Event Details

Speaker: Stefan Weigert (York)

Title: Mutually Unbiased Bases (or: Why six is the first odd integer)

Abstract: Consider two orthonormal bases of a complex Hilbert space. Take any pair of vectors (not from the same basis), and calculate the modulus of their scalar product. If the values of all the moduli found in this way take a single value only, the bases are called "mutually unbiased." Quantum physicists are interested in this property since it can be understood as a quantitative expression of complementarity. The eigenstates of two orthogonal components of a quantum spin and the (generalized) eigenstates of position and momentum of a quantum particle provide well-known examples of such bases.

It is known that a d-dimensional complex Hilbert space can support at most (d+1) sets of pairwise mutually unbiased bases. Complete sets of (d+1) mutually unbiased bases have been constructed for spaces of dimension d given by a prime number, or by the power of a prime. To date, it remains unknown if complete sets of mutually unbiased bases exist in spaces with dimensions different from a prime power, i.e. in composite dimensions such as six or ten.

In this presentation I will introduce mutually unbiased bases, describe their main properties and the long-standing open existence problem of complete sets in composite dimensions (cf. arxiv.org/abs/2410.23997 ). Along the way, I will highlight some alternative characterisations of mutually unbiased bases involving complex Hadamard matrices, Lie algebras or the theory of designs, for example.