Matrix Product States: Entanglement, symmetries, and state transformations

TL;DR

This paper investigates the entanglement properties, symmetries, and state transformations of translationally-invariant matrix product states (MPS), providing criteria for state convertibility and classifying their symmetry structures.

Contribution

It introduces a criterion for SLOCC transformations between MPS, classifies SLOCC equivalence classes for simple MPS, and characterizes all global and local symmetries of MPS.

Findings

Established a criterion for SLOCC convertibility of MPS.

Classified SLOCC classes for basic MPS.

Characterized all symmetries of MPS, including inhomogeneous ones.

Abstract

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.

Tables

Table 1. Table 1 : The SLOCC classification of normal d = D = 2 𝑑 𝐷 2 d=D=2 MPS. See the SM sup for more details.

type	$\chi$	# symm.	# SLOCC classes
GHZ	$\ne -1,0$	$2$	$\infty$ (see main text)
	$-1$	$2^N$	$1$ (cluster set)
	0	$1$	$1$ (symmetryless set)
W	n.a.	$N$-dependent	$1$

Equations

$$|A⟩=\sum _{j=0}^{d-1}\sum _{\alpha ,\beta =0}^{D-1}{A}_{\alpha ,\beta }^j|j,\alpha ,\beta ⟩$$

$$|\Psi (A)⟩=\sum _{j_1,\dots ,j_N}\mathrm{Tr}(A^{j_1}\dots A^{j_N})|j_1,\dots ,j_N⟩.$$

$$G_A=\{h=g\otimes x\otimes y^T|h|A⟩=|A⟩\}$$

$$h_1\to h_2\to \dots \to h_k\to h_1$$

$$G_A=\{s_{x,y}\otimes x\otimes y^T|x,y\in GL(D,\mathbb{C})\},$$

$$S(x_1,\dots ,x_N)=s_{x_N^{-1},x_1}\otimes \dots \otimes s_{x_{N-1}^{-1},x_N}.$$

$$G_{A_A}=\{s_x\otimes x^{-1}\otimes x^T|x\in GL(2,\mathbb{C})\},$$

$$S(x)=s_x\otimes s_{g^{-1}xg}\otimes \dots \otimes s_{g^{-(N-1)}x{g}^{N-1}},$$

$$A\stackrel{N}{\to }B\text{iff}\underset{j=1}{\overset{N}{⨂}}{g}_j|{\Psi }_N(A)⟩=|{\Psi }_N(B)⟩$$

$$G_{A,B}=\left\{h=g\otimes x\otimes y^T|h|A⟩=|B⟩\right\},$$

$$(\mathcal{B}{\mathcal{A}}^{-1}{)}^{\otimes N}|\Psi (A)⟩=|\Psi (B)⟩.$$

$$G_{A_A,A_{A,g}}=\{s_x\otimes g{x}^{-1}\otimes x^T|x\in GL(\mathbb{C},2)\}.$$

$$A\stackrel{N}{\sim }B\text{iff}\underset{j=1}{\overset{N}{⨂}}{g}_j|\Psi (A)⟩=|\Psi (B)⟩$$

$$h_1\stackrel{b\to c}{\to }{h}_2\stackrel{b\to c}{\to }\cdots \stackrel{b\to c}{\to }{h}_k\stackrel{b\to c}{\to }{h}_1.$$

$$GH{Z}_b\stackrel{N}{\sim }GH{Z}_c\iff \chi (b)=\{\begin{array}{l}\chi (c)\text{or}\chi (c{)}^{-1}\text{ and N even}\\ \chi (c)\text{and N odd}.\end{array}$$

Full text

1. Introduction: Entanglement is a resource for numerous striking applications of quantum information theory Nielsen and Chuang (2010); Horodecki et al. (2009). Furthermore, it is key to comprehend many peculiar properties of quantum many-body systems Amico et al. (2008); Eisert et al. (2010) and has become increasingly important in areas like quantum field theory or quantum gravity Nishioka (2018); Witten (2018). Despite its relevance, entanglement is far from being fully understood; at least in the multipartite setting. State transformations play a crucial role as they define a partial order in the set of states. For instance, if a state $\Psi$ can be transformed into a state $\Phi$ deterministically by local operations and classical communication (LOCC), then $\Psi$ is at least as entangled as $\Phi$ Horodecki et al. (2009). If two states cannot even probabilistically be interconverted via local operations, i.e., by so-called stochastic LOCC (SLOCC) transformations, their entanglement is not comparable, as one or the other may be more useful for different informational tasks Bennett et al. (2000). Thus, the study of state transformations is crucial for the theory of entanglement.

