Constructing locally indistinguishable orthogonal product bases in an $m\otimes n$ system

Abstract

Recently, Zhang et al [Phys. Rev. A 92, 012332 (2015)] presented $4d-4$ orthogonal product states that are locally indistinguishable and completable in a $d\otimes d$ quantum system. Later, Zhang et al. [arXiv: 1509.01814v2 (2015)] constructed $2n-1$ orthogonal product states that are locally indistinguishable in $m\otimes n$ ($3\le m\le n$). In this paper, we construct a locally indistinguishable and completable orthogonal product basis with $4p-4$ members in a general $m\otimes n$ ($3\le m\le n$) quantum system, where $p$ is an arbitrary integer from $3$ to $m$, and give a very simple but quite effective proof for its local indistinguishability. Specially, we get a completable orthogonal product basis with $8$ members that cannot be locally distinguished in $m\otimes n$ ($3\le m\le n$) when $p=3$. It is so far the smallest completable orthogonal product basis that cannot be locally distinguished in a $m\otimes n$ quantum system. On the other hand, we construct a small locally indistinguishable orthogonal product basis with $2p-1$ members, which is maybe uncompletable, in $m\otimes n$ ($3\le m\le n$ and $p$ is an arbitrary integer from $3$ to $m$). We also prove its local indistinguishability. As a corollary, we give an uncompletable orthogonal product basis with $5$ members that are locally indistinguishable in $m\otimes n$ ($3\le m\le n$). All the results can lead us to a better understanding of the structure of a locally indistinguishable product basis in $m\otimes n$.

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• Local distinguishability based genuinely quantum nonlocality without entanglement
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• Locally stable sets with minimum cardinality
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• Alternative method for deriving nonlocal multipartite product states
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• Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis
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• More assistance of entanglement, less rounds of classical communication
• Quantum nonlocality without entanglement depending on nonzero prior probabilities in optimal unambiguous discrimination
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• Exploring strong locality : Quantum state discrimination regime and beyond
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• Genuinely entangled subspaces beyond strongly nonlocal unextendible biseparable bases
• Conclusive Local State Marking: More Nonlocality With No Entanglement
• Graph connectivity based strong quantum nonlocality with genuine entanglement
• Application of Ramsey theory to localization of set of product states via multicopies