Quantum coin-tossing in a Bayesian Jeffreys framework

Abstract

The Jeffreys' reparameterization-invariant prior for the classical problem of tossing a coin (with its possible binary outcomes designated ±1) is the arc-sine distribution, 1 π (1 − z 2) 1 2 , where z ∈ [−1, 1] is the (long-run) expected value intrinsic to the coin. It is demonstrated here that the analogous prior for the problem of measuring a two-level quantum system — represented by a 2 × 2 density matrix ϱ parameterized by points ( x, y, z) in the Bloch sphere (unit ball) — along two orthogonal ( Y and Z) axes is 1 2 π(1 − y 2 − z 2) 1 2 , where y is the expected value along the Y-axis and z that along the Z-axis. This contention rests upon the identification of ϱ as the covariance matrix of a bivariate complex normal distribution over the vectors (φ) of two-dimensional complex Hilbert space. The identification, in turn, is based upon quantum probabilistic arguments of Bach, supplemented by an invocation of the principle of maximum entropy. In addition, the Jeffreys' priors are presented for several sets of pairs (entangled and not) of two-level systems.