Tensors in CS and geometry workshop, Nov. 11 (2014) Tensors and complexity in quantum information science Akimasa Miyake 1 CQuIC, University of New Mexico Outline Subject: math of tensors <-> physics of entanglement Aim: talk by a quantum information theorist about elementary concepts/results and many open questions 1. Classification of multipartite entanglement - geometry of complex projective spaces - polynomial invariants as entanglement measures 2. Entanglement and its complexity - complexity of quantum computation in terms of tensors - “coarser” classification of multipartite entanglement Classification of multipartite entanglement Miyake, Phys. Rev. A 67, 012108 (2003):dual variety, hyperdeterminants Miyake, Verstraete, Phys. Rev. A 69, 012101 (2004): 2x2x4 case Stochastic LOCC Entanglement: quantum correlation, never increasing on average under Local Operations and Classical Communication(LOCC) k k single copy of a pure state (ray) in H = ! 1 ⊗!⊗ " n k 1, ,k 1 1− … n − Ψ Ψ = ψ i ⊗!⊗ i ∑ i1!in 1 n i1,…,in =0 Ψ! = M1 ⊗!⊗ M n Ψ Ψ! SLOCC: probabilistic MSLj = ()k j (invertible) CP map, followed by cf. LOCC: deterministic MUj = ()k j etc. the postselection of a successful outcome SLOCC classification: classification of orbits of SL(k1 ) ×!× SL(kn ) (in bipartite cases: classification by the Schmidt rank ) Bipartite entanglement under SLOCC two k-level qudits system (pure states) on H = !k ⊗ !k k−1 GM= 12⊗ M T Ψ = ψ ii⊗ ψψii####→ MM12 ii ∑ ii12 12 ( 12) ( 12 ) ii12,0= basic matrix transformation the rank (Schmidt local rank) rs is invariant under invertible G: k equivalence classes decreasing under noninvertible G: totally ordered monotonicity 2 !P k −1 Secant variety Segre variety PHYSICAL REVIEW A, VOLUME 62, 062314 Three qubits can be entangled in two inequivalent ways W. Du¨r, G. Vidal, and J. I. Cirac Institut fu¨r Theoretische Physik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria ͑Received 26 May 2000; published 14 November 2000͒ Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accord- ingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communication ͑LOCC͒ with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the Greenberger-Horne-Zeilinger state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher-dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success. PACS number͑s͒: 03.67.Hk, 03.65.Bz, 03.65.Ca I. INTRODUCTION two pure states ͉␺͘ and ͉␾͘ can be obtained with certainty from each other by means of LOCC if and only if they are The understanding of entanglement is at the very heart of related by local unitaries ͑LU͓͒5,4͔. But even in the simplest quantum information theory ͑QIT͒. In recent years, there has bipartite systems, ͉␺͘ and ͉␾͘ are typically not related by been an ongoing effort to characterize qualitatively and LU, and continuous parameters are needed to label all quantitatively the entanglement properties of multiparticle equivalence classes ͓6–10͔. That is, one has to deal with systems. A situation of particular interest in QIT consists of infinitely many kinds of entanglement. In this context, an several parties that are spatially separated from each other alternative, simpler classification would be advisable. and share a composite system in an entangled state. This One such classification is possible if we just demand that setting requires the parties—which are typically allowed to the conversion of the states is through stochastic local opera- communicate through a classical channel—to only act lo- tions and classical communication ͑SLOCC͓͒4͔; that is, cally on their subsystems. But even restricted to local opera- through LOCC but without imposing that it has to be tions assisted with classical communication ͑LOCC͒, the achieved with certainty. In that case, we can establish an parties can still modify the entanglement properties of the equivalence relation stating that two states ͉␺ and ͉␾ are system and in particular they can try to convert one en- ͘ ͘ equivalent if the parties have a nonvanishing probability of tangled state into another. This possibility leads to natural ways of defining equivalence relations in the set of entangled success when trying to convert ͉␺͘ into ͉␾͘, and also ͉␾͘ states—where equivalent states are then said to contain the into ͉␺͘ ͓11͔. This relation has been termed stochastic same kind of entanglement—as well as establishing hierar- equivalence in Ref. ͓4͔. Their equivalence under SLOCC chies between the resulting classes. indicates that both states are again suited to implement the For instance, we