For bipartite systems, state transformations are fully characterized and have led to a very clear picture Nielsen (1999); Vidal (1999), which is widely used in different areas of research. For more parties such a characterization is much more challenging. In general, there are infinitely many classes of states that can be interconverted via SLOCC, and only in few cases they can be characterized, like for symmetric states or for certain tripartite and four-partite states Bastin et al. (2015); Migdał et al. (2013); Dür et al. (2000); Verstraete et al. (2002); Briand et al. (2004); Chitambar et al. (2010). Moreover, for more than four parties of the same local dimensions almost no state can be transformed into an inequivalent state via deterministic LOCC, and the partial order induced by LOCC becomes trivial Gour et al. (2017); Sauerwein et al. (2018). This shows that generic states are not very interesting from the perspective of local transformations. Additionally, most of the states in the Hilbert space cannot be reached in polynomial time even if constant-size nonlocal gates are allowed Poulin et al. (2011). Hence, the study of state transformations can be reduced to families of non-generic, but physically relevant states.

In this Letter we present a systematic investigation of state transformations for Matrix Product States (MPS) that describe translationally invariant systems (with periodic boundary conditions) Fannes et al. (1992); Pérez-García et al. (2007). Ground states of gapped 1D local Hamiltonians or states generated sequentially by a source can be efficiently approximated by MPS Hastings (2007); Schön et al. (2005). Hence, these states play a very important role in both, quantum information theory and in many-body physics. Despite the fact that they describe a broad variety of phenomena, they have a simple description: tripartite states – the fiducial states of MPS – completely characterize the MPS. We give a criterion to estipulate when an SLOCC transformation between two such MPS exists, and further give criteria to determine the SLOCC classes dictated by such a relation. These classes build a finer structure on top of the SLOCC classification of the fiducial states, with the additional structure depending on the system size.

The methods introduced here also allow us to identify all local symmetries of MPS (not only corresponding to unitary representations Sanz et al. (2009); Singh et al. (2010))111By local (global) symmetries we mean (in-)homogeneous symmetries, i.e., that a different (the same) action operates on each physical system.. This is interesting on its own right in the theory of tensor networks, as it induces a classification of zero temperature phases of matter Chen et al. (2011); Schuch et al. (2011); Pollmann and Turner (2012). As we show, the problems we address can be mapped to finding out certain cyclic structures of operators acting on tripartite states. Thus, our results allow to answer questions like: Can an AKLT state be transformed into a cluster state? What are all the symmetries of these states? What are the SLOCC classes of MPS? As we show, the answers to these questions can be strongly size-dependent.

2. Matrix Product States: We consider here a chain of $N$ $d$-level systems in a translationally invariant MPS. One such state, $\Psi(A)$, is defined in terms of a tripartite fiducial state,

[TABLE]

as

[TABLE]

Here, $D$ denotes the bond dimension and $A^{j}$ a matrix with components $A^{j}_{\alpha,\beta}$. The corresponding tensor is called injective if those matrices span the set of $D\times D$ matrices. The matrix $\mathcal{A}=\sum_{j,\alpha,\beta}A_{\alpha,\beta}^{j}|j\rangle\langle\alpha,\beta|$ then has a left inverse $\mathcal{A}^{-1}$ Pérez-García et al. (2007). This does not occur generically since it requires that $d\geq D^{2}$. We consider here normal tensors instead, which are generic and are those that become injective after blocking $L\leq 2D^{2}(6+\log_{2}(D))$ sites Michałek and Shitov . Furthermore, we consider $N\geq 2L+1$, so that we can apply the fundamental theorem of MPS Molnar et al. (2018). We call ${\cal N}_{N,D}$ the set of normal, translationally invariant MPS with bond dimension $D$ and $N\geq 2L+1$ sites. Note also that we only consider states with full local ranks as we could otherwise map the problem to smaller local dimensions.

