Artur Scherer (2005) Decoherent Histories and Quantum Maps.
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Discrete maps play an important role in the investigation of dynamical features of complex classical systems, especially within the theory of chaos. Similarly, quantum maps have proven to be a very useful mathematical tool within the study of complex quantum dynamical systems. Using the decoherent histories formulation of quantum mechanics we consider a particular framework for studying quantum maps which is motivated by the method of classical symbolic dynamics. Symbolic dynamics is known to be a very powerful method specifically invented for the purpose of representing classical dynamical systems by a discrete model that is suitable for information theoretic studies. Our framework uses the decoherent histories formalism which, similarly to classical symbolic dynamics, allows one to introduce information theoretic quantities with respect to system dynamics. Our research within this framework can be viewed as a contribution towards the development of a general theory of "quantum symbolic dynamics". We start by considering a special but very important example for a quantum map: the quantum baker's map, invented for the theoretical investigation of quantum chaos. Here we use the decoherent histories formalism to examine the coarse-grained evolution of this map with regard to the question of how classical predictability of the evolution depends on the character of coarse-graining. %is affected by the choice of coarse-graining. We demonstrate that hierarchical coarse-grainings display interesting features with respect to this question. We proceed with the investigation of decoherence properties of arbitrary unitary quantum maps. A number of interesting results is obtained within the framework of arbitrarily long histories constructed from a fixed projective partition of a finite dimensional Hilbert space. In particular, we derive simple necessary decoherence conditions, which employ only a single iteration of a given unitary quantum map. Furthermore, a surprising result is obtained with regard to the fundamental question of how the choice of the initial state affects decoherence of histories. Within the considered framework we show that if decoherence is established for arbitrary history lengths and all initial states from the smallest natural set of states that can be associated with the framework, then we get decoherence of such histories for arbitrary initial states. Finally, we make first steps towards proving analogous results for approximate decoherence and suggest various questions for future research.
This is a Published version This version's date is: 28/07/2005 This item is peer reviewed
https://repository.royalholloway.ac.uk/items/90805606-364f-bf88-c138-3cae30aeac19/1/
Deposited by () on 13-Jul-2010 in Royal Holloway Research Online.Last modified on 10-Dec-2010
[1] A. Scherer, A. N. Soklakov and R. Schack, “A simple necessary decoherencecondition for a set of histories”, Phys. Lett. A 326, 307 (2004).
[2] A. Scherer and A. N. Soklakov, “Initial states and decoherence of histories”,J. Math. Phys. 46, 042108 (2005).
[3] A. Scherer and A. N. Soklakov, “Decoherence properties of arbitrarily long histories”,in Proceedings of the 7th International Conference on Quantum Communication,Measurement and Computing (QCMC’04), edited by Stephen M. Barnett(AIP Press, Melville, NY, 2004).
[4] A. N. Soklakov and A. Scherer, “Information capacity and the fate of the Universe”,submitted to Phys. Rev. Lett..
[5] A. Scherer, A. N. Soklakov and R. Schack, “Classical predictability and coarsegrained evolution of the quantum baker’s map”, to be published.
[6] V. M. Alekseev and M. V. Yakobson, “Symbolic dynamics and hyperbolic dynamicsystems”, Phys. Reports 75, 287 (1981).
[7] R. Badii and A. Politi, Complexity (Cambridge University Press, Cambridge,1997).
[8] J. Ford, Phys. Today 36(4), 40 (1983)
[9] R. Schack and C. M. Caves, “Chaos of Lioville probability densities”, Phys. Rev. E 53, 3387 (1996).
[10] R. Schack and C. M. Caves, “Information-theoretic characterization of quantum chaos”, Phys. Rev. E 53, 3257 (1996).
[11] R.O. Vallejos and M. Saraceno, “The construction of a quantum Markov partition”,J. Phys. A: Math. Gen. 32, 7273 (1999)
[12] R. Schack and C. M. Caves, “Shifts on a finite qubit string: A class of quantum baker’s maps”, Applicable Algebra in Engineering, Communication and Computing (AAECC) 10, 305 (2000).
[13] A. N. Soklakov, “Measures of dynamical complexity”, PhD thesis, available at http://www.ma.rhbnc.ac.uk/techreports/ (Tech Report RHUL-2001-1).
[14] A. N. Soklakov and R. Schack, “Classical limit in terms of symbolic dynamics for the quantum baker’s map”, Phys. Rev. E 61, 5108 (2000).
[15] A. N. Soklakov and R. Schack, “Decoherence and linear entropy increase in the quantum baker’s map”, Phys. Rev. E 66, 036212 (2002).
[16] R. Griffiths, J. Stat. Phys. 36, 219 (1984).
[17] R. Omn`es, J. Stat. Phys. 53, 893, 933, 957 (1988).
[18] M. Gell-Mann and J. B. Hartle, “Quantum mechanics in the light of quantumcosmology”, in Complexity, Entropy, and the Physics of Information, edited byW. H. Zurek (Addison Wesley, Redwood City, CA, 1990).
[19] H. F. Dowker and J. J. Halliwell, “Quantum mechanics of history: The decoherence functional in quantum mechanics”, Phys. Rev. D 46, 1580 (1992).
[20] M. Gell-Mann and J. B. Hartle, “Classical equations for quantum systems”, Phys.Rev. D 47, 3345 (1993).