could agree in identifying any two states same tasks of QIT, althoughEncounter this time the probability with of a tensors which can be obtained from each other with certainty by successful performanceWhat is of the the SLOCC task may classification differ from ͉␾͘ ofto 2x2x2 (3-qubit) case? means of LOCC. Clearly, this criterion is interesting in QIT ͉␺͘. Notice in addition that since LU are a particular case of because the parties can use these two states indistinguishably SLOCC, states equivalent under LU are also equivalent un- for exactly the same tasks. It is a celebrated result ͓1͔ that, der SLOCC, the[Gelfand new classification, Kapranov being, Zelevinsky a coarse Discriminants, graining of Resultants, and when applied to many copies of a state, this criterion leads to the previous one.Multidimensional Determinants (1994), Chapter 14, Example 4.5] identifying all bipartite pure-state entanglement with The main aim of this work is to identify and characterize that of the Einstein-Podolsky-Rosen ͑EPR͒ state all possible kinds[rediscovery of pure-state in QIS: entanglement Dür, Vidal, of threeCirac qubits, PRA 62, 062314 (2000); (1/ͱ2)(͉00͘ϩ͉11͘) ͓2͔. That is, the entanglement of any under SLOCC.over Unentangled 1300 citations] states, and also those which are pure state ͉␺͘AB is asymptotically equivalent, under deter- product in one party while entangled with respect to the re- ministic LOCC, to that of the EPR state, the entropy of en- maining two, appear as expected,PHYSICAL to be trivial REVIEW cases. A, More VOLUME 62, 062314 tanglement E(␺AB)—the entropy of the reduced density ma- surprising is the fact that there are two different kinds of trix of either system A or B—quantifying the amount of EPR genuine tripartite entanglement.Three qubits Indeed, can be we entangled will show that in two inequivalent ways entanglement contained asymptotically in ͉␺͘AB . In contrast, any ͑nontrivial͒ tripartite entangled state can be converted, recent contributions have shown that in systems shared by ¨ by means of SLOCC, into one ofW. twoW. Du standardr, DU¨ G.R, Vidal, G. VIDAL,forms, and J. AND I. Cirac J. I. CIRAC PHYSICAL REVIEW A 62 062314 three or more parties, there are several inequivalent forms of namely either theInstitut GHZ state fu¨r Theoretische͓12͔ Physik, Universita¨t Innsbruck, A-6020 Innsbruck, Austria entanglement under asymptotic LOCC ͓3,4͔. ͑Received 26 May 2000; published 14 November 2000͒ This paper is essentially concerned with the entanglement or else a second state1 tively, some needed results related to SLOCC, the monoto- properties of a single copy of a state, and thus asymptotic Invertible local transformations of a multipartite system are used to define equivalence classes in the set of nicity of the 3-tangle under LOCC, and the fact that ͉W͘ results do not apply here. For single copies it is known that entangled states.͉GHZ͘ Thisϭ͑1/ classificationͱ2͉͒͑000͘ concernsϩ͉111͘) the entanglement͉W͑͘ϭ1͒͑ properties1/ͱ3͉͒͑001 of a͘ singleϩ͉010 copy͘ϩ of͉100 the͘ state.), Accord-͑2͒ retains optimally bipartite entanglement when one qubit is ingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other traced out. by means of local operations and classical communication ͑LOCC͒ with nonzero probability. When applied to 1050-2947/2000/62͑6͒/062314͑12͒/$15.0062 062314-1 ©2000 The Americanand Physical that this Society splits the set of genuinely trifold entangled pure states of a three-qubit system, thisstates approach into reveals two the sets existence which of are two unrelated inequivalent under kinds LOCC. of genuine That tripartite entanglement, for which the Greenberger-Horne-Zeilinger state and a W state appear as remarkable II. KINDS OF ENTANGLEMENT UNDER is, we will see that if ␺ can be converted into the state representatives. In particular, we show that the W state retains maximally͉ ͘ bipartite entanglement when any one STOCHASTIC LOCC of the three qubits is traced out. We generalize͉GHZ͘ ourin Eq. results͑1͒ bothand to͉␾ the͘ casecan of be higher-dimensional converted into the subsystems state ͉W͘ and also to more than three subsystems,in for Eq. all͑ of2͒, which then we it is show not that, possible typically, to transform, two randomly even chosen with pure only In this work we define as equivalent the entanglement of states cannot be converted into each othera very by means small of LOCC, probability not even of with success, a small͉ probability␺͘ into ͉ of␾͘ success.nor the two states ͉␺͘ and ͉␾͘ of a multipartite system iff local other way round.