Unknown quantum states: The quantum de Finetti representation

Journal of Mathematical Physics

Volumn 43, Issue 9, 2002, Pages 4537-4559

  Caves, Carlton M ^a   Fuchs, Christopher A ^b   Schack, Rüdiger ^c  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b LUCENT TECHNOLOGIES   (United States)

^c UNIVERSITY OF LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035981736     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1494475     Document Type: Article

References (106)

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2. See, for instance, Bohr's Como lecture, N. Bohr, "The Quantum Postulate and the Recent Development of Atomic Theory," in Atti del Congresso Internazionale dei Fisici 11-20 Settembre 1927, Como-Pavia-Roma, Volume Secondo (Nicola Zanichelli, Bologna, 1928), p. 565. This along with several iterations of the manuscript are reprinted in Niels Bohr: Collected Works, Volume 6, Foundations of Quantum Physics I (1926-1932), edited by J. Kalckar (North-Holland, Amsterdam, 1985).
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28. Few have said this more elegantly than J.B. Hartle, in Am. J. Phys. 36, 704 (1968). Here we quote the most important passage in that regard. Self-servingly, we have inserted the word "experimental" and substituted the phrase "probabilities of occurrence" for "truth values" to push it into a mildly better agreement with our own point of view: The state of a classical system is an objective property of the system and therefore changes only by dynamical laws. A quantum-mechanical state, being a summary of the observers' information about an individual physical system, changes both by dynamical laws and whenever the observer acquires new information about the system through the process of measurement. The existence of two laws for the evolution of the state vector by the Schrödinger equation on the one hand and by the process of measurement (sometimes described as the "reduction of the wave packet") on the other, is a classic subject for discussion in the quantum theory of measurement. The situation becomes problematical only if it is believed that the state vector is an objective property of the system. Then, the state vector must be required to change only by dynamical law, and the problem must be faced of justifying the second mode of evolution from the first. If, however, the state of a system is defined as a list of [experimental] propositions together with their [probabilities of occurrence], it is not surprising that after a measurement the state must be changed to be in accord with the new information ... . The "reduction of the wave packet" does take place in the consciousness of the observer, not because of any unique physical process which takes place there, but only because the state is a construct of the observer and not an objective property of the physical system.
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75. N.D. Mermin, Phys. Today 43(6), 9 (1990); N.D. Mermin, Am. J. Phys. 58, 731 (1990). Mermin was the first to point out the interesting properties of this three-system state, following the lead of D.M. Greenberger, M. Home, and A. Zeilinger, "Going beyond Bell's Theorem," in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 69, where a similar four-system state was proposed. We call attention, however, to the pronunciational perils of calling the state a "GHZM state" and therefore defer to the now standard label GHZ. See also D.M. Greenberger, M.A. Home, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990).
76. N.D. Mermin, Phys. Today 43(6), 9 (1990); N.D. Mermin, Am. J. Phys. 58, 731 (1990). Mermin was the first to point out the interesting properties of this three-system state, following the lead of D.M. Greenberger, M. Home, and A. Zeilinger, "Going beyond Bell's Theorem," in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 69, where a similar four-system state was proposed. We call attention, however, to the pronunciational perils of calling the state a "GHZM state" and therefore defer to the now standard label GHZ. See also D.M. Greenberger, M.A. Home, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990).
77. N.D. Mermin, Phys. Today 43(6), 9 (1990); N.D. Mermin, Am. J. Phys. 58, 731 (1990). Mermin was the first to point out the interesting properties of this three-system state, following the lead of D.M. Greenberger, M. Home, and A. Zeilinger, "Going beyond Bell's Theorem," in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 69, where a similar four-system state was proposed. We call attention, however, to the pronunciational perils of calling the state a "GHZM state" and therefore defer to the now standard label GHZ. See also D.M. Greenberger, M.A. Home, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990).
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87. Indeed, this kind of attitude appears to be the point of departure for all the previous proofs and discussions of the quantum de Finetti theorem in the literature. See Refs. 38, 39, and 58 and also Ref. 69.
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89. D. Petz, Prob. Th. Rel. Fields 85, 1 (1990); A. Bach, Europhys. Lett. 16, 513 (1991); L. Accardi and Y.G. Lu, Ann. Prob. 21, 1478 (1993); A. Bach, Indistinguishable Classical Particles (Springer, Berlin, 1997), Lecture Notes in Mathematics, New Series, Vol. m44; R.L. Hudson, "Some Properties of Reduced Density Operators," Int. J. Quantum Chem. 74, 595 (1999).
90. D. Petz, Prob. Th. Rel. Fields 85, 1 (1990); A. Bach, Europhys. Lett. 16, 513 (1991); L. Accardi and Y.G. Lu, Ann. Prob. 21, 1478 (1993); A. Bach, Indistinguishable Classical Particles (Springer, Berlin, 1997), Lecture Notes in Mathematics, New Series, Vol. m44; R.L. Hudson, "Some Properties of Reduced Density Operators," Int. J. Quantum Chem. 74, 595 (1999).
91. D. Petz, Prob. Th. Rel. Fields 85, 1 (1990); A. Bach, Europhys. Lett. 16, 513 (1991); L. Accardi and Y.G. Lu, Ann. Prob. 21, 1478 (1993); A. Bach, Indistinguishable Classical Particles (Springer, Berlin, 1997), Lecture Notes in Mathematics, New Series, Vol. m44; R.L. Hudson, "Some Properties of Reduced Density Operators," Int. J. Quantum Chem. 74, 595 (1999).
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97. For a small sampling of more recent considerations, see G.M. D'Ariano, L. Maccone, and M.G.A. Paris, J. Phys. A 34, 93 (2001); S. Weigert, Phys. Rev. Lett. 84, 802 (2000); V. Bužek, G. Drobný, R. Derka, G. Adam, and H. Wiedemann, Chaos, Solitons Fractals 10, 981 (1999).
98. For a small sampling of more recent considerations, see G.M. D'Ariano, L. Maccone, and M.G.A. Paris, J. Phys. A 34, 93 (2001); S. Weigert, Phys. Rev. Lett. 84, 802 (2000); V. Bužek, G. Drobný, R. Derka, G. Adam, and H. Wiedemann, Chaos, Solitons Fractals 10, 981 (1999).
99. This question appears to have been considered much earlier than the current interest: W. Band and J.L. Park, Found. Phys. 1, 133 (1970); J.L. Park and W. Band, ibid. 1, 211 (1971); W. Band and J.L. Park, ibid. 1, 339 (1971).
100. This question appears to have been considered much earlier than the current interest: W. Band and J.L. Park, Found. Phys. 1, 133 (1970); J.L. Park and W. Band, ibid. 1, 211 (1971); W. Band and J.L. Park, ibid. 1, 339 (1971).
101. This question appears to have been considered much earlier than the current interest: W. Band and J.L. Park, Found. Phys. 1, 133 (1970); J.L. Park and W. Band, ibid. 1, 211 (1971); W. Band and J.L. Park, ibid. 1, 339 (1971).
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103. One can get a feeling for this from the large review article, D.J. Aldous, "Exchangeability and Related Topics," in École d'Été de Probabilités de Saint-Flour XIII-1983, edited by P.L. Hennequin, Lecture Notes in Mathematics Vol. 1117 (Springer-Verlag, Berlin, 1985), pp. 1-198.
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106. D. Gottesman, private communication, January 2000.