# 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

### Abstract

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.

### ► References

[1] M. Falk, J. Hüsler, and R.-D. Reiss, Laws of Small Numbers: Extremes and Rare Events, 3rd ed. (Birkhäuser, Basel, 2011).
https:/​/​doi.org/​10.1007/​978-3-0348-0009-9

[2] D. L. Snyder and M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
https:/​/​doi.org/​10.1007/​978-1-4612-3166-0

[3] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).
https:/​/​doi.org/​10.1103/​RevModPhys.84.621

[4] A. S. Holevo, Quantum Systems, Channels, Information, 2nd ed. (de Gruyter, Berlin, 2019).
http:/​/​www.degruyter.com/​view/​product/​180320

[5] M. Tsang, Physical Review Letters 107, 270402 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.270402

[6] M. Tsang, R. Nair, and X.-M. Lu, Physical Review X 6, 031033 (2016a).
https:/​/​doi.org/​10.1103/​PhysRevX.6.031033

[7] M. Tsang, R. Nair, and X.-M. Lu, in Proc. SPIE, Quantum and Nonlinear Optics IV, Vol. 10029 (SPIE, Bellingham, WA, 2016) p. 1002903.
https:/​/​doi.org/​10.1117/​12.2245733

[8] M. Tsang, Contemporary Physics 60, 279 (2019a).
https:/​/​doi.org/​10.1080/​00107514.2020.1736375

[9] W. Larson and B. E. A. Saleh, Optica 5, 1382 (2018).
https:/​/​doi.org/​10.1364/​OPTICA.5.001382

[10] M. Tsang and R. Nair, Optica 6, 400 (2019).
https:/​/​doi.org/​10.1364/​OPTICA.6.000400

[11] W. Larson and B. E. A. Saleh, Optica 6, 402 (2019).
https:/​/​doi.org/​10.1364/​OPTICA.6.000402

[12] Z. Hradil, J. Řeháček, L. Sánchez-Soto, and B.-G. Englert, Optica 6, 1437 (2019).
https:/​/​doi.org/​10.1364/​OPTICA.6.001437

[13] K. Liang, S. A. Wadood, and A. N. Vamivakas, Optica 8, 243 (2021).
https:/​/​doi.org/​10.1364/​OPTICA.403497

[14] 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).
https:/​/​doi.org/​10.1364/​OE.427734

[15] Z. Hradil, D. Koutný, and J. Řeháček, Optics Letters 46, 1728 (2021).
https:/​/​doi.org/​10.1364/​OL.417988

[16] S. De, J. Gil-Lopez, B. Brecht, C. Silberhorn, L. L. Sánchez-Soto, Z. Hradil, and J. Řeháček, Physical Review Research 3, 033082 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.033082

[17] S. Kurdzialek, arXiv:2103.12096 [physics, physics:quant-ph] (2021).
arXiv:2103.12096

[18] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
http:/​/​www.sciencedirect.com/​science/​bookseries/​00765392/​123

[19] X.-M. Lu, H. Krovi, R. Nair, S. Guha, and J. H. Shapiro, npj Quantum Information 4, 64 (2018).
https:/​/​doi.org/​10.1038/​s41534-018-0114-y

[20] D. Gottesman, T. Jennewein, and S. Croke, Physical Review Letters 109, 070503 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.070503

[21] E. T. Khabiboulline, J. Borregaard, K. De Greve, and M. D. Lukin, Physical Review Letters 123, 070504 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.070504

[22] J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, 2018).
https:/​/​doi.org/​10.1017/​9781316848142

[23] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
https:/​/​doi.org/​10.1017/​CBO9781139644105

[24] A. W. van der Vaart and J. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics (Springer-Verlag, New York, 1996).
https:/​/​doi.org/​10.1007/​978-1-4757-2545-2

[25] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (Birkhäuser, Basel, 1992).
https:/​/​doi.org/​10.1007/​978-3-0348-0566-7

[26] M. Leitz-Martini, Quantum Stochastic Calculus using Infinitesimals, Ph.D. thesis, University of Tübingen, Tübingen, Germany (2001).

