Poisson Quantum Information

Mankei Tsang

Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583
Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551

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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.

The Poisson law is a fundamental rule of probability that describes the number of rare events over a long period of time. For centuries, scientists have used the Poisson law to describe many phenomena, from the number of Prussian soldiers killed by accidental horse-kicks in the 19th century to the random arrivals of particles of light, called photons, from stars at a telescope.

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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Cited by

[1] Cosmo Lupo, "Poisson states in quantum information", Quantum Views 5, 59 (2021).

[2] Xiao-Jie Tan and Mankei Tsang, "Quantum limit to subdiffraction incoherent optical imaging. III. Numerical analysis", Physical Review A 108 5, 052416 (2023).

[3] Mankei Tsang, "Efficient Superoscillation Measurement for Incoherent Optical Imaging", IEEE Journal of Selected Topics in Signal Processing 17 2, 513 (2023).

[4] Ben Wang, Liang Xu, Hongkuan Xia, Aonan Zhang, Kaimin Zheng, and Lijian Zhang, " Quantum-limited resolution of partially coherent sources", Chinese Optics Letters 21 4, 042601 (2023).

[5] Stanislaw Kurdzialek, "Back to sources – the role of losses and coherence in super-resolution imaging revisited", Quantum 6, 697 (2022).

[6] Giacomo Sorelli, Manuel Gessner, Mattia Walschaers, and Nicolas Treps, "Quantum limits for resolving Gaussian sources", Physical Review Research 4 3, L032022 (2022).

[7] Ilaria Gianani, Luis L. Sánchez-Soto, Aaron Z. Goldberg, and Marco Barbieri, "Efficient line shape estimation by ghost spectroscopy", Optics Letters 48 12, 3299 (2023).

[8] Mankei Tsang, "Quantum limit to subdiffraction incoherent optical imaging. II. A parametric-submodel approach", Physical Review A 104 5, 052411 (2021).

[9] Jasminder S. Sidhu, Michael S. Bullock, Saikat Guha, and Cosmo Lupo, "Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination", Quantum 7, 1025 (2023).

[10] Zixin Huang and Cosmo Lupo, "Quantum Hypothesis Testing for Exoplanet Detection", Physical Review Letters 127 13, 130502 (2021).

[11] 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).

[12] 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).

[13] Junyan Li and Shengshi Pang, "Quantum-limited superresolution of two incoherent point sources with unknown photon numbers", arXiv:2403.13752, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-07-15 20:35:03) and SAO/NASA ADS (last updated successfully 2024-07-15 20:35:04). The list may be incomplete as not all publishers provide suitable and complete citation data.

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