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arXiv:1905.01824 · Physical Review A

Multipartite states under elementary local operations

Frank E. S. Steinhoff

Published Aug 15, 2019• 3 citations
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Abstract

Multipartite pure states are equivalent under stochastic local operations and classical communication (SLOCC) whenever they can be mapped into one another by invertible local operations. It is shown that this is equivalent to the interconvertibility through finite sequences of elementary local operations. A multipartite version of the Gauss-Jordan elimination strategy is then obtained, enabling the analytical SLOCC classification of previously unknown examples. It is argued that the problem of SLOCC classification is equivalent to the problem of classifying the multipartite fully reduced forms that the coefficient matrix of a state can assume after being subjected to this Gauss-Jordan elimination procedure. The method is applied to examples of so-called locally maximally entangleable and hypergraph states, showing different SLOCC equivalences. Moreover, possible physical implications for states with a Dicke-like structure are sketched.

DOI
10.1103/physreva.100.022317

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Multipartite pure states are equivalent under stochastic local operations and classical communication (SLOCC) whenever they can be mapped into one another by invertible local operations. It is shown that this is equivalent to the interconvertibility through finite sequences of elementary local operations. A multipartite version of the Gauss-Jordan elimination strategy is then obtained, enabling the analytical SLOCC classification of previously unknown examples. It is argued that the problem of SLOCC classification is equivalent to the problem of classifying the multipartite fully reduced forms that the coefficient matrix of a state can assume after being subjected to this Gauss-Jordan elimination procedure. The method is applied to examples of so-called locally maximally entangleable and hypergraph states, showing different SLOCC equivalences. Moreover, possible physical implications for states with a Dicke-like structure are sketched.
Abstract We study the stochastic local operation and classical communication (SLOCC) equivalence for arbitrary dimensional multipartite quantum states. For multipartite pure states, we present a necessary and sufficient criterion in terms of their coefficient matrices. This condition can be used to classify some SLOCC equivalent quantum states with coefficient matrices having the same rank. For multipartite mixed state, we provide a necessary and sufficient condition by means of the realignment of matrix. Some detailed examples are given to identify the SLOCC equivalence of multipartite quantum states.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.
Absolutely maximally entangled (AME) states are maximally entangled for every bipartition of the system. They are crucial resources for various quantum information protocols. We present techniques for verifying that two AME states are equivalent concerning stochastic local operations and classical communication (SLOCC). The conjecture that for a given multipartite quantum system all AME states are SLOCC equivalent is proven false. We also show that the existence of AME states with minimal support of six or more particles results in the existence of infinitely many such non-SLOCC-equivalent states. Moreover, we present AME states which are not SLOCC equivalent to the existing AME states with minimal support.
We show that a single polynomial entanglement measure is enough to verify equivalence between generic nnn-qubit states under stochastic local operations with classical communication (SLOCC). SLOCC operations may be represented geometrically by M\"obius transformations on the roots of the entanglement measure on the Bloch sphere. Moreover, we show how the roots of the three-tangle measure classify four-qubit generic states, and propose a method to obtain the normal form of a four-qubit state, which bypasses the possibly infinite iterative procedure.
Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n\ensuremath{\ge}5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.
Quantum entanglement plays a central role in quantum information processing. A main objective of the theory of quantum entanglement is to classify different types of entanglement according to their inter-convertibility through manipulations that do not require quantum communication. While bipartite entanglement is well understood in this framework, the classification of entangle-ments among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2 ⊗ m ⊗ n. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two 2⊗m ⊗ n states are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC equivalence classes in 2 ⊗ 3 ⊗ n systems, we also determine the hierarchy between these classes.