Understanding quantum measurement from the solution of dynamical models. (English) Zbl 1296.81013

Allahverdyan, Armen E.; Balian, Roger; Nieuwenhuizen, Theo M.

Phys. Rep. 525, No. 1, 1-166 (2013).

Summary: The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the \(z\)-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including \(N\gg 1\) spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrix \(\hat {\mathcal D}(t)\). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final state \(\hat {\mathcal D}(t_f)\) that involves correlations between the system and the indications of the pointer, thus ensuring registration. Although \(\hat {\mathcal D}(t_f)\) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrix \(\hat {\mathcal D}(t_f)\) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on models are suggested, while the statistical interpretation is promoted for teaching.

MSC:

81P15	Quantum measurement theory, state operations, state preparations
81P05	General and philosophical questions in quantum theory
81V22	Unified quantum theories
81-02	Research exposition (monographs, survey articles) pertaining to quantum theory

Keywords:

quantum measurement problem; statistical interpretation; apparatus; pointer; dynamical models; ideal and imperfect measurements; collapse of the wavefunction; decoherence; truncation; reduction; registration

References:

[1]	Born, M., Z. Phys., 37, 863 (1926)
[2]	Heisenberg, W., Z. Phys., 43, 172 (1927)
[4]	Von Neumann, J., Mathematische Grundlagen der Quantenmechanik (1932), Springer: Springer Berlin, English translation by R. T. Beyer: Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955 · JFM 58.0929.06
[5]	Jammer, M., The Conceptual Development of Quantum Mechanics (1966), McGraw-Hill: McGraw-Hill New York, 2nd ed: New York: American Institute of Physics, 1989
[6]	Jammer, M., The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective (1974), Wiley-Interscience: Wiley-Interscience New York
[7]	Mehra, J.; Rechenberg, H., The Historical Development of Quantum Theory, Vol. 1, 2, 3, 4, 5, 6I and 6II (1982), Springer: Springer New York
[8]	Wheeler, J. A.; Zurek, W. H., Quantum Theory and Measurements (1983), Princeton University Press
[9]	Ballentine, L. E., Rev. Modern Phys., 42, 358 (1970), Sections I-IV
[10]	Blokhintsev, D. I., Quantum Mechanics: Lectures on Selected Questions (1988), Moscow State University: Moscow State University Moscow
[11]	Blokhintsev, D. I., Principal Questions of Quantum Mechanics (1988), Nauka Publishing House: Nauka Publishing House Moscow
[12]	Hepp, K., Helv. Phys. Acta, 45, 237 (1972)
[13]	Wigner, E. P., Am. J. Phys., 31, 6 (1963)
[14]	van Kampen, N. G., Physica A, 153, 97 (1988) · Zbl 0642.76105
[15]	Lupascu, A.; Saito, S.; Picot, T.; de Groot, P. C.; Harmans, C. J.P. M.; Mooij, J. E., Nat. Phy., 3, 119 (2007)
[16]	Ariano, G. M.D’; Sacchi, M. F., Optical von Neumann measurement, Phys. Lett., 231, 325 (1997) · Zbl 1053.81507
[17]	Bassi, A.; Ghirardi, G. C., Phys. Rep., 379, 257 (2003)
[18]	Bohm, D., Part I. Phys. Rev., 85, 166 (1952), Part II, Phys. Rev. (85) 180 · Zbl 0046.21004
[19]	Holland, P. R., The Quantum Theory of Motion (1993), Cambridge University Press: Cambridge University Press Cambridge
[20]	(Nieuwenhuizen, Th. M.; Mehmani, B.; Spicka, V.; Aghdami, M. J.; Khrennikov, A. Yu., Beyond The Quantum (2007), World Scientific Singapore)
[21]	Daneri, A.; Loinger, A.; Prosperi, G. M., Nuclear Phys., 33, 297 (1962)
[22]	Peres, A., Quantum Theory: Concepts and Methods (1993), Kluwer: Kluwer Dordrecht · Zbl 0820.00011
[23]	Einstein, A., Dialectica, 2, 320 (1948)
[24]	de Broglie, L., Nonlinear Wave Mechanics (1960), Elsevier: Elsevier Amsterdam
[25]	Everett, H., Rev. Modern Phys., 29, 454 (1957)
[26]	(Saunders, S.; Barrett, J.; Kent, A.; Wallace, D., Many Worlds? Everett, Quantum Theory and Reality (2010), Oxford University Press: Oxford University Press Oxford) · Zbl 1197.81035
[27]	Bell, J. S., Physics, 1, 195 (1964)
[28]	Park, J. L., Am. J. Phys., 36, 211 (1968)
[29]	Home, D.; Whitaker, M. A.B., Phys. Rep., 210, 223 (1992)
[30]	London, F.; Bauer, E., La théorie de la observation en mécanique quantique (1939), Hermann et Cie · JFM 65.1526.06
[31]	Muynck, W. M.de., Foundations of Quantum Mechanics, an Empiricist Approach (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1138.81305
[32]	Schlosshauer, M., Rev. Modern Phys., 76, 1267 (2004)
[33]	Zurek, W. H., Rev. Modern Phys., 75, 715 (2003)
[34]	Laloë, F., Am. J. Phys., 69, 655-701 (2001)
[35]	Laloë, F., Comprenons-nous vraiment la mécanique quantique (2011), EDP Sciences/CNRS Editions: EDP Sciences/CNRS Editions Paris
[36]	Bellac, M. Le., Quantum Physics (2006), Cambridge University Press: Cambridge University Press Cambridge
[37]	Greenberger, D. M.; Horne, M. A.; Zeilinger, A.
