for graphs, the joint neighborhood is simply the number of vertices adjacent to both: v and w, while section svw =1 or svw =0 depending if vertices v and w are connected. Furthermore, to simplify the notation, for a given excitation state |Gi and two selected subsystems v and w, the concurrence of the reduced state ρG , willbe denoted as vw Cvw := C(ρG ). (3.10) vw I obtained the following separabilitycriterion. Proposition 5. For ak-uniform hypergraph, the two-party concurrence Cvw reads 2 Cvw = maxn0, �nvw − psvwλo (3.11) |E| where λ = |E|− dv − dw + svw. Furthermore, I obtain the following result relating factorisation of the hypergraph with separabilityof the related state. Theproduct oftwo disjoint hypergraphs(V1,E1)t(V2,E2) isahypergraph (V1∪V2,E1tE2) withvertices beingthe unionV1 ∪ V2 of vertices sets and with edges of the following form: E1 t E2 := X e1 ∪ e2. e1∈E1,e2∈E2 Proposition 6. The excitation-state |Gi corresponding to a k-hypergraph is separable, |Gi = |GiV1 ⊗|GiV2 , i˙ it is a product hypergraph with respect to the division V1|V2. Corollary 1. (Only for graphs) The excitation-state |Gi is separable |Gi = |GiV1 ⊗|GiV2 i˙the relevant graph G is complete bipartite graph, G = KV1V2 . In fact, such a state forms a tensor product of two |W i-like states: |Gi = |W iV1 ⊗|W iV2 . We might reformulate separability criteria in terms of symmetries of the Dicke-like states. Corollary 2. A Dicke-like state |D2 iwith two excitations is separable NH across the bipartition N1|N2 i˙ its group of symmetry is equal to SN1 ×SN2 . Recall that the graph is called regular if the degree dv is constant for any vertex v. Moreover, it is called distance-1 regular graph, if it satisfes an additional condition: If vertices v and w are connected, they are connected with the samenumberofhyperedges. In otherwords, the section svw takes the samevaluesdepending on the distancebetweenvertices: (s for d(v, w)=1, svw = 0 for d(v, w) > 1. From Proposition5,Iderive an expression for the concurrenceCvw between subsystems corresponding to vertices of a distance-1 regular graph G. Proposition 7. For connected, and distance-1 regular graph G the concurrence Cvw between two nodes v and w reads, · ⎧⎪⎩ ⎨⎪ max{0,C} for d(v, w)=1 2 Cvw(G)= for d(v, w)=2 (3.12) nvw |E| 0 for d(v, w) > 2 where C = nvw − ps(|E|− 2d + s), d is the degree of each node, and s is a section for each adjacent vertices. Furthermore, an elementary calculations (presented in a detailed way in [BCZ21]) lead to the following result. Proposition 8. Square of the generalized concurrence C2 between the v|rest particle v and the rest of the system, expressed as a function of the number of edges |E| and the number of vertices N, reads dv�|E|− dv k(N − k) Cv2 |rest =4 |E|2 =4 N2 . (3.13) The second equation is valid under the assumption of the regularity of a khypergraph, i.e. the degrees of vertices are the same. 3.5 Examples of Dicke-like states In Sections 3.2 and 3.3,I presented two similar, but di˙erent, constructions of genuinely entangled states: excitation-states |Gi, related to a graph G, and Dicke-like states, determinedby a subgroup.A Dicke-like state canbe considered as a special case of the excitation-states, which exhibits a certain symmetry structure. We combine both representations and introduce examples of excitation-states related to the graphs given by highly symmetric objects, such as regular polygons, Platonic solids, and regular plane tilings. Such symmetric objects were already used in various contexts concerning multipartite entanglement including: quantifcation of entanglement ofpermutation-symmetric states[Mar11, BDGM15], identifcation of quantumnessof a state[GKG+20], search for the maximally entangled symmet
ric state [AMM10], or general geometrical quantifcation of entanglement [GKZ12, GBB10]especially among states with imposed symmetrieson the roots of Majoranapolynomial[MGB+10, BDGM15]. Results from Section 3.4, allow us to discuss their entanglement prop
erties. We computed the concurrence in two-partite subsystems for all presented examples. For excitation-states, the entanglement shared between a particular node v and the rest of the system depends only on the number of parties N, asitwas shownin Proposition8. Hence,we may defne the entanglement ratio Γv for the node v as: C2 =v Pi6v|i Γv := ∈ [0, 1] , (3.14) C2 v|rest which measures the ratio of entanglement sharedbetween particular parties inatwo-partitewayin comparisontothe amountofentanglement sharedin themulti-partiteway. Note that concurrence satisfes monogamyinequality (3.9), hence the parameterΓv takes values in the range [0, 1]. Example 1. Dicke states. The Dicke states |Dk i are excitation-states for N complete k-regularhypergraphs on N vertices. Their group of symmetry is the entirepermutation group SN . By Eqs.(3.12)and(3.13), and elementary calculation, we show that the concurrence in two-partite subsystems reads, N −1 N − 2  sN − 2 N − 2 ! Cv|w =2 − . kk − 1 kk − 2 The entanglement ratio Γv for a given node v is equal to N − 1 N − 2  sN − 2 N − 2 !2 Γv(DkN )= − . �N−1 �N−1  k − 1 kk − 2 kk−1 By Propositions7 and8, for the Dicke states |Dk N i the entanglement ratio at infnite dimension is nonzero, lim Γv(DNk )=2k − 1 − pk(k − 1). N→∞ Inparticular, for states related to graphs, k =1, we fnd √ lim Γv(DN 1 )=3 − 22. N→∞ Figure 3.3: Dicke state |Dk N i areassociated witha complete k-regularhypergraph with N vertices. The graph and the corresponding state is completely symmetric, so the labelsof the nodes canbe omitted. Dicke states |D2 N i with two excitations are related to complete graphs on N vertices. Atetrahedron represents the Dicke states |D14i, |D24i and |D43 i, depending on whether we consider the vertices, the edges or the faces of the tetrahedron. Example 2. Cyclic states. The simplest non-trivial subgroup of thepermutation group SN is a cyclic group CN . In general, the states |Dk iare N CN translationally invariant, i.e.invariant undera cyclicpermutationof qubits [VC06, WBG20]. Such a family of states is widely considered in several 1D models in condensed-matter physics, like XY model or the Heisenberg model. Note that such states |D2 iwith the number of excitations is N CN equal to k =2 can be constructed as an excitation-state. Indeed, consider the cyclic graph on N vertices. The relevant excitation-state, denoted by |CN i, matchesperfectly |D2 i. Note that the group of automorphisms of N CN a cyclic graph CN is a Dihedral group D2N , where the lower index stands for the order of the group |D2N | =2N. Hence we conclude that the group of symmetries of |D2 i= |CN i is a dihedral group D2N . Systems with dihe- N CN dral D2N symmetries were consider in a context of correlation theory of the chemical bond[DMD+21]. Molecules invariant under a rotation and inver
sion were investigated ibidem. The concurrence in two-partite subsystems takes the following value: (2/N for d(v, w)=2 Cv|w(CN )= , 0 otherwise with a small correction for N =4, where C2 =1 for distance two vertices. v|w The entanglement ratio Γv for a given node reads 1 Γv(CN )= , N − 2with the same correction in the case N =4, for which Γv =1. Example 3. Platonic states. Platonic solids were used to construct quantum statesinvarious contexts[GBB10, GKZ12,GKG+20]. Here, with any Platonicsolidweassociatetwoquantum excitation-states,bylookingatthe edges and faces of related solid. For instance, the tetrahedron is linked to the Dicke state |D24i (by reading edges) and |D34i (by reading faces), see Figure 3.3. We denote such state by |P eiN and |P f iN respectively. An elementary argument from the representation theory shows that group of symmetries of the Platonic solid, determinesthe symmetry of related states. In suchaway, we constructed states with symmetries S4, S4×S2, and A5×S2 respectively. In particular, in the case of dodecahedron the alternating symmetry A5 is observed, whichis not easy to achieve. The concurrence C12 and the entanglement ratio Γv for two-qubit systems obtained by partial trace of (N − 2) subsystems of distance-two of Platonic states determinedbythe edges of the solid is comparedbelow. Concurrence C12 |P ei4 0.333 |P ei6 0.667 |P ei8 0.333 |P ei12 0.133 |P ei20 0.067 Ent. Ratio Γv 0.333 0.500 0.444 0.160 0.074 Figure 3.5: On the left, Icosahedron, one of fve Platonic solids. Its edges (indicated by gray) determine the excitation state |P ei20 on 20 qubits. On the right, hexagonal tiling of a plane. The red rectangle indicates a fnite region for whichtheboundary left-right and up-downis glued together. Such a region determines the excitation state |Hi20 on 20 qubits. On both pictures, for a chosen node 1, the entanglement in the two-party subsystem C1v is vanishing except for subsystems corresponding to distance-two nodes, indicatedbygreen lines and green nodes. Example 4. Regular m-polytope families. There are three natural generalizations of Platonic solids in higher dimensions: the self-dual m-simplices and m-hypercubes with dual m-orthoplexes. Each of thesepolytopesprovides a set of k-uniformhypergraphs for 1 ≤ k ≤ m − 1 defnedbythe set of their k-dimensionalhyperedges.We denote thestates corresponding to the k-dimensional hyperedges of m-simplex as |Sk i and analogously |B2k m i for m m-hypercubes and |Ok i for m-orthoplexes, where the lower index stands 2m for the number of subsystems N in the state, note that N = m, 2m , 2m for m-simplex, m-hypercube, and m-orthoplexe respectively. The states related to the m-simplices are equivalent to the Dicke states |Sk i = |Dk i and hence mm exhibit the full symmetry. On the other hand, the symmetry group of |Bk 2m i and |Ok i is givenbythe non-trivialhyperoctahedral group Bm. The states 2m |Om−1 i are separable with respect to the partition 12|34|...|(2m−1)2m, with 2m eachpairofverticesbeing 2-distance or,in otherwords, lying ona common diagonal of the related m-orthoplex. Furthermore, one can easily calculate the concurrence Cvw for the states |O2 i connected to the 2-edges of the 2m m-orthoplex, Cvw (O2 2m )= ⎧⎪⎩ ⎨⎪ √ 2m2−4m+3−2m+4 max{0, 2 } for d(v, w)=1 2m−2m for d(v, w)=2 (3.15) 2/m 0 for d(v, w) > 2, and similarly the entanglement ratio Γv (O2 2m )= ⎧⎨ ⎩ 1/(m − 1) √ √ 2m2−6m+5+19 (3.16) 6m2−4m( 2m2−6m+5+5)+8 2(m−1)2 , where the frst case holds for m ≤ 3 and the second one for m> 3. With Propositions7and8at hand, one mayshow that the entanglement ratio for the states |O2 i converges to a nonzero value, 2m √ lim Γv(O2 )=3 − 22 ≈ 0.1716. (3.17) 2m n→∞ The situation is simpler for the hypercubic states |B22 m i, where the concurrenceoccurs onlybetween the distance-2 vertices, C12(B22m )=23−m/m, while the entanglement ratio reads 4(m − 1) Γv(B22 m )= , (2m − 2) m which asymptotically tends to zero. Example 5. Plane regular tilings. Regular and semi-regular tessellation of the plane areyet another highly symmetrical objects. Regular tilings correspond to Dicke-like states. The group of symmetry is given by a relevant wallpaper group, restricted to the chosen size of the tiling. Figure 3.5 presents the hexagonal tiling of the plane and relevantexcitation-state |HN i. For sucha tiling,the concurrencein bipartite subsystemstakesthe following value: (2/N for d(v, w)=2 Cv|w(HN )= , 0 otherwise for the tiling restricted to N nodes (minimal size of a cut is 3 × 3). Furthermore, the entanglement ratio Γv for a given node v reads, 41 Γv(HN )= . 