×

Properly quantized history-dependent Parrondo games, Markov processes, and multiplexing circuits. (English) Zbl 1242.81040

Summary: In the context of quantum information theory, “quantization” of various mathematical and computational constructions is said to occur upon the replacement, at various points in the construction, of the classical randomization notion of probability distribution with higher order randomization notions from quantum mechanics such as quantum superposition with measurement. For this to be done “properly”, a faithful copy of the original construction is required to exist within the new quantum one, just as is required when a function is extended to a larger domain. Here procedures for extending history-dependent Parrondo games, Markov processes and multiplexing circuits to their quantum versions are analyzed from a game theoretic viewpoint, and from this viewpoint, proper quantizations developed.

MSC:

81P45 Quantum information, communication, networks (quantum-theoretic aspects)
91A60 Probabilistic games; gambling
60J05 Discrete-time Markov processes on general state spaces
81S05 Commutation relations and statistics as related to quantum mechanics (general)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bleiler, S. A., A formalism for quantum games I - Quantizing mixtures (2008), Portland State University
[2] Meyer, D., Physical Review Letters, 82, 1052 (1999)
[3] Eisert, J.; Wilkens, M.; Lewenstein, M., Physical Review Letters, 83, 3077 (1999)
[4] Kummerer, B., Quantum Markov processes, (Coherent Evolution in Noisy Environments. Coherent Evolution in Noisy Environments, Lecture Notes in Physics, vol. 611 (2002), Springer-Verlag), 139-198, (Chapter 4) · Zbl 1178.46061
[5] Khan, F. S.; Perkowski, M. A., Theoretical Computer Science, 367, 3, 336 (2006)
[6] S.A. Bleiler, F.S. Khan, Quantum Markov processes - A game theoretic approach, Portland State University, in preparation.; S.A. Bleiler, F.S. Khan, Quantum Markov processes - A game theoretic approach, Portland State University, in preparation.
[7] Shende, V.; Bullock, S.; Markov, I., IEEE Transactions on Computer-Aided Design, 25, 6, 1000 (2006)
[8] Landsburg, S., Quantum game theory, available at · Zbl 1168.91310
[9] Abbott, D., Fluctuation and Noise Letters, 9, 1, 129 (2010)
[10] Lee, C. F.; Johnson, N., Parrondo games and quantum algorithms (2002)
[11] Lee, C. F.; Johnson, N., Quantum coherence, correlated noise and Parrondo games (2002)
[12] Parrondo, J. M.R.; Harmer, G. P.; Abbott, D., Physical Review Letters, 85, 24, 5226 (2000)
[13] Harmer, G. P.; Abbott, D., Fluctuation and Noise Letters, 2, 2, 71 (2002)
[14] Kay, R. J.; Johnson, N. F., Physical Review E, 67, 5, 056128 (2003)
[15] Aumann, R., Journal of Mathematical Economics, 1, 67 (1974)
[16] Myerson, R. B., Game Theory: Analysis of Conflict (1991), Harvard University Press · Zbl 0729.90092
[17] Flitney, A. P.; Ng, J.; Abbott, D., Physica A, 314, 35 (2002)
[18] Flitney, A. P.; Abbott, D., Physica A, 324, 1-2, 152 (2003)
[19] Brylinski, R. K.; Brylinski, J., Universal quantum gates, (Mathematics of Quantum Computation (2002), Chapman Hall/CRC), 101-113 · Zbl 0997.81015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.