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arXiv:1703.08773 · Quantum Information Processing

The local distinguishability of any three generalized Bell states

Yan-Ling Wang, Mao-Sheng Li, Shao-Ming Fei, Zhu‐Jun Zheng

Published Mar 29, 2017• 22 citations
arXivHTMLPDF
DOI
10.1007/s11128-017-1579-x

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Local distinguishability of sets of generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set of GBSs can be locally transformed to a certain type of GBS set that is easily distinguishable within local operations and one-way classical communication. We then show that if the space dimension ddd is a prime, these conditions are necessary and sufficient for sets of ddd GBSs in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ to be locally distinguishable. Thus we obtain a simple computable criterion for local distinguishability of sets of ddd GBSs in prime dimension ddd.
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.