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. protocols exist that allow the parties to convert each of the PACS number͑s͒: 03.67.Hk, 03.65.Bz, 03.65.CaThe previous result is based on the fact that, unlike the two states into the other one with some a priori probability GHZ state, not all entangled states of three qubits can be of success. In this approach, we follow the definition for expressed as a linear combination of only two product states. stochastic equivalence as given in ͓4͔.3 The underlying mo- I. INTRODUCTION two pure states ͉␺͘ and ͉␾͘ can be obtained with certainty tivation for this definition is that, if the entanglement of ␺ Remarkably enough,from each the other inequivalence by means under of LOCC SLOCC if and of only the if they are ͉ ͘ The understanding of entanglement is at the verystates heart͉GHZ of ͘ andrelated͉W by͘ can local alternatively unitaries ͑LU be͓͒5,4 shown͔. But fromeven in the the simplestand ͉␾͘ is equivalent, then the two states can be used to perform the same tasks, although the probability of a suc- quantum information theory ͑QIT͒. In recent years,fact there that has the 3-tanglebipartite͑residual systems, tangle͉␺͘ and͒, a͉ measure␾͘ are typically of tripartite not related by been an ongoing effort to characterize qualitativelycorrelations and introducedLU, and by continuous Coffman parameterset al. ͓14͔ are, does needed not in- to labelcessful all performance of the task may depend on the state that quantitatively the entanglement properties of multiparticlecrease on averageequivalence under LOCC, classes as͓6–10 we͔ will. That prove is, one here. has to dealis with being used. systems. A situation of particular interest in QIT consistsWe will of theninfinitely move to many the second kinds of main entanglement. goal of this In work, this context, an several parties that are spatially separated fromnamely each other the analysisalternative, of the simpler state classification͉W͘. It cannot would be be obtained advisable. A. Invertible local operators and share a composite system in an entangledfrom state. a This state ͉GHZ by means of LOCC and thus one could One͘ such classification is possible if we just demand thatSensible enough, this classification would remain useless setting requires the parties—which are typicallyexpect, allowed in to principle, that it has some interesting, characteris- the conversion of the states is through stochastic local opera-if in practice we would not be able to find out which states communicate through a classical channel—to onlytic properties. act lo- Recall that in several aspects the͑ GHZ state͓͔͒ tions and classical communication SLOCC 4 ; thatare is, related by SLOCC. Let us recall that, all in all, no prac- cally on their subsystems. But even restricted to localcan be opera- regardedthrough as the LOCC maximally but without entangled imposing state of that three it has to be tions assisted with classical communication ͑LOCC͒, the tical criterion is known so far that determines whether a ge- qubits. However,achieved if one with of the certainty. three qubits In that is case, traced we out, can the establish an parties can still modify the entanglement properties of the neric transformation can be implemented by means of remaining stateequivalence is completely relation unentangled. stating that two Thus, states the͉␺͘ en-and ͉␾͘ are system and in particular they can try to convert one en- LOCC. However, we can think of any local protocol as a tanglement propertiesequivalent of if the the partiesstate GHZ have aare nonvanishing very fragile probability of tangled state into another. This possibility leads to natural ͉ ͘ series of rounds of operations, where in each round a given under particle losses.success We when will trying prove to that,convert oppositely,͉␺͘ into ͉ the␾͘, en- and also ͉␾͘ ways of defining equivalence relations in the set of entangled party manipulates locally its subsystem and communicates tanglement of intoW is͉␺ maximally͘ ͓11͔. This robust relation under has disposal been termed of any stochastic states—where equivalent states are then said to contain the ͉ ͘ classically the result of its operation ͑if it included a mea- one of the threeequivalence qubits, in in the Ref. sense͓4͔. that Their the equivalence remaining under re- SLOCC same kind of entanglement—as well as establishing hierar- surement͒ to the rest of the parties. Subsequent operations chies between the resulting classes. duced densityindicates matrices2 that␳ both, ␳ states, and are␳ againretain, suited accord- to implement the AB BC AC can be made dependent on previous results and the protocol For instance, we could agree in identifying anying two to states severalsame criteria, tasks the of greatest QIT, although possible this amount time the of probability en- of a splits into several branches. This picture is useful because for which can be obtained from each other with certaintytanglement, by comparedsuccessful to performance any other of state the task of three may differ qubits, from ͉␾͘ to our purposes we need only focus on one of these branches. means of LOCC. Clearly, this criterion is interestingeither in pure QIT or mixed.͉␺͘. Notice in addition that since LU are a particular case of Suppose that state ͉␺͘ can be locally converted into state ͉␾͘ because the parties can use these two states indistinguishablyWe will finallySLOCC, analyze states entanglement equivalent under under LU are SLOCC also equivalent in un- with nonzero probability. This means that at least one branch for exactly the same tasks. It is a celebrated resultmore͓1͔ generalthat, multipartiteder SLOCC, the systems. new classification We will show being a that, coarse for graining of of the protocol does the job. Since we are concerned only when applied to many copies of a state, this criterionmost leads of these to systems,the previous there one. is typically no chance at all to identifying all bipartite pure-state entanglement with with pure states, we can always characterize mathematically transform locallyThe a given main aim state of into this worksome is other to identify if they and are characterize that of the Einstein-Podolsky-Rosen ͑EPR͒ state this branch as an operator which factors out as the tensor chosen randomly,all possible because kinds the space of pure-state of entangled entanglement pure states of three qubits (1/ͱ2)(͉00͘ϩ͉11͘) ͓2͔. That is, the entanglement of any under SLOCC. Unentangled states, and also those whichproduct are of a local operator for each party. For instance, in a depends on more parameters than those that can be modified pure state ͉␺͘AB is asymptotically equivalent, under deter- product in one party while entangled with respect to thethree-qubit re- case we would have that ͉␺͘ can be locally con- by acting locally on the subsystems. ministic LOCC, to that of the EPR state, the entropy of en- maining two, appear as expected, to be trivial cases.verted More into ͉␾͘ with some finite probability iff an operator The paper is organized as follows. In Sec. II, we charac- tanglement E(␺AB)—the entropy of the reduced density ma- surprising is the fact that there are two different kindsA  ofB  C exists such that trix of either system A or B—quantifying the amountterize of mathematically EPR genuine the tripartite equivalence entanglement. relation Indeed, established we will by show that entanglement contained asymptotically in ͉␺͘AB .stochastic In contrast, conversionsany ͑nontrivial under͒ LOCC,tripartite and entangled illustrate state its can perfor- be converted, ͉␾͘ϭA  B  C͉␺͘, ͑3͒ recent contributions have shown that in systemsmance shared by by applyingby means it to of the SLOCC, well-known into one bipartite of two case. standard In forms, three or more parties, there are several inequivalentSec. forms III, of we movenamely to either consider the GHZ a system state of͓12 three͔ qubits, for where operator A contains contributions coming from any entanglement under asymptotic LOCC ͓3,4͔. which we prove the existence of six classes of states under round in which party A acted on its subsystem, and similarly This paper is essentially concerned with the entanglementSLOCC, including the two genuinely tripartite ones. Section for operators B and C.4 Carrying on with the three-qubit properties of a single copy of a state, and thusIV asymptotic is devoted to studying the endurance of the entanglement example, let us now consider for simplicity that both states results do not apply here. For single copies it isof known the state that ͉W͘ against particle͉GHZ losses.͘ϭ͑1/ͱ In2͒ Sec.͉͑000 V,͘ϩ more͉111͘) gen- ͑1͒ eral multipartite systems are considered. Section VI contains 1050-2947/2000/62͑6͒/062314͑12͒/$15.00some conclusions.62 062314-1 Finally, Appendices ©2000 A–C The prove, American respec- Physical Society 3Stochastic transformations under LOCC had been previously analyzed in ͓15,16͔. 4In practice, the constraints A†A, B†B, C†Cр1 should be ful- 1An experimental realization using photons of such a state was filled if the invertible operators A,B,C are to come from local already proposed in ͓13͔. POVMs. In this work, we do not normalize them in order to avoid 2 The reduced density matrix ␳AB of a pure tripartite state ͉␺͘ is introducing unimportant constants to the equations. Instead, both defined as ␳ABϵtrC(͉␺͗͘␺͉). the initial and final states are normalized.

