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arXiv:1407.4634

Comment on: Local Unitary Equivalence of Multipartite Pure Qubits States

Ana María Guerra Martins

Published Jul 17, 2014• 0 citations
arXivHTMLPDF

Abstract

A Comment on the Letter by B. Kraus {\it Phys. Rev. Lett.}{\bf 104}, 020504 (2010).
DOI
10.48550/arxiv.1407.4634

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Abstract. We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors (L. De Lathauwer, B. De Moor, and J. Vandewalle, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253–1278). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus’s algorithm (B. Kraus, Phys. Rev. Lett., 104 (2010), 020504) used the HOSVD to compute normal forms of almost all [Formula: see text]-qubit pure states under the action of the local unitary group. Taking advantage of the double cover [Formula: see text], we produce similar algorithms (distinguished by the parity of [Formula: see text]) that compute normal forms for almost all [Formula: see text]-qubit pure states under the action of the group of stochastic local operations and classical communication.
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.