Uncertainty relations and indistinguishable particles

We show that for fermion states, measurements of any two finite outcome particle quantum numbers (e.g., spin) are not constrained by a minimum total uncertainty. We begin by defining uncertainties in terms of the outputs of a measurement apparatus. This allows us to compare uncertainties between multiparticle states of distinguishable and indistinguishable particles. Entropic uncertainty relations are derived for both distinguishable and indistinguishable particles. We then derive upper bounds on the minimum total uncertainty for bosons and fermions. These upper bounds apply to any pair of particle quantum numbers and depend only on the number of particles $N$ and the number of outcomes $n$ for the quantum numbers. For general $N$, these upper bounds necessitate a minimum total uncertainty much lower than that for distinguishable particles. The fermion upper bound on the minimum total uncertainty for $N$ an integer multiple of $n$ is zero. Our results show that uncertainty limits derived for single particle observables are valid only for particles that can be effectively distinguished. Outside this range of validity, the apparent fundamental uncertainty limits can be overcome.

Figure 1

Comparison of bounds on uncertainties $H_N\left(\mathcal{A}\right)$ and $H_N\left(\mathcal{B}\right)$ for distinguishable particles (21) and indistinguishable particles [relation (58) for fermions and relation (42) for bosons] for varying particle number $N$. The $q$ numbers are chosen to be complementary with $n=6$ distinct outcomes. This gives $c=1/\sqrt{6}$.