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arXiv:1401.7920 · Journal of Physics A Mathematical and Theoretical

The structure of qubit unextendible product bases

Nathaniel Johnston

Published Oct 9, 2014• 49 citations
arXivHTMLPDF

Abstract

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’.

DOI
10.1088/1751-8113/47/42/424034

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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’.
Unextendible product bases (UPBs) play an important role in quantum information theory. However, very little is known about UPBs in Hilbert space of local dimension more than three. In this paper, we study the UPBs in qutrit-ququad system and find that there only exist six, seven and eight-state UPBs. We completely characterize the six-state and seven-state UPBs. For eight-state UPBs, seven classes of UPBs are found. As auxiliary results, we study the distinguishability of qutrit-ququad UPBs by separable measurements, and find that there exists a UPB that cannot be distinguished.
Unextendible product bases (UPB) is an important concept and has extensive applications in various fields of quantum information theory. The construction of UPBs is closely related to combinatorics. Alon and Lovász (2001) first used a series of graph theoretic tools to characterize the necessary and sufficient conditions when the size of a UPB can attain the trivial lower bound. Later Feng (2006) brought the 1-factorization of graphs into the study of UPBs. In this paper, we further apply some graph theoretic tools to analyze the minimum size of a UPB under certain parameters and obtain a series of new results. Moreover, we have an almost complete characterization of all the potential sizes of UPBs in C2⊗C2⊗Ck\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^kC2⊗C2⊗Ck.
An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we construct a 7-qubit UPB of size 11. It solves an open problem in [Quantum Information Processing 19:185 (2020)]. Next, we graph-theoretically show that the UPB is locally indistinguishable in the bipartite systems of two qubits and five qubits, respectively. It turns out that the UPB corresponds to a complete graph with 11 vertices constructed by three sorts of nonisomorphic graphs. Taking the graphs as product vectors, we show that they are in three different orbits up to local unitary equivalence. Moreover, we also present the number of sorts of nonisomorphic graphs of complete graphs of some known UPBs and their orbits.
The construction of multiqubit unextendible product bases (UPBs) has been a long-standing problem. We give a systematic construction of multiqubit UPBs by studying their unextendible orthogonal matrices (UOMs) with graph-theoretic methods. Row decompositions of small UOMs produce the main intervals. Com- pletion graphs settle the remaining small UPB sizes, except for one size obtained from a good column structure. Nonexistence of the other UPB sizes follows from counts of fibres and row pairs. The computer is used only for a few finite UOM checks, whose exact tests are stated in the certificate appendix. These arguments determine the complete multiqubit UPB size spectrum in every dimension.
The 4-qubit unextendible product bases (UPBs) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. Based on this study we show that there is only one UPB of size 6 and six UPBs of size 9 in H=C2⊗C2⊗C4, three UPBs of size 9 in K=C4⊗C4, and no UPB of size 7 in H and K. Furthermore we construct a new family of 4-qubit positive-partial-transpose (PPT) entangled states of rank seven, and show that they are also PPT entangled states in H and K, respectively. We analytically derive a geometric measure of a special PPT entangled states.