We investigate the problem of constructing unextendible product bases in the qubit case - that is, when each local dimension equals 2. The cardinality of the smallest unextendible product basis is known in all qubit cases except when the number of parties is a multiple of 4 greater than 4 itself. We construct small unextendible product bases in all of the remaining open cases, and we use graph theory techniques to produce a computer-assisted proof that our constructions are indeed the smallest possible.
@InProceedings{johnston:LIPIcs.TQC.2013.93, author = {Johnston, Nathaniel}, title = {{The Minimum Size of Qubit Unextendible Product Bases}}, booktitle = {8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)}, pages = {93--105}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-55-2}, ISSN = {1868-8969}, year = {2013}, volume = {22}, editor = {Severini, Simone and Brandao, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.93}, URN = {urn:nbn:de:0030-drops-43173}, doi = {10.4230/LIPIcs.TQC.2013.93}, annote = {Keywords: unextendible product basis; quantum entanglement; graph factorization} }
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