The capacity of the photon counting channel

Abstract

Previous work has shown that the capacity of the single-mode photon-counting channel in the region in which quantum effects are most pronounced is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/(kT \ln 2)</tex> bit/J. Here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> is the quantum noise temperature, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> is Boltzmann's constant. However, to achieve this, the number of photons per second must be very low, so this does not determine the capacity of the average power-limited photon channel in bits per second. Here the previous work is extended and the capacity is determined in bits per second. This includes a determination of the optimum counting time. The capacity in bits per second is for low <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> asymptotic to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{s}/(kT \ln 2)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{s}</tex> is the average signal power. This is high enough to suggest that photon counting be seriously considered for applications such as a deep space to near Earth link. Some related problems are also considered including a peak power limitation.