On a General Method of Treating Uncomplete Systems in Quantum Mechanics

H.J. Groenewold

Xiao-Fan Zhen, Hui-Juan Zuo, Fei Shi +1 more

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.

Hilbrand Johannes Groenewold

Fei Shi, Mao-Sheng Li, Xiande Zhang +1 more

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.

Saronath Halder, Alexander Streltsov

In this work, we explore the notions unextendible product basis and uncompletability for operators which remain positive under partial transpose. Then, we analyze their connections to the ensembles which are many-copy indistinguishable under local operations and classical communication (LOCC). We show that the orthogonal complement of any bipartite pure entangled state is spanned by product states which form a nonorthogonal unextendible product basis (nUPB) of maximum cardinality. This subspace has one to one correspondence with the maximum dimensional subspace where there is no orthonormal product basis. Due to these, the proof of indistinguishability of a class of ensembles under LOCC in many-copy scenario becomes simpler. Furthermore, it is now clear that there are several many-copy indistinguishable ensembles which are different construction-wise. But if we consider the technique of proving their indistinguishability property under LOCC, then, for many of them it can be done using the general notion of unextendible product basis. Explicit construction of the product states, forming nUPBs is shown. Thereafter, we introduce the notion of positive partial transpose uncompletability to unify different many-copy indistinguishable ensembles. We also report a class of multipartite many-copy indistinguishable ensembles for which local indistinguishability property increases with decreasing number of mixed states.

Hilbrand Johannes Groenewold