[38]	Pan, J. W.; Bouwmeester, D.; Daniell, M.; Weinfurter, H.; Zeilinger, A., Nature, 403, 515 (2000)
[39]	Schrödinger, E., Naturwissenschanften, 23, 807 (1935)
[40]	Giulini, D.; Joos, E.; Kiefer, C.; Kupsch, J.; Stamatescu, O.; Zeh, H. D., Decoherence and the Appearance of the Classical World in Quantum Theory (1996), Springer: Springer Berlin · Zbl 0855.00003
[41]	Hammerer, K.; Sorensen, A. S.; Polzik, E. S., Rev. Modern Phys., 82, 1041 (2010)
[42]	Galindo, A.; Martin-Delgado, M. A., Rev. Modern Phys., 74, 347 (2002)
[43]	Nakamura, Y.; Pashkin, Yu. A.; Tsai, J. S., Nature (London), 398, 786 (1999)
[44]	Nakamura, Y., Phys. Rev. Lett., 88, 047901 (2002)
[45]	Vion, D.; Aassime, A.; Cottet, A.; Joyez, P.; Pothier, H.; Urbina, C.; Estève, D.; Devoret, M. H., Science, 296, 886-889 (2002)
[46]	Jost, J. D.; Home, J. P.; Amini, J. M.; Hanneke, D.; Ozeri, R.; Langer, C.; Bollinger, J. J.; Leibfried, D.; Wineland, D. J., Nature, 459, 683 (2009)
[47]	Busch, P.; Lahti, P. J.; Mittelstaedt, P., The Quantum Theory of Measurement (1996), Springer: Springer Berlin · Zbl 0868.46051
[48]	Ballentine, L. E., Quantum Mechanics (1990), Prentice Hall, Quantum Mechanics: A Modern Development, World Scientific, 1998 · Zbl 0203.27801
[49]	D’ariano, G. M.; Yuen, H. P., Phys. Rev. Lett., 76, 2832 (1996)
[50]	Gleason, A. M., J. Math. Mech., 6, 885 (1957)
[51]	Einstein, A.; Podolsky, B.; Rosen, N., Phys. Rev., 47, 777 (1935)
[52]	Balian, R., Am. J. Phys., 57, 1019 (1989)
[53]	Balian, R.; Vénéroni, M., Ann. Phys., 174, 229 (1987)
[54]	Mayer, J. E.; Mayer, M. G., (Statistical Mechanics (1977), John Wiley & Sons: John Wiley & Sons New York), pp. 145-154
[55]	Krylov, N. S., Works on the Foundations of Statistical Mechanics (1979), Princeton University Press: Princeton University Press Princeton
[56]	Callen, G. B., Thermodynamics and an Introduction to Thermostatistics (1960/1985), Wiley: Wiley New York · Zbl 0989.80500
[57]	Balian, R., Poincaré seminar, 2, 121 (2003), Birkhhaüser
[58]	Balian, R., Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys., 36, 323-353 (2005)
[59]	Allahverdyan, A. E.; Nieuwenhuizen, Th. M., Phys. Rev. E, 73, 066119 (2006)
[60]	Hahn, E. L., Phys. Rev., 80, 580 (1950)
[61]	Brewer, R. G.; Hahn, E. L., Sci. Am., 251, 6, 42 (1984)
[62]	Rhim, W.-K.; Pines, A.; Waugh, J. S., Phys. Rev. Lett., 25, 218 (1970)
[63]	Rhim, W.-K.; Pines, A.; Waugh, J. S., Phys. Rev. B, 3, 684 (1971)
[64]	Waugh, J. S., (Bagguley, G. G., Pulsed Magnetic Resonance: NMR, ESR and Optics (A Recognition of E. L. Hahn) (1992), Clarendon: Clarendon Oxford), p. 174
[65]	Creswick, R. J., Phys. Rev. Lett., 93, 100601 (2004)
[67]	Lüders, G., Ann. Phys. (Leipzig), 8, 322 (1951), English translation: K.A. Kirkpatrick, Ann. Phys. (Leipzig) 15 2006 (663)
[68]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., Europhys. Lett., 61, 453 (2003)
[70]	Shannon, C. E.; Weaver, W., The Mathematical Theory of Communication (1949), University of Illinois Press: University of Illinois Press Urbana · Zbl 0041.25804
[71]	Jaynes, E. T., Phys. Rev., 108, 171 (1957)
[72]	Lindblad, G., Non-Equilibrium Entropy and Irreversibility (1983), Reidel: Reidel Dordrecht · Zbl 0519.60099
[73]	Balian, R., From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, vol. I, II (2007), Springer: Springer Berlin · Zbl 1188.82001
[74]	Brillouin, L., Science and Information Theory (2004), Dover: Dover New York
[75]	Balian, R., Am. J. Phys., 67, 1078 (1999)
[76]	Yanase, M. M., Phys. Rev., 123, 666 (1961)
[77]	Allahverdyan, A. E.; Hovhannisyan, K., Phys. Rev. A, 81, 012312 (2010)
[78]	Balian, R., Eur. J. Phys., 10, 208 (1989)
