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On the heat channel and its capacity

The heat channel is defined by an analog filter and a subsequent inaccurate measurement of the filter output signal. The filter is related to the solution of the heat equation and to the heat kernel of the quantum mechanical harmonic oscillator, so the name of the channel. The channel is modeled as an infinite-dimensional vector Gaussian channel and the capacity in terms of average energy of the input signal is derived. The relation to rate distortion theory is investigated by calculating the rate distortion function of a closely connected Gaussian process. Then a second formula for the capacity of the heat channel based on average energy of the measured filter output signal is given. The result is interpreted in context of estimation theory and a parallel to a famous formula connecting mutual information with minimum mean-square error in Gaussian channels is presented.