Discrete-time classical and quantum Markovian evolutions: Maximum entropy problems on path space

Journal of Mathematical Physics

Volumn 51, Issue 4, 2010, Pages

  Pavon, Michele ^a   Ticozzi, Francesco ^a  

^a UNIVERSITY OF PADOVA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77953337161     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3372725     Document Type: Article

References (57)

1. Alicki R. Lendi K. Quantum Dynamical Semigroups and Applications 1987, 286. 10.1007/3-540-18276-4_5, Lecture Notes in Physics (Springer-Verlag, New York).
2. Azencott R. Grandes déviations et applications. Ecole d'Eté de Probabilitès de Saint Flour VIII 1980774. 10.1007/BFb0089623,Lecture Notes in Mathematics Vol. (Springer-VerlagBerlin).
3. Barnum H. Knill E. Reversing quantum dynamics with near-optimal quantum and classical fidelity. J. Math. Phys. 200243:2097. JMAPAQ0022-248810.1063/1.1459754
4. Beghi A. On the relative entropy of discrete-time Markov processes with given end-point densities. IEEE Trans. Inf. Theory 199642:1529. IETTAW0018-944810.1109/18.532893
5. Beghi A. Ferrante A. Pavon M. How to steer a quantum system over a Schrödinger bridge. Quantum Inf. Process. 20021:183. QIPUAT1570-075510.1023/A:1019873024970
6. Bernstein S. Sur les liaisons entre les grandeurs aléatoires. 1932288-309. Proc. Int. Congr. of Math.ZürichSwitzerland,pp.
7. Beurling A. An automorphism of product measures. Ann. Math. 196072:189. ANMAAH0003-486X10.2307/1970151
8. Bjelaković I. Deuschel J.-D. Krüger T. Seiler R. Siegmund-Schultze R. Szkola A. A quantum version of Sanov's theorem. Commun. Math. Phys. 2005260:659-671. CMPHAY0010-361610.1007/s00220-005-1426-2
9. Brémaud P. Markov Chains. Gibbs FieldsMonte Carlo Simulationand Queues 1999(Springer-VerlagNew York).
10. Breuer H.P. Petruccione F. The Theory of Open Quantum Systems 2002and(Oxford University PressOxford).
11. ∑kMk†Mk=I.
12. Carmichael H. An Open Systems Approach to Quantum Optics 1991Lecture Notes in Physics New Series M (Springer-VerlagNew York).
13. Cover T. Thomas J. Elements of Information Theory 2006and,2nd ed. (WileyNew York).
14. Crooks G. Quantum operation time reversal. Phys. Rev. A 200877:034101. PLRAAN1050-294710.1103/PhysRevA.77.034101
15. Dawson D. Gorostiza L. Wakolbinger A. On Schrödinger processes and large deviations. J. Math. Phys. 199031:2385. JMAPAQ0022-248810.1063/1.528840
16. Dembo A. Zeitouni O. Large Deviations Techniques and Applications 1993and(Jones and BartlettBoston).
17. Deuschel J.D. Stroock D. Large Deviations 1989and(AcademicBoston).
18. Doob J.L. A Markov chain theorem. Probability and Statistics (The H. Cramér Volume) 195950-57. (WileyNew York)pp.
19. Ellis R.S. EntropyLarge Deviations and Statistical Mechanics 1985(Springer-VerlagNew York).
20. Fleming W. Sheu S. Stochastic variational formula for fundamental solutions of parabolic PDE. Appl. Math. Optim. 198513:193. AMOMBN0095-461610.1007/BF01442207
21. Föllmer H. Random fields and diffusion processes. Ècole d'Ètè de Probabilitès de Saint-Flour XV-XVII 19881362:102-203. P. L. HennequininLecture Notes in Mathematics Vol.edited by (Springer-VerlagNew York)
22. Fortet R. Résolution d'un système d'equations de M. Schrödinger. J. Math. Pures Appl. 19409:83. JMPAAM0021-7824
23. Häggström O. Finite Markov Chains and Algorithmic Applications 200252. London Mathematical Society Student Texts(Cambridge University PressCambridge).
24. Holevo A.S. Statistical Structure of Quantum Theory 2001(Springer-VerlagBerlin).
25. Horn R.A. Johnson C.R. Matrix Analysis 1985and(Cambridge University PressCambridge).
26. Jamison B. Reciprocal processes. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 197430:65. ZWVGAA0044-371910.1007/BF00532864
27. Jamison B. The Markov processes of Schrödinger. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 197532:323. ZWVGAA0044-371910.1007/BF00535844
28. Karatzas I. Shreve S.E. Brownian Motion and Stochastic Calculus 1988and(Springer-VerlagNew York).