3 N − 2 Althoughthevalueofthetwo-partite concurrence Cv|w is the same as in the cyclic case, the parameter Γv takes a larger value. 3.6 Quantum circuits We presentbelowa quantum circuit eÿciently transforminga separable tensor product state into the excitations-state, related to any graph G. Presented constructionwas inspiredbysimilar circuits for Dicke states recently developed in[PBcv10, BE19]. As we showed in[BCZ21], our scheme uses between ∼ 4|V | and ∼ 10|E| CNOT gates, depending on the structure of a graph. First step. Choose arbitrary vertex v, and consider all adjacent vertices v1,...,vd, where d is the degree of v. Suppose that each vi is related to the ith particle, while v is related to d +1 particle. For i =1,...,d − 2, we consecutively apply the following three-qubit gates U(1) on parties {i,d,d+1} i, d, d +1: √ √ �|101i + i − 1 |011i 7−→ |011i (3.18) 1 i |110i 7−→ |110i |100i 7−→ |100i |010i 7−→ |010i |001i 7−→ |001i |000i 7−→ |000i , √ N 2mthe ratio Γv tends to a nonzero value 3 − 22 ≈ 0.17 (indicated by the horizontal dashed line). For families of states in whichthe local degree does not scale with the sizeofa graph(|B22m i, |CN i, |HN i), the complexityof Γv is O(1/N). where the action on the remaining subspace is arbitrary. Application of U(1) operation is relevant to the graphical operation of deleting an edge {i,d,d+1} e = {i, d +1}, see Figure 3.7. Secondly, we apply the following three-qubit operator U(2) on {d−1,d,d+1} parties d − 1, d, d +1 √ √ �|101i + d − 2 |001i 7−→ |001i (3.19) 1 d − 1 |011i 7−→ |011i |100i 7−→ |100i |010i 7−→ |010i |000i 7−→ |000i , where the action on the remaining subspace is arbitrary. This is related to the graphical operation of deleting an edge e = {d − 1,d +1}, see Figure 3.7. Finally we apply the following two-qubit operator U(3) on parties {d,d+1} d, d +1: √ √ �|11i + d − 1 |01i 7−→ |01i (3.20) 1 d |00i 7−→ |00i |10i 7−→ |10i where the action on the remaining subspace is uniquely determined. This is relevant to deleting the only remaining edge: e = {d, d +1}, see Figure 3.7. Thereisa simplelogicbehind thosethreeoperations. We combineallterms having excitations onposition d +1 into a single term with an excitation on this position. After applying these operations, the state takes the form of the superposition: 1 √ d |1i ⊗|0 ... 0i + |0i⊗ X |ψei  , vv p|E| e∈E\v where E \ v denotes a set of edges which do not contain vertex v. Second and the next steps. Consider the graph G0 =(V \ v, E \ v) with deleted vertex v and all adjacent edges. We repeat the procedure from the frst step for an arbitrary vertex w from the graph G0 . We repeat this procedure further, for G00 =(V \v, w; E \v, w), until we delete N −1 vertices, which fully separates the initial graph. Final step. After applying presented procedure iteratively N − 1 times, the state takes the form: 1 p|E| Xpd0 |1i ⊗|0 ... 0i vv vc , v∈V where d0 denotes the degree of the vertex v of the graph reduced according v to the procedure. The state above is similar to the state 1 |WN i := √ �|10 ··· 0i + ··· + |0 ··· 01i . N The separation of |WN i stateisawell-known procedure[JHJ+08], and might be obtainedbyperformingtwo-qubit gates U(4) : 1i √ √√ d0 +...+d0 |10i + d0 |01i 7−→d0 +...+d0 |10i v1 vi−1 vi v1 vi |00i 7−→ |00i on particles 1 and i. We refer to[BCZ21] for the precise estimation of the computational costfor the presented procedure. We simulated the simplest nontrivial cyclic state|C5i, in order to demonstratethe viabilityoftheprovided construction. We referto[BCZ21]fora detailed description. The overall circuit requires 22 CNOT operations, where the topology of the quantum computer is not taken into account. Such a circuit can be realised on the state-of-the-art 5-qubit quantum computers providedbyIBM – linear-topology Santiago andAthens with quantumvolumes(QV)of32andVigowithQVof16withTtopology.In totalweused more than 740 000 samples over all three computers, which gives distribution with signifcant values for all expected computational states proceeding from cyclicpermutations of the state |00011i with the probability 0.487 of fnding the system in one of them, see Figure 3.8. Further analises and the comparisonof obtainedresultstoamodelof noisemightbe fndin[BCZ21]. 3.7 Hamiltonians The Dicke states were introduced as a ground states of the Hamiltonians with a well–defned number of excitations. The excitation-states can be also considered as ground states of analogous Hamiltonians, with the same property of a well defned number of excitations. The single-mode Dicke Hamiltonian (known also as Tavis-Cummings model or generalized Jaynes-Cummings model) has the following form[EB03, Gar11]: † H = ω0 Jz + ωa† a + √ λ (a + a)(J− + J+) , N | interaction {z } where Jα are the collective operators: NNJz ≡ X σi ,J± ≡ X σi z±. i=1 i=1 (1) (2) three-qubit operations Uon parties 1, 3, 4, secondly Uon parties {1,3,4}{2,3,4} 2, 3, 4, and fnallythetwo-qubit gate U(3) on parties 3, 4. Resulting state is {3,4} separable with respect to the 4-th particle. Thisisgraphically representedby deleting all adjacent edges. In the consecutive steps,we apply thisprocedure iteratively to the remaining vertices. By fxing interaction, λ =0, the coupling term vanishes. Thus eigenstates have the simple tensor product form with a factor representing the Fock states of the feld and the other factor as an eigenstate of the collective angular momentum operator Jz. Operator Jz has the following degenerated eigenvalues: −N/2, −N/2+1, . . . , N/2. The further partition of its eigenspaces mightbeperformedbythe square of the total angular momentum operator J2, which canbe written as J2 ≡ J+J− + Jz(Jz + I), in terms of the raising andloweringoperators. The Dicke states |j, miN (in the more classical notation) are eigenstates ofboth operators Jz and J2: Jz |j, miN = m |j, miN , J2 |j, miN = j(j + 1) |j, miN with m = −j, −j +1,...,j corresponding to the number of excitations, while j =1/2, 3/2, . . . , N/2 (for N odd)and j =0, 1, . . . , N/2 (for N even) related to the cooperation number. In general, atomic confgurations for N> 2 contain entanglement[WM02]and are degeneratedfor j like states of given symmetries: cyclic, dihedral, symmetry of Platonic solids. Figure 3.6 compares entanglementpropertiesofintroducedfamiliesof states. Moreover,I showtwo di˙erent methodsof constructingintroduced families of states: Section 3.7 presents excitation states as groundstates of Hamilto
nians with 3-body interactions, while Section 3.6 proposes relevant quantum circuits. The complexityofintroduced circuits is comparable with the complexityof quantumcircuits proposedfor Dicke states knownin the literature. Asimple example of a cyclic state of fve qubits is successfully simulated on available quantum computers:IBM -Santiago andAthens – see Figure 3.8. Chapter 4 Absolutely maximally entangled states Absolutely Maximally Entangled (AME) states of a multipartite quantum system are maximally entangled for every bipartition of the system. AME states are special cases of k-uniform states characterized by the property that all of their reductions to k parties are maximally mixed. Both classes of states are crucial resources for various quantum information protocols. In this chapter, I briefy recall correspondence between AME states and classical combinatorial designs. I focus my attention on the di˙erent linear structures of classical designs that a˙ect the structure of the related AME and k-uniform states. 4.1 AME and k-uniform states Amultipartite quantum state|ψi∈H⊗N of N parties with a local dimension d d each is called AME, if is maximally entangled for every of its bipartition, i.e. the partial trace trS |ψihψ|∝ Id, (4.1) for any subsystem S of |S| = dN/2e parties[Sco04]. AME states of N partise with the local dimension d are denoted as AME(N, d). The class of AME states isbeing applied in several branches of quantum information theory: in quantum secret sharing protocols[HCR+12], in parallel open
destination teleportation[HC13], in holographic quantum error correcting codes[PYHP15, MFG+20], among many others. The state AME(N, d)allows one to construct a pure quantum error correction code (pure QECC), whichsaturates the Singletonbound[HCR+12]. Particular attention is paid toAME statesofanevennumberof parties, thoseareequivalenttonotionsas perfecttensors[PYHP15]ormultiunitary matrices[GAR+15]. AME states are special type of k-uniform states. Defnition 5. Aquantum state|ψi∈H⊗N of N is k-uniform if its reduced d densitymatrices are maximally mixed, i.e. ρS(ψ) := trS |ψihψ|∝ Id for anysubsystem S of k parties(|S|=k) and the complementary subsystem S. It is known that the uniformity k cannot exceed bN/2c [Sco04]. States whichsaturate thisbound, i.e. bN/2c-uniform states, are called AME states. Example 6. Greenberger–Horne–Zeilinger (GHZ) state 1  |GHZi = √ |000i + |111i 2 is a 1-uniform state. Its natural generalization to N parties with d distinguishable energy levels: 1  |GHZdN i = √ |0 ··· 0i + ··· + |d − 1 ··· d − 1i d is also 1-uniform. Example 7. The following state of four qutrits 1 |AME(4,3)i = |0000i + |0121i + |0212i + 3 |1110i + |1201i + |1022i + |2220i + |2011i + |2102i  is 2-uniform, so it is an AME state of4 qutrits[Hel13]. It reveals larger entanglementproperties than the corresponding |GHZ43i state of four qutrits. Note that both states presented in Examples 6 and 7 might be written in simple closed formulas: d−1 1 |GHZNd i = √ X |i, . . . , ii , d i=0 d−1 1 |AME(4,d)i = X |i, j, i + j, 2i + ji . (4.2) d i,j=0 where the summation is understood modulo d. AME(N, d)states are maximizing entanglement properties among allNparties states, eachwith d levels[HCR+12]. There is no general construction of AME(N, d)state, for an arbitrary number of partiesN and an arbitrary number of enery levels d. Surprisingly, AME states do not exist for any numbers N and d. Indeed,itwas frst observedbyHiguchi and Sudberyin their study of bipartite entanglementthat AME state of four qubits does not exist[HS00]. Until today, more of such negative results are known[HGS17, HESG18]. 