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arXiv:0908.0886 · Physical Review A

Entanglement equivalence of N -qubit symmetric states

Pierre Mathonet, Stéphanie Krins, Michel Godefroid, Lucas Lamata, E. Solano, Thierry Bastin

Published May 14, 2010• 70 citations
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Abstract

We study the interconversion of multipartite symmetric $N$-qubit states under stochastic local operations and classical communication (SLOCC). We demonstrate that if two symmetric states can be connected with a nonsymmetric invertible local operation (ILO), then they belong necessarily to the separable, W, or Greenberger-Horne-Zeilinger (GHZ) entanglement class, establishing a practical method of discriminating subsets of entanglement classes. Furthermore, we prove that there always exists a symmetric ILO connecting any pair of symmetric $N$-qubit states equivalent under SLOCC, simplifying the requirements for experimental implementations of local interconversion of those states.

DOI
10.1103/physreva.81.052315

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We analyze entanglement classes for permutation-symmetric states for nnn qudits (i.e., ddd-level systems), with respect to local unitary operations (LU equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the Jordan form of such one-particle operators is a SLOCC invariant. We find, as representatives of those classes, a discrete set of entangled states that generalize the Greenberger-Horne-Zeilinger and WWW states for the many-particle qudit case. In the latter case, we introduce the excitation state as a natural generalization of the WWW state for d>2d>2d>2.
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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.