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arXiv:1711.06026 · Physical Review A

Local unitary classification for sets of generalized Bell states

Bujiao Wu, Jiaqing Jiang, Jialin Zhang, Guojing Tian, Xiaoming Sun

Published Aug 3, 2018• 14 citations
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Abstract

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 $p$ 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 $p>2$, 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.

DOI
10.1103/physreva.98.022304

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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⨂dd\ensuremath{\bigotimes}dd\ensuremath⨂d quantum system into \ensuremath∏j=1n(kj+1)\ensuremath−1{\ensuremath{\prod}}_{j=1}^{n}({k}_{j}+1)\ensuremath{-}1\ensuremath∏j=1n​(kj​+1)\ensuremath−1 LU-inequivalent pairs when the prime factorization of d=\ensuremath∏j=1npjkjd={\ensuremath{\prod}}_{j=1}^{n}{p}_{j}^{{k}_{j}}d=\ensuremath∏j=1n​pjkj​​. Moreover, all triples in the p\ensuremathα\ensuremath⨂p\ensuremathα{p}^{\ensuremath{\alpha}}\ensuremath{\bigotimes}{p}^{\ensuremath{\alpha}}p\ensuremathα\ensuremath⨂p\ensuremathα quantum system for prime ppp can be partitioned into (\ensuremathα+3)6p\ensuremathα+O(\ensuremathαp\ensuremathα\ensuremath−1)\frac{(\ensuremath{\alpha}+3)}{6}{p}^{\ensuremath{\alpha}}+O(\ensuremath{\alpha}{p}^{\ensuremath{\alpha}\ensuremath{-}1})6(\ensuremathα+3)​p\ensuremathα+O(\ensuremathαp\ensuremathα\ensuremath−1) LU-inequivalent triples; especially, when \ensuremathα=2\ensuremath{\alpha}=2\ensuremathα=2 and p>2p>2p>2, there are exactly \ensuremath⌊56p2\ensuremath⌋+\ensuremath⌊p\ensuremath−26+(\ensuremath−1)\ensuremath⌊p3\ensuremath⌋p3\ensuremath⌋+3\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\ensuremath⌊65​p2\ensuremath⌋+\ensuremath⌊6p\ensuremath−2​+(\ensuremath−1)\ensuremath⌊3p​\ensuremath⌋3p​\ensuremath⌋+3 LU-inequivalent triples.
Local unitary classification of sets of generalized Bell states in Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd
Two sets of quantum entangled states that are equivalent under local unitary transformations may exhibit identical effectiveness and versatility in various quantum information processing tasks. Consequently, classification under local unitary transformations has become a fundamental issue in the theory of quantum entanglement. The primary objective of this work is to establish a practical LU-classification for all sets of l (≥2)l\ (\geq 2)l (≥2) generalized Bell states (GBSs), high-dimensional generalizations of Bell states, in a bipartite system Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd with d≥3d\geq 3d≥3 . Based on this classification, we determine the minimal cardinality of indistinguishable GBS sets in C6⊗C6\mathbb{C}^{6}\otimes \mathbb{C}^{6}C6⊗C6 under one-way local operations and classical communication (one-way LOCC). We first propose two classification methods based on LU-equivalence for all sets of l GBSs (l-GBS sets). We then establish LU-classification for all 2-GBS, 3-GBS, 4-GBS and 5-GBS sets in C6⊗C6\mathbb{C}^{6}\otimes \mathbb{C}^{6}C6⊗C6 . Since LU-equivalent sets share identical local distinguishability, it suffices to examine representative GBS sets from equivalent classes. Notably, we identify a one-way LOCC indistinguishable 4-GBS set among these representatives, thereby resolving the case of d=6d = 6d=6 for the problem of determining the minimum cardinality of one-way LOCC indistinguishable GBS sets in (Yuan et al. in Quantum Inf Process. 18:145, 2019) or (Zhang et al. in Phys Rev A 91:012329, 2015).
In order to study a large number of generalized Bell states (GBSs) efficiently, it is important to classify them by local unitary (LU) equivalence. For example, there are C254(=12650) sets of four GBSs in C5⊗C5, which greatly exceed the C164(=1820) sets of four GBSs in C4⊗C4, and it is inconceivable to explore these sets directly without prior classification. In this paper, we consider the classification of GBS sets in C5⊗C5. First, two simple and useful characterizations of LU-equivalence of GBS sets are given. Then, one characterization is used to divide all 12 650 sets of four GBSs into eight equivalence classes including six locally distinguishable classes and two classes that cannot be discriminated by one-way local operations and classical communication. The second characterization is powerful and can be used to write a program, by using which all C255(=53130) sets of five GBSs in C5⊗C5 are divided into 21 equivalence classes including 9 locally distinguishable classes and 12 locally indistinguishable classes.
It is known that every (single-qudit) Clifford operator maps the full set of generalized Pauli matrices (GPMs) to itself under unitary conjugation, which is an important quantum operation and plays a crucial role in quantum computation and information. However, in many quantum information processing tasks, it is required that a specific set of GPMs be mapped to another such set under conjugation, instead of the entire set. We formalize this by introducing local Clifford operator, which maps a given nnn-GPM set to another such set under unitary conjugation. We establish necessary and sufficient conditions for such an operator to transform a pair of GPMs, showing that these local Clifford operators admit a classical matrix representation, analogous to the classical (or symplectic) representation of standard (single-qudit) Clifford operators. Furthermore, we demonstrate that any local Clifford operator acting on an nnn-GPM (n≥2n\geq 2n≥2) set can be decomposed into a product of standard Clifford operators and a local Clifford operator acting on a pair of GPMs. This decomposition provides a complete classical characterization of unitary conjugation mappings between nnn-GPM sets. As a key application, we use this framework to address the local unitary equivalence (LU-equivalence) of sets of generalized Bell states (GBSs). We prove that the 31 equivalence classes of 444-GBS sets in bipartite system C6⊗C6\mathbb{C}^{6}\otimes \mathbb{C}^{6}C6⊗C6 previously identified via Clifford operators are indeed distinct under LU-equivalence, confirming that this classification is complete.
The necessary and sufficient conditions for the equivalence of arbitrary nnn-qubit pure quantum states under local unitary (LU) operations derived in [B. Kraus, Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two nnn-qubit states (with n>2n>2n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide; however, the states are neither LU equivalent nor can be mapped into each other by general local operations and classical communication.
We present a necessary condition for two sets of maximally entangled states to be locally equivalent in terms of local invariants. With the help of these invariants, first, we are able to completely classify all the distinguishable and indistinguishable sets of four generalized Bell states in the $4\ensuremath{\bigotimes}4$ quantum system: there are ten inequivalent classes including three locally indistinguishable and seven locally distinguishable quadruples. All distinguishable quadruples in this case can be distinguished by using only one-way classical communication, implying that the two-way classical communication has no advantage over one-way classical communication, similar to the case of qubit lattice states. Second, we are able to completely classify all the triplets of generalized Bell states in the quantum systems of all prime dimensions as well as in some composite dimensions.