062314-2 Dual variety point: a state hyperplane: a set of states Φ , orthogonal to Ψ F(Ψ,Φ) = ψ ϕ = 0 ∑ i1…in i1…in i1,…,in Ψ∈dual setXxX∨ ⇔∃ ∈ s.t. hyperplane F ( ΨΦ , )= 0 is tangent at x X: set of completely separable states & k1−1,…,kn −1 ( (1) (n) F(Ψ,x) = ψi i xi !xi = 0 ( ∑ 1… n 1 n i , ,i 0 ' 1 … n = ( ∂ F(Ψ,x) = 0, ∀j = 1,…,n, i = 0,…,k −1 ∂x( j ) j j ( i j ) have at least a nonzero solution x = (x(1) ,…,x(n) ) ∈ X ⇔ X ∨ : DetΨ = 0 set of degenerate entangled states (if codim X ∨ =1, i.e., k −1≤ (k −1) ∀j) j ∑ j"≠ j j! Hyperdeterminat

Det22Ψ = det Ψ =ψψ 00 11− ψψ 01 10 deg. 2 22 22 22 22 Det222Ψ =+++ψψ 000 111 ψψ 001 110 ψψ 010 101 ψψ 100 011− 2( ψψψψ 000 001 110 111

++++ψψψψ000 010 101 111 ψψψψ 000 100 011 111 ψψψψ 001 010 101 110 ψψψψ 001 100 011 110

++ψψψψ010 100 011 101)4( ψψψψ 000 011 101 110 + ψψψψ 001 010 100 111 ) deg. 4 ! SLOCC invariant entanglement monotones for genuine multipartite component C 2 Det = 22Ψ concurrence for 2-qubit pure states

τ = 4 Det222Ψ 3-tangle for 3-qubit pure states ! the degree of n-qubit hyperdeterminants grows very rapidly n 2 3 4 5 6 7 8 9 10 … deg. 2 4 24 128 880 6,816 60,032 589,312 6,384,384 ...

! generating function for the entanglement structure singularities of the hypersurface ! degenerate entangled classes 5

∨ M can define Xcusp as 7 ∨ ∨ 6 X∨ Xcusp = Xcusp|xo · G. (13) 4 Xnode∨ (1) ∨ This Xcusp is already closed without taking the closure. GHZ W B1 3 IV. CLASSIFICATION OF MULTIPARTITE S=X ENTANGLEMENT B2 B3 Xcusp∨

According to Sec. II and III, we illustrate the classi- Xnode∨ (2) Xnode∨ (3) fication of multipartite pure entangled states for typical cases.3-qubit6 (2x2x2) SLOCC classification7 7 M X X∨ 3 FIG. 2: The onion-like classification of SLOCC orbits in A.X 3-qubit (format 2 )case 6 ∨ the 3-qubit case. We utilize a duality between the smallest 4 Xnode∨ (1) closed subvariety X and the largest closed subvariety X∨. The classification of the 3 qubits under SLOCC has The dual variety X∨ (zero hyperdeterminant) and its sin- beenGHZ already done in [5, 6]. Surprisingly, Gelfand et al. node W B1 gularities constitute SLOCC-invariant closed subvarieties, so considered the3 same mathematical problem by DetA3 that they classify the multipartite entangled states (SLOCC in Example 4.5 of [16]. Our idea is inspired by this orbits). S=X ∨ example. We complement the Gelfand et al.X’s result, X B2 B3 ∨ analyzing additionallyXcusp∨ the singularities of X in detail.