4.2 Orthogonal arrays Combinatorial mathematics deals with the existence and properties of designs composed of elements of a fnite set and arranged with certain symmetry and balance[Ce07].Asimple exampleofa combinatorial designisgiven by a singleOrthogonal array[ASH99]. Orthogonal arrayisa combinatorial arrangements, tables with entries satisfying given orthogonal properties [Kos96]. A tight connection betweenOAs and (quantum) error-correcting codes[ASH99], and maximally entangled states[GZ14]brought a new life for these combinatorial objects. Aswealready presentedin Section2.4,an orthogonal arrayOA (r, N, d, k) isa table composedby r rows, N columns with entries taken from 0,...,d−1 in such a waythat each subset of k columns contains allpossible combinations of symbols with the same amount of repetitions, see Figure 4.1. The number k is known as the strength of the OA. We assume that OAs are simple, i.e. all rows are distinct. Notice, that the minimal number of rows in OAis equal tor = dk,OAs saturatingthisbound are calledof Index Unity. Ingeneral the index of theOA r λ = dk is alwaysa natural number. AnOA (r, N, d, k) is said tobe an irredundant orthogonal array(IrOA)ifin anysubset of N − k columns all combinations of symbols are di˙erent[GZ14]. Usually, an orthogonal array is called linear if the set of r rows form a vector spaceover the Galois feldGF(d). The linearitycondition is equivalent to the following: for eachtwo rows R1 and R2 of anOA, and anytwo elements c1,c2 ∈ GF(d), c1R1 + c2R2 (4.3) is also a row in the OA. Such an OA can be represented by a N × s generator matrix, whose rows form a basis of aforementioned vector space, see Figure 4.1. Up to the isomorphism of a linear space, the generator matrix G can always be written in the standard form G =[Ids|A], where A is an (N − s) × s matrix[RGRA17]. On the one hand, the generator matrix greatly suppresses the notation of anOA but also indicates the internal structure ofOA. In fact, except for linearOAs, thereis another classofOAsthatmightbe writteninaformofa generator matrix. Indeed, in place of the fnite feld GF(d), one may consider a fnite ring R of order |R| = d. In such a way all linear combinations of rows of the generator matrix forms a module over the ring R. A practical selection for a ring R is cyclic group Zd or, more generally, simple sum of cyclic groups Zd1 ⊕ ... ⊕ Zds . In fact those are the only possibilities for commutative rings R. Aswe alreadyhave showin Chapter2,OAs mightbe successfully used to construct m-resistant states of N-qudits, for which the entanglement of the reduced state of N − m subsystems is fragile for the loss of any additional subsystem. In fact, states constructed in this method exhibit more exceptional propertyofbeing k-uniform, aswe describebelow. For any OA(r, N, d, k) one may associate a pure quantum state |ψi∈ H⊗N byreadingallrowsofOAand creatingasuperpositionofr terms in the d computational basis[GZ14, ASH99], see Figure 4.2. Moreover, for IrOAa of a strength k, the corresponding quantum state is k-uniform[GZ14, ASH99]. Consider nowa linearOAsover the module R, |R| = d with the N × s generator matrix of theform G =[Ids|A], where A =(aij) is an (N − s) × s array with elements from R. The corresponding quantum state might be conveniently presented as follows: 1 |ψi := √ X |G~vi (4.4) ds ~v∈R⊗s where the multiplication and addition hidden inside an expression G~v are determinedbythe structureofa ring R, and vector ~v ∈R⊗s has exactly ds elements[RGRA17]. Example 8. Consider the following generator matrix  10 12  G = , (4.5) 01 11 0 0 0 0 0 1 1 1 0 1 1 1 0 2 2 2 0 2 2 2 0 3 3 3 0 3 3 3 1 0 1 2 1 0 1 2 1 1 2 3 1 1 0 3 1 2 3 0 1 2 3 0 2 0 1 3 2 1 2 0 3 0 1 1 over←−− Z4  1 0 0 1 1 1 2 1  over− −−→ GF(4) 1 2 2 3 0 1 2 2 3 1 3 2 2 2 0 2 2 2 0 1 2 3 1 3 2 3 1 0 3 0 3 2 3 0 3 1 3 1 0 3 3 1 2 0 3 2 1 0 3 2 1 3 3 3 2 1 3 3 0 2 OA(16,4,3,1) OA(16,4,3,2) Figure 4.1: In the middle, a generator matrix of an orthogonal array OA�d2 , 4, d, kforaprimepower dimension =4d =22. Rows of anOA are givenbyall linear combinations c1 (1, 0, 1, 2)+c2 (0, 1, 1, 1) for c1,c2 ∈ GF(4) and Z4 respectively. Notea di˙erenceinOAs strength, k =1 and k =2 for GF(4) and Z4 respectively. Indeed, pairs of symbols 00 and 02 repeatstwice on the second and fourth column in OA �d2 , 4, d, 1, while combinations 01 and 03 do not appearin thosetwo columns. BothOAs are irredundant. with entries from the ring R = Zd. Eq.(4.4) associates the quantum state d−1 1 |ψi = X |i1i|i2i|i1 + i2i|2i1 + i2i , (4.6) d i1,i2=0 where addition and multiplication are considered over the ring Zd. For any odd dimension d (2 -d), the corresponding state |ψi∈H⊗4 is an AME (4,d) d state, as the correspondingOA �d2 , 4, d, 2has strength k =2, see Figure 4.2. 1201 |1201i + 2021 |2021i + 2102 |2102i + 221 0 |2210i − −−→ 1120 −−−−−−−−→ |1120i + 00 0 2 12 21 10 |0000i + 01|0111i + 02|0222i + 10corresponding |1012i +  10 12  over state 01 11 GF(3) |AME(4,3)i∝ Figure 4.2: The orthogonal array of unity index OA (9, 4, 3, 2) on the left and repeated in the center. Each subset consisting of two columns contains allpossible combinationsof symbols. Here,two suchsubsets are highlighted. The relevant quantum state is obtained as a superposition of states corresponding to consecutive rows of the array. 4.3 Di˙erent linear structures For dimensiond =4, Figure 4.1 presentstwoOAs corresponding to the ring structures R = GF(4), Z4. Two quantum states |ψiGF(4), and |ψiZ4 might be constructed simply by reading consecutive rows of respective OAs. As the correspondingOAs has di˙erent strength, quantum states |ψiGF(4), and |ψiZ4 are 1-, and 2-uniform respectively, see Figure 4.1. In general, states obtainedbythe linear actionofa Galois feldGF(d) exhibit larger uniformity k then those based on the other ring structures R, |R| = d. Once more, consider the generator matrix G defnedby Eq.(4.5). In dimension d =9, we may consider three di˙erent ring structures: R = GF(9), Z9, and Z3 ⊕ Z3. The ring Z9 consists of nine numbers 0,..., 8 with the usual multiplication and addition modulo 9. We denote the related quantum state(4.6)by |ψiZ9 . Elements of Z3 ⊕ Z3 are given by pairs of numbers (a, b), where a, b = 0, 1, 2, while themultiplication and additionis considered on respectivepositions (a, b)+(a 0,b0)=(a + a 0,b + b0), (a, b) · (a 0,b0)=(a · a 0,b · b0), modulo 3. Note that a pair (a, b) might be canonically associated with the number 3a + b ∈ Z9, which we further use for linking with a pure quantum Figure 4.3: Elements of the Galois feld GF(9) are givenbyninepolynomials ax + b, where a, b =0, 1, 2. Their multiplication is considered modulo an irreduciblepolynomial x2+1. Exact structureofamultiplicationispresented in the table, while the addition structure is the same as for the ring Z3 ⊕ Z3. Note that there are no zero divisors in GF(9), i.e pairs of non-zero elements which product is zero. Eachpolynomial ax + b mightbe later associated with a number 3a + b =0,..., 8. state(4.6)denotedby |ψiZ3⊕Z3 ∈H⊗4 written in the computational basis, 9 (a, b) → 3a + b →|3a + bi. The structure of a Galois feld GF(9) might be presented as an addition and multiplication of nine polynomials in x variable ax + b, where a, b =0, 1, 2. Addition and multiplication of aforementioned polynomials is considered modulo an irreducible degree2polynomial x2 +1. Although addition ofpolynomials ax+b matches the addition of pairs (a, b), the multiplication structure isdi˙erent, see Figure 4.3. Same asbefore, eachpolyno
mial ax + b mightbe associated with the naturalnumber 3a + b ∈ Z9, which defnes quantum state |ψiGF(9) via Eq.(4.6) written in the computational basis. Unlike as for rings R = GF(4), Z4, three arrays OA (81, 4, 9, 2) corresponding to three di˙erent ring structures R = GF(9), Z9 and Z3 ⊕ Z3 are all of strength 2. Therefore, the related quantum states |ψiZ9 , |ψiZ3⊕Z3 , and |ψiGF(9), are all AME. In Section 6.8, we discuss the problem of equivalence of such states. 4.4 Conclusions In thischapter,Irecallthewell-established notionofAbsolutely Maximally Entangled and k-uniform states(Defnition5) andtheir correspondence to classical combinatorial designs. I focus my attention on the di˙erent linear structures of classical designs that a˙ect the structure of the related AME and k-uniform states. This chapter does not contain any signifcant contributionbythe author and is intended to introduce some notions used in the consecutive chapters. Chapter 5 Thirty-six entangled oÿcers of Euler In this chapter, I discuss a quantum variant of the famous problem of 36 oÿcers of Euler. While the classical problem of Euler is known to have no solutions, I present an analytical form of an AME(4, 6) state, which might be seen as a quantum solution to the Euler’s problem. Moreover, I show a coarse-grained combinatorial structurebehind constructed state. Considerations of such a coarse-grained combinatorial structures might lead to a successful approach for constructing genuinely entangled states beyond the stabilizer approach. An extension of some parts of this chapter, in whichthe author’s contributuion was not substantial can be found in the joint work [RBB+21], attachedtothe dissertationinAppendixD. 5.1 Thirty-six oÿcers of Euler In 1779, Euler examined the now-famous oÿcer problem[Eul82]: “Six dif
ferent regiments have six oÿcers, each onebelonging to di˙erent ranks. Can these36 oÿcersbe arrangedina square formationso that each rowand column contains one oÿcer of each rank and one of each regiment?” As Euler observed, thepossibility of such an arrangement is equivalent to existence of Graeco-Latin squares of order 6. Single Latin square of order d is flled with d copies of d symbols arranged in a square in such a waythat no row or columnofthesquare containsthe samesymboltwice.Two orthogonal Latin squares (OLS), also called Graeco-Latin square, is an arrangementof ordered pairsofsymbols,for instance, one Greekcharacterand one Latin,into d × d square. Each symbol appears exactly once in each row and column, while eachpairofsymbolsappears exactly onceintheentireOLS, see Figure5.1. Aα Bβ Cγ Cβ Aγ Bα Bγ Cα Aβ = A« K¨ Q© Q¨ A© K« K© Q« A¨ = 0, 0 2, 1 1, 2 1, 1 0, 2 2, 0 2, 2 1, 0 0, 1 Figure 5.1: An example of Graeco-Latin square of order d =3, relevant to AME(4,3) state. In the middle, Greek and Latin letters are replaced by ranks and suitsof cards, onthe rightbypairofnumbers. In Chapter 4 we discussed a connection between combinatorial designs, such as OLS, and multipartite entangled states, and error-correcting codes. Inparticular, to anyOLS, one may associatea quantum state |ψi∈H⊗4 d d−1 1 |ψi = X Tijk` |ii|ji|ki|`i , (5.1) d i,j,k,`=0 where Tijk` =1 ifthe pair (k, `) is an entry in i-th rowand j-thcolumn, while Tijk` =0 otherwise. Conditions imposed on OLS translates onto fact that for the following bipartition of four indices into pairs: ij|k`, ik|j`, i`|jk, the j` jk corresponding matrices arepermutations, i.e. T ijk` ,T ,T arepermutation ik i` matrices. Unitary matrices are a natural generalization ofpermutation matrices. j` jk A tensor Tijk` for which each of matrices T k` ,T ,T is a unitary matrix ij iki` is called perfect. Furthermore, such a square matrix U = T ijk` is called 2T k` j` unitary, as the three matrices: U = , and related reshu˜ed UR = T ij ik , and partialy transposed (UR)Γ = T jk matrices are all unitary. Perfect tensor i` provides an isometry betweenanypairofitsindices. The partial traceofthe state |ψi relatedtoaperfect tensoris maximally mixedforanybipartition of the system, i.e. ρS ∝ Id for |S| =2, and hence is related to an AME(4,d) state[Sco04]. Two OLS of order d exist for d =3, 4, 5 and any natural number d ≥ 7 [JH01]. On the otherhand, OLS of order2 and 6 do not exist, which canbe easily observed for order 2,andwhatwasprovenby an exhaustive casestudy for order 6 [Tar00]. The only local dimensiond for which the existence of a quantum version of OLS was not decidedis six[HS00, AME, HW, YSW+21]. 