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.
 M. Tsang, Physical Review Letters 107, 270402 (2011).
 S. A. Wadood, K. Liang, Y. Zhou, J. Yang, M. A. Alonso, X.-F. Qian, T. Malhotra, S. M. Hashemi Rafsanjani, A. N. Jordan, R. W. Boyd, and A. N. Vamivakas, Optics Express 29, 22034 (2021).
 S. De, J. Gil-Lopez, B. Brecht, C. Silberhorn, L. L. Sánchez-Soto, Z. Hradil, and J. Řeháček, Physical Review Research 3, 033082 (2021).
 C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
 D. Gottesman, T. Jennewein, and S. Croke, Physical Review Letters 109, 070503 (2012).
 E. T. Khabiboulline, J. Borregaard, K. De Greve, and M. D. Lukin, Physical Review Letters 123, 070504 (2019).
 M. Leitz-Martini, Quantum Stochastic Calculus using Infinitesimals, Ph.D. thesis, University of Tübingen, Tübingen, Germany (2001).
 E. Nelson, Radically Elementary Probability Theory (Princeton University Press, Princeton, New Jersey, 1987).
 A. Uhlmann and B. Crell, in Entanglement and Decoherence, edited by A. Buchleitner, C. Viviescas, and M. Tiersch (Springer, Berlin, 2009) pp. 1–60.
 R. Nair and M. Tsang, Physical Review Letters 117, 190801 (2016).
 C. Lupo and S. Pirandola, Physical Review Letters 117, 190802 (2016).
 C. Oh, S. Zhou, Y. Wong, and L. Jiang, Physical Review Letters 126, 120502 (2021).
 R. Bemis, ``Light Bartlein Color Maps (https://www.mathworks.com/matlabcentral/fileexchange/17555-light-bartlein-color-maps),'' MATLAB Central File Exchange (online) (2016), retrieved May 30, 2021.
 Cosmo Lupo, "Poisson states in quantum information", Quantum Views 5, 59 (2021).
 Stanislaw Kurdzialek, "Back to sources – the role of losses and coherence in super-resolution imaging revisited", Quantum 6, 697 (2022).
 Giacomo Sorelli, Manuel Gessner, Mattia Walschaers, and Nicolas Treps, "Quantum limits for resolving Gaussian sources", Physical Review Research 4 3, L032022 (2022).
 Mankei Tsang, "Quantum limit to subdiffraction incoherent optical imaging. II. A parametric-submodel approach", Physical Review A 104 5, 052411 (2021).
 Zixin Huang and Cosmo Lupo, "Quantum Hypothesis Testing for Exoplanet Detection", Physical Review Letters 127 13, 130502 (2021).
 Syamsundar De, Jano Gil-Lopez, Benjamin Brecht, Christine Silberhorn, Luis L. Sánchez-Soto, Zdeněk Hradil, and Jaroslav Řeháček, "Effects of coherence on temporal resolution", Physical Review Research 3 3, 033082 (2021).
 Kevin Liang, S. A. Wadood, and A. N. Vamivakas, "Coherence effects on estimating general sub-Rayleigh object distribution moments", Physical Review A 104 2, 022220 (2021).
The above citations are from Crossref's cited-by service (last updated successfully 2022-12-08 08:00:39) and SAO/NASA ADS (last updated successfully 2022-12-08 08:00:40). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.