[79]	Hossein Partovi, M., Phys. Rev. Lett., 50, 1883 (1983)
[80]	Brillouin, L., J. Appl. Phys., 24, 1152 (1953)
[81]	Maruyama, K.; Nori, F.; Vedral, V., Rev. Modern Phys., 81, 1 (2009)
[82]	Lindblad, G., Comm. Math. Phys., 28, 245 (1972)
[83]	Bohr, N., Nature, 121, 78 (1928)
[84]	Bohr, N., Nature, 121, 580 (1928)
[85]	Landau, L. D.; Lifshitz, E. M., Quantum Mechanics (1958), Pergamon Press: Pergamon Press Oxford · Zbl 0081.22207
[86]	Rovelli, C., Internat. J. Theoret. Phys., 35, 1637 (1996)
[87]	Bohm, D., Quantum Theory (1991), Dover: Dover New York · Zbl 0048.21802
[88]	Jaeger, G., Entanglement, Information, and the Interpretation of Quantum Mechanics (2009), Springer: Springer Berlin
[89]	Omnès, R., Phys. Rev. A, 65, 052119 (2002)
[90]	Pearle, P., Phys. Rev. D, 13, 857 (1976)
[91]	Grigorenko, A. N., J. Phys. A: Math. Gen., 28, 1459 (1995)
[92]	Bell, J. S., Helv. Phys. Acta, 48, 93 (1975)
[93]	Busch, P.; Lahti, P. J., Found. Phys., 26, 875 (1996)
[95]	Willis, C. R.; Picard, R. H., Phys. Rev. A, 9, 1343 (1974)
[96]	Delos, J. B.; Thorson, W. R.; Knudson, S. K., Phys. Rev. A, 6, 709 (1972)
[97]	Stenholm, S., Phys. Rev. A, 47, 2523 (1993)
[98]	Diosi, L., Phys. Rev. A, 61, 022108 (2000)
[99]	Diosi, L., Emergence of classicality: from collapse phenomenologies to hybrid dynamics, (Blanchard, Ph.; Guilini, D.; Joos, E.; Kiefer, C.; Stamatescu, I. O., Decoherence: Theoretical, Experimental and Conceptual Problems. Decoherence: Theoretical, Experimental and Conceptual Problems, Lecture Notes in Physics, vol. 538 (2000), Springer-Verlag: Springer-Verlag Berlin, Heidelberg)
[100]	Boucher, W.; Traschen, J., Phys. Rev. D, 37, 3522 (1988)
[101]	Anderson, A., Phys. Rev. Lett., 74, 621 (1995)
[102]	Gindensperger, E.; Meier, C.; Beswick, J. A., J. Chem. Phys., 113, 9369 (2000)
[103]	Prezhdo, O. V.; Brooksby, C., Phys. Rev. Lett., 86, 3215 (2001)
[104]	Burghardt, I.; Parlant, G., J. Chem. Phys., 120, 3055 (2004)
[105]	Sherry, T. N.; Sudarshan, E. C.G., Phys. Rev. D, 20, 857 (1979)
[106]	Peres, A.; Terno, D. R., Phys. Rev. A, 63, 022101 (2001)
[107]	Caro, J.; Salcedo, L. L., Phys. Rev. A, 60, 842 (1999)
[108]	Salcedo, L. L., Phys. Rev. A, 54, 3657 (1996)
[109]	Hall, M. J.; Reginatto, M., Phys. Rev. A, 72, 062109 (2005)
[110]	Kiefer, C., Quantum Gravity (2004), Clarendon Press: Clarendon Press Oxford · Zbl 1053.83001
[111]	Terno, D. R., Found. Phys., 36, 102 (2006)
[112]	Hall, M. J., Phys. Rev. A, 78, 042104 (2008)
[113]	Elze, H. T.; Gambarotta, G.; Vallone, F., Int. J. Quant. Inf., 9, 203 (2011), Suppl. 1
[115]	Grangier, P., Eur. J. Phys., 23, 331 (2002)
[117]	Everitt, M. J.; Munro, W. J.; Spiller, T. P., Phys. Lett. A, 374, 2809 (2010)
[118]	Gisin, N.; Percival, I. C., J. Phys. A., 25, 5677 (1992)
[119]	Everitt, M. J.; Clark, T. D.; Stiffell, P. B.; Ralph, J. F.; Bulsara, A. R.; Harland, C. J., Phys. Rev. E, 72, 066209 (2005)
[120]	Blanchard, Ph.; Jadczyk, A., Phys. Lett. A, 175, 157 (1993)
[121]	Weiss, U., Quantum Dissipative Systems (1993), World Scientific, Singapore · Zbl 1137.81300
[122]	Gardiner, C. W., Quantum Noise (1991), Springer-Verlag, Berlin · Zbl 0773.60095
[123]	Thirring, W., (Quantum Mechanics of Large Systems. Quantum Mechanics of Large Systems, A Course in Mathematical Physics, vol. 4 (1980), Springer-Verlag: Springer-Verlag Wien)
[124]	Pechukas, P., Phys. Rev. Lett., 73, 1060 (1994)
[125]	Blanchard, Ph.; Jadczyk, A., Phys. Lett. A, 203, 260 (1995)
[126]	Kibble, T. W.B., Commun. Math. Phys., 65, 189 (1979)
[127]	Heslot, A., Phys, Rev. D, 31, 1341 (1985)
[128]	Gibbons, G. W., J. Geom. Phys., 8, 147 (1992)
[129]	Ashtekar, A.; Schilling, T. A.