The dimensions,Xnode∨ (2) representatives,Xnode∨ (3) names, and varieties of the orbits are summarized as follows. The basis vector seen in Sec. II A, of biseparable states between the 1st |i1⟩⊗|i2⟩⊗|i3⟩ is abbreviated to |i1i2i3⟩. party andcusp the rest of the parties. Its dimension is ∨ 1+3 = 4. Likewise, Xnode(j)forj =2, 3givesthe dim 7: |000⟩ + |111⟩, GHZ ∈ M(= CP 7)−X∨. FIG. 2: Two types of singularities of X∨. X∨ correspondsbiseparable to the bitangent class of X for,wherebothtangenciesareofthe the 2nd or 3rd party, respectively. FIG. 3: The onion-like classification of SLOCC orbits in the 3-qubit case. We∨ utilize∨ a dualitynode∨ between∨ the smallest ∨ ∨ dim 6: first|001 order.⟩+|010X⟩cusp∨+|100corresponds⟩, W ∈ X to the−X tangentsing = X at− anX inflectioncusp. pointSo, the of X class,whereitstangencyisofthesecondorder. of X −Xsing is found to be tripartite entan- closed subvariety X and the largest closeddim subvariety 4: |001⟩ +X|010∨.Thedualvariety⟩, |001⟩ + |100⟩, X|010∨ (zero⟩ + |100 hyperdeterminant)⟩, gled and states, its whose representative is W. We can intuitively singularities constitute SLOCC-invariant closed subvarieties so∨ that they classify the multipartite entangled states (SLOCC orbits). biseparable Bj ∈ Xnode(j)−X for j =1, 2, 3. see that, among genuine tripartite entangled states, W is ∨ C 1 C 3 Xnode(j)= Pj-th × P are three closed irreducible rare, compared to GHZ [5]. Finally, the intersection of ∨ ∨ C. Singularities of the hyperdeterminant∨ components of Xsing = Xcusp. Xnode(j)isthecompletelyseparableclassS,givenby dim 3: |000⟩, completely separable S the Segre variety X of dimension 3. Another intuitive ∨ ∈A.X = 3-qubitWe describe (formatX∨ the( singularj2)=3)caseCP locus1 ×CP of1 the×CP dual1. variety X .Thetechnicaldetailsaregivenin[19].Itisknownexplanation about this procedure is seen in Appendix A. that,j=1 for,2,3 thenode boundary format, the next largest closed subvariety X∨ is always an irreducible hypersurface ! Now wesing clarify the relationship of six classes by non- in X∨;incontrast,fortheinteriorone,X∨ has generally two closed irreducible components of codimension The classification of the 3 qubits underG SLOCC= GL2 has× beenGL2 alread× GLydonein[4,5].Surprisingly,Gelfand2 has the onion structuresing of invertibleet al. local operations. Because noninvertible local 1inX∨, node X∨ and cusp X∨ type singularities. The rest of this subsection can be skipped for the considered the same mathematical problemsix orbits by on DetMA3(seein Example Fig.node 2), by 4.5 excluding of [17].cusp Our the idea orbit is∅ inspired(= operations by this cause the decrease in local ranks [21], the X∨ (∅first)). reading. The dual It variety is also illustratedX∨ is given for the by Det 3-qubitA =0 case in∨ tpartiallyhe appendix ordered of [9]. structure of entangled states in the 3 example. We complement the Gelfand etnode al.’s result, analyzing∨ additionally the singularities3 of X in details. First, Xnode is the closure of the set of hyperplanes tangent to the Segre variety X at more than one The dimensions, representatives, names,(cf. and Eq. varieties (10)). Its of dimension the orbits∨ are is 7 summarized− 1=6.Theout- as follows. Thequbits, basis included in Fig.∨ 4, appears. Two inequivalent side ofpointsX∨ is (cf. generic Fig. 2). tripartiteXnode can entangled be composed class of of closed the irreducibletripartite subvarieties entangledX classes,node(J)labeledbythesubset GHZ and W, have the same vector |i1⟩⊗|i2⟩⊗|i3⟩ is abbreviated to |i1i2i3⟩. (1) (j) (n) maximalJ ⊂ dimension,{1,...,n}, whoseincluding representative∅.Indicatingthattwosolutions is GHZ. This localx =( ranksx ,...,x (2, 2, 2),...,x for each)ofEq.(7)coincidefor party so that they are not in- dim 7: |000⟩ + |111⟩, GHZ ∈ M(= CP 7)−X∨. ∨ suggestsj ∈ thatJ,thelabel almost anyJ distinguishes state in the the 3 qubits pattern can in these be lo- solutions.terconvertible In order to by re thewrite noninvertibleXnode(J), let local us pick operations up a (i.e., ∨ ∨ ∨ ∨ (j) dim 6: |001⟩ + |010⟩ + |100⟩, W ∈ X −Xsing = X o −Xcusp. cally transformedpoint x (J)suchthatitshomogeneouscoordinates into GHZ with a finite probability, andxij =generalδij ,0 for LOCC).j ∈J and Twoδij ,kj classesfor j/∈J hold.Itisconvenientto different physical prop- dim 4: |001⟩ + |010⟩, |001⟩ + |100⟩, |010⟩ + |100⟩, biseparable B ∈ X∨ (j)−X for j =1, 2, 3. vice versa.label Next, the positions we can of identifyj 1 in eachnodeX∨x(j) asbyX a∨ multi-index,whichis [i ,...,ierties]. For [5]; example, the GHZxo(1) representative is labeled by [0 state,k ,...,k has the] maxi- X∨ (j)=CP 1 ×CP 3 are three closed irreducible components of X∨ sing= X∨ cusp. 1 n 2 n node j-th the unionand ofxo three(1,...,n closed)isjustwrittenby irreduciblesing subvarietiesxocusp.WhenXX∨∨ is(j the) hyperplanemal amount tangent of generic to X at tripartitexo(J), its entanglement ”xo(J)-section” measured dim 3: |000⟩, completely separable S ∈ X = ∨ X∨ (j)=CP 1 ×CP 1 ×CP 1. node for j =1X, j2|=1x,o3[20](alsoseeAppendixA).Forexample,(,J2),3is givennode as by the 3-tangle τ ∝ |DetA3|,whiletheWrepresentative ∨ ! state has the maximal amount of (average) 2-partite en- G = GL2 × GL2 × GL2 has the onionXnode structure(1) means of by six definition orbits on that,M (see in addition Fig. 3), to by the excluding con- ′ the orbit′ ∨ ∨ ∨ ∨ (1) [i1,...,in]differs from [i1,...,in] ∅ (= X (∅)). The dual variety X is given by DetA =0(cf.Eq.(10)).Itsdimensionis7o ′ ′ −tanglement1=6. distributed over 3 parties (also [22]). Under node dition of X in Sec.3 IIB,X | therex (J) = existsA someall ai1 nonzero,...,i =0x s.t. o , (12) ∨ (2)! (3)" n of x (J)inatmostoneindex # The outside of X is generic tripartitesuch entangled that F (A, class x)=0 of the forany maximalx ,x dimens";i.e.,asetoflin-ion, whose representativeLOCC, is a state in these two classes can be transformed " GHZ. This suggests that almost any state in the 3 qubits can(1) be locally2 (2) transformed"(3) into GHZo withinto a finite any state in one of the three biseparable classes ear equationsin ordery thati2,i3 (x Eqs.(7))=( have∂ /∂ thex nontrivial∂x )F (A, solution x)=0x for(J). Then in terms of the hyperplane bitangent to X at ∨ ∨ i2 i3 B (j =1, 2, 3), where the j-th local rank is 1 and the probability, and vice versa. Next, we can identifyo Xsingo as Xcusp,whichistheunionofthreeclosedirreducible(1)∨ j ij =0, 1hasanontrivialsolutionx and x (J), we can definexXnode.Thisindicatesthat(J)as subvarieties X∨ (j)forj =1, 2, 3(cf.[19]).Forexample,X∨ (1) means by definition that, in additionothers to are 2. Three classes Bj never convert into each node the ”bipartite” matrix node ∨ (1) ∨ ∨(2) (3)other.∨ Likewise, a state in Bj can be locally transformed the condition for X in Sec.2, there exists some nonzero x such that F (A, x)=0X forany(J)=(Xx | ,xo ∩ X;i.e.,a| o ) · G, (13) node x into anyx (J) state in the completely separable class S of local set of linear equations a000 a001 a010 a011 (14) ranks (1, 1, 1). where G="GLa100k1+1a×101···×a110GLkan+1111acts# M on from the right and the bar stands for the closure. ∂2 ∨ (1) Second, Xcusp is the set of hyperplanes having a critical pointThis iswhich how is the not onion-likeasimplequadraticsingularity classification of SLOCC or- yi2,i3 (x )= F (A, x)=0 for ij =0, 1(15) ′ (2) (3) (j) (j ) never has the full rank (i.e., six 2×2minorsinEq.(14)are o bits reveals that multipartite′ 2 entangled classeso constitute ∂x(cf.i2 ∂ Fig.xi3 2). Precisely, the quadric part of F (A, x)atx is a matrix y(j,i ),(j ,i ′ ) =(∂ /∂x ∂x ′ )F (A, x ), zero). We can identify X∨ (1) as the set CP 1 × CP 3, the partially orderedj j structure.ij Itij indicates significant node ′ ′ 1st ′ ′ (1) where the pairs (j,∨ ij), (j ,ij )(1∨ ≤ ij ≤ kj, 1 ≤ ij ≤ kj )aretherow,columnindexrespectively.Denotingby ∨ o ∨ ∨ has a nontrivial solution x .Thisindicatesthatnotonlyo Xnode(1) ⊂ Xcusp,butthe”bipartite”matrix o Xcusp|x the variety of the Hessian det y =0 in the x -section X |x of Eq. (12), we can define Xcusp as