5.2 AME state of four quhex In[RBB+21], we present a solution for the quantum version of the Euler’s problem of 36 oÿcers. More specifcally, we provide there an analytical form of an AME(4, 6)state,whichcanbeassociatedwithanexampleof orthogonal quantum Latin squaresofordersix[MV16, GRDMZ18, Ric20,RBB+21]. In order to fnd a 2-unitary matrix of order 36, we used an iterative numerical technique based on nonlinear maps in the space of unitary matrices U(d2) [RAL20]. One of key challengeswas to fnd an appropriate seed matrix, which generates a numerical 2-unitary matrix of order d2 = 36. Details of the procedure and an example of a seed that leads to the 2-unitary solution mightbe fnd in[RBB+21] The author’s main contribution to the solution of this problem, was fnding an analytical form of the presented state, obtained with relation(5.1) from a 2-unitary matrix U36, which determines the tensor Tijk`. Any 2unitary matrix remains 2-unitary under a multiplication by local unitary operators. Using this freedom, we applied a searchalgorithm over the group U(6) ⊗ U(6) oflocal unitaryoperations,to orthogonalize certain rows and columns in a given numerical 2-unitary matrix U and its rearrangements UR , (UR)Γ. The particular choice of the orthogonalityrelations corresponds to the block structure of the eventually obtained analytical solution. These tools canbegeneralizedto constructmulti-unitaryoperatorsand corresponding AME states in other local dimensions and number of parties, and can potentially yield states that are not createdbypresently known techniques. In order to present our solution for the problem of 36 entangled oÿcers of Euler, we express coeÿcients of a quantum state |ψi∈H⊗4 (four qudits 6 state) via a four–index tensor Tijk` as it is presented on Eq. 5.1. Nonvanishing terms of Tijk` might be conveniently written in form of a table, see Fig. 5.2. Provided construction is based on root of unity of order 20, denotedby ω = exp(iπ/10). There are three non-zero amplitudes: √√ a =[ 2(ω + ω)]−1 =[5+ 5]−1/2 ' 0.3717 √√ b =[ 2(ω3 + ω3)]−1 = [(5 + 5)/20]1/2 ' 0.6015 √ c =1/ 2 ' 0.7071 22 which might be expressed in terms of ω. Two relations a+ b2 = c=1/2 √ and b/a = ϕ =(1+ 5)/2 the golden ratio, determine uniquely amplitudes appearing in the solution and explain why the constructed AME state deserves tobe called the golden AME state. Checking the propertyofbeing an AME state comes down to verifcation of several equations involving roots of unityof order20,whichI discussina detailedwayin Section5.4. 5.3 Structure of the AME(4,6) state Recall that a classical OLS corresponds to a 2-unitarypermutation matrix. Since there is no solution to the original Euler’s problem, the 2-unitarypermutation matrix of size 36 does not exist. Nonetheless, the 2-unitary matrix presented on Figure 5.2 has the structure of nine 4 × 4 blocks. Moreover, the block structure is preserved under reshu˜ing and the partial transpose of the matrix. In that sense, we found the invariant block structure in the original 36 × 36 with respect to reshu˜ing and partial transpose operations. In fact, the problem of fnding 36 entangled oÿcers of Euler splits into two sub-problems: fnding such a block invariant structure at frst, and then select adequate non-zero elements inside the chosen structure. Such an approach indicates the plausible direction of search for other AME and k-uniform states. In fact, the notion of such an invariant structure might be formalized via certain combinatorial designs. Grouping symbols of indices k and ` in the presentedperfect tensor Tijk` in pairs: 1, 2 → A/α, and 3, 4 → B/β, and 5, 6 → C/γ for k/` respectively, results in coarse-grained OLS, which reveals the described block structure, see Figure 5.3. Aα Aβ Cγ Cα Bβ Bγ Cα Cβ Bγ Bα Aβ Aγ Bγ Bα Aβ Aγ Cα Cβ Aγ Aα Cβ Cγ Bα Bβ Cβ Cγ Bα Bβ Aγ Aα Bβ Bγ Aα Aβ Cγ Cα Figure 5.3: A coarse-grained OLS of order 6, which reveals the block structureofaperfect tensor Tijk`. Indices of non-vanishing elements of the tensor Tijk` are presented: i in row, j in column, while a pair of coarse-grained indices k, ` in relevant entry. Eachpairof symbolsrepeats exactlyfour times on the grid. Moreover, each symbol on eachposition repeats exactlytwice in each row and column. Furthermore, presented AME(4, 6)statemightbe writteninthe following from, which reveals its block structure 6 |Ψi = X |i, ji⊗|ψi,ji , (5.2) i,j=1 ∼ where each of states |ψi,ji∈H36 = H3 ⊗H3 ⊗H2·2 has the followingform:  |ψi,ji = |2jj k+2i +1i⊗|j +2ji k+1i ⊗Id ⊗Uij  √ 1 |00i + √ 1 |11i 22 22 | H{z 3 }| H{z 3 } |{z} H2·2 (5.3) where i, j =1,..., 6 and Uij are unitary matrices of order 2, which form mightbe read from Figure 5.4. Note that pairs of indices 2jj k+2i +1 ,j +2ji k+1 (5.4) 22 are in accordance with the entries of 6 × 6 coarse-grainedOLS presented on Figure 5.3 after associating Greek character and one Latin characters with numbers: A, α → 0; B, β → 1; C, γ → 2. Most of the examples of quantum error correction codes belong to the class of, so-called, additive (stabilizer) states[CSSZ09, AR21]. In particular all hitherto known AME states, are either stabilizer states[AR21], or might  0 γ  matrix reads U12 = .. γω10 0 be derived from the stabilizer construction[BR20].We examined all stabilizer sets of four quhex in their standard form and did not fnd an AME(4, 6) state, hence the presented state is not a stabilizer state. Nonadditive quantum codes are in general more diÿcult to construct, however, in many cases, they outperform the stabilizer codes[YCLO08]. Thus far, the stabilizer approachpractically contained the combinatorial approach to constructing AME and k-uniform states. As we demonstrated, the consideration of coarse-grained combinatorial structures might be successful in constructing genuinely entangled states and have advantages over the stabilizer approach. It is tempting to believe that the quantum design presented here will trigger further research on quantum combinatorics. Even though OLS of order six does not exist, we present a coarse-grained OLS of that order, which structure standsbehind the provided AME state. This sheds some light on how to construct quantum nonadditive AME states, and more generally QECCs, when the stabilizer approach fails. 5.4 Verifcation of 2-unitarity j` jk In this section we show that each of three matrices T ijk` ,T ,T related to ik i` the tensor Tijk` is unitary. In order to simplify the notation, we introduce a matrix U of size 36 with entries, determinedbythe tensor T jk , in the follow i` ing way Up,s = Tijkl, with p = j + 6(i − 1) and s = `(k − 1), i.e. the matrix corresponding to the fattening T jk . Similarly,by UR and (UR)Γ, we denote i` j` jk matrices corresponding to the fattening T and T respectively. Opera ik i` tions R and Γ are known as reshu˜ing and the partial transposerespectively, compare with Section 5.1. To show the tensor Tijk` is perfect, we shall verify that three related matrices: U, UR and (UR)Γ are unitary. In fact, each of those matrices has the block structure with nine 4 × 4 sub-matrices. Hence our task simplifes to verifcation that constituent blocks are unitary matrices. Consider the matrix U. Interestingly, except for one blockcomponent in the matrix U,alleightremainingblocksareequivalent(uptoamultiplication of rows and columns by adequate phases) to the following 4 × 4 unitary matrix: ⎡ ⎢⎢ ⎤ ⎥⎥ a a b b 0 0 c −c c −c 0 0 b b −a −a V = ⎣ . ⎦ of antipodal points on the complex plane, for example, the orthogonality between the frst two rows reads bc�1 − 1=0. (5.5) The right top corner in Figure 5.6 presents the exceptional block of matrix U. Six orthogonalityrelationsbetween rows read a 2�ω8 + ω−8+ b2�ω4 + ω−4=0, (5.6) ab�1+ ω2 + ω−8 − 1=0, (5.7) ab�ω−2 + ω2 + ω−8 + ω8=0, up to a phase factor, where ω = exp(iπ/10). Each equation might be presented as a unitarity rectangle -a constellation of fourpoints on the complex plane which sum up to zero, as it is shown on Fig. 5.6. Observe that the secondandthird equationsabove arebothrelatedtotwopairsofantipodal points on the complex plane. Geometric interpretation of the amplitudesa and b is shown in Fig. 5.5. √ 1+ 5 ϕ = . 2 Althoughboth matrices UR and (UR)Γ enjoythe structure of nine 4 × 4 blocks, similar to U, the particular arrangement inside their blocks is signifcantly di˙erent from the U matrix. Blocks in UR and (UR)Γ are of four distincttypesuptomultiplicationof theirrowsand columnsbyphase fac-tors, see Fig. 5.6. Orthogonality relations between rows of both matrices refect their complex structure. In particular, we distinguished fve addiFigure 5.6: Non-vanishing elements of three unitary matrices U, UR and (UR)Γ of order 36 are presented on top. The modulus of a non-vanishing element is representedbythe intensityofthe background color, wheregreen, orange and red are related to constants a, b, and c respectively. Eachmatrix has the structure of nine 4 × 4 blocks, the sub-structure within the blocks is di˙erent. Orthogonalityrelationsbetween pairs of rows in block matrices may be presented asa constellationofpoints on the complex plane which sum up to zero. Constellations related to Eqs.(5.5-5.12) are indicated. tional orthogonalityrelations givenbythe following equations: a 2ω4 + ab�ω10 + ω−4+ b2ω−4 =0, (5.8) a 2ω−3 + ab�ω5 + ω3+ b2ω−7 =0, (5.9) ab�ω−4 + ω−6+ bcω5 =0, (5.10) ab�ω−8 + ω−2+ acω5 =0, (5.11) 2 a + b2ω4 + bcω−7 =0. (5.12) Related constellations are presented on Fig. 5.6. The above-listed equalities provide orthogonalitybetween rows in the three matrices U, UR and (UR)Γ , hence they imply that the matrix U is 2-unitary. Interestingly, in the presented solution each oÿcer is entangled with at most three other oÿcers (out of remaining 35). This implies the matrix U is sparse. Amplitudes a, b, c which appear in the presented construction might be defned as the unique solution of the following three algebraic equations: √ 2 + b22 a= c=1/2 and b/a = ϕ = (1+ 5)/2, see Fig. 5.5. Notice the similaritiesbetweenalgebraic equations which leadtovalues a, b, c, and the algebraic equations which lead to the amplitudes in an another AME state, in the heterogenous 2 × 3 × 3 × 3 system presented in[HESG18]. The phases of the coeÿcients shown in Figure 5.2, being multiples of ω = exp(iπ/10), are chosen in such a waythat all 36 quantum states |ψiji, each represented by a single row of the matrix U, are equivalent to the standard,two-qubitBell state. Thisis easytosee,forany state formedby √ two coeÿcients of moduli, |c| =1/ 2, as states |ψ11i or |ψ56i represented in the second and the third row in the upper left block in Figure 5.2. One can show that this property holds also for other states. For example, the state |ψ63i correspondingtothe frst lineofthe aforementionedblock canbe written in the product basis as |ψ63i = aω10 |11i + bω15 |12i + bω5 |21i + a |22i . 22 Thus the partial trace reads, TrB |ψ63ihψ63| = diag(a+ b2,a+ b2)= I/2, whichproves that |ψ63i is locally equivalent to the maximally entangled Bell state. A similar reasoning works for all other states consisting of four terms and representedinFigure5.2bygreenandyellowelements.Insuchaway,all 36 states, corresponding to 36 entangled oÿcers of Euler, canbe considered as maximally entangled, two qubit states. 