We use several examples of some particularly relevant states of bond dimension 2 to illustrate our results. They are generated by fiducial states $|X_{b}\rangle=({\openone}\otimes b\otimes{\openone})|X\rangle$, where $X$ is one of the following states:

1. (i)

the W state $|W\rangle=|100\rangle+|010\rangle+|001\rangle$; 2. (ii)

the GHZ state $|GHZ\rangle=|000\rangle+|111\rangle$; 3. (iii)

the cluster state $|GHZ_{H}\rangle$, where $H=\sum_{ij}(-1)^{ij}|i\rangle\langle j|$; 4. (iv)

the state $|A_{A}\rangle=\sqrt{2}|010\rangle-|100\rangle+|111\rangle-\sqrt{2}|201\rangle$, which generates the AKLT state; 5. (v)

the state $|VB\rangle=\sum_{ij}|k_{ij}ij\rangle$ with $d=4$ and $k_{ij}=2i+j$ generating the valence bond state.

The W and GHZ states play a central role in entanglement theory Dür et al. (2000); Greenberger et al. (1989), the cluster state in measurement-based quantum computation Raussendorf and Briegel (2001), and the AKLT Affleck et al. (1987) and the valence bound state are paradigmatic examples that appear in condensed matter physics. The latter is, furthermore, injective and the fixed point of a renormalization procedure Verstraete et al. (2005).

3. Symmetries: Global symmetries, of the form $u^{\otimes N}$, of MPS were considered in Sanz et al. (2009); Singh et al. (2010), and have led to the classification of phases of MPS in spin chains Chen et al. (2011); Schuch et al. (2011); Pollmann and Turner (2012). Here we extend those results in two ways by considering: (i) non-unitary symmetries and (ii) local symmetries for which the operators acting on different spins can be different Note (1). That is, given $\Psi(A)\in{\cal N}_{N,D}$ we look for all operators $g=\bigotimes_{j=1}^{N}g_{j}$ such that $|\Psi(A)\rangle=g|\Psi(A)\rangle$.

In order to solve this problem, we define

[TABLE]

where $T$ denotes the transponse in the standard basis. We say that $h_{1},h_{2}\in G_{A}$ with $h_{i}=g_{i}\otimes x_{i}\otimes y_{i}^{T}$, can be concatenated and write $h_{1}\to h_{2}$ if $y_{1}x_{2}\propto{\openone}$. For $k\in\mathbb{N}$ we call a sequence $\{h_{i}\}_{i=1}^{k}\subseteq G_{A}$ with

[TABLE]

a $k-$cyle. Then we have:

The symmetry of the state corresponding to the cycle $h_{1}\to h_{2}\to\ldots\to h_{N}\to h_{1}$ is $g_{1}\otimes\dots\otimes g_{N}$. The trivial symmetry with $g={\openone}$ always exists. The proof is based on the fundamental theorem of MPS Molnar et al. (2018) and is given in the Supplemental Material (SM) sup . Hence, one simply has to determine $G_{A}$ and find all of its $N$-cycles to characterize the local symmetries of $\Psi(A)$. It suffices to find all minimal cycles of $G_{A}$ from which all others can be obtained by concatenation. For example, a 2-cycle can always be concatenated with itself to an $N$-cycle if $N$ is even. The global symmetries are defined in terms of 1-cycles, and thus require $g\otimes x^{-1}\otimes x^{T}|A\rangle=|A\rangle$. For $g$ unitary we therefore recover previous results Sanz et al. (2009); Singh et al. (2010). The novelty relies on the fact that one may also have local symmetries, with different $g_{j}$. In the following we illustrate this fact and the dependence of the symmetries on the system size.