a000 a001 a010 a011 ∨ ∨ Xcusp = Xcusp|xo · G.(16) (14) " a100 a101 a110 a111 # This X∨ is already closed without taking the closure. cusp ∨ never has the full rank (i.e., six 2 × 2minorsinEq.(16)arezero).WecanidentifyXnode(1) as the set C 1 C 3 P1st × P ,seeninSec.2,ofbiseparablestatesbetweenthe1stpartyand the rest of the parties. Its ∨ dimension is 1 + 3 = 4. Likewise, Xnode(j)forj =2, 3givesthebiseparableclassforthe2nd,3rdparty,IV. CLASSIFICATION OF MULTIPARTITE ENTANGLEMENT ∨ ∨ respectively. So, the class of X −Xsing is found to be tripartite entangled states, whose representative is W. We can intuitively see that, among genuineAccording tripartite to Sec.2 entangled and Sec.3, states, we W illustrate is rare, the compared classificati to GHZon of multipartite[4]. pure entangled states for typical ∨ Finally, the intersection of Xnode(j)isthecompletelyseparableclasscases. S,givenbytheSegrevarietyX of dimension 3. Another intuitive explanation about this procedure is seen in the appendix of [9]. Now we clarify the relationship of six classes by noninvertible local operations. Because noninvertible local operations cause the decrease in local ranks [20], the partially ordered structure of entangled states in the 3 2x2x4 SLOCC classification

! SLOCC orbits are added outside.

! Duality suggests one subsystem is too large, compared to the rest. 22××nn( ≥ 4); 224 ××

! LOCC conversions of distributed 4 qubits. parties 2 3 4 classes 4 9 ∞

coarser finer Entanglement measures in 2x2x4 case

1,1,n− 1 Ψ = ψ ijk⊗⊗ ∑ ijk ABC ijk,,= 0 tr By calculating local ranks (,rrr123 , )ofρiji= ∀≠ ΨΨ ! (2,2,4) generic class (dim 15)

! (2,2,3) Clares local operation: ∀ ψ ijk = 0(k ≥ 3)

ψψψψψψ000 010 100 010 100 110 ψψψψψψ 000 010 110 000 100 110 "≠ 0 dim14 Det 223Ψ = ψψψψψψ 001 011 101 011 101 111− ψψψψψψ 001 011 111 001 101 111 # $= 0 dim13 ψψψψψψ002 012 102 012 102 112 ψψψψψψ 002 012 112 002 102 112

! (2,2,2) Clares local operation: ∀ ψ ijk = 0(k ≥ 2) 22 22 22 22 Det222Ψ =+++ψψ 000 111 ψψ 001 110 ψψ 010 101 ψψ 100 011− 2( ψψψψ 000 001 110 111