[130]	Kryukov, A. A., Found. Phys., 37, 3 (2007)
[131]	Ballentine, L. E.; Yang, Yumin; Zibin, J. P., Phys. Rev. A, 50, 2854-2859 (1994)
[132]	Habib, S.; Jacobs, K.; Mabuchi, H.; Ryne, R.; Shizume, K.; Sundaram, B., Phys. Rev. Lett., 88, 040402 (2002)
[133]	Wiebe, N.; Ballentine, L. E., Phys. Rev. A, 72, 022109 (2005)
[134]	Peña, L.de.la.; Cetto, A. M., The Quantum Dice: an Introduction to Stochastic Electrodynamics (1996), Kluwer Academic: Kluwer Academic Dordrecht
[135]	Peña, L.de.la.; Cetto, A. M., Found. Phys., 31, 1703 (2001)
[136]	Nieuwenhuizen, Th. M., Classical Phase Space Density for the Relativistic Hydrogen Atom, AIP Conf. Proc., 810, 198 (2006) · Zbl 1116.81037
[137]	Nieuwenhuizen, Th. M., (The Pullback Mechanism in Stochastic Electrodynamics. The Pullback Mechanism in Stochastic Electrodynamics, AIP Conf. Proc., vol. 962 (2007)), 148 · Zbl 1208.81038
[139]	Khrennikov, A. Yu., EPL, 88, 40005 (2009)
[140]	Khrennikov, A. Yu., EPL, 90, 40004 (2010)
[143]	Mermin, N. D., Phys. Today, 38, 4, 38 (1985)
[144]	Aspect, A.; Grangier, P.; Roger, G., Phys. Rev. Lett., 49, 91 (1982)
[145]	Aspect, A.; Dalibard, J.; Roger, G., Phys. Rev. Lett., 49, 1804 (1982)
[146]	Ou, Z. Y.; Mandel, L., Phys. Rev. Lett., 61, 50 (1988)
[147]	Shih, Y. H.; Alley, C. O., Phys. Rev. Lett., 61, 2921 (1988)
[148]	Rarity, J. G.; Tapster, P. R., Phys. Rev. Lett., 64, 2495 (1990)
[149]	Kwiat, P. G.; Steinberg, A. M.; Chiao, R. Y., Phys. Rev. A, 47, R2472 (1993)
[150]	Accardi, L., Phys. Rep., 77, 169 (1981)
[152]	Khrennikov, A., What does probability theory tell us about Bell’s inequality?, Adv. Sci. Lett., 2, 488 (2009)
[153]	Nieuwenhuizen, Th. M., (Accardi, L.; Adenier, G.; Fuchs, C. A.; Jaeger, G.; Khrennikov, A. Yu.; Larsson, J. A.; Stenholm, S., AIP Conf. Proc., vol. 1101. AIP Conf. Proc., vol. 1101, Foundations of Probability and Physics - 5 (2009), Am. Inst. Phys: Am. Inst. Phys Melville, NY), pp 127
[154]	Nieuwenhuizen, Th. M., Found. Phys., 41, 580 (2010)
[155]	Bub, J., Found. Phys., 18, 701 (1988)
[156]	Emch, G. G., Algebraic Methods in Statistical Mechanics and Quantum Field Theory (1972), J. Wiley-Interscience: J. Wiley-Interscience New York · Zbl 0235.46085
[157]	Whitten-Wolfe, B.; Emch, G. G., Helv. Phys. Acta, 49, 45 (1976)
[158]	Landsman, N. P., Rev. Math. Phys., 20, 1173 (2008)
[159]	Nakazato, H.; Pascazio, S., Phys. Rev. Lett., 70, 1 (1993)
[160]	Sewell, G. L., Rep. Math. Phys., 56, 271 (2005)
[161]	Sewell, G. L., Markov Process. Related Fields, 13, 425 (2007)
[163]	Green, H. S., Il Nuovo Cim., 9, 880 (1958)
[164]	Cini, M., Il Nuovo Cim., 73B, 27 (1983)
[165]	Requardt, M., Z. Naturforsch., 39, A, 1147 (1984)
[166]	Gaveau, B.; Schulman, L. S., J. Stat. Phys., 58 (1990)
[167]	Merlin, R., Europhys. Lett., 76, 541 (2006)
[169]	Fioroni, M.; Immirzi, G.
[170]	Pankovic, V.; Predojevic, M.
