Abstract:

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.

Authors:

• Nathaniel Johnston

Download:

• Official publication from LIPIcs
• Preprint from arXiv:1302.1604 [quant-ph]
• Local preprint [pdf]
• Slideshow presentation #1 [pdf]
• Slideshow presentation #2 [pdf]

Cite as:

• 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

Supplementary material:

• Code for proving that no UPB of size 4k + 3 exists on 4k qubits – C script for completing the proof of Lemma 6

Related publications:

• N. Johnston. The structure of qubit unextendible product bases. E-print: arXiv:1401.7920 [quant-ph], 2014.
• J. Chen and N. Johnston. The minimum size of unextendible product bases in the bipartite case (and some multipartite cases). E-print: arXiv:1301.1406 [quant-ph], 2013.