An Information-Theoretic Measure of Uncertainty due to Quantum and Thermal Fluctuations

TL;DR

This paper introduces an information-theoretic measure of uncertainty for quantum systems that captures both quantum and thermal fluctuations, generalizes the uncertainty principle, and analyzes its evolution in non-equilibrium dynamics.

Contribution

It defines a new measure of uncertainty based on phase space probability distributions, relates it to entropy at high temperatures, and derives bounds for its evolution in linear quantum systems.

Findings

The measure coincides with von Neumann entropy at high temperatures.

It remains non-zero at zero temperature, capturing quantum fluctuations.

A lower bound for the measure's evolution is established for linear systems.

Abstract

We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information $I$ of the phase space probability distribution $\la z | \rho | z \ra $, where $|z \ra $ are coherent states, and $\rho$ is the density matrix. The uncertainty principle is expressed in this measure as $I \ge 1$. For a harmonic oscillator in a thermal state, $I$ coincides with von Neumann entropy, $- \Tr(\rho \ln \rho)$, in the high-temperature regime, but unlike entropy, it is non-zero at zero temperature. It therefore supplies a non-trivial measure of uncertainty due to both quantum and thermal fluctuations. We study $I$ as a function of time for a class of non-equilibrium quantum systems consisting of a distinguished system coupled to a heat bath. We derive an evolution equation for $I$. For the harmonic oscillator, in the Fokker-Planck regime, we show that $I$ increases monotonically. For more general Hamiltonians, $I$ settles down to monotonic increase in the long run, but may suffer an initial decrease for certain initial states that undergo ``reassembly'' (the opposite of quantum spreading). Our main result is to prove, for linear systems, that $I$ at each moment of time has a lower bound $I_t^{min}$, over all possible initial states. This bound is a generalization of the uncertainty principle to include thermal fluctuations in non-equilibrium systems, and represents the least amount of uncertainty the system must suffer after evolution in the presence of an environment for time $t$.