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arXiv:2207.04763 · New Journal of Physics

Unextendible and uncompletable product bases in every bipartition

Fei Shi, Mao-Sheng Li, Xiande Zhang, Qi Zhao

Published Nov 1, 2022• 11 citations
arXivHTMLPDF

Abstract

Abstract Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging from quantum nonlocality to quantum cryptography. A generalized concept called uncompletable product basis also attracts much attention. In this paper, we find some unextendible product bases that are uncompletable product bases in every bipartition, which answers a 19 year-old open question proposed by DiVincenzo et al (2003 Commun. Math. Phys. 238 379–410). As a consequence, we connect such unextendible product bases to local hiding of information, positive-partial-transpose entangled states and genuinely entangled states. Furthermore, we give a sufficient condition for the existence of an unextendible product basis that is still unextendible in every bipartition, and the existence of such a UPB is another open question proposed by Demianowic et al (2018 Phys. Rev. A 98 012313). Our results advance the understanding of the geometry of unextendible product bases.

DOI
10.1088/1367-2630/ac9e14

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Abstract Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging from quantum nonlocality to quantum cryptography. A generalized concept called uncompletable product basis also attracts much attention. In this paper, we find some unextendible product bases that are uncompletable product bases in every bipartition, which answers a 19 year-old open question proposed by DiVincenzo et al (2003 Commun. Math. Phys. 238 379–410). As a consequence, we connect such unextendible product bases to local hiding of information, positive-partial-transpose entangled states and genuinely entangled states. Furthermore, we give a sufficient condition for the existence of an unextendible product basis that is still unextendible in every bipartition, and the existence of such a UPB is another open question proposed by Demianowic et al (2018 Phys. Rev. A 98 012313). Our results advance the understanding of the geometry of unextendible product bases.
In 2003, DiVincenzo et al. put forward the question of whether there exists an unextendible product basis (UPB) that is an uncompletable product basis in every bipartition [DiVincenzo et al. Commun. Math. Phys. 238, 379–410 (2003)]. Recently, Shi et al. presented a UPB in tripartite systems that is also a strongly uncompletable product basis (SUCPB) in every bipartition [Shi et al. New J. Phys. 24, 113025 (2022)]. However, whether there exist UPBs that are SUCPBs in only one or two bipartitions remains unknown. We provide a sufficient condition for the existence of SUCPBs based on a quasi U-tile structure. We analyze all possible cases about the relationship between UPBs and SUCPBs in tripartite systems. In particular, we construct a UPB with smaller size d3 − 3d2 + 1 in Cd⊗Cd⊗Cd, which is an SUCPB in every bipartition and has a smaller cardinality than the existing one.
Unextendible product bases (UPBs) provide a versatile tool with various applications across different areas of quantum information theory. Their comprehensive characterization is thus of great importance and has been a subject of vital interest for over two decades now. An open question asks about the existence of UPBs, which are genuinely unextendible, i.e., they are not extendible even with biproduct vectors. In other words, the problem is to verify whether there exist genuinely entangled subspaces (GESs), subspaces composed solely of genuinely multiparty entangled states, complementary to UPBs. We solve this problem in the negative for many sizes of UPBs in different multipartite scenarios. In particular, in the all-important case of equal local dimensions, we show that there are always forbidden cardinalities for such UPBs, including the minimal ones corresponding to GESs of the maximal dimensions.
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.
Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. A famous open question is whether there exist genuinely unextendible product bases (GUPBs), namely multipartite product bases that are unextendible with respect to every possible bipartition. Here we shed light on this question by providing a characterization of UPBs and GUPBs in terms of orthogonality graphs. Building on this connection, we develop a method for constructing UPBs in low dimensions, and we derive a lower bound on the size of any 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. Finally, we discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.
A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving local measurement can be performed to this set. In general, it is difficult to show that such an orthogonality-preserving local measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully show the existence of unextendible product bases (UPBs) that are locally irreducible in every bipartition in d ⊗ d ⊗ d for any d ≥ 3 , and 3 ⊗ 3 ⊗ 3 achieves the minimum dimension for the existence of such UPBs. These UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Our result solves an open question given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. It also sheds new light on the connections between UPBs and strong quantum nonlocality.
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.