[171]	Ne’eman, Y., Found. Phys., 16, 361 (1986)
[172]	Zimanyi, G. T.; Vladar, K., Found. Phys. Letters, 1, 175 (1988)
[173]	Ullersma, P., Physica (Utrecht), 32, 90 (1966)
[174]	Möhring, K.; Smilansky, U., Nucl. Phys., A338, 227 (1980)
[175]	Caldeira, A. O.; Leggett, A. J., Ann. Phys., 149, 374 (1983)
[176]	Nieuwenhuizen, Th. M.; Allahverdyan, A. E., Phys. Rev. E, 66, 036102 (2002)
[177]	Van wezel, J.; Van den brink, J.; Zaanen, J., Phys. Rev. Lett., 94, 320401 (2005)
[178]	Haake, F.; Walls, D. F., Phys. Rev. A, 36, 730 (1987)
[179]	Haake, F.; Zukowski, M., Phys. Rev. A, 47, 2506 (1993)
[180]	Venugopalan, A., Phys. Rev. A, 61, 012102 (2000)
[181]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., Phys. Rev. A, 64, 032108 (2001)
[182]	Spehner, D.; Haake, F., J. Phys. A: Math. Theor., 41, 072002 (2008)
[183]	Spehner, D.; Haake, F., Phys. Rev. A, 77, 052114 (2008)
[184]	Mozyrsky, D.; Privman, V., Mod. Phys. Letters B, 14, 303 (2000)
[186]	Jordan, P., Phil. Sci., 16, 269 (1949)
[187]	Ludwig, G., Z. Phys., 135, 483 (1953)
[188]	Pessoa, O.; Freire, O.; Greiff, A. De., Phys. Perspect., 10, 138 (2008)
[189]	Jauch, J. M.; Wigner, E. P.; Yanase, M. M., Nuovo Cimento B, 48, 144 (1967)
[190]	Renninger, M., Z. Phys., 136, 251 (1953)
[191]	Fehrs, M. H.; Shimony, A., Phys. Rev. A, 9, 2317 (1974)
[192]	Shimony, A., Phys. Rev. A, 9, 2321 (1974)
[193]	Busch, P.; Shimony, A., Studies. History Philos. Modern Phys. B, 27, 399 (1997)
[194]	Fine, A., Phys. Rev. D, 2, 2783 (1970)
[197]	Breuer, H.-P.; Petruccione, F., The Theory of Open Quantum Systems (2002), Oxford University Press: Oxford University Press Oxford
[198]	Allahverdyan, A. E.; Mahler, G., Europhys. Lett., 84, 40007 (2008)
[199]	(Blanchard, Ph.; Guilini, D.; Joos, E.; Kiefer, C.; Stamatescu, I. O., Decoherence: Theoretical, Experimental and Conceptual Problems. Decoherence: Theoretical, Experimental and Conceptual Problems, Lecture Notes in Physics, vol. 538 (2000), Springer-Verlag, Berlin: Springer-Verlag, Berlin Heidelberg)
[200]	Walls, D. F.; Collett, M. J.; Milburn, G. J., Phys. Rev. D, 32, 3208 (1985)
[201]	Walls, D. F.; Milburn, G. J., Quantum Optics (1994), Springer: Springer Berlin · Zbl 1138.81586
[202]	Braun, D.; Haake, F.; Strunz, W. T., Phys. Rev. Lett., 86, 2913 (2001)
[203]	Slichter, C. P., Principles of Magnetic Resonance (1990), Springer: Springer Berlin
[204]	Abragam, A., The Principles of Nuclear Magnetism (1961), Clarendon: Clarendon Oxford
[205]	Abragam, A.; Goldman, M., Nuclear Magnetism: Order and Disorder (1982), Clarendon: Clarendon Oxford
[206]	Klimontovich, Yu. L., Statistical Theory of Open Systems (1997), Kluwer: Kluwer Amsterdam · Zbl 1001.92510
[207]	Ballentine, L. E., Phys. Rev. A, 70, 032111 (2004) · Zbl 1227.81217
[208]	Zurek, W. H., Phys. Rev. Lett., 90, 120404 (2003)
[209]	Zurek, W. H., Phys. Rev. A, 71, 052105 (2005)
[210]	Schlosshauer, M.; Fine, A., Found. Phys., 35, 197 (2005)
[211]	Smolin, L., Found. Phys., 42, 1239 (2012)
[214]	(Cushing, J. T.; Fine, A.; Goldstein, S., Bohmian Mechanics and Quantum Theory: An Appraisal (1996), Kluwer: Kluwer Dordrecht)
[215]	Ghirardi, G. C.; Rimini, A.; Weber, T., Phys. Rev. D, 34, 470 (1986)
[216]	Adler, S. L., Quantum Theory as an Emergent Phenomenon (2004), Cambridge University Press
[217]	Pearle, P., (Breuer, H. P.; Petruccione, F., Open Systems and Measurements in Relativistic Quantum Field Theory. Open Systems and Measurements in Relativistic Quantum Field Theory, Lecture Notes in Physics, vol. 526 (1999), Springer: Springer Berlin)
[218]	Brody, D. C.; Hughston, L. P., J. Phys. A, 39, 833 (2006)
[219]	Bohm, D.; Bub, J., Rev. Modern Phys., 38, 453 (1966)
[220]	Bialynicki-Birula, I.; Mycielski, J., Ann. Phys., 100, 62 (1976)
[221]	Gisin, N., J. Phys. A: Math. Gen., 14, 2259 (1981) · Zbl 0474.35040
[222]	Gisin, N., Helv. Phys. Acta, 62, 363 (1989)
[223]	Weinberg, S., Ann. Phys., 194, 336 (1989)
[225]	Shull, C. G., Phys. Rev. Lett., 44, 765 (1980)
[226]	Bollinger, J. J., Phys. Rev. Lett., 63, 1031 (1989)
[227]	De Raedt, H.; De Raedt, K.; Michielsen, K., Europhys. Lett., 69, 861 (2005)
