An 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 $\text{la}z\mathrm{\mid }\rho \mathrm{\mid }z\text{ra}$, where $\mathrm{\mid }z\text{ra}$ are coherent states, and $\rho$ is the density matrix. The uncertainty principle is expressed in this measure as $I \ge 1$\mathrm{.}Foraharmonicoscillatorinathermalstate,$I$ coincides with von Neumann entropy, $-\text{Tr}(\rho \ln \rho )$, 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 $t$.