# Quasi-Quantization: Classical Statistical Theories with an Epistemic Restriction

## Abstract

A significant part of quantum theory can be obtained from a single innovation relative to classical theories, namely, that there is a fundamental restriction on the sorts of statistical distributions over physical states that can be prepared. This is termed an “epistemic restriction” because it implies a fundamental limit on the amount of knowledge that any observer can have about the physical state of a classical system. This article provides an overview of *epistricted theories*, that is, theories that start from a classical statistical theory and apply an epistemic restriction. We consider both continuous and discrete degrees of freedom, and show that a particular epistemic restriction called *classical complementarity* provides the beginning of a unification of all known epistricted theories. This restriction appeals to the symplectic structure of the underlying classical theory and consequently can be applied to an arbitrary classical degree of freedom. As such, it can be considered as a kind of *quasi-quantization* scheme; “quasi” because it generally only yields a theory describing a subset of the preparations, transformations and measurements allowed in the full quantum theory for that degree of freedom, and because in some cases, such as for binary variables, it yields a theory that is a distortion of such a subset. Finally, we propose to classify quantum phenomena as weakly or strongly nonclassical by whether or not they can arise in an epistricted theory.

## Notes

### Acknowledgments

I acknowledge Stephen Bartlett and Terry Rudolph for discussions on the quadrature subtheory of quantum mechanics, Jonathan Barrett for suggesting to define the Poisson bracket in the discrete case in terms of finite differences, and Giulio Chiribella, Joel Wallman and Blake Stacey for comments on a draft of this article. Much of the work presented here summarizes unpublished results obtained in collaboration with Olaf Schreiber. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