29. Knill E. Laflamme R. A theory of quantum error correcting codes. Phys. Rev. A 199755:900. PLRAAN1050-294710.1103/PhysRevA.55.900
30. Knill E. Laflamme R. Viola L. Theory of quantum error correction for general noise. Phys. Rev. Lett. 200084:2525. PRLTAO0031-900710.1103/PhysRevLett.84.2525
31. Kolmogoroff A. Zur Umkehrbarkeit der statistiscen Naturgesetze. Math. Ann. 1937113:766. MAANA30025-583110.1007/BF01571664
32. Kraus K. StatesEffects and Operations: Fundamental Notions of Quantum Theory 198310.1007/3-540-12732-1(SpringerNew York).
33. ∫VdP for every boundedcontinuous function f on V.
34. Levy B.C. Krener A.J. Kinematics and dynamics of reciprocal diffusions. J. Math. Phys. 199334:1846. JMAPAQ0022-248810.1063/1.530141
35. Levy B.C. Krener A.J. Stochastic mechanics of reciprocal diffusions. J. Math. Phys. 199637:769. JMAPAQ0022-248810.1063/1.531414
36. Lindblad G. Completely positive maps and entropy inequalities. Commun. Math. Phys. 197540:147. CMPHAY0010-361610.1007/BF01609396
37. Maassen H. Quantum probability applied to the damped harmonic oscillator. Quantum Probability Communications 200312:23-58. S. AttalJ. M. Lindsayinedited by and (World ScientificSingapore)Vol.
38. Mitzenmacher M. Upfal E. Probability and Computing: Randomized Algorithms and Probabilistic Analysis 2005and(Cambridge University PressCambridge,).
39. Mohseni M. Rezakhani A.T. Lidar D.A. Quantum process tomography: Resource analysis of different strategies. Phys. Rev. A 200877:032322. PLRAAN1050-294710.1103/PhysRevA.77.032322
40. Del Moral P. Feynman-Kac Formulae 2004(Springer-VerlagBerlin).
41. Nagasawa M. Can the Schrödinger equation be a Boltzmann equation?. Diffusion Processes and Related Problems in Analysis 199022:155-200. M. PinskyinProgress in Probability edited by (BirkhäuserBoston)
42. Nelson E. Dynamical Theories of Brownian Motion 1967(Princeton University PressPrinceton).
43. Nelson E. The adjoint Markov process. Duke Math. J. 195825:671. DUMJAO0012-709410.1215/S0012-7094-58-02561-4
44. Neveu J. Discrete-Parameter Martingales 1975(North-HollandAmsterdam/American ElsevierNew York).
45. Nielsen M.A. Chuang I.L. Quantum Computation and Quantum Information 2000and(Cambridge University PressCambridge).
46. von Neumann J. Mathematical Foundations of Quantum Mechanics 1955(Princeton University PressPrinceton).
47. Pavon M. Directions in Mathematical Systems Theory and Optimization 2003227-238. A. RantzerC. I. Byrnes10.1007/3-540-36106-5_17inedited by and (Springer-VerlagBerlin)
48. Pavon M. Wakolbinger A. On free energystochastic controland Schrödinger processes. ModelingEstimation and Control of Systems with Uncertainty 1991334-348. G. B. Di MasiA. GombaniA. Kurzhanskiand edited by,and (BirkhäuserBoston)
49. Pelaggi C.M. Principi variazionali stocastici per funzionali della densità e per la funzione importanza. 1988Laurea (Master) thesisUniversity of Rome La Sapienza.
50. Sakurai J.J. Modern Quantum Mechanics 1994(Addison-WesleyNew York).
51. Schrödinger E. Sur la théorie relativiste de l'électron et l'interpretation de la mécanique quantique. Ann. Inst. Henri Poincare 19322:269. AIHPA20365-320X
52. Schrödinger E. Über die Umkehrung der Naturgesetze. Sitzungsber. Preuss. Akad. Wiss.Phys. Math. Kl. 19311:144. SPWPAI0371-2435
53. Ticozzi F. Pavon M. On time-reversal and space-time harmonic processes for Markovian quantum channels. ande-print arXiv:0811.0929v2.
54. Vedral V. The role of relative entropy in quantum information theory. Rev. Mod. Phys. 200274:197. RMPHAT0034-686110.1103/RevModPhys.74.197
55. Wakolbinger A. A simplified variational characterization of Schrödinger processes. J. Math. Phys. 198930:2943. JMAPAQ0022-248810.1063/1.528481
56. Wakolbinger A. Schrödinger Bridges from 1931 to 1991. 1990E. Cabañain Proceedings of the Fourth Latin American Congress in Probability and Mathematical Statisticsedited byMexico City.
57. Zambrini J.C. Stochastic mechanics according to E. Schrödinger. Phys. Rev. A 198633:1532. PLRAAN1050-294710.1103/PhysRevA.33.1532