Explicit class field theory from quantum measurements Speaker(s): Jon Yard
Abstract:
It is easy to prove that ddimensional complex Hilbert space can contain at most d^2 equiangular lines. But despite considerable evidence and effort, sets of this size have only been proved to exist for finitely many d. Such sets are relevant in quantum information theory, where they define optimal quantum measurements known as SICPOVMs (Symmetric Informationally Complete Positive OperatorValued Measures). They also correspond to complex projective 2designs of the minimum possible cardinality. Numerical evidence points to their existence for all d as orbits of finite Heisenberg groups, the current record being d=844 [ScottGrassl '17]. However, to date, they are only proven to exist for finitely many d (the current record being d=323 [SG17]) via computerassisted calculations in number fields of degree increasing with d. In this talk, I will discuss the structure of these number fields, which turn out to be specific abelian extensions of specific real quadratic number fields [Appleby, Flammia, McConnell, Y. 1604.06098]. Such fields are known to exist by general theorems of class field theory, but until now, had never been found 'explicitly' in Nature. This contrasts the classical situation for abelian extensions of CM fields, which are generated by the torsion points of abelian varieties with complex multiplication. All known Heisenbergcovariant SICPOVMs have unitary symmetries under the associated Weil representation that are intimately related to the structure of the underlying number fields. A proper understanding of this relationship may ultimately lead to a general proof of their existence in all dimensions, rather than the finite number of examples currently proved to exist. Date: 16/10/2017  2:00 pm
Series: Mathematical Physics
