Matrix Product States: Entanglement, symmetries, and state transformations David Sauerwein∗ Max-Planck-Institut f¨urQuantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany and Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21A, 6020 Innsbruck, Austria Andras Molnar† and J. Ignacio Cirac‡ Max-Planck-Institut f¨urQuantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany and Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen,Germany Barbara Kraus§ Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21A, 6020 Innsbruck, Austria (Dated: January 24, 2019) We analyze entanglement in the family of translationally-invariant matrix product states (MPS). We give a criterion to determine when two states can be transformed into each other by SLOCC transformations, a central question in entanglement theory. We use that criterion to determine SLOCC classes, and explicitly carry out this classification for the simplest, non-trivial MPS. We also characterize all symmetries of MPS, both global and local (inhomogeneous). We illustrate our results with examples of states that are relevant in different physical contexts. 1. Introduction: Entanglement is a resource for numer- formations can be reduced to families of non-generic, but ous striking applications of quantum information theory physically relevant states. [1, 2]. Furthermore, it is key to comprehend many pecu- In this Letter we present a systematic investigation of liar properties of quantum many-body systems [3, 4] and state transformations for Matrix Product States (MPS) has become increasingly important in areas like quantum that describe translationally invariant systems (with pe- field theory or quantum gravity [5, 6]. Despite its rele- riodic boundary conditions) [19, 20]. Ground states of vance, entanglement is far from being fully understood; gapped 1D local Hamiltonians or states generated se- at least in the multipartite setting. State transforma- quentially by a source can be efficiently approximated by tions play a crucial role as they define a partial order MPS [21, 22]. Hence, these states play a very important in the set of states. For instance, if a state Ψ can be role in both, quantum information theory and in many- transformed into a state Φ deterministically by local op- body physics. Despite the fact that they describe a broad erations and classical communication (LOCC), then Ψ is variety of phenomena, they have a simple description: tri- at least as entangled as Φ [2]. If two states cannot even partite states – the fiducial states of MPS – completely probabilistically be interconverted via local operations, characterize the MPS. We give a criterion to estipulate i.e., by so-called stochastic LOCC (SLOCC) transforma- when an SLOCC transformation between two such MPS tions, their entanglement is not comparable, as one or exists, and further give criteria to determine the SLOCC the other may be more useful for different informational classes dictated by such a relation. These classes build a tasks [7]. Thus, the study of state transformations is finer structure on top of the SLOCC classification of the crucial for the theory of entanglement. fiducial states, with the additional structure depending For bipartite systems, state transformations are fully on the system size. characterized and have led to a very clear picture [8, 9], The methods introduced here also allow us to iden- which is widely used in different areas of research. For tify all local symmetries of MPS (not only corresponding more parties such a characterization is much more chal- to unitary representations [23, 24])[25]. This is interest- arXiv:1901.07448v2 [quant-ph] 23 Jan 2019 lenging. In general, there are infinitely many classes of ing on its own right in the theory of tensor networks, as states that can be interconverted via SLOCC, and only it induces a classification of zero temperature phases of in few cases they can be characterized, like for symmet- matter [26–28]. As we show, the problems we address ric states or for certain tripartite and four-partite states can be mapped to finding out certain cyclic structures of [10–15]. Moreover, for more than four parties of the same operators acting on tripartite states. Thus, our results local dimensions almost no state can be transformed into allow to answer questions like: Can an AKLT state be an inequivalent state via deterministic LOCC, and the transformed into a cluster state? What are all the sym- partial order induced by LOCC becomes trivial [16, 17]. metries of these states? What are the SLOCC classes