By taking a Poisson limit for a sequence of rare quantum objects, I derive simple formulas for the Uhlmann fidelity, the quantum Chernoff quantity, the relative entropy, and the Helstrom information. I also present analogous formulas in classical information theory for a Poisson model. An operator called the intensity operator emerges as the central quantity in the formalism to describe Poisson states. It behaves like a density operator but is unnormalized. The formulas in terms of the intensity operators not only resemble the general formulas in terms of the density operators, but also coincide with some existing definitions of divergences between unnormalized positive-semidefinite matrices. Furthermore, I show that the effects of certain channels on Poisson states can be described by simple maps for the intensity operators.
Quantum information, on the other hand, is a burgeoning research field that investigates the ultimate amount of information conveyed by quantum objects, such as photons. As quantum mechanics says that all objects must observe some rules of probability, the amount of information is usually limited. For example, a typical question in quantum information is "how much can I learn about the stars by observing the photons from them?"
Quantum information theory is tough. To solve problems in this research area, we have to use quantum mechanics, probability, statistics, and tons of mathematics. In view of the success of the Poisson law, this paper raises an interesting question: "Can the Poisson law help us simplify quantum information theory?" The answer is yes, and this paper shows that, when considering a stream of rare quantum objects, such as photons from a star, we can use the Poisson law to produce some neat formulas regarding the ultimate amount of information in the objects.
Admittedly, this work is quite theoretical, and its main goal for now is to simplify quantum information theory and make the lives of scientists in this area a bit easier. Time will tell whether the theory will lead to new insights that can improve telescopes, microscopes, and other quantum applications. But considering that the Poisson law has been so useful in so many areas of science and engineering, we may expect that its partnership with quantum information will be a pretty successful one too.
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