Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem

Physical Review Letters

Volumn 91, Issue 12, 2003, Pages 1204031-1204034

  Busch, P ^a  

^a UNIVERSITY OF HULL   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL OPERATORS; PROBABILITY DISTRIBUTIONS; THEOREM PROVING;

QUANTUM STATES;

QUANTUM THEORY;

EID: 0142188672     PISSN: 00319007     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (18)

1. n ∈ ℰ (script H sign).
2. A. M. Gleason, J. Math. Mech. 6, 885 (1957).
3. An early version of this Letter (P. Busch, e-print quant-ph/9909073) aimed at a "resurrection" von Neumann's infamous no-hidden variables theorem. The simplified proof of Gleason's theorem on which the argument was based attracted considerable interest For this reason, the focus of the present Letter is on this result concerning the most general quantum mechanical probability measures.
4. E. B. Davies, Quantum Theory of Open Systems (Academic Press, New York, 1976).
5. J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932).
6. The statement of the theorem was given without proof in P. Busch, P. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer-Verlag, Berlin, 1996), 2nd ed. To the author's knowledge, the credit for this result is due to the late Słlawek Bugajski, who envisaged (unpublished) it at a time when the relevance of effects and POVMs for quantum physics was not yet well established A proof similar to the present one was published in the abstract mathematical context of base norm and order unit spaces by E. Beltrametti and S. Bugajski, J. Math. Phys. (N.Y.) 38, 3020 (1997).
7. The statement of the theorem was given without proof in P. Busch, P. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer-Verlag, Berlin, 1996), 2nd ed. To the author's knowledge, the credit for this result is due to the late Słlawek Bugajski, who envisaged (unpublished) it at a time when the relevance of effects and POVMs for quantum physics was not yet well established A proof similar to the present one was published in the abstract mathematical context of base norm and order unit spaces by E. Beltrametti and S. Bugajski, J. Math. Phys. (N.Y.) 38, 3020 (1997).
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10. For a review of early criticism of this assumption, cf. M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974), Sec. 7.4. According to Jammer, the first to raise this point was German philosopher of physics Grete Hermann, Abhandlungen der Fries'schen Schule 6, 75-152 (1935). For a more recent account, see N. D. Mermin, Rev. Mod. Phys. 65, 803 (1993).
11. For a review of early criticism of this assumption, cf. M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974), Sec. 7.4. According to Jammer, the first to raise this point was German philosopher of physics Grete Hermann, Abhandlungen der Fries'schen Schule 6, 75-152 (1935). For a more recent account, see N. D. Mermin, Rev. Mod. Phys. 65, 803 (1993).
12. For a review of early criticism of this assumption, cf. M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974), Sec. 7.4. According to Jammer, the first to raise this point was German philosopher of physics Grete Hermann, Abhandlungen der Fries'schen Schule 6, 75-152 (1935). For a more recent account, see N. D. Mermin, Rev. Mod. Phys. 65, 803 (1993).
13. An interpretation of objective quantum probabilities (in pure states and for projections) in terms of propensities is discussed by N. Gisin, J. Math. Phys. (N.Y.) 25, 2260 (1984). Here the term "objective" probability refers to the indeterminateness of a property whose probability is not 1 or 0 (in a pure state). Under appropriate conditions one can extend the concept of propensity to mixed states and POVMs, namely, in the case where the effects of the POVM do not commute with the density operator. A propensity interpretation thus understood was envisaged by W. Heisenberg who used terms such as disposition, potentiality, or tendency of actualization. The use of the "propensity" interpretation of probability in the context of quantum mechanics was advocated by K. R. Popper to describe situations of indeterminacy. A dispositional theory of probability was already proposed by C. S. Peirce who used the term "would be." I am indebted to A. Shimony for this piece of information on the history of the propensity interpretation. For references, see M. Jammer, The Philosophy of Quantum Mechanics (Ref. [9]), p. 449, footnote 44.
14. An interpretation of objective quantum probabilities (in pure states and for projections) in terms of propensities is discussed by N. Gisin, J. Math. Phys. (N.Y.) 25, 2260 (1984). Here the term "objective" probability refers to the indeterminateness of a property whose probability is not 1 or 0 (in a pure state). Under appropriate conditions one can extend the concept of propensity to mixed states and POVMs, namely, in the case where the effects of the POVM do not commute with the density operator. A propensity interpretation thus understood was envisaged by W. Heisenberg who used terms such as disposition, potentiality, or tendency of actualization. The use of the "propensity" interpretation of probability in the context of quantum mechanics was advocated by K. R. Popper to describe situations of indeterminacy. A dispositional theory of probability was already proposed by C. S. Peirce who used the term "would be." I am indebted to A. Shimony for this piece of information on the history of the propensity interpretation. For references, see M. Jammer, The Philosophy of Quantum Mechanics (Ref. [9]), p. 449, footnote 44.
15. T. Breuer, Phys. Rev. Lett. 88, 240402 (2002).
16. J. Bub, Interpreting the Quantum World (Cambridge University Press, Cambridge, 1997).
17. A. Kent, Phys. Rev. Lett. 83, 3755 (1999).
18. D. M. Appleby, Phys. Rev. A 65, 022105 (2002).