++++ψψψψ000 010 101 111 ψψψψ 000 100 011 111 ψψψψ 001 010 101 110 ψψψψ 001 100 011 110

++ψψψψ010 100 011 101)4( ψψψψ 000 011 101 110 + ψψψψ 001 010 100 111 )"≠ 0 GHZ (dim11) # = 0 W (dim10) ! (1,2,2), (2,1,2), (2,2,1) Biseparable 1,2,3 (dim 8,8,6) $ ! (1,1,1) Separable (dim 5) SLOCC conversions among different classes unique maximally entangled class, noninvertible lying on the top. local operations Its representative is 2 Bell pairs: ( 00+ 11) ⊗( 00+ 11 ) AC12 BC C

A B

5 graded partial order of 9 entangled classes 4-qubit hyperdeterminant 2 2 2 3 2 2 3 3 3 2 2 4 Det2222Ψ = c1 c2 c3 − 4c0c2c3 − 4c1 c2c4 − 4c1 c3 − 6c0c1 c3 c4 +16c0c2 c4 3 3 2 4 4 2 2 2 2 2 +18c0c1c2c3 +18c1 c2c3c4 − 27c0 c3 − 27c1 c4 −80c0c1c2 c3c4 −128c0 c2 c4 2 2 2 2 2 2 3 3 +144c0c1 c2c4 +144c0 c2c3 c4 −192c0 c1c3c4 + 256c0c4 deg. 24 1 ∂4 c j = 4− jjDet 222Ψ ψψ00iiixx+ 11 iii (4− jj )! ! xx ( 123 123 ) ∂∂01 4-qubit generic entangled states: αβ( 0000+++ 1111 0011 1100) +( 0000 + 1111−− 0011 1100 ) γδ( 0101+++ 1010 0110 1001) +( 0101 + 1010−− 0110 1001 ) 2 Det222222222222 0 2222Ψ = ((αβαγαδβγβδγδ−−−−−−≠)( )( )( )( )( )) (Vandermonde determinant)2

! infinitely many SLOCC orbits in generic entanglement ! segmentation on the top of the partial order

n ! GHZ and W are not generic (i.e., Det2 Ψ = 0) for n >= 4 qubits Asymptotic bipartite case Classification of entanglement when there are infinitely many identical copies A B A B

LOCC

⊗M ⊗M ⊗ N ⊗ N Ψ+ = ( λ 00 + 1− λ 11 ) Φ+ = ( 00 + 11 ) LOCC is reversible when N = ME(Ψ) = M(−λ logλ − (1− λ)log(1− λ)) Entanglement entropy (uniqueness of entanglement measures) embedding into a symmetric subspace (cf. Veronese mapping) Asymptotic case and MREGS Classification of multipartite entanglement with infinitely many identical copies is still open.

Minimally Reversibly Entanglement Generating Set (MREGS)?

$ ⊗n ( 000 111 ) 1 & + & ⊗n2 ⊗M & 001 + 010 + 100 Ψ ↔ % ( ) & ( 00 + 11 )⊗n3 & '& ? • Does it exist? • If so, is it made by finite generators? 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 Entanglement and its complexity

Van den Nest, Miyake, Dür, Briegel, Phys. Rev. Lett. 97, 150504 (2006) Van den Nest, Dür, Miyake, Briegel, New. J. Phys. 9, 204 (2007) quantum circuit as tensor composition entanglement can be generated by a quantum circuit

a 1-qubit gate: SU(2) unitary map !2 " !2 a 2-qubit gate: SU(4) unitary map !2 ⊗ !2 " !2 ⊗ !2 changing the quality of entanglement 2-qubit Controlled-Z gate

CZ = 00 00 + 01 01 + 10 10 − 11 11 = 0 0 ⊗1+ 1 1 ⊗σ z C ={1, 2} : ++=( 01 +) + Bell state 01 0011 → ++ −≡ zx+ zx C ={1, 2, 3} : GHZ state +++=+( 01 +) + 01000111 → +++−−≡xzx+ xzx

C ={1, 2, 3, 4} : 1D cluster state ++++=( 01 +) +++

01 000111 (not GHZ) → ( zz+ σ z ) ( xzxxzx+ ) 0000 0111 1000 1111 =++zxzx zxzx zxzx− zxzx Multipartite entanglement as graphs [ review: Hein et al., quant-ph/0602096] 1. For a graph G, vertices = qubits, edges = Ising-type interaction pattern. degree is the number of edges from a vertex. G CZ(,)abm , where CZ= diag(1,1,1, 1) () a () b =+∏ −∝σσzz ⊗ (,)ab∈ edges 1 01 +=2 ( + ) 2. joint eigenstate of N commuting correlation operators for N qubits. ()ab () KGaaxz== G, K σσ⊗ bN∈ a 3. any stabilizer state can be written as a graph state, up to local unitaries.

2D cluster state GHZ state 7-qubit codeword Universality of quantum computation

Using single-qubit projective measurements and classical feedforward of measurement outcomes (LOCC), • it’s capable to simulate any unitary gate operation on a (known) n-qubit input state deterministically • it’s capable to produce any corresponding output state ψ !!LOC!C → φ measured ! ! !##"##$ N n N -n

2D cluster states { Ck k ,1,2,... = } were shown to be universal Definition of Universal resource (without encoding): A set of states Ψ = ψ , ψ , . . . , is universal for QC, if { 1 2 } ∃ ψ ""LOC"C →∀ φ measured !k ! !##"##$ N n N -n Criterion for universality Idea: if any significant entanglement feature exhibited by a set of universal resource states (ex. the 2D cluster states) is not available from another set Ψ, then it cannot be universal

EE()φφ≥≥ (" )whenever φφLOCC '

A set of states Ψ cannot be universal for MQC if

sup∀∈ΨφψEE (φψ )> sup ( )

sup∀ φ EEC (φ )= supkk ( )

