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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 thedensity matrix. The uncertainty principle is expressed in this measure as $I\ge 1$. For a harmonic oscillator in a thermal state, $I$ coincides with vonNeumann entropy, $- \Tr(\rho \ln \rho)$, in the high-temperature regime, butunlike entropy, it is non-zero at zero temperature. It therefore supplies anon-trivial measure of uncertainty due to both quantum and thermalfluctuations. We study $I$ as a function of time for a class of non-equilibriumquantum systems consisting of a distinguished system coupled to a heat bath. Wederive an evolution equation for $I$. For the harmonic oscillator, in theFokker-Planck regime, we show that $I$ increases monotonically. For moregeneral 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 toprove, 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 generalizationof the uncertainty principle to include thermal fluctuations in non-equilibriumsystems, and represents the least amount of uncertainty the system must sufferafter evolution in the presence of an environment for time $t$.Comment: 36 pages (revised uncorrupted version), Report IC 92-93/2

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