Konrad Szymański, Junior Fellows Seminar: Finite-dimensional position and momentum and their covariance matrices

Speaker
Konrad Szymański, Slovak Academy of Sciences

Abstract
In this talk, I introduce self-adjoint matrices \(Q\) and \(P\) related via discrete Fourier transform, whose exponentials generate the Weyl-Heisenberg group over \(\mathbb{Z}/d\mathbb{Z}\). They play the role of finite-dimensional analogues to position and momentum operators; the aim is to characterize the allowed 2×2 covariance matrices \(\Gamma_{ij} = \langle A_i A_j \rangle – \langle A_i \rangle \langle A_j \rangle\) with \((A_1,A_2) = (Q,P)\). Drawing from the theory of numerical ranges, it is shown that the problem has a fourfold symmetry induced by the discrete Fourier transform constraining the extremizers of the trace of covariance matrices, and — by a classical result of Au-Yeung and Poon — that for every mixed state there exists a state of maximum rank 2 with the same covariance matrix. Numerical observations about the structure of the trace and determinant (the two unitary invariants) are reflected in the geometry of the relevant numerical ranges as well. The results are compared with the infinite-dimensional case.