Jonathan Jedwab; How many mutually unbiased bases can exist in complex space of dimension d?

*Monday* Nov 21, 2016, 3:30pm.
LOW 117
Jonathan Jedwab, Department of Mathematics, Simon Fraser University

Abstract: A set of m mutually unbiased bases in C^d comprises md unit vectors in C^d, partitioned into m orthonormal bases for C^d such that the pairwise angle between all vectors from distinct bases is arccos(1/sqrt(d)). Schwinger noted in 1960 that no information can be obtained when a quantum system is prepared in a state belonging to one of the bases of such a set but is measured with respect to any other one of the bases. This property can be exploited in secure quantum key exchange, quantum state determination, quantum state reconstruction, and detection of quantum entanglement.

The central problem is to determine the largest number \mu(d) of mutually unbiased bases that can exist in C^d. It has been known for 40 years that \mu(d) <= d + 1, but a construction achieving the upper bound d + 1 is known only when d is a prime power. Despite considerable effort and a huge literature, there has been little progress in determining \mu(d) when d is not a prime power, with the notable exception of Weiner’s 2013 result that \mu(d) never equals d. Even the smallest non-prime-power case d = 6 remains baffling: all that is known is that \mu(6) ∈ {3, 4, 5, 7}.

I shall give an overview of the motivation and current state of knowledge for this problem, and describe some new insights involving combinatorial designs. No prior knowledge will be assumed.

This is joint work with Lily Yen of Capilano University.