! entanglement width: distinguish 1D and 2D cluster states ! geometric measure ! Schmidt measure (logarithm of tensor rank) ! localizable entanglement Entanglement width

maximal entropy of entanglement associated with certain bipartitions

Ewd(ψψ )=minT maxeTE e ( ) ∈ AT

subcubic edge e tree T

! non-increasing under LOCC (i.e. fulfills (P1))

! equivalent to the rank-width in graph theory

Ewd(Ckkk× )≥ log2 (+ 2)− 1 Any universal resource must have a diverging entanglement width. Criterion by localizable entanglement (a,b) EL ()ψ [ Verstraete et al., PRL 92, 027901 (2004); Gour et al., PRA 72, 042329 (2005) ] entanglement (as measured by the concurrence) that can be at most created between two qubits (a,b) by LOCC

! non-increasing under LOCC (fulfills (P1))

(,)ab ! maximum number NLE of qubits such that EabNLLE()1,(,)ψ = ∀∈

(,)ab ECL ()1,(,)kk× = ∀ ab 2 : unbounded NCLE ()kk× == km

Non-universal resource states:

! all states such that EL < 1 (e.g. W states) ! all states such that EL decays with distance (NLE is bounded ) Non-universal graph states

Any set of the graph states whose entanglement-width is bounded for the number of qubits N is not a universal resource. “interaction geometry determines the value as a resource”

! 1D cluster states (Ewd = 1, SM=N/2)

! GHZ (tree) states, fully connected graphs

! graphs with bounded tree-width or clique-width

complementary to classical simulatibility (note universal QC is simply believed not to be classically simulatable efficiently) •Measurement-based computation on some resource states (e.g. 1D cluster states and GHZ states) [Nielsen 05; Markov & Shi 05; Jozsa 06; Van den Nest et al. 06; …] Tensor network as quantum computation SIAM J. COMPUT. c 2008 Society for Industrial and Applied Mathematics Vol. 38, No. 3, pp. 963–981 ⃝

SIMULATING QUANTUM COMPUTATION BY CONTRACTING TENSOR NETWORKS ∗

IGOR L. MARKOV† AND YAOYUN SHI†

Abstract. The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has a treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(log T ). Among many implications, we show efficient simulations for log-depth circuits whose gates applySIAM J. to C nearbyOMPUT qubits. only, a natural constraint satisfiedc 2010 by Society most for physical Industrial implementations. and Applied Mathematics We also showVol. 39, that No.one-way 7, pp. 3089–3121 quantum computation of Raussendorf⃝ and Briegel (Phys. Rev. Lett., 86 (2001), pp. 5188–5191), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidthQUANTUM graph COMPUTATION with a constant maximum AND degree. THE (The EVALUATION requirement on the OF maximum TENSOR degree was removed in [I. L. Markov and Y. Shi, preprint:quant-ph/0511069].) NETWORKS∗

Key words. quantum computation,ITAI ARAD computational† AND ZEPH complexity, LANDAU † treewidth, tensor network, clas- sical simulation, one-way quantum computation Abstract. We present a quantum algorithm that additively approximates the value of a tensor networkAMS to subject a certain classifications. scale. When combined81P68, with 68Q05, existing 68Q10, results, 05C83, this 68R10 provides a complete problem for quantum computation. The result is a simple new way of looking at quantum computation in

whichDOI. unitary10.1137/050644756 gates are replaced by tensors and time is replaced by the order in which the tensor network is “swallowed.” We use this result to derive new quantum algorithms that approximate the partition function of a variety of classical statistical mechanical models, including the Potts model. 1. Introduction. The recent interest in quantum circuits is motivated by several complementaryKey words. tensor considerations. networks, quantum Quantum algorithms, information statistical processing mechanical is models rapidly becoming a reality as it allows the manipulation matter on an unprecedented scale. Such ma- AMS subject classification. 68Q12 nipulations may create particular entangled states or implement specific quantum

evolutions—theyDOI. 10.1137/080739379 find uses in atomic clocks, ultra-precise metrology, high-resolution lithography, optical communication, etc. On the other hand, engineers traditionally simulate1. Introduction. new designs beforeThe discoveryimplementing by Peter them. Shor Such in 1994simulation of a quantum may identify algorithm subtle design flaws and save both costs and effort. It typically uses well-understood host for factoring n digit numbers in poly(n) steps stimulated a large amount of interest hardware,in the power e.g., of one quantum can simulate computation a quantum [Sho97]. circuit Since on then, a commonly-used the search for conventional quantum computer.algorithms that provide exponential speedup over the best known classical algorithms hasMore yielded ambitiously, a number of quantum results: algorithms circuits compete for a number with conventional of group and computing number theo- and communication.retic problems that, Quantum-mechanical like Shor’s algorithm, effects use the may quantum potentially Fourier lead transform to computational as the

speed-ups,essential ingredient more secure (e.g., or [Wat01, more Kup03, efficient vDHI03, communication, Hal07]); an better algorithm keeping for an of oracle secrets, + etc.graph To problem this end, that one uses seeks the new notion circuits of a quantum and algorithms random with walk revolutionary [CCD 03]; and behav- re- iorcently, as in algorithms Shor’s work for approximating on number-factoring, combinatorial or provable and topological limits on quantitiespossible behaviors. such as Whilethe Jones proving polynomial abstract and limitations the Tutte polynomial on the success [FKW02, of unknown FKLW02, algorithms AJL06, AAEL07]. appears

moreThese di lastfficult, algorithms a common related line to of the reasoning Jones and for Tutte such polynomials results is based are fundamentally on simulation.