5.5 Conclusions In this chapter I presented the famous combinatorial problem of 36 oÿcers wasposedby Euler, and its quantumversion. I present an analytical form of an AME(4, 6) state of four subsystems with six levels each, which might be seen as a quantum solution to Euler’s problem of fnding two orthogonal Latin squares of order six. The existence of such a state is equivalent to the existence of a 2-unitary matrix of order 36 or aperfect tensor with four indices, each running from one to six. This result allows us to optimally encodea single quhexintoa three quhex state.Furthermore,Ishowa coarsegrained combinatorial structurebehind the constructed state. Further analysis of such coarse-grained combinatorial structures mightpotentially lead to the construction of other genuinely entangled statesbeyond the stabilizer approach. Chapter 6 Classifcation of absolutely maximally entangled states In this chapter of the thesis, I present techniques for verifying whether twoAME statesareequivalentwithrespecttoStochasticLocalOperations and Classical Communication (SLOCC). I falsify the conjecture that for a given multipartite quantum system all AME states are SLOCC-equivalent. I also show that the existence of AME states with minimal support of 6 or more particles results in the existence of infnitely many such nonSLOCC-equivalent states. Moreover, I present AME(5,d)states which are not SLOCC-equivalent to the existing AME states with minimal support. Proofs of the statements presented in this Chapter, as well as an extension of some parts, in which the author’s part was not substantial can be found inthejoint work[BR20], attachedtothedissertationinAppendixB. 6.1 Local equivalence of AME and k-uniform states The initialmotivation addressedinthischapteristhequestion whetherdifferent constructions of AME states are equivalent by any local transformation. Note, that for AME states all reduced densitymatrices are maximally mixed. Therefore, the classical methodforverifcation oflocal equivalence, which is comparison of Schmidt rank and coeÿcients, will obviously fail [HW08]. Contrary to the AME states, it was already shown that some k-uniform states are notlocally equivalent[RGA20], this resultwas indeed based ona comparison of Schmidt ranks of reduced densitymatrices. Nevertheless, the aforementioned rank argument is never conclusive for SLOCC-verifcation of two k-uniform states of minimal support. In this Chapter, we tackle those two cases, i.e k-uniform states of the minimal support and AME states, for which thebound on thenumber k is saturated, and provide general techniques of SLOCC-equivalence verifcation between such states. Webegin with theformal defnitionof SLOCC-and LU-equivalence.Two N-qudit states |ψi and |φi are LU-equivalent if one canbe transformed into anotherbylocal unitary operators, i.e |φi = U1 ⊗· ··⊗ UN |ψi . (6.1) Similarly, two states |ψi and |φi are SLOCC-equivalent if and only if there exists a local invertible operator connecting those states[DVC00]: |φi = O1 ⊗· ··⊗ ON |ψi . (6.2) Since LU-and SLOCC-equivalences are equivalence relations, the state space mightbenaturallypartitioned into LU classes and SLOCC classes respectively. Asa consequence of Kempf-Ness theorem[KN06],two AME states, or more generally k-uniform states are SLOCC-equivalent if and only if they are LU-equivalent[GW10]. 6.2 Local equivalences, case 2k 1 and 2 4 particles is equivalent to the classifcation of relevant orthogonal arrays, OA(d2,N,d,2),uptopermutationof indicesoneachposition. Potentially, for N =4 two AME(4,d) states of minimal support might be in the same SLOCC class, even though thecorrespondingOA are notequivalent. In literature, the classifcation of OAs is usually considered up to permutations of rows and columns[HSS97, YMJ08]. Note thatpermutation of columns resembles thephysicaloperationof exchanging subsystems. Hence, while verifyingthe SLOCC-equivalence one should always indicate whether such operations are considered[VDDMV01, GGM18]. In particular,bythe classifcation of OAs, there is at most one OA(dk ,N,d,k) for dimensions d =2,..., 17 and for any strength k and any number N [Slo]. Therefore, 2-uniform state with minimal support and the local dimension d =2,..., 17 are always SLOCC equivalent or SLOCC equivalent after permutation of parties. Conjecture 2. All 2-uniform states of minimal support are LU-equivalent, and hence represent the same SLOCC class. 6.6 Three-uniform states In Section 6.5, we have shown that the number of LU/SLOCC classes for 2-uniform states of minimal support greatly coincides with the number of relatedOAs which are non-isomorphic. In particular,two states whichdi˙er onlybyphases areLU-and SLOCC-equivalent.Aswe shall see, thisisina strong contrast to the 3-uniform states. We shall illustrate it on the example of AME(6,d)states.Note, that for any d ≥ 4 there exists an AME(6,d)state with minimal support[RGRA17], its preciseformmightbeobtainedbyreading consecutiverowsofthe corre-spondingOA(d3,6,d,3) from theOAs table[Slo]. One mayenhance successive terms of an AME(6,d)state with anyphase factor |ω| =1. Such family of states: d−1 1 |AME(6,d)ωi = √ X ωi,j,k |i, j, ki⊗|ψi,j,ki ! dd i,j,k=0 is obviously a family of AME(6,d)states. We focus our attention on states with all phases are equal to unity ωi,j,k =1 with one exception: ω0,0,0 = α. We denote them as|ψαi. In[BR20],Ishowed that states |ψ eiφ1 i and |ψ iφ2 i e are not LU-equivalent unless φ1 = φ2 + πt for some integer t. In such a way, I obtained a continuous family of non-LU-equivalent AME(6,d)states with minimal support. Corollary 6. With the above notation, the AME(6,d)states,d ≥ 4: ! 1 |AME(6,d)iφ i := √ e iφ |000000i + X |i, j, ki⊗|ψi,j,ki (6.4) e dd i,j,k6 =(0,0,0) arepairwise in di˙erent LU-and SLOCC-classes for all phases φ ∈ [0,π). In fact, for any k-uniform state with minimal support where k> 2, the similar construction of continuous non LU-equivalent family might be provided. Corollary 7. If there exists a k-uniform state with minimal support |ψi where k> 2, then there are infnitely manypairwise non LU-and SLOCCequivalent k-uniform states 6.7 Number of non SLOCC-equivalent AME orbits In this section, we shortly summarize the number of non-SLOCC-equivalent AME states (the number of orbits) and AME states with minimal support. The existence of AME states with minimal support for N,d < 8 was analyzed[Ber17]basedonthetableofOAsand similarcombinatorialdesigns. According to the discussion presented in the previous sections, if N ≥ 6 existence of AME(N, d)state with minimal support persuade to infnitely manynon-SLOCC-equivalent such states, see Corollary7. inTable 6.3 and Table 6.2 respectively. Note, that verifcation of the existence of AME states which are not necessarily of the minimal support is more complex problem. Several results concerning this problem[HS00, PPGB09, HGS17, Ber17, HESG18] are summarizes in the tables of AME states[HW]. Although the exact classifcation ofAME statesupto SLOCC-equivalenceisnotknown,in somespecifccases non-trivial lowerbound is given. 6.8 Further discussion and open problems As we have shown, the LU-and SLOCC-classifcation of k-uniform states, evenofminimal support,isin facta complex project,whichinvolves many open mathematical problems, in particular: 1. Existence and extensionofmutually orthogonal Latinhypercubes. 2. Classifcation of Hadamard matrices of Butson type B(d, d). 3. Classifcation/uniquenessofOAsof index unity(withoutpermutation). among others. In addition, we discuss some open problems regarding LU-and SLOCC-classifcation of k-uniform states with minimal support in a detailed way. Table 6.3: The minimalnumberof non-SLOCC-equivalentAME orbits. The question markby zerovalue suggests thatthe existenceofthe relevant state is dubitative, while 0 itself emphasizes that the relevant state certainly does not exist. Note that the AME(4, 6)was constructed in Chapter5. Firstly, consider any two k-uniform states of minimal support |ψi and |ψ0i with all phases equal to 1. We have showed that both states are LUequivalent if and only if there exist localpermutation matrices relating |ψi and |ψ0i. On the other hand,both states are in one-to-one correspondence withOAsof index unity. In such away, the existenceoflocalpermutation isequivalenttoan isomorphismbetweentwoOAsof indexunity,and hence LU-classifcationis equivalent to the classifcation ofOAs of index unity. ClassifcationofOAsof index unityis anopen mathematical problem.For small number of parties N, uniformity k, and the local dimension d is small d< 9,itis known thatallOAsof index unity areisomorphic[BM08, WW92, ST07, SGZ18]. Conjecture 3. All OAs of index unity with small local dimension d< 9 are isomorphic by permutations of symbols on each level. Equivalently, all k-uniform states with minimal support and all terms phases equal are LUequivalent. Secondly, in Proposition9 shows that any LU-operatorbetweentwo k-uniform states of minimal support consists of products of phase (diagonal) and permutation matrices (for 2k 2. The precise description of SLOCC classes, however, is not given. Thirdly, Section6.3 discusses the signifcantdi˙erencebetween k-uniform states of minimal support with 2klary 3). Furthermore, when uniformity achieves its upper bound, i.e. k = 2N, such equivalenceis always providedby a Butson-type matrix or monomial matrix (Proposition 11). This restriction is valid, however, only for small local dimensions d and number of parties N (Remark 1), in particu
lar for arbitrary N and d< 9. I illustrate the usefulness of the provided criteria on various examples. Firstly, I show that the existence of AME states with minimal support of 6 or more particles yields the existence of infnitely many such non-SLOCC-equivalent states (Corollary7). Secondly, I show that some AME states cannot be locally transformed into existing AME states of minimal support (Proposition 10). Chapter 7 Roots ofpolynomial invariants In this chapter, I provide necessary and suÿcient conditions for generic Nqubit statestobe equivalent under SLOCCoperations usinga singlepolynomial entanglement measure. More precisely,Iinvestigate how the roots of the entanglement measurebehave under SLOCCoperations.I demonstrate that SLOCC operations maybe represented geometricallybyMius transformations on the roots of the entanglement measure on the Bloch sphere. I show that if the states are SLOCC-equivalent, then the roots of thepolynomial entanglement measure for each state must be related by a Mius transformation, whichis straightforwardtoverify. I use thisprocedureto show that the roots of the 3-tangle measure classify 4-qubit generic states. Moreover, I propose an alternative method to obtain the normal form of a 4-qubit state whichbypasses thepossibly infnite iterative procedure. An extension of presented results in which the author’s part was not substantial canbe foundinthejoint work[BQA21], attachedtothe dissertationin AppendixE. 7.1 Polynomial Invariant Measures An entanglement measure is anyfunction E(|ψi) defned for all pure states of N qubits which vanishes for all separable states. As we discussed in Introduction, a particularly desired feature of an entanglement measure is invariance under SLOCCoperations. Apart fromthe normalizationof states, aSLOCC operation acting on N-qubit state |ψi can be represented by the action of a local invertible operator with a determinant equal to one, O∈ ~ N SL(2, C)⊗N , where SL(2, C)is the special linear group of complex matrices of order 2. In this chapter, we consider only the subclass of invertible matrices with determinant one, unlike in Chapter6, see Eq.(6.2). To distinguish the subclass of invertible matrices with determinant one from the class of all invariable matrices, we shall denote them with the calligraphic font O in contrast to the notation O used in Chapter6 for all invertible matrices. Agiven entanglementmeasure E(|ψi) defned on the system of N qubits is called a SL-invariantpolynomial of homogeneous degree h if it ispolynomial in the coeÿcients of a pure state |ψi and satisfes E�κ O|ψi = κhE�|ψi  (7.1) ~N for each realconstant κ> 0 and operator O ~N ∈ SL(2, C)⊗N [DVC00, GW13, ES14]. Note that Eq.(7.1) relates the entanglement measure of any nonnormalized state with the entanglement measure of the related normalized statebyfxingoperator O ~N asanidentityoperatorandchoosingan appropri ate constant κ.A SL-invariantpolynomial (SLIP) measure of homogeneous degree h will be denoted as SLIP hN , where the upper index indicates the degree h ofthepolynomialandthelower indexisrelatedtothenumberof qubits N. Among various approaches to the problem of quantifcation and classifcationofentanglement,theoneviaSLIP measuresturnedouttobeaparticularly useful. The most famous examples of such measures are concurrence and three-tangle, which measure the 2-body and 3-body quantum correlationsofthesystem respectively[Woo98, CKW00]. SLIP measuresprovidea convenient methodfor entanglement classifcation and its practicaldetection atthe same time.For example,itwasdemonstrated that almostall SLOCC equivalence classes canbe distinguishedbyratiosof SLIPmeasures[GW13]. Moreover, any given two states are SLOCC-equivalent if a complete set of SLIP measures achieves the same values forbothof them[VES11]. How
