Uncertainty Relations and Indistinguishable Particles

TL;DR

This paper investigates uncertainty relations for indistinguishable particles, revealing that fermion states can have arbitrarily low total uncertainty for certain measurements, challenging traditional limits based on single-particle observables.

Contribution

It derives entropic uncertainty relations for indistinguishable particles and establishes conditions under which uncertainty bounds can be surpassed.

Findings

Fermion states can have zero total uncertainty for certain quantum number pairs.

Upper bounds on uncertainty depend on particle number and outcomes, and can be much lower than for distinguishable particles.

Traditional uncertainty limits do not hold for indistinguishable particles in specific regimes.

Abstract

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 multi-particle 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.