Matrix Product States: Entanglement, Symmetries, and State Transformations

Sauerwein, David; Molnár, András; Cirac, J. I.; Kraus, Barbara

doi:10.1103/physrevlett.123.170504

Cited by 17 publications

(55 citation statements)

References 43 publications

(48 reference statements)

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“…It is the symmetries of the state which allow the transformation to be deterministic. In [36] the symmetries and SLOCC transformations of translationally invariant matrix product states (which include certain network structures) have been characterized. In the following, we characterize the symmetries of arbitrary networks of bipartite sources which distribute maximally entangled two-qubit states.…”

Section: Arxiv:210501090v1 [Quant-ph] 3 May 2021mentioning

confidence: 99%

Entangled Pure State Transformations via Local Operations Assisted by Finitely Many Rounds of Classical Communication

et al. 2017

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We consider generic pure n-qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems. We characterize those states which can be reached from some other state via local operations assisted by finitely many rounds of classical communication (LOCC_{N}). For n qubits with n>3, we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of LOCC_{N}. Moreover, we identify a class of states for which any LOCC_{N} protocol can be realized via a concatenation of deterministic steps. We show, however, that in general there exist state transformations which require a probabilistic step within the protocol, which highlights the difference between bipartite and multipartite LOCC.

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Section: Arxiv:210501090v1 [Quant-ph] 3 May 2021mentioning

confidence: 99%

Entangled Pure State Transformations via Local Operations Assisted by Finitely Many Rounds of Classical Communication

et al. 2017

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“…Finally, let us remark that 2 × m × n-states find application as fiducial states in the context of matrix product states [59]. In particular, properties of 2 × m × n-states can be used to characterize properties of the matrix product state they give rise to, such as local symmetries, or SLOCC classes.…”

Section: Resultsmentioning

confidence: 99%

A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems

Słowik

Hebenstreit

Kraus

et al. 2020

Quantum

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Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.

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“…This lattice is also complete, that is, the supremum and infimum exist for arbitrary subsets of probability vectors [20][21][22]. We remark that the lattice structure of majorization, beyond its partial order, has recently been found useful for different quantum information problems, namely the study of majorization uncertainty relations [23][24][25][26], entanglement transformations [27][28][29][30], and optimal common resource in majorization-based quantum resources theories [22], among others [31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

Extremal elements of a sublattice of the majorization lattice and approximate majorization

Massri¹,

Bellomo²,

Holik³

et al. 2020

J. Phys. A: Math. Theor.

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Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B p (x) with p ≥ 1 formed by the probability vectors whose p -norm distance to the center x is less than or equal to a radius . Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls B p (x) with 1 < p < ∞. On the other hand, we show that B ∞ (x) is a complete sublattice of the majorization lattice. As a consequence, this ball has also extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the 1 -norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.

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Matrix Product States: Entanglement, Symmetries, and State Transformations

Cited by 17 publications

References 43 publications

Entangled Pure State Transformations via Local Operations Assisted by Finitely Many Rounds of Classical Communication

Entangled Pure State Transformations via Local Operations Assisted by Finitely Many Rounds of Classical Communication

A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems

Extremal elements of a sublattice of the majorization lattice and approximate majorization

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