For injective MPS with $D=d^{2}$ it is straightforward to show that sup

[TABLE]

where $s_{x,y}=\mathcal{A}({x^{T}}^{-1}\otimes y^{-1})\mathcal{A}^{-1}$. These operators can be concatenated to infinitely many cycles of arbitrary length. The corresponding symmetries are parametrized via regular matrices $x_{1},\ldots,x_{N}$ as

[TABLE]

For \mathcal{A}=\mbox{1{\bf l}} we obtain the large local symmetry group of the injective valence bond state. Normal (but not injective) states have a much smaller set of symmetries. For the AKLT state

[TABLE]

where $s_{x}$ is a function given in the SM sup . Clearly, elements of $G_{A_{A}}$ can only be concatenated with themselves. Consequently, the local symmetry group of the AKLT state possesses only global symmetries of the form $s_{x}^{\otimes N}$. Moreover, this group is isomorphic to the projective linear group $PGL(2,\mathbb{C})$ and includes the well-known symmetries with $s_{x}\in SO(3)$ Affleck et al. (1987). For the AKLT-type states we have $G_{A_{A,g}}=({\openone}\otimes g\otimes{\openone})G_{A}({\openone}\otimes g^{-1}\otimes{\openone})$ and the local symmetries of $\Psi(A_{A,g})$ read

[TABLE]

where $x$ is such that $g^{-N}xg^{N}=x$. Hence, $G_{A_{A,g}}$ is generically smaller than $G_{A_{A}}$. Moreover, it consists of non-global symmetries which are $N$-dependent. The symmetries of the cluster state are also non-global and coincide with the $2^{N}$ so-called stabilizer symmetries Hein et al. (2006); sup . For W-generated states the set $G_{A_{W,g}}$ contains infinitely many elements. However, the only nontrivial minimal cycles are 2-cycles. Hence, the corresponding MPS has only the trivial symmetry for odd $N$ and infinitely many non-translationally invariant symmetries for even $N$.

4. State transformations: Here, we answer the question of when a state, $\Psi(A)$, can be converted into another one, $\Psi(B)$, by SLOCC. As both states correspond to normal tensors, they both belong to ${\cal N}_{N,D}$. We write $A\overset{N}{\rightarrow}B$ if the transformation is possible. Note that

[TABLE]

for some $g_{j}$. We distinguish also here between global (where all $g_{j}$ are equal) and local transformations. As for symmetries, we define the set

[TABLE]

where $x,y$ are regular, but not necessarily $g$. That is, we also consider the case where the physical dimensions, $d_{A}$ and $d_{B}$ do not coincide. It is straightforward to show that if $|B\rangle=h_{0}|A\rangle$ with $h_{0}=g_{0}\otimes x_{0}\otimes y_{0}^{T}$ then $G_{A,B}\supseteq h_{0}G_{A}$. For $d_{A}=d_{B}$, which is the case iff $g_{0}$ is regular, we have $G_{A,B}=h_{0}G_{A}$. Defining concatenations of elements in $G_{A,B}$ as well as $k$-cycles as before, we have:

The proof is given in the SM sup . Theorem 2 solves the state transformation problem. We can immediately make some simple statements about different possibilities that may occur. For instance, if $G_{A,B}$ only contains a 1-cycle, then $A\overset{N}{\rightarrow}B$ for all $N$ with just global transformations. However, if the only minimal cycle is a 2-cycle, then the transformation can only happen for even $N$. As we illustrate in the following, some transformations might require more sophisticated operations and one obtains a rich variety of situations.

An injective $\Psi(A)\in{\cal N}_{N,D}$ can be transformed to any $\Psi(B)\in{\cal N}_{N,D}$ via the global operation

[TABLE]