Fordifferent example, from if the the previous behavior algorithms. of a quantum The work circuit presented can be here faithfully began simulated as a study on of a the core features of these algorithms. conventional computer, then the possible speed-up achieved by the quantum circuit is limitedThis work by the presents cost of a simulation.simple new way Thus, of looking aside from at quantum sanity-checking computation. new designs The

forconsequences quantum information-processing are (a) new quantum algorithms, hardware, (b) more the e castingfficient of simulation the aforementioned can lead to Jones polynomial and Tutte polynomial results in a new light, and (c) a new geometric sharper bounds on all possible algorithms. view of quantum computation that will hopefully lead to more new algorithms. The fundamental object for this new view is a tensor network which we now ∗Received by the editors November 10, 2005; accepted for publication (in revised form) November 27,briefly 2007; describe published (a electronically precise description June 25, 2008. of tensor networks is given in section 2). A tensorhttp://www.siam.org/journals/sicomp/38-3/64475.html network T (G, ) is a graph G, a finite set of colors that can be used to label M the†Department edges of G,alongwithafinitearrayofdata of Electrical Engineering and ComputerMv Science,assigned University to of each Michigan, vertex 2260v Hay-V ward Street, Ann Arbor, MI 48109-2121 ([email protected],∈ M [email protected]). The∈ first

Downloaded 11/09/14 to 64.106.62.45. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php of the graph. This finite set of data Mv is of the following form: for each possible author’s research was supported in part by NSF 0208959, the DARPA QuIST program, and the Air Forcecoloring Research of the Laboratory. edges incident The tosecond the vertex author’sv,thevertexisassignedacomplexnumber. research was supported in part by NSF 0323555, 0347078,Thus for and a given 0622033. coloring l of all the edges of the graph, each vertex has an assigned value—we denote the product of these values by cl. The value of a tensor network 963 T (G, ) is defined to be the sum of cl over all possible labelings l of G. TheM notion of a tensor network geometrically captures many fundamental linear

algebra concepts such as inner product, matrix multiplication, the trace of a matrix,

∗Received by the editors October 30, 2008; accepted for publication (in revised form) March 9, 2010; published electronically June 24, 2010.

Downloaded 11/09/14 to 64.106.62.45. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. http://www.siam.org/journals/sicomp/39-7/73937.html †Department of Electrical Engineering and Computer Sciences, University of California at Berke- ley, Berkeley, CA 94720 ([email protected], [email protected])

3089

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. Examples of universal resources

graph states by 2D regular tilings are universal resources.

hexagonal lattice square lattice triangular lattice (2D cluster state)

degree = 3 degree = 4 degree = 6 minimum possible degree for uniform lattices to be universal

More by other tensor network states [Gross, Eisert PRL 98, 220503 (2007); Gross et.al., PRA 76, 052315 (2008)] 6

Tensor network states 6

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Workshops Spring 2014 FIG. 5. (Color online) Homogeneous tensor network states for the ground state in an infinite lattice in D =1spacial

