Abstract
In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent��s personalist Bayesian degrees of belief, or probabilities, concerning the results of measurements. These probabilities obey the usual probability rules as required by Dutch-book coherence, but quantum mechanics imposes additional constraints upon them. In this paper, we explore the question of deriving the structure of quantum-state space from a set of assumptions in the spirit of quantum Bayesianism. The starting point is the representation of quantum states induced by a symmetric informationally complete measurement or SIC. In this representation, the Born rule takes the form of a particularly simple modification of the law of total probability. We show how to derive key features of quantum-state space from (i) the requirement that the Born rule arises as a simple modification of the law of total probability and (ii) a limited number of additional assumptions of a strong Bayesian flavor.
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References
Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43, 4537 (2002)
Fuchs, C.A.: Notes on a Paulian Idea: Foundational, Historical, Anecdotal & Forward-Looking Thoughts on the Quantum. Växjö University Press, Växjö (2003). With foreword by N. David Mermin. Preprinted as arXiv:quant-ph/0105039v1 (2001)
Schack, R., Brun, T.A., Caves, C.M.: Quantum Bayes rule. Phys. Rev. A 64, 014305 (2001)
Fuchs, C.A.: Quantum mechanics as quantum information (and only a little more). arXiv:quant-ph/0205039v1 (2002); abridged version in: Khrennikov, A. (ed.) Quantum Theory: Reconsideration of Foundations, pp. 463–543. Växjö University Press, Växjö (2002)
Fuchs, C.A.: Quantum mechanics as quantum information, mostly. J. Mod. Opt. 50, 987 (2003)
Schack, R.: Quantum theory from four of Hardy’s axioms. Found. Phys. 33, 1461 (2003)
Fuchs, C.A., Schack, R.: Unknown quantum states and operations, a Bayesian view. In: Paris, M.G.A., Řeháček, J. (eds.) Quantum Estimation Theory, pp. 151–190. Springer, Berlin (2004)
Caves, C.M., Fuchs, C.A., Schack, R.: Subjective probability and quantum certainty. Stud. Hist. Philos. Mod. Phys. 38, 255 (2007)
Appleby, D.M.: Facts, values and quanta. Found. Phys. 35, 627 (2005)
Appleby, D.M.: Probabilities are single-case, or nothing. Opt. Spectrosc. 99, 447 (2005)
Timpson, C.J.: Quantum Bayesianism: a study. Stud. Hist. Philos. Mod. Phys. 39, 579 (2008)
Fuchs, C.A., Schack, R.: Quantum-Bayesian coherence. Rev. Mod. Phys. (2009, submitted). arXiv:0906.2187v1 [quant-ph]
Ramsey, F.P.: Truth and probability. In: Braithwaite, R.B. (ed.) The Foundations of Mathematics and Other Logical Essays, pp. 156–198. Harcourt Brace, New York (1931)
de Finetti, B.: Probabilismo. Logos 14, 163 (1931); transl., Probabilism. Erkenntnis 31, 169 (1989)
Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954)
de Finetti, B.: Theory of Probability. Wiley, New York (1990), 2 volumes
Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, Chichester (1994)
Jeffrey, R.: Subjective Probability. The Real Thing. Cambridge University Press, Cambridge (2004)
Logue, J.: Projective Probability. Oxford University Press, Oxford (1995)
Appleby, D.M., Flammia, S.T., Fuchs, C.A.: The Lie algebraic significance of symmetric informationally complete measurements. J. Math. Phys. (2010, submitted). arXiv:1001.0004v1 [quant-ph]
Appleby, D.M., Ericsson, Å., Fuchs, C.A.: Pseudo-QBist State Spaces. Found. Phys. (2009, accepted)
Skyrms, B.: Coherence. In: Rescher, N. (ed.) Scientific Inquiry in Philosophical Perspective, pp. 225–242. University of Pittsburgh Press, Pittsburgh (1987)
Zauner, G.: Quantum designs���foundations of a non-commutative theory of designs (in German). PhD thesis, University of Vienna (1999)
Caves, C.M.: Symmetric informationally complete POVMs. Unpublished (1999)
Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171 (2004)
Fuchs, C.A.: On the quantumness of a Hilbert space. Quantum. Inf. Comput. 4, 467 (2004)
Appleby, D.M.: SIC-POVMs and the extended Clifford group. J. Math. Phys. 46, 052107 (2005)
Appleby, D.M., Dang, H.B., Fuchs, C.A.: Physical significance of symmetric informationally-complete sets of quantum states. arXiv:0707.2071v1 [quant-ph] (2007)
Scott, A.J., Grassl, M.: SIC-POVMs: a new computer study. arXiv:0910.5784v2 [quant-ph] (2009)
Ferrie, C., Emerson, J.: Framed Hilbert space: hanging the quasi-probability pictures of quantum theory. New J. Phys. 11, 063040 (2009)
Wootters, W.K.: Quantum mechanics without probability amplitudes. Found. Phys. 16, 391 (1986)
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Fuchs, C.A., Schack, R. A Quantum-Bayesian Route to Quantum-State Space. Found Phys 41, 345–356 (2011). https://doi.org/10.1007/s10701-009-9404-8
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DOI: https://doi.org/10.1007/s10701-009-9404-8