of This shows that generic states are not very interesting MPS? As we show, the answers to these questions can be from the perspective of local transformations. Addition- strongly size-dependent. ally, most of the states in the Hilbert space cannot be 2. Matrix Product States: We consider here a chain reached in polynomial time even if constant-size nonlocal of N d-level systems in a translationally invariant MPS. gates are allowed [18]. Hence, the study of state trans- One such state, Ψ(A), is defined in terms of a tripartite 2 fiducial state, where T denotes the transponse in the standard basis. T We say that h1, h2 GA with hi = gi xi yi , can be d 1 D 1 ∈ ⊗ ⊗ − − j concatenated and write h1 h2 if y1x2 11. For k N A = Aα,β j, α, β (1) k → ∝ ∈ | i | i we call a sequence hi i=1 GA with j=0 α,β=0 { } ⊆ X X h h ... h h (4) as 1 → 2 → → k → 1 a k cyle. Then we have: Ψ(A) = Tr(Aj1 ...AjN ) j , . , j . (2) − | i | 1 N i j ,...,j Theorem 1. The local (global) symmetries of Ψ(A) 1X N ∈ N,D are in one-to-one correspondence with the N-cycles j N Here, D denotes the bond dimension and A a matrix (1-cycles) in GA. with components Aj . The corresponding tensor is α,β The symmetry of the state corresponding to the cycle called injective if those matrices span the set of D D j × h1 h2 ... hN h1 is g1 gN . The matrices. The matrix = j,α,β Aα,β j α, β then has → → → → ⊗ · · · ⊗ 1 A | ih | trivial symmetry with g = 11 always exists. The proof a left inverse − [20]. This does not occur generically A P2 is based on the fundamental theorem of MPS [30] and is since it requires that d D . We consider here normal given in the Supplemental Material (SM) [35]. Hence, one tensors instead, which are≥ generic and are those that be- 2 simply has to determine GA and find all of its N-cycles to come injective after blocking L 2D (6 + log2(D)) sites characterize the local symmetries of Ψ(A). It suffices to [29]. Furthermore, we consider ≤N 2L + 1, so that we ≥ find all minimal cycles of GA from which all others can can apply the fundamental theorem of MPS [30]. We call be obtained by concatenation. For example, a 2-cycle N,D the set of normal, translationally invariant MPS can always be concatenated with itself to an N-cycle if withN bond dimension D and N 2L + 1 sites. Note ≥ N is even. The global symmetries are defined in terms also that we only consider states with full local ranks 1 T of 1-cycles, and thus require g x− x A = A . For as we could otherwise map the problem to smaller local g unitary we therefore recover⊗ previous⊗ results| i | [23,i 24]. dimensions. The novelty relies on the fact that one may also have We use several examples of some particularly relevant local symmetries, with different gj. In the following we states of bond dimension 2 to illustrate our results. They illustrate this fact and the dependence of the symmetries are generated by fiducial states X = (11 b 11) X , | bi ⊗ ⊗ | i on the system size. where X is one of the following states: For injective MPS with D = d2 it is straightforward to (i) the W state W = 100 + 010 + 001 ; show that [35] (ii) the GHZ state| iGHZ| =i 000| +i 111| ; i | i | i | i T (iii) the cluster state GHZH , where H = GA = sx,y x y x, y GL(D, C) , (5) ( 1)ij i j ; | i { ⊗ ⊗ | ∈ } ij − | ih | T 1 1 1 (iv) the state A = √2 010 100 + 111 √2 201 , where sx,y = (x − y− ) − . These operators can P A A ⊗ A which generates| i the| AKLTi − state;| i | i − | i be concatenated to infinitely many cycles of arbitrary length. The corresponding symmetries are parametrized (v) the state VB = ij kijij with d = 4 and kij = 2i + j generating| i the valence| i bond state. via regular matrices x1, . , xN as The W and GHZ statesP play a central role in entangle- S(x1, . , xN ) = s 1 ... s 1 . (6) xN− ,x1 xN− 1,xN ment theory [12, 31], the cluster state in measurement- ⊗ ⊗ − based quantum computation [32], and the AKLT [33] For = 1l we obtain the large local symmetry group of and the valence bound state are paradigmatic examples the injectiveA valence bond state. Normal (but not injec- that appear in condensed matter physics. The latter tive) states have a much smaller set of symmetries. For is, furthermore, injective and the fixed point of a the AKLT state renormalization procedure [34]. 1 T GAA = sx x− x x GL(2, C) , (7) { ⊗ ⊗ | ∈ } N 3. Symmetries: Global symmetries, of the form u⊗ , where sx is a function given in the SM [35]. Clearly, of MPS were considered in [23, 24], and have led to the elements of GAA can only be concatenated with them- classification of phases of MPS in spin chains [26–28].