Tensor Networks and Simulationsdimensions. (i) A homogeneous MPS is characterized by a FIG. 6. (Color online) (i) Two sites x1 and x2 of the lat- Apr. 21 – Apr. 24, 2014 single tensor that is repeated infinitely many times through- tice L are connected through several paths within the tensor Program: Quantum Hamiltonian Complexityout the tensor network. (ii) A homogeneous scale invariant network. The geodesic corresponds to the shortest of such MERA is characterized by two tensors,Add a disentangler to Calendar and paths, where the length of a path is measured e.g. by the an isometry, repeated throughout the tensor network, which number of tensors in the path. In the example, the shortest FIG. 5. (Color online) Homogeneous tensor network states consists of infinitely many layers. path between sites x1 and x2 contains 6 tensors, and therefore for the ground state in an infinite lattice in D =1spacial View schedule & video » View abstracts » the length of the geodesic is D(x1,x2)=6.Eq.13relates dimensions. (i) A homogeneous MPS is characterized by a FIG. 6. (Color online) (i) Two sites x1 and x2 of thethe lat- length of geodesics with the assymptotic decay of corre- singleOrganizers: tensor that is repeated infinitely many times through- tice L are connected through several paths within the tensorlations in the tensor network (assuming that correlations are out the tensor network. (ii) A homogeneous scale invariant Ignacio Cirac (Max Planck Institute, Garching), Franking Verstraete we consider (Universitynetwork.generic of The Vienna)properties geodesic. corresponds of homogeneous to theMPS, shortest ofpredominantly such carried by the tensors in the geodesic). (ii) MERA is characterized by two tensors, a disentangler and PEPS andpaths, MERA where with the a finitelength bond of a path dimension is measuredχ.We e.g. byRegion the ΩA of the tensor network that contains (the indices an isometry, repeated throughout the tensor network, which consider statesnumber of of an tensors infinite in thelattice path.L, In see the Fig. example, 5, and the shortestcorresponding to) region A of the lattice L.Theboundary consistsThis of workshop infinitely will many focus layers. on tensor network (MPS, PEPS, MERA) based simulations of strongly correlated quantum focus mostlypath on between the asymptotic sites x1 andbehaviourx2 contains of such 6 tensors, proper- and therefore∂ΩA of region ΩA consists of the set of indices connecting many body systems, relation to area laws, quantum spin glasses,the as well length as related of the work geodesic in mathematics, is D(x ,x quantum)=6.Eq.13relates ties, namely in the decay of correlations at1 large2 distances ΩA with the rest of the tensor network. The number n(A) chemistry and quantum information theory. the length of geodesics with the assymptotic decay of corre-of such indices is interpreted as measure of the size |∂ΩA| of and scaling of entanglement entropy for large blocks of lations in the tensor network (assuming that correlationsthe are boundary ∂ΩA.Intheexample,n(A)=|∂ΩA| =3.An sites. ing weEnquiries consider maygeneric be sentproperties to the organizers of homogeneous at this address.MPS, predominantly carried by the tensors in the geodesic).upper (ii) bound to the entanglement entropy of a region A of the PEPS and MERA with a finite bond dimension χ.We Region ΩA of the tensor network that contains (the indiceslattice L is given in terms of n(A), Eq. 20. corresponding to) region A of the lattice L.Theboundary considerPoster states session: of an infinite lattice L, see Fig. 5, and IV.∂ CORRELATIONSΩA of region ΩA consists AND of GEODESICS the set of indices connecting focusThis mostly workshop on the willasymptotic include a posterbehaviour session of on such Wednesday, proper- April 23, from 4–5:30 p.m. To participate in the poster ties, namely in the decay of correlations at large distances ΩA with the rest of the tensor network. The number n(A) session, you must first be registered for the workshop. The Simonsof Institute such indices will provide is interpreted a 30 inch asx 40 measure inch board, of the an size |∂ΩbetweenA| of these sites is defined as the length of the shortest and scaling of entanglement entropy for large blocksThe of asymptotic decay of correlations has long been easel, and clips to attach the poster to the board; presenters mustthe print boundary and bring ∂theirΩA.Intheexample, own posters. To presentn(A)= a |∂ΩA| =3.Anpath connecting them, see Fig. 6(i). sites. known to be exponential in an MPS1–3 and polynomial upper bound to the entanglement entropy of a region A of theLet C(x ,x ) denote a correlation function between poster, email [email protected] by Tuesday,in April the scale15. invariant MERA16,18,19. In this section we 1 2 lattice L is given in terms of n(A), Eq. 20. positions x and x .Itturnsoutthatforboththe point out that such behaviour is dictated by the struc- 1 2 MPS and the scale invariant MERA, the decay of correla- Invited Participants: ture of geodesics in the geometry attached to each of IV. CORRELATIONS AND GEODESICS tions can be expressed in terms of the distance D(x ,x ) Bela Bauer (Microsoft Research Station Q), Michael Ben-Orthese tensor(Hebrew network University states. of Jerusalem), For an Cedric MPS, Beny the (Leibniz later is a 1 2 within the tensor network, Universität Hannover), Fernando Brandao (Universityrather College straightfoward London),between Xie theseChen statement; (UC sites Berkeley), is defined for Paul the as Christiano the MERA, length it(UC of was the shortest The asymptotic decay of correlations has long been 92 first notedpath by Swingle connecting. them, see Fig. 6(i). −αD(x1,x2) knownBerkeley), to be exponential Ignacio Cirac in(Max an Planck MPS 1–3Institute,and Garching), polynomial Jens Eisert (FU Berlin), David Pérez García (Universidad C(x1,x2) ≈ e , (13) Let C(x ,x ) denote a correlation function between in theComplutense scale invariant de Madrid), MERA Sevag16,18,19 Gharibian. In this (UC sectionBerkeley), we Jeongwan Haah (Massachusetts1 2 Institute of Technology), positions x and x .Itturnsoutthatforboththefor some positive constant α. This expression assumes pointJutho out thatHaegeman such (Ghent behaviour University), is dictated Sean Hallgren by the (Pennsylvania struc- State University),1 Matt2 Hastings (Microsoft MPS and the scale invariant MERA, the decay of correla-that the correlations between the two sites are mostly tureResearch), of geodesics Patrick in theHayden geometry (Stanford attached University), to Alexandra each of KollaA. (University Geodesics of Illinois, within Urbana-Champaign), a tensor network Vladimir tions can be expressed in terms of the distance D(x1carried,x2) through the tensors/links in the geodesic path theseKorepin tensor (Stony network Brook states. University), For anZeph MPS, Landau the (UC later Berkeley), is a Urmila Mahadev (UC Berkeley), Carl Miller (University within the tensor network, connecting them. It originates in the fact that for both ratherof straightfowardMichigan), Bruno Nachtergaele statement; for(UC the Davis), MERA, Daniel itNagajGiven was (University a tensor of networkVienna), Ashwin state forNayak the (University state |Ψ ⟩ofof a lat- the MPS (D = 1 dimensions) and the scale invariant 92 tice L, and two sites of L at positions x−αandD(x1x,x2,wecan) first notedWaterloo), by SwingleDidier Poilblanc. (University of Toulouse), Frank Pollmann (Max Planck Institute,C(x1,x Dresden),2) ≈ e 1 Anupam2 , MERA(13) (in any dimensions), the correlator C(x ,x )can define a notion of distance between these two sites within 1 2 Prakash (UC Berkeley), Tomaz Prosen (University of Ljubljana), Xiao-Liang Qi (Stanford University), Nicholas Read be obtained by evaluating an expression with the (ap- the tensorfor network some positiveas follows. constant Firstα we.notice This expression that the assumes (Yale University), Or Sattath (Hebrew University of Jerusalem), Norbert Schuch (RWTH Aachen), Leonard Schulman proximate) form A. Geodesics within a tensor network two sites arethat connected the correlations by paths between within the the two tensor sites net- are mostly work, wherecarried each through path consists the tensors/links of a list of tensors in the geodesic and path † D(x1,x2) C(x1,x2) ≈ ⃗vL · (T ) · ⃗vR, (14) links/indicesconnecting connecting them. the It tensors. originates To inany the such fact path, that for both Given a tensor network state for the state |Ψ⟩ of a lat- we then associatethe MPS a length, (D = as 1 givendimensions) by the number and the of scale ten- invariantthat is, a scalar product involving two vectors ⃗vL and tice L, and two sites of L at positions x1 and x2,wecan sors (or links)MERA in the (in any path. dimensions), Then the distance the correlatorD(x1,xC(2x) ,x⃗v)canR and the D(x1,x2)-th power of some transfer matrix within 1 2 define a notion of distance between these two sites be obtained by evaluating an expression with the (ap- the tensor network as follows. First we notice that the proximate) form two sites are connected by paths within the tensor net- work, where each path consists of a list of tensors and † D(x1,x2) C(x1,x2) ≈ ⃗v · (T ) · ⃗vR, (14) links/indices connecting the tensors. To any such path, L we then associate a length, as given by the number of ten- that is, a scalar product involving two vectors ⃗vL and sors (or links) in the path. Then the distance D(x1,x2) ⃗vR and the D(x1,x2)-th power of some transfer matrix Complexity of entanglement Universal resources are necessarily macroscopic entanglement! entropic entanglement width maximum size of unit localizable entanglement Schmidt-rank width universal resource families ON()→∞ classically ON(log( )) efficiently simulatable geometric measure Schmidt measure (logarithm of tensor rank) Summary: open questions 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 - “complete” characterization and “monogamy” constraints (cf. two previous talks) - asymptotic case: minimally generating set

2. Entanglement and its complexity - necessary and sufficient condition for graph states to be universal - classes of tensor networks which are efficiently computable