Kelly Betting with Quantum Payoff: a continuous variable approach

Salvatore Tirone1, Maddalena Ghio1, Giulia Livieri1, Vittorio Giovannetti2, and Stefano Marmi1

1Scuola Normale Superiore, I-56126 Pisa, Italy
2NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy

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The main purpose of this study is to introduce a semi-classical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the free-energy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all their initial resources into coherent state amplitude.

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[1] R. Bell and T. M. Cover. Game-theoretic optimal portfolios. Management Science, 34 (6): 724–733, 1988. 10.1287/​mnsc.34.6.724.

[2] G. N. Iyengar and T. M. Cover. Growth optimal investment in horse race markets with costs. IEEE Transactions on Information Theory, 46 (7): 2675–2683, 2000. 10.1109/​18.887881.

[3] L. V. Williams. Information efficiency in financial and betting markets. Cambridge University Press, 2005. 10.1017/​CBO9780511493614.

[4] J. L. Kelly. A new interpretation of information rate. The Bell System Technical Journal, 35 (4): 917–926, 1956. 10.1002/​j.1538-7305.1956.tb03809.x.

[5] L. C. MacLean, E. O. Thorp, and W. T. Ziemba. The Kelly capital growth investment criterion: Theory and practice, volume 3. World Scientific, 2011. 10.1142/​7598.

[6] A. S. Holevo. Quantum Systems, Channels, Information: A Mathematical Introduction. De Gruyter, 2012. 10.1515/​9783110273403.

[7] M. M. Wilde. Quantum Information Theory. Cambridge University Press, 2 edition, 2017. 10.1017/​9781316809976.

[8] A. S. Holevo and R. F. Werner. Evaluating capacities of bosonic gaussian channels. Physical Review A, 6 (032312), 2001. 10.1103/​PhysRevA.63.032312.

[9] F. Caruso, V. Giovannetti, and A. S. Holevo. One-mode bosonic gaussian channels: a full weak-degradability classification. New Journal of Physics, 8 (12): 310, 2006. 10.1088/​1367-2630/​8/​12/​310.

[10] V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo. Ultimate classical communication rates of quantum optical channels. Nature Photonics, 8: 796–800, 2014. 10.1038/​nphoton.2014.216.

[11] A. Serafini. Quantum Continuous Variables. CRC Press, 2017. 10.1201/​9781315118727.

[12] D. B. Hausch, V. S. Lo, and W. T. Ziemba. Efficiency of racetrack betting markets, volume 2. World Scientific, 2008. 10.1142/​6910.

[13] D. A. Meyer. Quantum strategies. Physical Review Letters, 82 (5): 1052, 1999. 10.1103/​PhysRevLett.82.1052.

[14] L. Goldenberg, L. Vaidman, and S. Wiesner. Quantum gambling. Physical Review Letters, 82 (16): 3356, 1999. 10.1103/​PhysRevLett.82.3356.

[15] J. Eisert, M. Wilkens, and M. Lewenstein. Quantum games and quantum strategies. Physical Review Letters, 83 (15): 3077, 1999. 10.1103/​PhysRevLett.83.3077.

[16] R. Alicki and M. Fannes. Entanglement boost for extractable work from ensembles of quantum batteries. Physical Review E, 87 (042123), 2013. 10.1103/​PhysRevE.87.042123.

[17] R. Alicki and R. Kosloff. Introduction to Quantum Thermodynamics: History and Prospects, pages 1–33. Springer International Publishing, 2018. 10.1007/​978-3-319-99046-0_1.

[18] M. N. Bera, A. Winter, and M. Lewenstein. Thermodynamics from information, pages 799–820. Springer International Publishing, 2018. 10.1007/​978-3-319-99046-0_33.

[19] W. Pusz and S. L. Woronowicz. Passive states and kms states for general quantum systems. Communications in Mathematical Physics, 58 (3): 273–290, 1978. 10.1007/​BF01614224.

[20] A. Lenard. Thermodynamical proof of the gibbs formula for elementary quantum systems. Journal of Statistical Physics, 19 (6): 575–586, 1978. 10.1007/​BF01011769.

[21] G. M. Andolina, M. Keck, A. Mari, M. Campisi, V. Giovannetti, and M. Polini. Extractable work, the role of correlations, and asymptotic freedom in quantum batteries. Physical Review Letters, 122 (047702), 2019. 10.1103/​PhysRevLett.122.047702.

[22] D. Farina, G. M. Andolina, A. Mari, M. Polini, and V. Giovannetti. Charger-mediated energy transfer for quantum batteries: An open-system approach. Physical Review B, 99 (035421), 2019. 10.1103/​PhysRevB.99.035421.

[23] W. Niedenzu, M. Huber, and E. Boukobza. Concepts of work in autonomous quantum heat engines. Quantum, 3: 195, October 2019. 10.22331/​q-2019-10-14-195.

