Multiqubit UPB: The method of formally orthogonal matrices

Abstract

We use formal matrices whose entries we view as vector variables taking unit vectors values in one-qubit Hilbert spaces of a multiqubit quantum system. We construct many unextendible product bases (UPBs) of new sizes in such systems and provide a new construction of UPBs of $n$ qubits of cardinality $n+1$ when $n\equiv 3(\mathrm{mod}4)$. We also give a new method of constructing multiqubit entangled states with all partial transposes positive.

19 pages, the example in Appendix B corrected

References in corpus (3)

• Scalable multi-particle entanglement of trapped ions
• Qubit-qudit states with positive partial transpose
• Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$

Cited by in corpus (6)

• Unextendible and strongly uncompletable product bases
• The construction and local distinguishability of multiqubit unextendible product bases
• Constructing unextendible product bases from multiqubit ones
• Constructing $2\times 2\times 4$ and $4\times 4$ unextendible product bases and positive-partial-transpose entangled states
• $4\times 4$ unextendible product basis and genuinely entangled space
• Progress in the study of the (non)existence of genuinely unextendible product bases