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

Abstract

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.

Rigidity and a common framework for mutually unbiased bases and k‐nets

“…Thus, Corollary 3 implies that the uncompletability part of the result still holds if one of the MUBs is discarded. Previously, many systems of MUBs have been shown to satisfy a weaker form of unextendibility [16].…”

Complementary decompositions and unextendible mutually unbiased bases

“…A natural idea is to try to find a subgroup H ≤ SL 2 (p) of order (p + 1)(p − 1), since then there is no element of order p in H, as the order of an element divides the order of the group. In the paper [15] these constructions are called Galois MUBs. A similiar theorem as Proposition 6 and the same MUBs also appear in [16].…”

“…Mutually unbiased bases are then equivalent to subalgebras whose traceless parts are orthogonal to each other with respect to the Hilbert-Schmidt scalar product of matrices. The same kind of orthogonality relationship can be studied also for other kind of subalgebras, for example factors are of a special interest [7,10,17].Recently there was some interest in non-complete MUBs that cannot be extended to a complete set [5,4,15]. Here we will use complementary decompositions to study the problem.…”

“…Recently there was some interest in non-complete MUBs that cannot be extended to a complete set [5,4,15]. Here we will use complementary decompositions to study the problem.…”

Maximal Partial Spreads of Polar Spaces

“…Recently maximal partial spreads of symplectic polar spaces received particular attention due to their applications in quantum information theory. In fact they correspond to so-called weakly unextendible mutually unbiased bases [13], [18]. In this paper we are interested in maximal partial spreads of H(2n + 1, q 2 ), Q + (4n − 1, q), Q(4n − 2, q), n 2, for any q and of W(2n + 1, q), n 2, when q is even.…”