Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

Abstract

We consider questions posed in a recent paper of Mandayam, Bandyopadhyay, Grassl and Wootters [10] 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 [19] 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 $\mathbb{F}_d$. We then supply alternative and short proofs of results obtained in [10], as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of [10], and speculate on variations of this conjecture.