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arXiv:0808.2989 · Physical Review A

Multiparty quantum states stabilized by the diagonal subgroup of the local unitary group

David W. Lyons, Scott N. Walck

Published Oct 16, 2008• 14 citations
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Abstract

We classify, up to local unitary equivalence, the set of $n$-qubit states that is stabilized by the diagonal subgroup of the local unitary group. We exhibit a basis for this set, parametrized by diagrams of nonintersecting chords connecting pairs of points on a circle, and give a criterion for when the stabilizer is precisely the diagonal subgroup and not larger. This investigation is part of a larger program to partially classify entanglement type (local unitary equivalence class) via analysis of stabilizer structure.

DOI
10.1103/physreva.78.042314

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Local unitary equivalence plays a crucial role in quantifying and categorizing the entanglement of quantum states. When it comes to two isospectral diagonal bipartite quantum states, determining whether they are locally unitary equivalent turns out to be a rather challenging problem. Intuitively speaking, we conjecture that for two diagonal quantum states, local unitary equivalence and local permutation equivalence between them are the same. By taking into account the cosets of the fixed point subgroup and the tensor decomposability of permutation matrices, we present the classifications of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text] under local permutation equivalence, respectively. Furthermore, relying on the classification results of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text], we demonstrate that if two diagonal quantum states are not locally permutation equivalent, then they cannot be locally unitary equivalent. This offers significant backing for the conjecture that two general bipartite diagonal quantum states are locally unitary equivalent if and only if they are locally permutation equivalent.
Local unitary (LU) equivalent states are crucial for understanding the structure of quantum entanglement. We investigate the LU equivalence based on the LU Bargmann invariants (LUBIs). We consider for which states the LUBIs are sufficient to ensure the LU equivalence. We focus on a class of multipartite states with complete local commutativity, called CLC states. We first show that for multipartite CLC states, all the LUBIs are sufficient to ensure the local permutation equivalence, if their single-party marginals are diagonal and non-degenerate. By virtue of the fact that the CLC property remains unchanged under local operations, we verify that the LUBIs are also sufficient to ensure the LU equivalence of multipartite CLC states whose single-party marginals are non-degenerate. Furthermore, when the single-party marginals are degenerate, we propose concrete examples of two-qudit and three-qubit systems, showing that there are states which share all the LUBIs but are not LU equivalent.