Measuring Multiparticle Indistinguishability with the Generalized Bunching Probability

TL;DR

The paper introduces generalized bunching probability $b(S|U,\rho)$ as a partial diagnostic for multiparticle boson indistinguishability in passive linear optics, addressing the challenge of full visible-state tomography. It analyzes two randomization strategies—permuting occupied visible modes and applying Haar-random unitaries—and proves monotonicity of $b$ with respect to distinguishability under a refined representation-theoretic framework, including a connection to Lieb's permanental-dominance conjecture. A key result is that the weak generalized-bunching conjecture implies that the perfectly indistinguishable state maximizes mean and refined bunching probabilities, with consequences such as Schur-convexity for uniform hidden states and a thermometry method for Gibbs states. These findings offer a practical, information-rich partial tomography-style approach to benchmarking indistinguishability in experiments, with potential applications in cold-atom thermometry and deeper ties to immanants and Schur-Weyl duality.

Abstract

The indistinguishability of many bosons undergoing passive linear transformations followed by number basis measurements is fully characterized by the visible state of the bosons. However, measuring all the parameters in the visible state is experimentally demanding. In this work, we seek to perform partial characterization of the visible state by measuring properties of it that are available after randomization. First we study the case where the occupied visible modes are randomly permuted, and second we study the case where Haar random linear optical unitaries are applied. In each case, we find that the generalized bunching probability -- which is the probability that all the input bosons arrive in a given subset of the output modes -- obeys monotonicity with respect to some partial order of distinguishability of the input bosons. As an intermediate result, we show that Lieb's permanental-dominance conjecture for immanants is equivalent to the following statement: for states that are invariant under permutations of the occupied visible modes, the generalized bunching probability is maximized when the bosons are perfectly indistinguishable. We also prove that a consequence of the monotonicity of the generalized bunching probability after Haar averaging is that this average is maximized when the bosons are perfectly indistinguishable. Finally, we discuss applications of our results to thermometry of cold-atom systems.

Figures (2)

• Figure 1: Interferometer with $n$ input bosons, with at most one in each of $m$ spatial modes. The interferometer acts only on the spatial degree of freedom, but the bosons may have some other hidden DOFs, such as time. We perform number-basis measurements at the end, which do not resolve the hidden DOFs. The generalized bunching probability shchesnovichUniversalityGeneralizedBunching2016 is the probability that all the input photons arrive in a given subset of the visible output modes.
• Figure 2: A schematic description of refinement of partitions. $\mu$ is obtained by partitioning the parts of $\lambda$.

Theorems & Definitions (123)

• Definition 2.1: Linear-optical action
• Definition 3.1: Singly-occupied subspace
• Definition 3.2: Permutation matrix on $\bm{i}$
• Definition 3.3: Permutation-invariant state
• Conjecture 3.4: Weak Generalized Bunching
• Definition 3.5
• Conjecture 3.6: Lieb's Permanental-Dominance Conjecture for Immanants
• Theorem 3.7
• Definition 3.8: Partially labelled state
• Definition 3.9: Refinement partial order