[27] E. Nelson, Radically Elementary Probability Theory (Princeton University Press, Princeton, New Jersey, 1987).

[28] A. Uhlmann and B. Crell, in Entanglement and Decoherence, edited by A. Buchleitner, C. Viviescas, and M. Tiersch (Springer, Berlin, 2009) pp. 1–60.
https:/​/​doi.org/​10.1007/​978-3-540-88169-8_1

[29] M. Hayashi, Quantum Information Theory: Mathematical Foundation, 2nd ed. (Springer, Berlin, 2017).
https:/​/​doi.org/​10.1007/​978-3-662-49725-8

[30] R. Bhatia, T. Jain, and Y. Lim, Expositiones Mathematicae 37, 165 (2019).
https:/​/​doi.org/​10.1016/​j.exmath.2018.01.002

[31] K. M. R. Audenaert, J. Calsamiglia, R. Muñoz-Tapia, E. Bagan, L. Masanes, A. Acin, and F. Verstraete, Physical Review Letters 98, 160501 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.160501

[32] M. Nussbaum and A. Szkoła, The Annals of Statistics 37, 1040 (2009).
https:/​/​doi.org/​10.1214/​08-AOS593

[33] A. Wehrl, Reviews of Modern Physics 50, 221 (1978).
https:/​/​doi.org/​10.1103/​RevModPhys.50.221

[34] G. Lindblad, Communications in Mathematical Physics 33, 305 (1973).
https:/​/​doi.org/​10.1007/​BF01646743

[35] I. S. Dhillon and J. A. Tropp, SIAM Journal on Matrix Analysis and Applications 29, 1120 (2007).
https:/​/​doi.org/​10.1137/​060649021

[36] S.-i. Amari, Information Geometry and Its Applications (Springer Japan, Tokyo, 2016).
https:/​/​doi.org/​10.1007/​978-4-431-55978-8

[37] T. Kailath, IEEE Transactions on Communication Technology 15, 52 (1967).
https:/​/​doi.org/​10.1109/​TCOM.1967.1089532

[38] A. Cichocki and S.-i. Amari, Entropy 12, 1532 (2010).
https:/​/​doi.org/​10.3390/​e12061532

[39] M. Tsang, Physical Review A 99, 012305 (2019b).
https:/​/​doi.org/​10.1103/​PhysRevA.99.012305

[40] R. Nair and M. Tsang, Physical Review Letters 117, 190801 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.190801

[41] C. Lupo and S. Pirandola, Physical Review Letters 117, 190802 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.190802

[42] J. H. Eberly, American Journal of Physics 40, 1374 (1972).
https:/​/​doi.org/​10.1119/​1.1986858

[43] Y. L. Len, C. Datta, M. Parniak, and K. Banaszek, International Journal of Quantum Information 18, 1941015 (2020).
https:/​/​doi.org/​10.1142/​S0219749919410156

[44] C. Lupo, Physical Review A 101, 022323 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.022323

[45] C. Oh, S. Zhou, Y. Wong, and L. Jiang, Physical Review Letters 126, 120502 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.126.120502

[46] P. A. Meyer, Quantum Probability for Probabilists, 2nd ed. (Springer-Verlag, Berlin Heidelberg, 1995).
https:/​/​doi.org/​10.1007/​BFb0084701

[47] M. G. Genoni and T. Tufarelli, Journal of Physics A: Mathematical and Theoretical 52, 434002 (2019).
https:/​/​doi.org/​10.1088/​1751-8121/​ab3fe0

[48] L. Peng and X.-M. Lu, Physical Review A 103, 042601 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.042601

[49] 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.
https:/​/​www.mathworks.com/​matlabcentral/​fileexchange/​17555-light-bartlein-color-maps

### Cited by

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

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

[3] 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).

[4] 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).

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

The above citations are from Crossref's cited-by service (last updated successfully 2022-01-25 01:10:39) and SAO/NASA ADS (last updated successfully 2022-01-25 01:10:40). The list may be incomplete as not all publishers provide suitable and complete citation data.