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arXiv:quant-ph/0401156 · Physical Review A

Entanglement bases and general structures of orthogonal complete bases

Zai-Zhe Zhong

Published Oct 14, 2004• 1 citation
arXivHTMLPDF

Abstract

In quantum mechanics and quantum information, to establish the orthogonal bases is a useful means. The existence of unextendible product bases impels us to study the `entanglement bases' problems. In this paper, the concepts of entanglement bases and exact-entanglement bases are defined, and a theorem about exact-entanglement bases is given. We discuss the general structures of the orthogonal complete bases. Two examples of applications are given. At last, we discuss the problem of transformation of the general structure forms.

DOI
10.1103/physreva.70.044302

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In quantum mechanics and quantum information, to establish the orthogonal bases is a useful means. The existence of unextendible product bases impels us to study the `entanglement bases' problems. In this paper, the concepts of entanglement bases and exact-entanglement bases are defined, and a theorem about exact-entanglement bases is given. We discuss the general structures of the orthogonal complete bases. Two examples of applications are given. At last, we discuss the problem of transformation of the general structure forms.
Unextendible product bases (UPBs) have been shown to have many important uses in quantum information theory, particularly in the qubit case. However, very little is known about their mathematical structure beyond three qubits. We present several new results about qubit UPBs, including a complete characterization of all four-qubit UPBs, which we show there are exactly 1446 of. We also show that there exist p -qubit UPBs of almost all sizes less than 2 p . This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.
Finite-dimensional entanglement for pure states has been used extensively in quantum information theory. Depending on the tensor product structure, even a set of separable states can show non-intuitive characters. Two situations are well studied in the literature, namely, the unextendible product basis (UPB) by Bennett et al.4, and completely entangled subspaces explicitly given by Parthasarathy 22. More recently, Boyer et al.6, Boyer and Mor7 and Liss et al.21 studied spaces which have only finitely many pure product states. We carry this further and consider the problem of perturbing different spaces, such as the orthogonal complement of an UPB and also Parthasarathy’s completely entangled spaces, by taking linear spans with specified product vectors. To this end, we develop methods and theory of variations and perturbations of the linear spans of certain UPBs, their orthogonal complements, and also Parthasarathy’s completely entangled subspaces. Finally, we give examples of perturbations with infinitely many pure product states.
Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled but genuinely entangled. Here we explore the connection between UPBs and such genuinely entangled subspaces (GESs) and provide classes of nonorthogonal UPBs that lead to GESs for any number of parties and local dimensions. We then show how these subspaces can be immediately utilized for a simple general construction of genuinely entangled states in any such multipartite scenario.
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.