[24] E. Smith and D. K. Foley. Classical thermodynamics and economic general equilibrium theory. Journal of economic dynamics and control, 32 (1): 7–65, 2008. 10.1016/​j.jedc.2007.01.020.

[25] D. H. Kim and S. Marmi. Distribution of asset price movement and market potential. Journal of Statistical Mechanics: Theory and Experiment, 2015 (7): P07001, 2015. 10.1088/​1742-5468/​2015/​07/​P07001.

[26] W. M. Saslow. An economic analogy to thermodynamics. American Journal of Physics, 67 (12): 1239–1247, 1999. 10.1119/​1.19110.

[27] D. Orrell. A quantum model of supply and demand. Physica A: Statistical Mechanics and its Applications, 539: 122928, 2020a. ISSN 0378-4371. 10.1016/​j.physa.2019.122928.

[28] D. Orrell. The value of value: A quantum approach to economics, security and international relations. Security Dialogue, 51 (5): 482–498, 2020b. 10.1177/​0967010620901910.

[29] D. S. Wiersma. The physics and applications of random lasers. Nature Physics, 4 (5): 359–367, May 2008. 10.1038/​nphys971.

[30] D. F. Walls and G. J. Milburn. Quantum Optics. Springer-Verlag, 2008. 10.1007/​978-3-540-28574-8.

[31] A. E. Allahverdyan, R. Balian, and Th. M. Nieuwenhuizen. Maximal work extraction from finite quantum systems. Europhysics Letters, 67 (4): 565, 2004. 10.1209/​epl/​i2004-10101-2.

[32] E. G. Brown, N. Friis, and M. Huber. Passivity and practical work extraction using gaussian operations. New Journal of Physics, 18, 2016. 10.1088/​1367-2630/​18/​11/​113028.

[33] T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, 2012. 10.1002/​047174882X.

[34] V. M. Markushev, V. F. Zolin, and Ch. M. Briskina. Powder laser. Zhurnal Prikladnoi Spektroskopii, 45: 847–850, 1986.

[35] N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain. Laser action in strongly scattering media. Nature, 368 (6470): 436–438, March 1994. 10.1038/​368436a0.

[36] V.S. Letokhov. Generation of light by a scattering medium with negative resonance absorption. Soviet Physics JETP, 26 (4): 835, April 1968. URL https:/​/​​abs/​1968JETP...26..835L/​abstract.

[37] D. S. Wiersma and A. Lagendijk. Light diffusion with gain and random lasers. Physical Review E, 54: 4256–4265, October 1996. 10.1103/​PhysRevE.54.4256.

[38] R. Bhattacharya and M. Majumdar. Random dynamical systems: theory and applications. Cambridge University Press, 2007. 10.1017/​CBO9780511618628.

[39] M. Nicol, N. Sidorov, and D. Broomhead. On the fine structure of stationary measures in systems which contract-on-average. Journal of Theoretical Probability, 15 (3): 715–730, 2002. 10.1023/​A:1016224000145.

[40] P. Diaconis and D. Freedman. Iterated random functions. SIAM review, 41 (1): 45–76, 1999. 10.1137/​S0036144598338446.

[41] S. Gouezel. Méthodes entropiques pour les convolutions de bernoulli (d'après hochman, shmerkin, breuillard, varju). Asterisque, (414): 251–287, 2019. 10.24033/​ast.1086.

[42] M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J. Schreiber. Pisot and Salem Numbers. Birkhäuser Basel, 1992. 10.1007/​978-3-0348-8632-1.

[43] K. Falconer. Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd, 2 edition, 2003. 10.1002/​0470013850.

[44] M. Misiurewicz and A. Rodrigues. Real $3x+1$. Proceedings of the American Mathematical Society, 133 (4): 1109–1118, 2005. 10.1090/​S0002-9939-04-07696-8.

[45] V. Bergelson, M. Misiuriewicz, and S. Senti. Affine actions of a free semigroup on the real line. Ergodic Theory and Dynamical Systems, 26 (5): 1285–1305, 2006. 10.1017/​S014338570600037X.

[46] Y. Demichel. Renormalization of the hutchinson operator. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 14: 085, 2018. 10.3842/​SIGMA.2018.085.

[47] C. J. G. Evertsz and B. B. Mandelbrot. Multifractal measures, volume 1092, pages 921–953. Springer-Verlag, New York, 1992. 10.1007/​978-1-4757-4740-9.

Cited by

[1] Salvatore Tirone, Raffaele Salvia, and Vittorio Giovannetti, "Quantum Energy Lines and the Optimal Output Ergotropy Problem", Physical Review Letters 127 21, 210601 (2021).

[2] Andrés F. Ducuara and Paul Skrzypczyk, "Characterization of Quantum Betting Tasks in Terms of Arimoto Mutual Information", PRX Quantum 3 2, 020366 (2022).

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