However, using that $G_{A,B}\supseteq(\mathcal{B}\mathcal{A}^{-1}\otimes{\openone}\otimes{\openone})G_{A}$, with $G_{A}$ given in (5), we find that $G_{A,B}$ also contains infinitely many $N$-cycles that lead to non translationally invariant operators that transform $|\Psi(A)\rangle$ into $|\Psi(B)\rangle$. As a special case, we obtain the well-known result that the injective valence bond state can be transformed to any MPS in ${\cal N}_{N,2}$. Since injective MPS are generic in $\mathcal{N}_{N,D}$ for $d=D^{2}$, a randomly selected MPS of these dimensions can be transformed into any other state of the same dimensions. In contrast to that, transformations from normal (but not injective) states are much more restricted. For example, we show below that for $d=D=2$ any two randomly selected states $\Psi(A),\Psi(B)\in\mathcal{N}_{N,2}$ cannot be transformed into each other for any $N$. For the AKLT state and the cluster state, $G_{A_{A},A_{Cl}}$ contains only 2-cycles. Hence, the AKLT state can be transformed into the cluster state iff $N$ is even (see SM sup ). The reverse transformation is impossible since the physical dimension cannot be increased. Particularly sophisticated transformations are necessary to transform the AKLT state into certain AKLT-type states, $\Psi(A_{A,g})$, for which

[TABLE]

Using Theorem 2 it is easy to show that $A_{A}\overset{N}{\rightarrow}A_{A,g}$ iff $g^{N}\propto{\openone}$. Hence, the feasibility of this transformation is highly size dependent. Moreover, for any $M\in\mathbb{N}$ there exists a regular $g$ such that $A_{A}{\overset{M}{\rightarrow}}A_{A,g}$ is not possible for any $M<N$, but it is for $M=N$.

5. Equivalence classes under SLOCC transformations: SLOCC classes give a coarse but very useful classification of entanglement in many-body systems. We show now how these classes can be obtained for ${\cal N}_{N,D}$.

We write $A\overset{N}{\sim}B$ if $\Psi(A)$ is SLOCC equivalent to $\Psi(B)$, i.e.,

[TABLE]

for some regular $g_{j}$. Thus, we can reduce the study of SLOCC classes to that of the tripartite fiducial states.

In order to simplify this task, let us make some observations. First, $A\overset{N}{\sim}B$ iff $A\overset{N}{\rightarrow}B$ and $B\overset{N}{\rightarrow}A$. Because of Theorem 2 the equivalence $A\overset{N}{\sim}B$ thus requires $G_{A,B}\neq\emptyset$ and therefore that $|A\rangle$ and $|B\rangle$ themselves belong to the same tripartite SLOCC class. Hence, the equivalence relation $\overset{N}{\sim}$ induces a classification that is finer than the SLOCC classification of tripartite states. Second, for any regular $g$, $g^{\otimes N}|\Psi(A)\rangle$ is obviously in the same class as $|\Psi(A)\rangle$ and for any regular $x$ we trivially have $|A\rangle\overset{N}{\sim}({\openone}\otimes x^{-1}\otimes x^{T})|A\rangle$, since both states correspond to the same MPS. That is, $A\overset{N}{\sim}B$ trivially holds if the relation $|B\rangle=(g\otimes x^{-1}\otimes x^{T})|A\rangle$ holds (i.e., there exists a $1$-cycle). We get rid of this trivial case by restricting the SLOCC classes to the quotient set induced by that relation. Hence, it only remains to consider states of the form $|A_{b}\rangle=({\openone}\otimes b\otimes{\openone})|A\rangle$. This observation leads to the following procedure to characterize SLOCC classes of normal MPS (see Fig. 2): (i) for each tripartite SLOCC class, choose a representative, $A$; (ii) consider all states $|A_{b}\rangle=({\openone}\otimes b\otimes{\openone})|A\rangle$ corresponding to a normal tensor; (iii) determine the classes among those states according to the relation $\overset{N}{\sim}$. We now show how this procedure can be carried out.

According to Theorem 2, $A_{b}\overset{N}{\sim}A_{c}$ requires the existence of an $N$-cycle in $G_{A_{b},A_{c}}$ (or, equivalently, in $G_{A_{c},A_{b}}$). The fact that $G_{A_{b},A_{c}}=({\openone}\otimes c\otimes{\openone})G_{A}({\openone}\otimes b^{-1}\otimes{\openone})$ motivates the following definition (analogous to (4)). We say that $h_{1},h_{2}\in G_{A}$, with $h_{i}=g_{i}\otimes x_{i}\otimes y_{i}^{T}$, can be $(b\to c)$-concatenated, if $y_{1}bx_{2}\propto c$ and then write $h_{1}\xrightarrow{b\to c}h_{2}$. A sequence $\{h_{i}\}_{i=1}^{k}\subseteq G_{A}$ is called a $(b\to c)$-$k$-cycle if

[TABLE]

We obtain the following corollary to Theorem 2.

