Abstract:

A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovász in 2001, but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.

Authors:

• Jianxin Chen
• Nathaniel Johnston

Download:

• Preprint from arXiv:1301.1406 [quant-ph]
• Official publication from Springer
• Local preprint [pdf]
• Slideshow presentation [pdf]

Cite as:

• J. Chen and N. Johnston. The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases). Communications in Mathematical Physics, 333(1):351-365, 2015.

Supplementary material:

• Code for Computing Some Unextendible Product Bases – MATLAB scripts for explicitly constructing matrices that satisfy the conditions of Lemmas 4, 5, and 6. These matrices can then be used to construct UPBs of the size described by the paper.

Related publications:

• N. Johnston. The minimum size of qubit unextendible product bases. In Proceedings of the 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC), 2013. doi: 10.4230/LIPIcs.TQC.2013.93