Unextendible Mutually Unbiased Bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

Abstract

We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of “unextendible mutually unbiased bases.” We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set of nonidentity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d a prime, and the geometry of non-degenerate alternating bilinear forms of rank N over finite fields F d . We then supply alternative and short proofs of results obtained in Mandayam et al. (2014), as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of Mandayam et al. (2014) and speculate on variations of this conjecture