[228]	Michielsen, K.; Jin, F.; Raedt, H. De., J. Comp. Theor. Nanosci., 8, 1052 (2011)
[229]	Griffiths, R. B., J. Stat. Phys., 36, 219 (1984)
[230]	Aharonov, Y.; Bergmann, P. L.; Lebowitz, J., Phys. Rev., 134:B, 1410 (1964)
[231]	Griffiths, R. B., Consistent Quantum Theory (2002), Cambridge University Press · Zbl 1140.81304
[232]	Gell-Mann, M.; Hartle, J. B., Phys. Rev. D, 47, 3345 (1993)
[233]	Hohenberg, P., Rev. Modern Phys., 82, 2835 (2010)
[234]	Omnès, R., Understanding Quantum Mechanics (1999), Princeton University Press · Zbl 1043.81500
[235]	Kent, A., Phys. Rev. Lett., 78, 2874 (1997)
[236]	Diosi, L., Phys. Rev. Lett., 92, 170401 (2004)
[237]	Griffiths, R. B.; Hartle, J., Phys. Rev. Lett., 81, 1981 (1998)
[238]	Griffiths, R. B., Phys. Rev. A, 57, 1604 (1998)
[239]	Kent, A., (Breuer, H. P.; Petruccione, F., Relativistic Quantum Measurement and Decoherence. Relativistic Quantum Measurement and Decoherence, Lect. Not. in Phys., vol. 559 (2000)), 93
[240]	Bassi, A.; Ghirardi, G. C., Phys. Lett. A, 257, 247 (1999)
[241]	Bassi, A.; Ghirardi, G. C., J. Stat. Phys., 98, 457 (2000)
[242]	Griffiths, R. B., J. Stat. Phys., 99, 1409 (2000)
[243]	Bassi, A.; Ghirardi, G. C., Phys. Lett. A, 265, 153 (2000)
[244]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., Atti d. Fond. G. Ronchi, An. LVIII, N., 6, 719-727 (2003)
[245]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., (Khrennikov, A., AIP Conf. Proc., vol. 750 (2005), Am. Inst. Phys: Am. Inst. Phys Melville, NY), 26-34
[246]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., Physica E, 29, 261-271 (2005)
[247]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., (Adenier, G.; Khrennikov, A. Yu.; Nieuwenhuizen, Th. M., AIP Conf. Proc. vol. 810 (2006), Am. Inst. Phys.,: Am. Inst. Phys., Melville, NY), 47-58
[248]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., (Nieuwenhuizen, Th. M.; Mehmani, B.; Spicka, V.; Aghdami, M. J.; Khrennikov, A. Yu., Beyond the Quantum (2007), World Scientific), 53-65
[249]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., Physica E, 42, 339 (2010)
[250]	Landau, L. D.; Lifshitz, E. M., Statistical Physics, vol. 1 (1978), Pergamon Press: Pergamon Press Oxford
[251]	Lavis, D. A.; Bell, G. M., Statistical Mechanics of Lattice Systems, vol. 1 (1999), Springer-Verlag: Springer-Verlag Berlin
[252]	Nagaev, E., Magnets with Complicate Exchange Interaction (1988), Nauka, Moscow
[253]	van Kampen, N. G., Stochastic Processes in Physics and Chemistry (1992), North Holland: North Holland Amsterdam · Zbl 0511.60038
[254]	Risken, H., The Fokker-Planck Equation (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0546.60084
[255]	Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1972), Dover: Dover New York · Zbl 0543.33001
[256]	Baclawski, K.; Donsker, M. D., Mark Kac: Probability, Number Theory, and Statistical Physics, Mathematicians of Our Time 14 (1979), Cambridge: Cambridge Mass.-London
[257]	Mehta, M. L., Random Matrices (2004), Elsevier: Elsevier London
[258]	Suzuki, M.; Kubo, R., J. Phys. Soc. Jap., 24, 51 (1968)
[259]	Bolton, H. C.; Leng, C. A., J. Phys. C, 6, 3737 (1973)
[260]	Goldstein, J. C.; Scully, M. O., Phys. Rev. B, 7, 1084 (1973)
[261]	Wang, Y. K., Phys. Rev. B, 8, 5199 (1973)
[262]	Palmer, R. G., Adv. Phys., 31, 669 (1982)
[263]	Mézard, M.; Parisi, G.; Virasoro, M. A., Spin Glass Theory and Beyond (1987), World Scientic: World Scientic Singapore · Zbl 0992.82500
[264]	Leuzzi, L.; Nieuwenhuizen, Th. M., Thermodynamics of the Glassy State (2008), Taylor and Francis: Taylor and Francis New York · Zbl 1181.82001
[265]	Suzuki, M., Progr. Theoret. Phys., 56, 77 (1976)
[266]	Suzuki, M., J. Stat. Phys., 16, 11 (1977)
[267]	Caroli, B.; Caroli, C.; Roulet, B.; Saint-James, D., Physica A, 108, 233 (1981)
[268]	Bunde, A.; Gouyet, J.-F., Physica A, 132, 357 (1985) · Zbl 0587.60080
[269]	Hu, G., Phys. Lett. A, 116, 361 (1986)
[270]	Ohira, T.; Pearle, P., Am. J. Phys., 56, 692 (1988)
[271]	Ozawa, M., Phys. Rev. Lett., 88, 050402 (2002)
[272]	Meister, B. K.