Note that this corollary requires that the $N$-cycle contains at least two different elements of $G_{A}$. This fact can be used to simplify the procedure. For instance, if one wants to determine the $A_{c}$ connected by 2-cycles, one can take arbitrary $\alpha,\beta$ and impose $y_{\alpha}cx_{\beta}=b\propto y_{\beta}cx_{\alpha}$, from which one can eliminate $b$. Then, the condition can be mapped into the eigenvalue equation $M\vec{c}=\lambda\vec{c}$, where $M=y_{\alpha}^{-1}y_{\beta}\otimes(x_{\alpha}x_{\beta}^{-1})^{T}$. Thus, by choosing all possible pairs in $G_{A}$ one can identify all classes corresponding to 2-cycles. Corollary 1 solves also straightforwardly the equivalence problem of MPS under local unitary operations (see Verstraete et al. (2005) for global unitary operations).

The procedures above can be carried out whenever the tripartite SLOCC classes are known, as is the case for $d=2,D\geq 2$ Chitambar et al. (2010). Here, we determine the classes for the simplest non-trivial MPS, i.e., those with $d=D=2$ (see Table 1). The fiducial states are either SLOCC equivalent to the GHZ or the W state Dür et al. (2000). Hence, the corresponding SLOCC classes separate into GHZ- and W-generated ones. All W-generated MPS are SLOCC equivalent. As explained before, it, hence, remains to consider states of the form $|GHZ_{b}\rangle={\openone}\otimes b\otimes{\openone}|GHZ\rangle$. The resulting classes can be coarse grained into three sets according to the value of $\chi(b)\equiv\frac{b_{00}b_{11}}{b_{01}b_{10}}$, where $b_{ij}$ denote the entries of $b$: (i) the generic set ($\chi\neq-1,0$); (ii) the cluster set ($\chi=-1$); (iii) the symmetryless set ($\chi=0$). The generic set is of full measure in the set of all MPS with $d=D=2$ and is comprised of states whose local symmetries are of the form $\{{\openone},s^{\otimes n}\}$. For two such states we have

[TABLE]

Thus, there are infinitely many, $N$-dependent classes. The cluster set coincides with the set of states which are SLOCC equivalent to the cluster state. They possess $2^{N}$ local symmetries. The states in the symmetryless set are also all SLOCC equivalent and have only the trivial local symmetry. Combined with the class of the non-normal MPS generated by the GHZ state these classes constitute the SLOCC classification of entangled MPS with $d=D=2$.

6. Summary and outlook: We solved the problem of when an MPS generated by a normal tensor can be transformed into another via SLOCC and showed how local symmetries of normal MPS can be characterized. In contrast to other results we considered all, in particular non translationally invariant and non unitary, operations. This revealed interesting features of many, particularly relevant states. Furthermore, we provided a procedure to characterize SLOCC classes of normal MPS and explicitly determined them for $d=D=2$. We believe that these results can be extended to non-normal MPS and (certain) Projected Entangled Pair States. Furthermore, one can also determine the SLOCC classes of MPS with higher dimensions and their corresponding symmetries. The theory presented here also serves as a basis to study deterministic LOCC transformations. Finally, our characterization of all local symmetries may be relevant in the study of phases of matter for 1D systems.

*Acknowledgments: — * This research was supported by the Austrian Science Fund (FWF) through Grants No. Y535-N16 and No. DK-ALM:W1259-N27, the Austrian Academy of Sciences’ Innovation Fund “Research, Science and Society”, by the European Research Council through the ERC Advanced Grant QENOCOBA under the EU Horizon 2020 program (grant agreement 742102) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2111 - 390814868.