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.