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Derivation of the Rules of Quantum Mechanics from Information-Theoretic Axioms

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Abstract

Conventional quantum mechanics with a complex Hilbert space and the Born Rule is derived from five axioms describing experimentally observable properties of probability distributions for the outcome of measurements. Axioms I, II, III are common to quantum mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first noted by von Neumann (in Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955) and independently by Turing (Teuscher and Hofstadter, Alan Turing: Life and Legacy of a Great Thinker, Springer, Berlin, 2004), in which the increase in entropy resulting from a measurement is reduced by a suitable intermediate measurement. This is shown to be impossible for local hidden variable theories. Axiom IV, together with the first three, almost suffice to deduce the conventional rules but allow some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom V recognizes a property of the distribution of outcomes of random measurements on qubits which holds only in the complex Hilbert space model. It is then shown that the five axioms also imply the conventional rules for any finite dimension.

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Notes

  1. Thanks to W. Wooters for this observation.

References

  1. Mielnik, B.: Commun. Math. Phys. 15, 1–46 (1969)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Birkhoff, G.D., von Neumann, J.: Ann. Math. 37, 823 (1936)

    Article  Google Scholar 

  3. Hardy, L.: Quantum theory from five reasonable axioms. arXiv:quant-ph/0101012v4

  4. Chiriella, G., D’Ariano, G.M., Perinotti, P.: Phys. Rev. 81, 062348 (2010). See for a recent presentation of this approach

    Article  Google Scholar 

  5. Gleason, A.M.: J. Math. Mech. 6, 885 (1957)

    MathSciNet  MATH  Google Scholar 

  6. Sykora, S.: J. Stat. Phys. 11(1), 17–27 (1974)

    Article  MathSciNet  ADS  Google Scholar 

  7. Wootters, W.: Found. Phys. 20, 1365 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  8. Stotland, A., Pomeransky, A., Bachmat, E., Cohen, D.: Europhys. Lett. 67, 700–706 (2004). arXiv:quant-ph/0401021v2

    Article  MathSciNet  ADS  Google Scholar 

  9. Teuscher, C., Hofstadter, D.: Alan Turing: Life and Legacy of a Great Thinker, p. 54. Springer, Berlin (2004). ISBN 3540200207

    MATH  Google Scholar 

  10. von Neumann, J.: Mathematical Foundations of Quantum Mechanics, p. 365. Princeton University Press, Princeton (1955). German edition 1932

    MATH  Google Scholar 

  11. Sierpinski, W.: Introduction to General Topology. U. Toronto Press, Toronto (1934)

    Google Scholar 

  12. Fivel, D.I.: Phys. Rev. Lett. 67, 285 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Bumcrot, R.J.: Modern Projective Geometry (1969). Holt, Reinhart, Winston, Inc. p. 23

    MATH  Google Scholar 

  14. Fivel, D.I.: Phys. Rev. A, Gen. Phys. 50(1), 2108 (1994). arXiv:hep-th/9405042v1

    Article  ADS  Google Scholar 

Download references

Acknowledgements

I am gratefull to Prof. Philip Goyal and the Perimeter Institute for hosting the “Reconstructing Quantum Mechanics” conference in 2009 which led me to revisit my 1994 paper and resulted in significant simplifications in the argument. Conversations with Prof. Wm. Wootters at the conference led me to think further about the problem of eliminating exotic alternatives to conventional quantum mechanics which together with the paper of Stotland et al. stimulated the approach taken in this paper.

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  1. University of Maryland, College Park, MD, USA

    Daniel I. Fivel

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Appendix

Appendix

For conventient reference we here reproduce the derivation [12, 14] of the relationship between p and the d-metric in the conventional model and local hidden variable theories.

In a local hidden variable theory one has a set Λ with a measure μ such that

$$p(x,y) = \mu(\varLambda (x)\cap \varLambda (y)),\quad \mu(\varLambda (x)) = 1, \forall x.$$
(A.3)

To evaluate the d-metric we must compute the supremum over z of. |μ(Λ(x)∩Λ(z))−μ(Λ(y)∩Λ(z))|. But we note that the contribution coming from any overlap of Λ(x) and Λ(y) will cancel. Hence one can compute the z maximizing the expression as if the sets are disjoint. This occurs when either z=x or z=y and gives 1−μ(Λ(x)∩Λ(y)) whence

$$d(x,y) = 1 - p(x,y)$$
(A.4)

which disagrees with (4.22).

In the CRQM we have (reverting to Dirac notation):

$$p(x,y) = |\langle x|y\rangle|^2 = \mathrm{Tr}(\pi(x)\pi(y)),\quad \pi(z) \equiv |z\rangle \langle z|,$$
(A.5)

whence

$$d(x,y) = \sup_z|Tr(\pi(x)\pi(z)) - \mathrm{Tr}(\pi(y)\pi(z))| = \sup_z|\langle z|\pi(x) -\pi(y)|z \rangle|.$$
(A.6)

But this is just the largest eigenvalue of π(x)−π(y). Since the π’s are projectors:

$$(\pi (x) - \pi (y))^3 = (1 - |\langle x|y \rangle|^2)(\pi(x) - \pi(y))$$
(A.7)

and one reads off the largest eigenvalue to obtain (4.22).

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Fivel, D.I. Derivation of the Rules of Quantum Mechanics from Information-Theoretic Axioms. Found Phys 42, 291–318 (2012). https://doi.org/10.1007/s10701-011-9603-y

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