Retrodictive distributions induced by proper and improper mixture constructions

Determine whether the Mai–Alquier proper-mixture construction and the induced improper-mixture construction generate distinct probability distributions over retrodiction maps, characterize how those distributions depend on the system dimension D and environment dimension K, and quantify any resulting differences in quantum-state recovery or other noisy inverse problems.

Background

The paper establishes an exact equality in distribution between induced random mixed states obtained by tracing out a K-dimensional environment from a Haar-random bipartite pure state and Mai–Alquier mixtures of K independent Haar-random pure states with symmetric Dirichlet weights. Although these constructions produce identically distributed density operators, the paper notes that they represent different physical or epistemic realizations: the Mai–Alquier ensemble corresponds to a proper classical mixture, whereas the induced ensemble arises as an improper mixture from partial tracing an entangled state.

The unresolved issue concerns prior-extended Petz maps and related retrodictive procedures, for which proper and improper realizations of the same reduced density operator may lead to different retrodictive updates. The problem is to determine whether the two specific constructions yield different distributions over retrodiction maps, establish the dependence of any difference on D and K, and measure its consequences for quantum-state recovery or other noisy inverse problems.