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arXiv:quant-ph/0401023 · International Journal of Quantum Information

MULTIPARTITE ENTANGLEMENT UNDER STOCHASTIC LOCAL OPERATIONS AND CLASSICAL COMMUNICATION

Akimasa Miyake

Published Mar 1, 2004• 27 citations
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Abstract

Stochastic local operations and classical communication (SLOCC), also called local filtering operations, are a convenient, useful set of quantum operations in grasping essential properties of entanglement. We give a quick overview of the characteristics of multipartite entanglement in terms of SLOCC, illustrating the 2-qubit and the rest (2×2×n) quantum system. This not only includes celebrated results of 3-qubit pure states, but also has implications to 2-qubit mixed states.

DOI
10.1142/s0219749904000080

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Stochastic local operations and classical communication (SLOCC), also called local filtering operations, are a convenient, useful set of quantum operations in grasping essential properties of entanglement. We give a quick overview of the characteristics of multipartite entanglement in terms of SLOCC, illustrating the 2-qubit and the rest (2×2×n) quantum system. This not only includes celebrated results of 3-qubit pure states, but also has implications to 2-qubit mixed states.
We study multipartite entanglement under stochastic local operations and classical communication (SLOCC) and propose the entanglement classification under SLOCC for arbitrary-dimensional multipartite (nnn-qudit) pure states via the rank of coefficient matrix, together with the permutation of qudits. The ranks of the coefficient matrices have been proved to be entanglement monotones. The entanglement classification of the 2⊗2⊗2⊗42 \otimes 2 \otimes 2 \otimes 42⊗2⊗2⊗4 system is discussed in terms of the generalized method, and 22 different SLOCC families are found.
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.
Classification and quantification of bipartite qubit, tripartite qubit, and bipartite qutrit entanglement are studied in detail based on the entanglement measure of multipartite pure states. The results show that different ways of pure state entanglement can be characterized by extremal values of the corresponding entanglement measure under the stochastic local operation and classical communication(SLOCC). Detailed analysis shows that there are three different kinds of entanglement under the SLOCC in both tripartite qubit and bipartite qutrit pure state systems. Comparison of the measure with the recently proposed entropy-product measure is made and briefly discussed.
The coefficient matrix is an efficient tool in entanglement classification under stochastic local operation and classical communication (SLOCC). In this work, we take all the ranks of the coefficient matrices into account in the method of entanglement classification, and give the entanglement classification procedure for arbitrary-dimensional multipartite pure states under SLOCC. As a main application, we study the entanglement classification for quantum systems in the Hilbert space H = C2 ⊗ C2 ⊗ C2 ⊗ Cd in terms of the ranks of the coefficient matrices.
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.
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.