ever, the size of such a set grows exponentially with the number of qubits N, making it intractable to use this approachto decide SLOCC-equivalence between states with more than four qubits[LBS+07]. Any SLIP measure E might be further extended to the set of mixed states by determining the largest convex function on such set which coincides with E onthe setof pure states[Uhl98]. Despite the simple defnition of a convex roof extension, its evaluation requires non-linear minimization procedure, and through this,isachallenging taskforageneral density matrix[OV06, RDMLA14, TMG15,ROA16, SG18]. An auspicious attempt to address this taskwas carried outbyintroducing the so-called zero-polytope, the convexhullof pure states withvanishing E measure[OSU08, LOSU06, Ost16, GO18]. In the simplest case of rank-2 density matrices ρ, the zeropolytope canbe representedasaconvexpolytope inscribedinaBlochsphere, spanned by the roots of E [LOSU06, RA16a]. I adapt this approach, and focus only on the vertices of the zero-polytope, equivalently the roots of polynomial invariant. 7.2 System of roots We begin with the general discussion of roots of a SLIP measure, which adapts approachpresentedin[LOSU06, RA16a]. Firstly, considera (N +1)partite qubit state |ψi. Such a state canbe uniquelly written as |ψi = |0i|ψ0i + |1i|ψ1i , (7.2) which provides the canonical decomposition of its reduced densitymatrix ρ = |ψ0ihψ0| + |ψ1ihψ1| obtained by tracing out the frst qubit. Notice that both states |ψ0i and |ψ1i are in general neither normalized nor orthogonal. Secondly, consider the following family of non-normalized states |ψzi = z |ψ0i + |ψ1i , (7.3) where z ∈ C ˆis taken from the extended complex plane Cˆ, which contains all complex numbers plus infnity. We shall refer to such a representation as the extended plane representation. Furthermore, consider any SLIPh measure E defned on the set of N N partite pure qubit states. By the defnition, E ispolynomial in the coeÿcients of |ψzi, henceitis alsopolynomialinthe complexvariable z [OSU08]. In such away, thepolynomial E(z |ψ0i + |ψ1i) has exactly h roots: ζ1,...,ζh (whichmaybe degenerated and/or at infnity), in accordance with thedegree of E. Using the complexnumber z, the states|ψzi canbemapped to the surface of a sphere via the standard stereographic projection (θ, φ) := (arctan 1/|z|, −arg z) writtenin spherical coordinates. Furthermore, apoint on the unit 2-sphere (θ, φ) canbe associated with the following quantum state θθ | ψezi := cos |ψ0i + sin e iφ |ψ1i (7.4) 22 with z = ctg(θ/2) e−iφ. Note that |ψ0i lies in the Northpole and |ψ1i lies in the Southpole, see Figure7.1.We shall referto such a representationasthe Bloch sphere representation. Note the following proportionality of states: | ψezi∝|ψzi, and that neither of these states is normalized, since |ψ0i and |ψ1i are not normalized in general either. −iφ relatedbythe stereographic projection z = ctg(θ/2) e. 7.3 Local operations on the system of roots Each linear invertible operator O = �ab , might be associated with a cd 0 az+b Mius transformation z 7→ z:= , which maps the extended complex cz+d plane C ˆinto itself[BZ17, Aul12]. The compositionoftwo such transforma
tions is related to the multiplication of the associated operators. Further 0 dz−b more, z 7→ z:= is an inverse Mius transformation related with −cz+a O−1 = � d −b . Although Mius transformations aretypically represented −ca on the extended complex plane, one mayalso representthem as transformations on the Bloch sphere via the stereographic projection. This correspondencebetween invertible operators and Mius transformations represented on the Bloch sphere was already successfully used for SLOCC classifcation ofpermutation-symmetric states[BKM+09,RM11, KL20]. Studies of e˙ects of SLOCC operations on the system of roots might be summarized in the theorembelow. Theorem 2. Consider an (N +1)-partite pure quantum state |ψi = |0i|ψ0i+ |1i|ψ1i. The roots ζi of any SLIPh entanglement measure associated to the N partial trace of the frst qubit: 1. are invariant under invertible operators, i.e. invariant under 1⊗O ∈ ~ N SL(2, C)⊗N operators; dζi−b 2. transform via an inverse Mius transformation ζ0 = w.r.t the i −cζi+a O = �ab ⊗ 1N ∈ SL(2, C)operator. cd Before we proceed with the proof of Theorem 2, we shall emphasize that normalizing the states |ψ0i and |ψ1i after the action of the local operator O∈ SL(2, C)⊗N+1, as is the case in existing related works[OSU08, N~+1 LOSU06, GO18, RA16a, RA16b], spoils the consistency of the stereographic projection with the roots transformations. Asa consequence, the action of SLOCC operators on the states |ψzi would no longerbe givenbythe associated Mius transformation, and the statements presented in Theorem 2 would no longer hold true. Proof of Theorem2. As we discussed, a N +1 partite qubit state |ψi∈ ⊗(N+1) Hmightbe written as 2 |ψi = |0i|ψ0i + |1i|ψ1i . (7.5) This form provides the canonical decomposition of the reduced density matrix ρ = |ψ0ihψ0| + |ψ1ihψ1| for the two non-normalized states |ψ0i , |ψ1i∈H⊗N . Consider now a re 2 versible operator O = �ab ∈ SL(2, C)acting on the frst qubit. Under the cd action of above operator, the state |ψi is transformed into  |ψ0i := O|ψi = |0i a |ψ0i+b |ψ1i +|1i c |ψ0i+d |ψ1i = |0i|ψ00 i+|1i|ψ10 i (7.6) where |ψ00 i := a |ψ0i + b |ψ1i , (7.7) |ψ10 i := c |ψ0i + d |ψ1i . (7.8) Consider now any superposition of states |ψ00 i and |ψ10 i. In particular, observe that |ψ0 i := z |ψ00 i + |ψ10 i = za |ψ0i + b |ψ1i + c |ψ0i + d |ψ1i z =(az + b) |ψ0i +(cz + d) |ψ1i az + b ∝|ψ0i + |ψ1i . cz + d In other words, we have az + b 0 O|ψzi = |ψz0 i ,z = , (7.9) cz + d i.e. the operator O transforms states in the extended plane representation by applying a Mbius transformation on the index z. Suppose that ζi is a complex rootofapolynomial function E, i.e. E(ζi |ψ0i + |ψ1i)=0. Acting on the frst qubit with operator O, the density matrix obtained by tracing out the frst qubitbecomes |ψ00 ihψ0 | + |ψ10 ihψ10 |, and hence the entanglement 0 measure E will vanish for a roots ζ0 , such that E(ζ0 |ψ00 i + |ψ10 i)=0. Using ii Eqs.(7.7)-(7.8), the later equation transforms into aζi 0 + b  E (cζi 0 + d) |ψ0i + |ψ1i =0 (7.10) cζi 0 + d where the factor (cζi 0 + d) is not relevant. Comparing with the equation for the roots before the action of O, we conclude that the roots transform according to the inverse Mius transformation as dζi − b ζ0 = , (7.11) i −cζi + a under the action of the operator O. Consequently, the roots of the zeropolytope transform with respect to the inverse Mius transformation associated to the operator O = �ab . cd Furthermore, considermulti-localoperatorsO = O1 ⊗ ... ⊗ON acting ~ N on the remaining qubits of the state |ψi from Eq.(7.5). The state |ψi transforms accordingly as |ψ0i := O ~|ψi = |0iO ~|ψ0i + |1iO ~|ψ1i . (7.12) NN N | {z i } | :={z |ψ0 i } :=|ψ0 01 After the action of O ~, a value of entanglement measure E reads N  Ez |ψ00 i + |ψ10 i = EO ~�z |ψ0i + |ψ1i  . N However, since E is SL(2, C)⊗N invariant function, one may conclude that E(z |ψ0i + |ψ1i)=0 i˙ E(z |ψ00 i + |ψ10 i)=0. Hence the roots of both polynomial equations are the same. As a consequence, the roots of the the zero-polytope remain unchanged under the action of O ~. This concludes N the proof of Theorem2. 7.4 States discrimination The decomposition(7.2) canbeperformed with respect to any other sub �ab  system, each with its own system of roots. Any local operator Ok = cd acting on the k-th subsystem will infuence independently the corresponding k-th system of roots and the associated zero-polytope according to the dζi−b Mius transformation ζi 7→ . A global action of a local operator −cζi+a O1 ⊗···⊗ON+1 shall a˙ect all roots and thus related zero-polytopes. Since a Mius transformations are bijections of the Blochsphere, the total numberof roots will alwaysbe preserved[RA16b].Furthermore, since Mius transformations are fully classifed, the existence of a local transformation betweentwogiven statesbecomes straightforwardtoverify. Inthisway, Theorem2providesnovelsolutionfortheproblemof discrim
