Abstract:

Unextendible product bases 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 unextendible product bases, including a complete characterization of all four-qubit unextendible product bases, which we show there are exactly 1446 of. We also show that there exist p-qubit UPBs of almost all sizes less than \(2^p\).

Authors:

• Nathaniel Johnston

Download:

• Official publication from Journal of Physics A: Mathematical and Theoretical
• Preprint from arXiv:1401.7920 [quant-ph]
• Local preprint [pdf]

Cite as:

• N. Johnston. The structure of qubit unextendible product bases. Journal of Physics A: Mathematical and Theoretical, 47:424034, 2014.

Supplementary material:

• 4 qubit UPBs – A complete list of all 1446 different UPBs on 4 qubits
• 8 state UPBs – A list of all UPBs with 8 or fewer states on any number of qubits
• Code for finding qubit UPBs – The C and Maple code that was used to find the 4-qubit and 8-state UPBs above, as well as the 6-qubit UPB from the paper
• In search of a 4-by-11 matrix – A blog post about the problem of searching for qubit UPBs

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