Marcus Appleby

From Quantum Probabilities to Hilbert’s Twelfth Problem

The talk concerns SIC POVMs (“symmetric informationally complete measurements positive operator valued measures”), or SICs for short. These structures were originally proposed in quantum information and quantum foundations, where they lead to a particularly appealing representation of the quantum state in probabilistic terms. Numerous examples have been constructed, the current record being an exact SIC in dimension 1299, and an approximate one in dimension 2208. This encourages the conjecture that SICs exist in every finite dimension. However, in spite of much theoretical work over more than twenty years, the question remains open. In this talk I will describe a connection between this physics problem and some major open problems in algebraic number theory. The connection is particularly intriguing because, although there are numerous examples of cross-fertilization between physics and pure mathematics, connections between physics and number theory are not so common. No specialist knowledge will be assumed of either quantum mechanics or number theory.