inating (N + 1)-qubit states up to the SLOCC-equivalence[VES11, ZZH16, GW11,BR20].Toverifyiftwo pure states are SLOCC-equivalent, one can thus use the following procedure: 1) Choose anySLIPh entanglementmeasure of degree h ≥ 3 and calculate N its roots for each subsystem forboth states. Note thatageneric state will always have h roots for each subsystem. 2) Focus on one subsystem i, where 1 ≤ i ≤ N +1 and choose 3 of the total h roots from each state. 3) Write the unique Mius transformation between the two triplets of roots. Derive thelocal operator Oi associated to suchtransformation. 4) Choose a di˙erent set of 3 roots of the second state and repeat step 3). Repeat it for all 3!�h possibilities. 3 5) Repeat steps 3) and 4) for all other subsystems. Consider the tensor products of all the local operators obtained. This results in a fnite set of operators of the form O1 ⊗· ··⊗ON+1. 6) If the two given (N + 1)-qubit states are SLOCC-equivalent, one of obtainedoperatorsmusttransformone stateintotheother. Otherwise, they are not SLOCC-equivalent. Presented procedurehastwo mainimportant features. Firstly,it factorizes the problem of fnding SLOCC-equivalence. Indeed, local operations are determined separately for eachsubsystem. Secondly, it discretizes the initial discrimination task. Indeed, there are at most (3!�h )N+1 local operators 3 which might provide SLOCC equivalencebetween initial states. √ state 1/ 2(|0i|GHZi + |1i|Wi). This system of fourpoints canbe mapped intoa normal system (i.e. symmetrically relatedpoints z, −z, 1/z, −1/z)by a Mius transformation. Similar local transformations can be performed with respect to other subsystems, transforming the states into a state in the normal form. 7.5 Normal system of roots and cross-ratio Any three distinctpoints on the sphere canbetransformed onto anyother three distinct points via a unique Mius transformation. This is not the casefor fourpointsonthe sphere.Foragivenfourpointsonthe extended complex plane z1,z2,z3,z4, one may associate a so-called cross-ratio z3 − z1 z4 − z2 λ�z1,z2,z3,z4:= , (7.13) z3 − z2 z4 − z1 which is preserved under any Mius transformations[RM11, BZ17]. Two systems of four distinct points are related via Mius transformations if and only if their associated cross-ratios are the same. The cross-ratio is not invariant underpermutations ofpoints, however, and depending on the ordering taken for the fourpoints,it takes six relatedvalues[RM11]: 11 λ − 1 λ λ, , 1 − λ, , , . λ1 − λλλ − 1 Aparticular interesting set of fourpoints is one of the formz, 1/z, −z, −1/z, which we call a normal system. 1 Consider such a set of four related complexpoints Φ= {z, , −z, −1 }. zz Itis convinient to associate with them the cuboid spannedbyeightpoints: 1 11 1 Φ ∪ ¯z, , ¯, −¯, Φ= n, −z, − z, z, − o z zz ¯z ¯ as it is presented on Figure 7.3. Notice that all six faces of the cuboid are parallel to one of the canonical planes: XZ, XY , or YZ. This property is equivalenttothe initial assumptionthatthesetofpoints Φ is in normal form. Clearly,allrotationsoftheBlochball(associatedtothe unitaryoperations) preserve the form of the cuboid. Nevertheless, only a special subgroup of all rotations leave faces of the cuboid parallel to XZ, XY , or YZ. This special subgroup G24 contains exactly 24 elements and is generatedbythree rotations of π/2 around X, Y , and Z axis:  cos π/4 −i sin π/4 1 1 −i Rx(π/2) =−i sin π/4 cos π/4 = √ � −i 1 , (7.14) 2 cos π/4 −sin π/4 1 �1 −1 Ry(π/2)== √ , (7.15) sin π/4 cos π/4 11 2 e−iπ/4 0 1 �1−i 0 Rz(π/2) =iπ/4 = √ . (7.16) 0 e0 1+i 2 Notice that G24 is a group of rotations preserving the regular cube (the group of orientable cube symmetries). All rotations in the G24 group preserve the normal-form structure of Φ. Therefore, the normal form is uniquelly determined up to 24 rotations in the G24 group. Weshowthatanysetoffourpointsmaybemappedintoa normal system, for which z, 1/z, −z, −1/z. Proposition12. Any system of non-degeneratedfourpointsz1,z2,z3,z4 on 1 the Bloch sphere can be transformed onto the normal form z, , −z, −1 via zz some Mius transformation T . Transformation T is uniquely defned up to 24 rotations in the group G24. Figure 7.3:Anormal systemof fourpoints z, 1/z, −z, −1/z together with the conjugate points ¯z, −¯z span the cuboid whose faces are parallel z, 1/¯z, −1/¯to the XZ, XY and YZ planes. There are exactly 24 rotations of the Bloch sphere whichpreservethis property, composing the elements of the group G24. Two of them: rotation by a π/2 angle around X and Y axes are presented on theleft and right respectively. The system of roots transforms according 0 z−i 00 z−1 to Eqs.(7.14-7.16), which gives z 7→ z:= and z 7→ z:= for the −iz+1 z+1 two presented rotations. Proof. Consider a complex number λ. There exists a complex number z, such that the cross-ratio of the fourpointsis equal to λ, i.e.  z, 1; −z, − 1= λ. (7.17) zz Indeed, the cross-ratio on the left side equals 4z2/(1 + z2)2, andthe equation 4z2/(1 + z2)2 = λ has exactly four complex solutions √ 4 − 2λ +1 − λ 11 z0 = ,, −z0, − . (7.18) 2λz0 z0 In suchaway, foragivenvalue λ there exists a unique λ-normal system, such 1 that the cross-ratio of its vertices is given by (z0, ; −z0, − 1 )= λ. Note z0 z0 that replacement of the vertex z0 byanyother vertex z0, 1/z0, −z0, or −1/z0 1 does not change the value of the cross-ratio (z0, ; −z0, − 1 )= λ. Further z0 z0 more, there is a unique Mius transformation T whichmaps z1,z2,z3 onto z0, 1/z0, −z0, with the remaining z4 mapped onto −1/z0. Observe that the value z0 is unique up to its inverse, opposite and opposite inverse elements, according to Eq.(7.18), with the correspondingMius transformations associated to the matrices T,σxT,σyT and σzT . Each of those transformations maps the entire set ofpoints {z1,z2,z3,z4} onto the same set ofpoints {z0, 1/z0, −z0, −1/z0}, (the exact bijection between those two sets of roots isdi˙erent for each transformation). Depending on the chosen order of four points {z1,z2,z3,z4}, the cor1 λ−1 responding cross-ratio takes one of six values: λ, λ 1 , 1 − λ, and 1−λ , λ λ . For each value, there is a corresponding set of solutions of the form λ−1 {z0, 1/z0, −z0, −1/z0} via Eq.(7.18) with related four Mius transformations. hence, in total there are 24 Mius transformations that map any four non-degeneratedpoints ontoa normal system. Each transformation is related to an element of the group G24 which has exactly 24 elements. 7.6 SLOCC classifcation of four qubit states In this section I discuss how one may use the normal system of roots for SLOCC-classifcation of small systems. Firstly, consider the three-qubit case. Genuinely entangled pure threequbit states are SLOCC-equivalentto oneoftwo states[Sud01] 1 |GHZi = √ (|000i + |111i), 2 1 |Wi = √ (|001i + |010i + |100i), 3 as we discussed in Introduction. Consider the 2-tangle τ(2) [Woo98]as the entanglement measure, and its roots as we discussed so far. One may use the roots of τ(2) to distinguishbetween thetwo classes. Indeed, all reduced density matrices of the |Wi state have a single root, while there are always two distinctrootsfor the |GHZi state[RA16a]. Secondly, consider the four-qubit case. Contrary to the case of three qubits, thereare infnitely manySLOCC classesof four qubitstates[Sud01]. Nevertheless, four-qubit states were successfully divided into nine families, most of which contains an infnite number of SLOCC-classes[VDDMV01, CD07, SdVK16]. The family with the most degrees of freedom, so called Gabcd family,is representedby states of the following form[VDDMV01]: a + da − d |Gabcdi = �|0000i + |1111i + �|0011i + |1100i  22 b + cb − c + �|0101i + |1010i + �|0110i + |1001i , 22 2 6b2 = 2 6d2 where a= 6c= are pairwise di˙erent. Note that any generic 4qubit states,i.e. 4-qubit states with random coeÿcientsbelonging to the Gabcd family,sinceit has the most degreesoffreedom. Choosing the 3-tangle τ(3) [CKW00]as the entanglement measure, we computed roots of states |Gabcdi. As we shall see, each state has four nondegenerate roots already in the normal form. Indeed, the state |Gabcdi has the following decomposition with respect to the frst subsystem |Gabcdi = |0i|ψ0i + |1i|ψ1i, where a + da − db + cb − c |ψ0i = |000i + |011i + |101i + |110i , 2 222 a + da − db + cb − c |ψ1i = |111i + |100i + |010i + |001i . 2 222 Suppose that the three-tangle vanishes, τ(3)(ζ |ψ0i + |ψ1i)=0. Since τ (3) is a SL(2, C)⊗3 invariant measure, for anylocal operators O1, O2, O3 we have τ (3)(O1 ⊗O2 ⊗O3)�ζ |ψ0i + |ψ1i =0 . Thus |ψ0i =(σx ⊗ σx ⊗ σx) |ψ1i , |ψ1i =(σx ⊗ σx ⊗ σx) |ψ1i , where σx,σy,σz arePauli matrices. Hence, taking all local operators O1, O2, and O3 equal to σx, we conclude that 1 0= τ(3)(σx ⊗ σx ⊗ σx)�ζ |ψ0i + |ψ1i = ζ |ψ1i + |ψ0i∝ |ψ0i + |ψ1i , ζ (7.19) and 1/ζ is another root of τ(3). Similarly, considering (σy ⊗ σy ⊗ σy) and (σz ⊗ σz ⊗ σz), one mayfnd another two roots −ζ, −1/ζ of a measure τ (3). Furthermore, it shows that the roots ofτ(3) evaluated on anystate from the Gabcd family are symmetrical with respect to rotations around X, Y, Z axes by the angle π. Writting τ(3)(z |ψ0i + |ψ1i)=0 explicitely, we obtain the equation 2 τ(3)(z |ψ0i + |ψ1i)= Az4 − 2(2B + A)z + A =0, (7.20) where A =(b2 −c2)(a2 −d2) and B =(c2 −d2)(a2 −b2). The above equation 22 is non-degenerated i˙ A, B, A +2B 6a= b2 = c6 =0, whichhappens i˙ 66= d2 are pairwise di˙erent. Recall that the normal form of roots is unique up to the group of rotations G24. In such a way, the problem of SLOCC-equivalence of states |Gabcdi becomes solvable, with a discrete amount of solutions. Indeed, two states in the Gabcd class are SLOCC equivalent i˙ one canbe obtained from the other bythe actionofan elementofthe fniteclassofoperatorsO∈G⊗4 . We thus 24 fnd that exactly 192 states of the form |Gabcdi are SLOCC-equivalent. Proposition 13. Two states |Gabcdi and |Ga0b0c0d0 i are SLOCC-equivalent i˙ theircoeÿcients arerelatedby the following threeoperations: 1. multiplication by a phase factor (a0,b0,c0,d0)= eiφ(a, b, c, d), 2. andpermutation σ ∈ S4 of coeÿcients (a0,b0,c0,d0)= σ(a, b, c, d), 3. and change of sign in front of two coeÿcients from a, b, c, d. Before we proceed with the proof, note that the symmetry in Proposition13isgivenbytheWeylgroupof Cartantype D4. Such symmetry has alreadybeen observedamong generatorsof four-qubitpolynomialinvariants exhibit thistypeof symmetry[LT03, CD07, HLP14].Asaconsequence, this resultconstitutes anew relationbetween 4-qubitinvariants and the convex roof extension of 3-tangle τ(3). This may shed some light on the problem of generalizing the CKW inequality[CKW00]for four qubit states[GW10, EOS09, RDMLA14, ES14,ROA16], andbeyond[EOS09, GW10, ES15]. Lemma 1. Any local operator O = O1 ⊗O2 ⊗O3 ⊗O4 ∈ SL(2, C)⊗4 which 22 transforms states |Gac0d0 i ∝ O|Gabcdi with a= 6c= d2, is of the 0b06b2 = 6form Oi ∈G24. Proof. Consider a local operator O1 acting on the frst qubit which transforms the state |Gabcdi onto |Ga0b0c0d0 i. Operator O1 also transforms corresponding systems of roots denoted as Λ and Λ0, respectively, via the action of the related Mius transformation. Note that both systems Λ and Λ0 are in the normal form, therefore, according to Proposition 12, we have that Oi ∈G24. Similar analysis with respect to all other qubits shows that O2, O3, O4 ∈G24. Proof of Proposition 13. Lemma1 shows that the search fora general form of an SLOCC-equivalencebetween the states |Gabcdi and |Ga0b0c0d0 i mightbe restricted to the search within the fnite class of operators O∈G⊗4 Note 24 . that the group G24 has only 24 elements, and hence one may numerically verify that there are exactly 8 × 24 = 192 states in the Gabcd family which are SLOCC-equivalent to |Gabcdi by O∈G⊗4 For example, the following 24 . tensor operation Rx π ⊗ Rx π ⊗ Rx π ⊗ Rx π  (7.21) 2222 transforms state |Gabcdi into |G−b−acdi. This might be written as a transformation of a tuples of indices: the tuple (a, b, c, d) is transformed into the tuple (−b, −a, c, b). Similarly, the operators showed on the following right hand sides provide the corresponding transformations of the tuple (a, b, c, d) on the left side: Ry π 2 ⊗ Ry π 2 ⊗ Ry π 2 ⊗ Ry π 2  Rz π 2 ⊗ Rz π 2 ⊗ Rz π 2 ⊗ Rz π 2  ←→ ←→ (a, d, c, b), (−d, b, c, −a), Ry(π) ⊗ Ry(π) ⊗ 1⊗ 1 ←→ (a, −b, −c, d), Rx(π) ⊗ Rx(π) ⊗ 1⊗ 1 ←→ (a, b, −c, −d), Ry(π) ⊗ 1⊗ Ry(π) ⊗ 1 ←→ (d, c, b, a), Rx(π) ⊗ 1⊗ Rx(π) ⊗ 1 ←→ (b, a, d, c) . In addition, the tuples (a, b, c, d) and (−a, −b, −c, −d) represent the same state, since states are defned up to the global phase. Note that any com-position of the above operations also providesSLOCC equivalencesbetween states of the form |Gabcdi. The eight aforementioned transformations of tuples generate allpossiblepermutations of the a, b, c, d indices, together with thechangeofasignofanytwoorallfour indices. Notethat thereare exactly in exactly 1+ � 4ways.Thisgives tuplesrepresentingSLOCC +1=8 192 2 24 permutationsin totalandfor eachpermutationthesigns canbe matched equivalent states, whichperfectly matches thenumerical result. Noteyet another trivial manipulation with indices a, b, c, d comes from multiplyingby a global phase. Thisoperation transformsthe indices as iθ (e a,e iθb, eiθ c, e iθd) ∼ (a, b, c, d) , resulting in the same quantum state for any real number θ ∈ [0, 2π). In particular, for θ = π, the system of opposite indices determines the same state as the initial one, i.e. (−a, −b, −c, −d) ∼ (a, b, c, d). Finally,we showa linkbetween normal systemsof roots and the normal form of pure states of four qubits in the Gabcd class. We say that a state is in its normal form if all its reductions are maximally mixed[VDDM03]. Theprocessto determinethe normalformofa state(ifit exists)is straightforward, although, it may also turn out to be an infnite iterative process [VDDM03]. However, presented results of Theorem 2 applied to the four qubits states in the Gabcd family show that this diÿculty canbeavoided. Indeed, consider the representative state of the Gabcd family is in normal form [VDDM03]and the corresponding roots associated with the three-tangleτ (3) measure also forma normal system. One maycalculate the roots associated with anystate in the Gabcd class, and transform themintoa normal systemof roots using a Mius transformation. Furthermore, the associated SLOCC operator to convert the initial state into a state in the normal form. We illustrate this procedure by transforming the widely discussed four-partite 1 state √ (|0i|GHZi + |1i|Wi) [GO18, LOSU06] into its normal form, see 2 Figure 7.2. Without presented technique, the standard wayof obtaining the normal form would result in an infnite iterative procedure. 