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arXiv:2402.11785

The construction of multiqubit unextendible product bases

Caohan Cheng, Lin Chen

Published Feb 19, 2024• 0 citations
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Abstract

The unextendible orthogonal matrices (UPBs) can be used for various problems in quantum information. We provide an algorithm to check if two UPBs are non-equivalent to each other. We give a method to construct UPBs and we apply this method to find all $5$-qubit UPBs of size eight. We apply the algorithm to check if the $5$-qubit UPBs of size eight are non-equivalent to each other. Based on all the $5$-qubit UPBs of size eight, we propose a theorem for constructing a new UPB non-equivalent to a given one.

DOI
10.48550/arxiv.2402.11785

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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.
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