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arXiv:1406.6171 · Physical Review A

Local unitary equivalence of multiqubit mixed quantum states

Ming Li, Tinggui Zhang, Shao-Ming Fei, Xianqing Li‐Jost, Naihuan Jing

Published Jun 23, 2014• 27 citations
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Abstract

We present a computable criterion for completely classifying multiqubit quantum states under local unitary operations. The criterion can be used to detect whether two quantum states in multiqubit systems are locally unitarily equivalent or not. Once a positive answer is obtained, we are further able to compute the corresponding unitary operators precisely. Since the scheme is based on the mean values of some quantum mechanical observables, it supplies an experimental way to judge the local equivalence of quantum states.

DOI
10.1103/physreva.89.062325

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We present a computable criterion for completely classifying multiqubit quantum states under local unitary operations. The criterion can be used to detect whether two quantum states in multiqubit systems are locally unitarily equivalent or not. Once a positive answer is obtained, we are further able to compute the corresponding unitary operators precisely. Since the scheme is based on the mean values of some quantum mechanical observables, it supplies an experimental way to judge the local equivalence of quantum states.
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We study the local unitary equivalence of arbitrary dimensional multipartite quantum mixed states. We present a necessary and sufficient criterion of the local unitary equivalence for general multipartite states based on matrix realignment. The criterion is shown to be operational, even for particularly degenerated states, by detailed examples. In addition, explicit expressions of the local unitary operators are constructed for locally equivalent states. Complementary to the criterion, an alternative approach based on the partial transposition of matrices is also given, which makes the criterion more effective in dealing with generally degenerated mixed states.
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.
In this paper, we study the local unitary (LU) classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we first introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all pairs in the $d\ensuremath{\bigotimes}d$ quantum system into ${\ensuremath{\prod}}_{j=1}^{n}({k}_{j}+1)\ensuremath{-}1$ LU-inequivalent pairs when the prime factorization of $d={\ensuremath{\prod}}_{j=1}^{n}{p}_{j}^{{k}_{j}}$. Moreover, all triples in the ${p}^{\ensuremath{\alpha}}\ensuremath{\bigotimes}{p}^{\ensuremath{\alpha}}$ quantum system for prime ppp can be partitioned into $\frac{(\ensuremath{\alpha}+3)}{6}{p}^{\ensuremath{\alpha}}+O(\ensuremath{\alpha}{p}^{\ensuremath{\alpha}\ensuremath{-}1})$ LU-inequivalent triples; especially, when $\ensuremath{\alpha}=2$ and p>2p>2p>2, there are exactly $\ensuremath{\lfloor}\frac{5}{6}{p}^{2}\ensuremath{\rfloor}+\ensuremath{\lfloor}\frac{p\ensuremath{-}2}{6}+{(\ensuremath{-}1)}^{\ensuremath{\lfloor}\frac{p}{3}\ensuremath{\rfloor}}\frac{p}{3}\ensuremath{\rfloor}+3$ LU-inequivalent triples.