[273]	Arthurs, E.; Goodman, M. S., Phys. Rev. Lett., 60, 2447 (1988)
[274]	Martens, H.; Muynck, W.de., Found. Phys., 20, 255 (1990)
[275]	Appleby, D. M., Int. J. Theor. Phys., 37, 1491 (1998)
[276]	Luis, A., Phys. Rev. Lett., 88, 230401 (2002)
[277]	Maccone, L., EPL, 77, 40002 (2007)
[278]	Busch, P., Phys. Reports, 452, 155 (2007)
[279]	Allahverdyan, A. E.; Balian, R.; Nieuwenhuizen, Th. M., Phys. Rev. Lett., 92, 120402 (2004)
[280]	Mehmani, B.; Allahverdyan, A. E.; Nieuwenhuizen, Th. M., Phys. Rev. A, 77, 032122 (2008)
[281]	Aquino, G.; Giraldi, F., Phys. Rev. A, 78, 062115 (2008)
[282]	Wigner, E. P., Z. Physik, 133, 101 (1952)
[283]	Araki, H.; Yanase, M. M., Phys. Rev., 120, 622 (1960)
[284]	Stenholm, S., Ann. Phys., 218, 233 (1992)
[285]	Sokolovski, D.; Sherman, E. Ya., Phys. Rev. A, 84, 030101(R) (2011)
[286]	Clauser, J. F.; Horne, M. A.; Shimony, A.; Holt, R. A., Phys. Rev. Lett., 23, 880 (1969)
[287]	Jaeger, G.; Viger, C.; Sarkar, S., Phys. Lett. A, 210, 5 (1996)
[288]	Hess, K.; Michielsen, K.; Raedt, H. De., Europhys. Lett., 87, 60007 (2009)
[289]	Jaynes, E. T., (Rosenkrantz, R. D., Papers On Probability, Statistics and Statistical Physics (1983), D. Reidel: D. Reidel Dordrecth, Holland)
[290]	Katz, A., Principles of Statistical Mechanics: the Information Theory Approach (1967), Freeman: Freeman San Francisco
[291]	Lenard, A., J. Stat. Phys., 19, 575 (1978)
[292]	Bassett, I. M., Phys. Rev. A, 18, 2356 (1978)
[293]	Allahverdyan, A. E.; Nieuwenhuizen, Th. M., Physica A, 305, 542 (2002) · Zbl 0986.82028
[294]	Allahverdyan, A. E.; Nieuwenhuizen, Th. M., Phys. Rev. E, 71, 046107 (2005)
[295]	Allahverdyan, A. E.; Hovhannisyan, K. V., Europhs. Lett., 95, 60004 (2011)
[296]	Gemmer, J.; Michel, M.; Mahler, G., Quantum Thermodynamics (2004), Springer: Springer NY · Zbl 1090.80001
[297]	Parrondo, J. M.R.; Van den broeck, C.; Kawai, R., New J. Phys., 11, 073008 (2009)
[298]	Ebeling, W., Physica A, 182, 108 (1992)
[299]	Alter, O.; Yamamoto, Y., Quantum Measurement of a Single System (2001), Wiley: Wiley NY
[300]	Plotnitsky, A., Found. Phys., 41, 466 (2011)
[301]	Balazs, N. L.; Jennings, B. K., Phys Rep., 104, 347 (1984)
[302]	Hillery, M.; O’Connell, R. F.; Scully, M. O.; Wigner, E. P., Phys. Rep., 121, 106 (1984)
[303]	Genovese, M., Phys. Rep., 413, 319 (2007)
[304]	Kofler, J.; Brukner, C., Phys. Rev. Lett., 101, 090403 (2008)
[305]	O’Connell, A. D., Nature, 464, 697-703 (2010)
[306]	Balian, R.; Vénéroni, M., Nuclear Phys. B, 408, 445 (1993)
[307]	Nagourney, W.; Sandberg, J.; Dehmelt, H., Phys. Rev. Lett., 56, 2797 (1986)
[308]	Leibfried, D.; Blatt, R.; Monroe, C.; Wineland, D., Rev. Modern Phys., 75, 281 (2003)
[309]	Venn, J., The Logic of Chance (1866), Macmillan: Macmillan London, reprinted by Chelsea, New York, 1962 · Zbl 0122.13106
[310]	von Mises, R., Probability, Statistics and Truth (1957), Macmillan: Macmillan London · Zbl 0080.33801
[311]	Cox, R. T., Am. J. Phys., 14, 1 (1946)
[312]	de Finetti, B., Theory of Probability (1990), Wiley: Wiley New-York
[313]	Balian, R.; Balazs, N. L., Ann. Phys., 179, 97 (1987)
[315]	Khrennikov, A. Yu., Found. Phys., 32, 1159 (2002)
[316]	Caves, C. M.; Fuchs, C. A.; Schack, R., Phys. Rev. A, 65, 022305 (2002)
[317]	Rivas, A.; Luis, A., Phys. Rev. A, 77, 022105 (2008)
[318]	d’Espagnat, B., Veiled Reality, An Analysis of Present-Day Quantum Mechanical Concepts (1995), Addison-Wesley: Addison-Wesley Reading, Massachusetts, Originally published in French 1994 · Zbl 1044.81008
[319]	Sayrin, C.; Dotsenko, I.; Zhou, X., Nature, 477, 73 (2011)
[320]	Uffinck, J., Studies in Hist. and Phil. of Mod. Phys., 26B, 223 (1995)
[321]	Blankenbecler, R.; Partovi, M. H., Phys. Rev. Lett., 54, 373 (1985)
[322]	Campa, A.; Dauxois, T.; Ruffo, S., Phys. Rep., 480, 57 (2009)
[323]	Rozonoer, L. I., Probl. Peredachi Inf., 35, 84 (1999)
[324]	Allahverdyan, A. E.; Nieuwenhuizen, Th. M., Phys. Rev. E, 71, 066102 (2005)
[325]	Cohen, O., Phys. Rev. A, 60, 80 (1999)
[326]	Fratini, F.; Hayrapetyan, A. G., Phys. Scr., 84, 035008 (2011)
[327]	Terno, D. R., Phys. Rev. A, 63, 016101 (2000)
[328]	Bodor, A.; Diosi, L.