[21] J.J. Halliwell, “Decoherent Histories for spacetime domains”, in: Time in quantum mechanics, ed. by J.G. Muga, R. Sala Mayato, and I.L. Egusquiza (Springer,Berlin, 2002). p 153.
[22] H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York,1950).
[23] R. Bowen, Equilibrium states and the ergodic theory of Anosov Diffeomorphisms,Lecture notes in mathematics Vol 470 (Springer, Berlin, 1975).
[24] A. Peres Quantum Theory: Concepts and Methods (Kluwer Academic Publishers,Dordrecht, 1995).
[25] M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros, Ann. Phys. 122, 26 (1979).
[26] M. Saraceno, “Classical structures in the quantized baker’s transformation”, Ann.Phys. 199, 37 (1990).
[27] V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin,New York, 1968).
[28] N. L. Balazs and A. Voros, “The quantized baker’s transformation”, Ann. Phys.190, 1 (1989).
[29] J. B. Hartle, “Quantum Cosmology: Problems for the 21st Century”, in Proceedings of the 11th Nishinomiya-Yukawa Symposium, edited by K. Kikkawa, H.Kunitomo, and H. Ohtsubo (World Scientific, Singapore, 1998).
[30] J. J. Halliwell, Decoherent histories and hydrodynamic equations”, Phys. Rev. D 58, 105015 (1998).
[31] D. Poulin, “Classicality of quantum information processing”, Phys. Rev. A 65,42319 (2002).
[32] O. Rudolph, “Consistent Histories and Operational Quantum Physics Int. J.Theor. Phys. 35, 1581 (1996)
[33] O. Rudolph, “On the Consistent Effect Histories Approach to Quantum Mechanics”,J. Math. Phys. 37, 5368 (1996)
[34] H. F. Dowker and A. Kent, “On the Consistent Histories Approach to QuantumMechanics”, J. Stat. Phys. 82, 1575 (1996).
[35] J. B. Hartle, “Quantum past and the utility of history”, Physica Scripta T76, 67(1998).
[36] T. A. Brun and J. B. Hartle, “Entropy of classical histories”, Phys. Rev. E 59,6370 (1999).
[37] M. Gell-Mann and J. B. Hartle, “Strong decoherence”, in Quantum-ClassicalCorrespondence: Proceedings of the 4th Drexel Conference on Quantum Non-Integrability, edited by D.-H. Feng and B.-L. Hu (International Press, Cambridge,Massachusetts, 1997), pp. 3 – 35.
[38] L. Diosi, “Anomalies of weakened decoherence criteria for quantum histories”,Phys. Rev. Lett. 92, 170401 (2004).
[39] T. A. Brun and J. B. Hartle, “Classical dynamics of the quantum harmonicchain”, Phys. Rev. D 60, 123503 (1999).
[40] J. M. Jauch, Foundations of Quantum Mechanics. AddisonWesley, Reading, Massachusetts,1968, page 92.
[41] J.R.Waldram, The theory of thermodynamics. Cambridge University Press, Cambridge,1985, pages 293-298.
[42] A. Valentini, “Signal-locality, uncertainty, and the subquantum H-theorem”,Phys. Lett. A 156, 5 (1991).
[43] M. V. Berry, in Chaos and Quantum Physics, edited by M.-J. Giannoni, A. Voros,and J. Zinn-Justin (North-Holland, Amsterdam, 1991).
[44] H. F. Dowker and A.Kent, “Properties of Consistent Histories”, Phys. Rev. Lett.75, 3038 (1995).
[45] A. Kent, “Quasiclassical dynamics in closed quantum systems”, Phys. Rev. A 54,4670 (1996).
[46] D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. D. Zeh, Decoherence and the Appearance of a Classical World in Quantum Theory, (Springer-Verlag, Berlin Heidelberg, 1996).
[47] J. J. Halliwell, “Information Dissipation in Quantum Cosmology and the Emergence of Classical Spacetime”, in Complexity, Entropy, and the Physics of Information,edited by W. H. Zurek (Addison Wesley, Redwood City, CA, 1990).
[48] J. W. Cassels, An Introduction to Diophantine Approximation. Cambridge Tracts in Mathematics and Mathematical Physics No. 45 (Cambridge University Press,Cambridge, 1957), Theorem VI, page 13.
[49] L. Diosi, “On the maximum number of decoherent histories”, Phys. Lett. 203 A,267-268 (1995).
[50] W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?”, Phys. Rev. D 24, 1516 (1981).
[51] W. H. Zurek, “Environment-induced superselection rules”, Phys. Rev. D 26, 1862(1981).
[52] W. H. Zurek, “Decoherence and the Transition from Quantum to Classical”, Phys.Today 44(10), 36 (1991).
[53] W. H. Zurek, Prog. Theor. Phys. 89, 281 (1993); in Physical origins of TimeAsymmetry, edited by J. Halliwell et al. (Cambridge University press, Cambridge,England, 1993)
[54] J. P. Paz and W. H. Zurek, “Environment-induced decoherence, classicality, and consistency of quantum histories”, Phys. Rev. D 48, 2728 (1993).
[55] W. H. Zurek, “Negotiating the tricky border between quantum and classical”,Phys. Today 46(4), 36 (1993).
[56] D. Aharonov and M. Ben-Or, in The 37th IEEE Annual Symposium on FOCS(IEEE, New York, 1996), p.46.