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arXiv:2109.13537 · Journal of Mathematical Physics

Unextendible product operator basis

Mengyao Hu, Lin Chen, Fei Shi, Xiande Zhang, Jordi Tura

Published Dec 1, 2022• 3 citations
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Abstract

Quantum nonlocality is associated with the local indistinguishability of orthogonal states. Unextendible product basis (UPB), a widely used tool in quantum information, exhibits nonlocality, which is the powerful resource for quantum information processing. In this work, we extend the definitions of nonlocality and genuine nonlocality from states to operators. We also extend UPB to the notions of unextendible product operator basis, unextendible product unitary operator basis (UPUOB), and strongly UPUOB. We construct their examples and show the nonlocality of some strongly UPUOBs under local operations and classical communications. We study the phenomenon of these operators acting on quantum states. As an application, we distinguish the two-dimensional strongly UPUOB, which only consumes three ebits of entanglement. Our results imply that such UPUOBs exhibit nonlocality as UPBs, and the distinguishability of them requires entanglement resources.

DOI
10.1063/5.0097918

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Quantum nonlocality is associated with the local indistinguishability of orthogonal states. Unextendible product basis (UPB), a widely used tool in quantum information, exhibits nonlocality, which is the powerful resource for quantum information processing. In this work, we extend the definitions of nonlocality and genuine nonlocality from states to operators. We also extend UPB to the notions of unextendible product operator basis, unextendible product unitary operator basis (UPUOB), and strongly UPUOB. We construct their examples and show the nonlocality of some strongly UPUOBs under local operations and classical communications. We study the phenomenon of these operators acting on quantum states. As an application, we distinguish the two-dimensional strongly UPUOB, which only consumes three ebits of entanglement. Our results imply that such UPUOBs exhibit nonlocality as UPBs, and the distinguishability of them requires entanglement resources.
Abstract A set of multipartite orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. It is known that unextendible product bases (UPBs) can show the phenomenon of quantum nonlocality without entanglement. Thus it is interesting to investigate the strong quantum nonlocality for UPBs. Most of the UPBs with the minimum size cannot demonstrate strong quantum nonlocality. In this paper, we construct a series of UPBs with different large sizes in d A ⊗ d B ⊗ d C and d A ⊗ d B ⊗ d C ⊗ d D for d A , d B , d C , d D ⩾ 3, and we also show that these UPBs have strong quantum nonlocality, which answers an open question given by Halder et al (2019 Phys. Rev. Lett. 122 040403) and Yuan et al (2020 Phys. Rev. A 102 042228) for any possible three and four-partite systems. Furthermore, we propose an entanglement-assisted protocol to locally discriminate the UPB in 3 ⊗ 3 ⊗ 4, and it consumes less entanglement resource than the teleportation-based protocol. Our results build the connection between strong quantum nonlocality and UPBs.
An unextendible product basis (UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled state. We exhibit a tripartite $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e., all three corresponding $2\ifmmode\times\else\texttimes\fi{}4$ bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local positive operator valued measurements and classical communication.
As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.
It is known that the states in an unextendible product basis (UPB) cannot be distinguished perfectly when the parties are restricted to local operations and classical communication (LOCC). Previous discussions of such bases have left open the following question: What entanglement resources are necessary and/or sufficient for this task to be possible with LOCC? In this paper, I present protocols which use entanglement more efficiently than teleportation to distinguish certain classes of UPB's. The ideas underlying my approach to this problem offer rather general insight into why entanglement is useful for such tasks.
Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. Here we provide an equivalent characterization of UPBs in graph-theoretic terms. Different from previous graph-theoretic investigations of UPBs, which focused mostly on the orthogonality relations between different product states, our characterization includes a graph-theoretic reformulation of the unextendibility condition. Building on this characterization, we develop a constructive method for building UPBs in low dimensions and shed light on the open question of whether there exist genuinely unextendible product bases (GUPBs), that is, multipartite product bases that are unextendible with respect to every possible bipartition. We derive a lower bound on the size of any candidate GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs and discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension. Finally, we apply our characterization to the problem of distinguishing UPB states under local operations and classical communication, deriving a necessary condition for reliable discrimination in the asymptotic limit.
An unextendible biseparable basis (UBB) is a set of orthogonal pure biseparable states which span a subspace of a given Hilbert space while the complementary subspace contains only genuinely entangled states. These biseparable bases are useful to produce genuinely entangled subspace in a multipartite system. Such a subspace could be more beneficial for information theoretic applications if we were able to extract distillable entanglement across every bipartition from each state of this subspace. In this paper, we have derived a rule for constructing such a class of UBB which exhibits the phenomenon of strong quantum nonlocality. This result positively answers the open problem raised by Agrawal et al. [Agrawal, Halder, and Banik, Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition, Phys. Rev. A 99, 032335 (2019)], that there exists a UBB which can demonstrate the phenomenon of strong quantum nonlocality in the perspective of the local irreducibility paradigm.