Measuring Multiparticle Indistinguishability with the Generalized Bunching Probability

Abstract: The indistinguishability of many bosons undergoing passive linear transformations followed by number basis measurements is fully characterized by its visible state. However, measuring all of the parameters in the visible state is experimentally demanding. We argue that the generalized bunching probability -- which is the probability that all the input bosons arrive in a given subset of the output modes -- provides useful partial information about the indistinguishability of the input bosons, by establishing that it is monotonic with respect to certain partial orders of distinguishability of the bosons. As an intermediate result, we prove that if Lieb's permanental-dominance conjecture holds, then among states that are invariant under permutations of the occupied visible modes, the generalized bunching probability is maximized when the bosons are perfectly indistinguishable. As a corollary, we show that if Lieb's conjecture holds, then the generalized bunching probability is monotonic with respect to the refinement partial order on what we refer to as partially labelled states. We also prove, unconditionally, that for states such that the single-particle density matrix is the same for each particle, the Haar average of the generalized bunching probability is Schur convex with respect to the eigenvalues of said single-particle density matrix. As an application of the Schur-convexity, we show that when the single-particle density matrix is a Gibbs state, the mean generalized bunching probability serves as a thermometer.