Local unitary classification for sets of generalized Bell states

Abstract

In this paper, we study the local unitary classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we firstly 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 of pairs in $d\otimes d$ quantum system into ${\prod }_{j=1}^n(k_j+1)$ LU-inequivalent pairs when the prime factorization of $d={\prod }_{j=1}^n{p}_j^{k_j}$. Moreover, all of triples in $p^{\alpha }\otimes p^{\alpha }$ quantum system for prime $p$ can be partitioned into $\frac{(\alpha +3)}{6}{p}^{\alpha }+O(\alpha p^{\alpha -1})$ LU-inequivalent triples, especially, when $\alpha =2$ and $p>2$, there are exactly $\lfloor \frac{5}{6}{p}^2\rfloor +\lfloor \frac{p-2}{6}+(-1{)}^{\lfloor \frac{p}{3}\rfloor }\frac{p}{3}\rfloor +3$ LU-inequivalent triples.

13 pages, 2 figures

References in corpus (3)

• Quantum secret sharing based on local distinguishability
• Local Unitary Equivalence of Multi-qubit Mixed quantum States
• Bell inequalities for multipartite qubit quantum systems and their maximal violation

Cited by in corpus (9)

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• Stochastic Local Operations with Classical Communication of Absolutely Maximally Entangled States
• Simple criterion for local distinguishability of generalized Bell states in prime dimension
• Construnctions of LOCC indistinguishable set of generalized Bell states
• Finding out all locally indistinguishable sets of generalized Bell states
• Anyonic quantum multipartite maskers in the Kitaev model
• Exchange-Symmetrized Qudit Bell Bases and Bell-State Distinguishability
• Local unitary classification of sets of generalized Bell states in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$
• Classical representation of local Clifford operators