## References

- 1.R.W. Spekkens, Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A
**75**(3), 032110 (2007)ADSCrossRefGoogle Scholar - 2.S.D. Bartlett, T. Rudolph, R.W. Spekkens, Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction. Phys. Rev. A
**86**(1), 012103 (2012)ADSCrossRefGoogle Scholar - 3.M. Born, E. Wolf,
*Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light*(Cambridge University Press, Cambridge, 1999)CrossRefzbMATHGoogle Scholar - 4.M.F. Pusey, Stabilizer notation for Spekkens’ toy theory. Found. Phys.
**42**(5), 688–708 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 5.S.J. Van Enk, A toy model for quantum mechanics. Found. Phys.
**37**(10), 1447–1460 (2007)ADSMathSciNetCrossRefGoogle Scholar - 6.D. Gross, Hudson’s theorem for finite-dimensional quantum systems. J. Math. Phys.
**47**, 122107 (2006)Google Scholar - 7.O. Schreiber, R.W. Spekkens, The power of epistemic restrictions in reconstructing quantum theory: from trits to qutrits, unpublished, 2008. R.W. Spekkens, The power of epistemic restrictions in reconstructing quantum theory, Talk, Perimeter Institute, http://pirsa.org/09080009/, 10 August 2008
- 8.T.H. Boyer,
*Foundations of Radiation Theory and Quantum Electrodynamics, Chapter A Brief Survey of Stochastic Electrodynamics*(Plenum, New York, 1980)Google Scholar - 9.C.M. Caves, C.A. Fuchs, Quantum information: how much Information in a state vector? (1996). arXiv:quant-ph/9601025
- 10.J.V. Emerson, Quantum chaos and quantum-classical correspondence. Ph.D. thesis, Simon Fraser University, Vancouver, Canada (2001)Google Scholar
- 11.L. Hardy, Disentangling nonlocality and teleportation (1999). arXiv:quant-ph/9906123
- 12.K.A. Kirkpatrick, Quantal behavior in classical probability. Found. Phys. Lett.
**16**(3), 199–224 (2003)MathSciNetCrossRefGoogle Scholar - 13.W.K. Wootters, W.H. Zurek, A single quantum cannot be cloned. Nature
**299**(5886), 802–803 (1982)ADSCrossRefGoogle Scholar - 14.C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W.K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett.
**70**(13), 1895 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 15.C.H. Bennett, G. Brassard et al., Quantum cryptography: public key distribution and coin tossing, in
*Proceedings of IEEE International Conference on Computers, Systems and Signal Processing*, vol. 175, (New York 1984)Google Scholar - 16.R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys.
**81**(2), 865 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 17.C.H. Bennett, D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, W.K. Wootters, Quantum nonlocality without entanglement. Phys. Rev. A
**59**(2), 1070 (1999)ADSMathSciNetCrossRefGoogle Scholar - 18.C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, B.M. Terhal, Unextendible product bases and bound entanglement. Phys. Rev. Lett.
**82**(26), 5385 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 19.M.D. Choi, Completely positive linear maps on complex matrices. Linear Algebra Appl.
**10**, 285–290 (1975)MathSciNetCrossRefzbMATHGoogle Scholar - 20.A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys.
**3**, 275–278 (1972)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 21.M.A. Naimark. Izv. Akad. Nauk SSSR, Ser. Mat.
**4**:277–318 (1940)Google Scholar - 22.W.F. Stinespring, Positive functions on C*-algebras. Proc. Am. Math. Soc.
**6**(2), 211–216 (1955)MathSciNetzbMATHGoogle Scholar - 23.J.S. Bell, On the Einstein Podolsky Rosen paradox. Physics
**1**(3), 195–200 (1964)Google Scholar - 24.S. Kochen, E.P. Specker, The problem of hidden variables in quantum mechanics. J. Math. Mech.
**17**, 59 (1967)MathSciNetzbMATHGoogle Scholar - 25.R.W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A
**71**(5), 052108 (2005)ADSMathSciNetCrossRefGoogle Scholar - 26.Y.C. Liang, R.W. Spekkens, H.M. Wiseman, Specker’s parable of the overprotective seer: a road to contextuality, nonlocality and complementarity. Phys. Rep.
**506**(1), 1–39 (2011)ADSMathSciNetCrossRefGoogle Scholar - 27.N. Harrigan, R.W. Spekkens, Einstein, incompleteness, and the epistemic view of quantum states. Found. Phys.
**40**, 125 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 28.C.M. Caves, C.A. Fuchs, R. Schack, Quantum probabilities as Bayesian probabilities. Phys. Rev. A
**65**(2), 022305 (2002)ADSMathSciNetCrossRefGoogle Scholar - 29.C.A. Fuchs, Quantum mechanics as quantum information, mostly. J. Mod. Opt.
**50**(6–7), 987–1023 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 30.C.A. Fuchs, R. Schack, Quantum-Bayesian coherence. Rev. Mod. Phys.
**85**(4), 1693 (2013)ADSCrossRefGoogle Scholar - 31.M.F. Pusey, J. Barrett, T. Rudolph, On the reality of the quantum state. Nat. Phys.
**8**(6), 475–478 (2012)CrossRefGoogle Scholar - 32.P.G. Lewis, D. Jennings, J. Barrett, T. Rudolph, Distinct quantum states can be compatible with a single state of reality. Phys. Rev. Lett.
**109**(15), 150404 (2012)ADSCrossRefGoogle Scholar - 33.R. Colbeck, R. Renner, Is a system’s wave function in one-to-one correspondence with its elements of reality? Phys. Rev. Lett.
**108**(15), 150402 (2012)ADSCrossRefGoogle Scholar - 34.M.S. Leifer, R.W. Spekkens, Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Phys. Rev. A
**88**(5), 052130 (2013)ADSCrossRefGoogle Scholar - 35.C.J. Wood, R.W. Spekkens, The lesson of causal discovery algorithms for quantum correlations: causal explanations of bell-inequality violations require fine-tuning (2012). arXiv:1208.4119
- 36.A. Zeilinger, A foundational principle for quantum mechanics. Found. Phys.
**29**(4), 631–643 (1999)MathSciNetCrossRefGoogle Scholar - 37.T. Paterek, B. Dakić, Č. Brukner, Theories of systems with limited information content. New J. Phys.
**12**(5), 053037 (2010)ADSCrossRefGoogle Scholar - 38.B. Coecke, B. Edwards, R.W. Spekkens, Phase groups and the origin of non-locality for qubits. Electron. Notes Theor. Comput. Sci.
**270**(2), 15–36 (2011)CrossRefGoogle Scholar - 39.S. Mansfield, T. Fritz, Hardy’s non-locality paradox and possibilistic conditions for non-locality. Found. Phys.
**42**(5), 709–719 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 40.S. Abramsky, L. Hardy, Logical bell inequalities. Phys. Rev. A
**85**(6), 062114 (2012)ADSCrossRefGoogle Scholar - 41.B. Schumacher, M.D. Westmoreland, Modal quantum theory. Found. Phys.
**42**(7), 918–925 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 42.J. Barrett, Information processing in generalized probabilistic theories. Phys. Rev. A
**75**(3), 032304 (2007)ADSCrossRefGoogle Scholar - 43.L. Hardy, Quantum theory from five reasonable axioms (2001). arXiv:quant-ph/0101012
- 44.B. Coecke, E.O. Paquette, Categories for the practising physicist, in
*New Structures for Physics*. Lecture Notes in Physics, ed. by B. Coecke (Springer, Berlin, 2009), pp. 173–286Google Scholar - 45.B. Coecke, B. Edwards, Toy quantum categories. Electron. Notes Theor. Comput. Sci.
**270**(1), 29–40 (2011)CrossRefGoogle Scholar - 46.G. Chiribella, G.M. DAriano, P. Perinotti, Probabilistic theories with purification. Phys. Rev. A
**81**(6), 062348 (2010)ADSCrossRefGoogle Scholar - 47.E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev.
**40**(5), 749 (1932)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 48.C. Gardiner, P. Zoller,
*Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics*, vol. 56 (Springer, New York, 2004)zbMATHGoogle Scholar - 49.K.S. Gibbons, M.J. Hoffman, W.K. Wootters, Discrete phase space based on finite fields. Phys. Rev. A
**70**(6), 062101 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 50.R.W. Spekkens, Negativity and contextuality are equivalent notions of nonclassicality. Phys. Rev. Lett.
**101**(2), 20401 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 51.N.D. Mermin, Hidden variables and the two theorems of John Bell. Rev. Mod. Phys.
**65**(3), 803 (1993)ADSMathSciNetCrossRefGoogle Scholar - 52.D.M. Greenberger, M.A. Horne, A. Zeilinger, Going beyond Bell’s theorem,
*Bell’s Theorem, Quantum Theory and Conceptions of the Universe*(Springer, New York, 1989), pp. 69–72CrossRefGoogle Scholar - 53.D. Gottesman, The Heisenberg representation of quantum computers (1998). arXiv:quant-ph/9807006