On indistinguishability of quantum states (Q2905806)

The paper under review is concerned with the study of indistinguishability of quantum states (understood as density matrices on a Hilbert space \(H\)) with respect to a given quantum measurement \(M\), i.e., a semi-spectral measure on \(\mathbb{R}\) built of positive operators in \(B(H)\). The relevant equivalence class of a state \(\rho\) is explicitly characterized if \(M\) has countably many outcomes (i.e., is a semi-spectral measure built of countably many positive operators) and either \(\rho\) is pure or \(H\) is finite-dimensional. The method of proving the result is based on first considering the case of a pure state and a genuine spectral measure and then purifying the state and dilating the measurement. Further, some explicit non-symmetric informationally complete (distinguishing all states) measurements on an \(n\)-dimensional Hilbert space built of \(n^2\) elements are constructed and an analogous technique is shown to lead to the existence of informationally complete measurements which have countably many outcomes on an infinite-dimensional separable Hilbert space.