7.7 Conclusions Inthischapter,Ishowedhowa single SL-invariantpolynomialentanglement measure SLIPh defned in N-quibit system canbe used to derive necessary N and suÿcientconditions for anytwo (N +1)-qubit states tobe equivalentunder SLOCCoperations. Thiswaspossiblebyshowing that the rootsof any SLIPh measure transform via Mius transformations under the SLOCC N operationsperformed on the subsystems (Theorem2). In thatway, SLOCC equivalence between two states is implied by the easily verifable existence of a Mius transformation relating the aforementioned roots for each sub-system, as it is described in a detailed way in the procedure presented in τ (3) is Section 7.4. Furthermore, I demonstrated that the 3-tangle measure enoughto classify 4-qubit generic states(Proposition13). Lastly,Ipresent a procedure of determining the normal form of states in the Gabcd family that circumvents thepossibility of an infnite iterative process of the standard procedure. Chapter 8 Concluding remarks 8.1 Conclusions This dissertation covers various aspects of the quantifcation and classifcation of entanglement in multipartite states andthe role of symmetry in such systems.The main resultsof the researchin this thesis canbe summarized in the following. 1. Chapter2introduces the notion of m-resistant states of N-partite systems and presents itby analogy with topological links (Defnition1). I present two general methods of constructing m-resistant states, one based on the Majorana representation of symmetric states (Proposition1), the other on thecombinatorialobjects known as orthogonal arrays (Propositions2 and3). 2. In Chapter 3, I introduce the notion of H-symmetric states, where H is an arbitrary subgroup ofpermutation group – see Defnition2. Proposition4provides an elementaryexampleofa quantum state with the given group of symmetry. 3. I introduce two families of states with remarkable symmetric properties: in Defnition4 excitation-states |Gi, related to a graph G, and in Defnition 3 Dicke-like states, determined by an embedding of a subgroup H< SN . Furthermore,I investigate entanglement properties of excitation states related to regular graphs, in particular the form of 2-partieconcurrence (Proposition7), and generalized concurrence (Proposition8)of such states. 4. I present two di˙erent methods of constructing excitation states: in Section 3.7 excitation states are obtained as ground states of Hamiltonians with 3-body interactions, while in Section 3.6 as an outcome of a quantum circuit, with satisfactory complexity. A simple example of fvequbit excitationis successfully simulated onavailable quantum computers: IBM -Santiago andAthens (Figure 3.8). 5. In Chapter5,Ipresentthe famous combinatorial problem of 36 oÿcers thatwasposedbyEuler, and its quantumversion.Ishow an analytical form of an AME(4, 6) state,whichmightbe seenasaquantum solution to the Euler’s problem. Moreover, I present a coarse-grained combinatorial structure behind constructed AME(4, 6) state, and elaborate how such coarse-grained combinatorial structures might lead to the construction of other genuinely entangled statesbeyond the stabilizer approach. 6. Chapter 6 presents general techniques of verifcation whether two k-uniform or AME states are SLOCC-equivalent. In particular, I show that SLOCC equivalencebetweentwo k-uniform states of N particles with the minimal support has one of the following forms: if k< 2N,it is providedbylocal monomial matrix (Proposition9and Corollary3), while if k =2N bya Butson-type complex Hadamard matrix or monomial matrix (Proposition11). 7. I apply general results concerning the SLOCC equivalence of AME states. In particular, I show that some AME states cannot be locally transformed into existing AME states of minimal support (Proposition 10), which falsifes the conjecture that for a given multipartite quantum system all AME states are locally equivalent. Furthermore, Ishow that the existence of AME states with minimal support of6or more particles yields the existence of infnitely many suchnon-SLOCCequivalent states (Corollary7). As an immediate consequence,I show that not all AME statesbelong to the classof stabilizer states. 8. In Chapter7,I showhowa single entanglement measure canbe used to derive necessary and suÿcient conditions for any two multipartite quantum qubit states tobe equivalent under SLOCC operations. Precisely, I show that under an SLOCC operation the roots of any appropriate measure transform via related Mius transformation (Theorem 2). SLOCC equivalence between two states is implied by the easilyverifable existenceofa Mius transformation relatingthe afore mentioned roots for each subsystem, as it is precisely described in the procedure presented in Section 7.4. 9. I apply general results described in Chapter 7 to 4-qubit systems. In particular,I demonstrate that the 3-tangle measure is enough to classify 4-qubit generic states (Proposition13). Lastly,Ipresenta pro
cedure of determining the normal form of a generic 4-qubit state that circumvents thepossibilityof an infnite iterative process of the standard procedure. 8.2 Open problems As we have demonstrated so far, the quantifcation and classifcation of entanglement for multipartite states is an ambitious long-term project. We hope that the results presented in this dissertation will contribute to the development of these complex issues. In addition, we outline related problems and important open questions for future research. A. Aswe elaborated in Chapter2,it is not clear if m-resistant states of N-qubit do exists for any m =0, 1,...,N − 2, especially among symmetric states. One may also pose the more general problem of verifcation whether, for any number of parties N and number m, there exists an m-resistant N-qudit state, see Conjecture1. B. Admissible groups of symmetries in multipartite quantum systems of N-qudits. For anysubgroup of symmetric group H< SN , what is the minimal local dimension d for which there exists a H-symmetric state |ψi∈H⊗N , see Question1? In particular, whichgroups of symmetries d are admissible in systems of N qubits? C. Chapter 5 presents a coarse-grained combinatorial structure behind constructed AME(4, 6) state. Is there any general method for constructing suchobjectsinanydimension?Basedon such coarse-grained combinatorial structures,isitpossibletoprovidea general construction of other genuinely entangled statesbeyond the stabilizer approach? D. Uniqueness of AME states in small dimensions. Verify if all OAs of index unity with small local dimension d< 9 are isomorphicbypermutations of symbols on each level (Conjecture 3). From this would follow that all related k-uniform states with minimal support and all phases equal are LU-equivalent. E. PreciseformofSLOCC-equivalencebetweenAME(2k,d)states. Which subclass of Butson-type complex Hadamard matrices might appear in the LU-equivalencebetween to AME(2k,d)states of the minimal support? Verify if all AME(2k,d)states with minimal support are LUequivalent if and only if they are LM-equivalent (Conjecture4). F. Linear structures of orthogonal arrays and corresponding k-uniform and AME states. Verify if AME states related to di˙erent linear structures of classical combinatorial designs are not equivalent. In particular, verify whether three AME(4,9) states presented in Conjecture 5 are SLOCC-equivalent. G. CKW-equalities[CKW00]. Proposition 13 constitutes a new relation between 4-qubit invariants and the convex roof extension of 3-tangle. Isitpossibleto understandpresented relationin contextof generalized CKW(in)equalities of four qubit states? H. Classifcation of large multipartite systems. Chapter 7 shows how generic 4-qubit states may be classifed by the 3-tangle measure. Is itpossible to provide such classifcation for generic N-qubit states, N> 4, via a single entanglement measure? Bibliography [AB06] S. Anders and H. J. Briegel. Fast simulation of stabilizer circuits using a graph-state representation. Phys. Rev. A, 73:022334, 2006. [AC13] L. Arnaud and N. Cerf. Exploring pure quantum states with maximally mixed reductions. Phys. Rev.A, 87:012319, 2013. [ADS13] M. Amin,N. Dickson, andPeterM. Smith.Adiabatic quantum optimization with qudits. Quantum Information Processing, 12:1819–1829, 2013. [AI07] G. Adesso and F. Illuminati. Strong monogamy of bipartite and genuine multipartite entanglement: The Gaussian case. Phys. Rev.Lett.,99:150501, 2007. [AME] List of Open Quantum Problems, Problem 35, IQOQI Vienna. https://oqp.iqoqi.univie.ac.at/
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Raissi, Stochastic local operations with classical communication of absolutely maximally entangled states, Phys. Rev. A 102,022413 (2020). [C] A. Burchardt, J. Czartowski, and K. łyczkowski, Entanglement in highly symmetric multipartite quantum states, Phys. Rev. A 104, 022426 (2021). [D] S. Rather∗ , A. Burchardt∗, W. Bruzda, G. Rajchel-Mieldzio¢, A. Lakshminarayan, K. łyczkowski, Thirty-six entangledoÿcers of Euler: Quantum solution to a classically impossible problem, ArXiv: 2104.05122 (2021). ∗Contributed equally [E] A. Burchardt, G. M. Quinta, R. André, Entanglement Classifcation via Single Entanglement Measure, ArXiv: 2106.00850 (2021). A Cut-resistantlinks and multipartite entanglement resistant to particle loss B Stochastic local operations with classical communication of absolutely maximally entangled states C Entanglement in highly symmetric multipartite quantum states D Thirty-six entangled oÿcers of Euler: Quantum solution to a classically impossible problem E Entanglement Classifcation via Single Entanglement Measure