Information-theoretic measure of uncertainty due to quantum and thermal fluctuations

We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information I of the phase-space probability distribution 〈z‖ρ‖z〉, where ‖z〉 are coherent states and ρ is the density matrix. As shown by Lieb I≥1, and this bound represents a strengthened version of the uncertainty principle. For a harmonic oscillator in a thermal state, I coincides with von Neumann entropy, -Tr(ρlnρ), in the high-temperature regime, but unlike entropy, it is nonzero (and equal to the Lieb bound) at zero temperature. It therefore supplies a nontrivial measure of uncertainty due to both quantum and thermal fluctuations. We study I as a function of time for a class of nonequilibrium 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, if the width of the coherent states is chosen to be the same as the width of the harmonic oscillator ground state. For other choices of the width, and for more general Hamiltonians, I settles down to a 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 ${\mathit{I}}_{\mathit{t}}^{\min }$, over all possible initial states.

This bound is a generalization of the uncertainty principle to include thermal fluctuations in nonequilibrium systems, and represents the least amount of uncertainty the system must suffer after evolution in the presence of an environment for time t. ${\mathit{I}}_{\mathit{t}}^{\min }$ is an envelope, equal for each time t, to the time evolution of I for a certain initial state, which we calculate to be a nonminimal Gaussian. ${\mathit{I}}_{\mathit{t}}^{\min }$ coincides with the Lieb bound in the absence of an environment, and is related to von Neumann entropy in the long-time limit. The form of ${\mathit{I}}_{\mathit{t}}^{\min }$ indicates that the thermal fluctuations become comparable with the quantum fluctuations on a time scale equal to the decoherence time scale, in agreement with earlier work of Hu and Zhang. Our results are also related to those of Zurek, Habib, and Paz, who looked for the set of initial states generating the least amount of von Neumann entropy after a fixed period of nonunitary evolution.