[329]	Molmer, K., Phys. Rev. A, 55, 3195 (1997)
[330]	Schrödinger, E., Proc. Cambridge Philos. Soc., 32, 446 (1936)
[331]	Rudolph, T.; Sanders, B. C., Phys. Rev. Lett., 87, 077903 (2001)
[332]	Kok, P.; Braunstein, S. L., Phys. Rev. A, 61, 042304 (2000)
[333]	Khinchin, A. Y., Sov. Phys. Usp., 9, 141 (1929)
[334]	Compagner, A., Am. J. Phys., 59, 700 (1991)
[335]	Jaynes, E. T., IEEE Trans. Syst. Science & Cyb., 4, 227 (1968)
[337]	Skilling, J., Synthese, 63, 1 (1985)
[338]	Hughston, L. P.; Jozsa, R.; Wootters, W. K., Phys. Lett. A, 183, 14 (1993)
[339]	’tHooft, G., Nuclear Phys. B, 72, 461 (1974)
[340]	Van hove, L., Physica, 23, 441 (1957)
[341]	Mello, P. A.; Pereyra, P.; Kumar, N., J. Stat. Phys., 51, 77 (1998)
[345]	Balian, R., Nuovo Cimento, 57B, 183 (1968)
[346]	Bohr, N., Discussion with Einstein on epistomological problems in atomic physics, (Schilp, P. A., Albert Einstein: Philosopher-Scientist. Albert Einstein: Philosopher-Scientist, The Library of Living Philosophers (1949), Evanston: Evanston IL), 200-241, Reprinted in N. Bohr, Physics Uspekhi, 66, 571 (1958). See also N. Bohr, Dialectica 2, 312, 1948
[347]	Murdoch, D., Niels Bohr’s Philosophy of Physics (1987), Cambridge University Press: Cambridge University Press Cambridge
[348]	Petersen, A., The philosophy of Niels Bohr, The Bulletin of the Atomic Scientists, September (1963)
[349]	Laughlin, R. B., A Different Universe: Reinventing Physics from the Bottom Down (2005), Basic Books: Basic Books New York
[350]	Leonhardt, U., Measuring the Quantum State of Light (1997), Cambridge University Press: Cambridge University Press Cambridge
[351]	Boas, H. P.
[353]	Penrose, R., The Emperor’s New Mind (1999), Oxford University Press: Oxford University Press UK, pp. 475-481
[354]	Bell, J. S., Speakable and Unspeakable in Quantum Mechanics (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1239.81001
[355]	Fuchs, C. A.; Peres, A., Phys. Today, 53, 70 (2000)
[356]	Arndt, M.; Nairz, O.; Vos-Andreae, J.; Keller, C.; Van der zouw, G.; Zeilinger, A., Nature, 401, 680 (1999)
[357]	Romero-Isart, O.; Juan, M. L.; Quidant, R.; Cirac, J. I., New J. Phys., 12, 033015 (2010)
[358]	Blanchard, Ph.; Lugiewicz, P.; Olkiewicz, R., Phys. Lett. A, 314, 29 (2003)
[359]	Rubin, M. A., Found. Phys. Lett., 17, 323 (2004)
[360]	Rubin, M. A., Found. Phys., 33, 379 (2003)
[361]	Palacios-Laloy, A., Nature Physics, 6, 442 (2010)
[362]	Gleyzes, S.; Kuhr, S.; Guerlin, C., Nature, 446, 297 (2007)
[363]	Clerk, A. A.; Girvin, S. M.; Stone, A. D., Phys. Rev. B, 67, 165324 (2003)
[364]	Bushev, P.; Rotter, D.; Wilson, A., Phys. Rev. Lett., 96, 043003 (2006)
[365]	Aharonov, Y.; Albert, D. Z.; Vaidman, L., Phys. Rev. Lett., 60, 1351 (1988)
[366]	Hofmann, H. F., Phys. Rev. A, 81, 012103 (2010)
[367]	Wiseman, H. M., New J. Phys., 9, 165 (2007)
[368]	Johansen, L. M., Phys. Lett. A, 366, 374 (2007)
[369]	Tollaksen, J.; Aharonov, Y.; Casher, A.; Kaufherr, T.; Nussinov, S., New J. Phys., 12, 013023 (2010)
[370]	Bauer, M.; Bernard, D., Phys. Rev. A, 84, 044103 (2011)
[372]	Lundeen, J. S.; Sutherland, B.; Patel, A.; Stewart, C.; Bamber, C., Nature, 474, 188 (2011)
[373]	Kocsis, S.; Braverman, B.; Ravets, S., Science, 332, 1170-1173 (2011)
[374]	Landau, L. D., Z. Phys., 45, 430 (1927)
[375]	Partovi, H. M., Phys. Lett. A, 137, 440 (1989)
[376]	Mityugov, V. V., Phys. Usp., 170, 681 (2000)
[377]	Brailovskii, A. B.; Vaks, V. L.; Mityugov, V. V., Phys. Usp., 166, 795 (1996)