There has been interest in the dynamics of noninteracting bosons because of the boson sampling problem. These dynamics can be difficult to predict because of the complicated interference patterns that arise due to their indistinguishability. However, if there are unobserved, hidden degrees of freedom, the indistinguishability can be disrupted in the observations. The generalized bunching probability is defined to be the probability that noninteracting bosons undergo linear optical evolution and all arrive in a subset of sites. The strong generalized bunching conjecture states that among models where the hidden state of the bosons is separable, the generalized bunching probability is maximized when the bosons are perfectly indistinguishable. This conjecture was shown to be false [Seron, et. al., (2023)]. We show that among models where the hidden state is permutation invariant, the generalized bunching probability is maximized by perfectly indistinguishable bosons if and only if Lieb’s permanental dominance conjecture for immanants is true. We discuss applications of this conjecture to measuring the temperature of bosons trapped in an optical lattice.
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