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International Journal of Theoretical Physics

, Volume 45, Issue 7, pp 1189–1215 | Cite as

Toy Model for a Relational Formulation of Quantum Theory

  • David Poulin
Article

Abstract

In the absence of an external frame of reference—i.e., in background independent theories such as general relativity—physical degrees of freedom must describe relations between systems. Using a simple model, we investigate how such a relational quantum theory naturally arises by promoting reference systems to the status of dynamical entities. Our goal is twofold. First, we demonstrate using elementary quantum theory how any quantum mechanical experiment admits a purely relational description at a fundamental. Second, we describe how the original “non-relational” theory approximately emerges from the fully relational theory when reference systems become semi-classical. Our technique is motivated by a Bayesian approach to quantum mechanics, and relies on the noiseless subsystem method of quantum information science used to protect quantum states against undesired noise. The relational theory naturally predicts a fundamental decoherence mechanism, so an arrow of time emerges from a time-symmetric theory. Moreover, our model circumvents the problem of the “collapse of the wave packet” as the probability interpretation is only ever applied to diagonal density operators. Finally, the physical states of the relational theory can be described in terms of “spin networks” introduced by Penrose as a combinatorial description of geometry, and widely studied in the loop formulation of quantum gravity. Thus, our simple bottom-up approach (starting from the semiclassical limit to derive the fully relational quantum theory) may offer interesting insights on the low energy limit of quantum gravity.

Keywords

relational quantum theory quantum gravity quantum information 

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References

  1. Aharonov, Y. and Kaufherr, T. (1984). Quantum frames of reference. Physical Review D 30, 368.CrossRefADSMathSciNetGoogle Scholar
  2. Aharonov, Y. and Susskind, L. (1967). Charge superselecion rule. Physical Review 155(5), 1428.CrossRefADSGoogle Scholar
  3. Baez, J. (1995). Baiz, J. (1995). Spin networks in nonperturbative quantum gravity. In L. H. Kauffman, (ed.) The Interface of Knots and Physics, American Mathematical Society, Hattiesburg Mississippi. p. 167.Google Scholar
  4. Baez, J. C. (1996). Spin network states in gauge theory. Advances in Mathematics 117, 253.CrossRefMathSciNetMATHGoogle Scholar
  5. Baez, J. C. (2000). An introduction to spin foam models of quantum gravity and BF theory. Lecture Notes in Physics 543, 25.Google Scholar
  6. Bartlett, S. D., Rudolph, T., and Spekkens, R. W. (2004). Decoherence-full subspaces and the cryptographic power of a private shared reference frame. Physical Review A 70, 32307.CrossRefADSGoogle Scholar
  7. Bartlett, S. D., Rudolph, T., and Spekkens, R. W. (2005). Dialogue concerning two views on quantum coherences: factist and fictionist. quant-ph/0507214.Google Scholar
  8. Busch, P. and Singh, J. (1998). Luders theorem for unsharp quantum effects. Physics Letters A 249, 10–24.CrossRefADSGoogle Scholar
  9. Caves, C. M., Fuchs, C. A., and Schack, R. (2002). Quantum probabilities as bayesian probabilities. Physsical Review A 65, 022305.CrossRefADSMathSciNetGoogle Scholar
  10. DeWitt, B. S. (1967). Quantum theory of gravity I: the canonical theory. Physical Review 160, 1113.CrossRefADSMATHGoogle Scholar
  11. Freidel, L. and Livine, E. R. (2003). Spin networks for noncompact groups. Journal of Mathematical Physics 44, 1322.CrossRefADSMathSciNetMATHGoogle Scholar
  12. Fuchs, C. A. (2002). Quantum foundations in the light of quantum information. In A. Gonis, (ed.) 2001 NATO Advanced Research Workshop “Decoherence and its implications in quantum computation and information transfer,” Mikonos, Greece.Google Scholar
  13. Girelli, F. and Livine, E. R. (2005). Reconstructing quantum geometry from quantum information: spin networks as harmonic oscillators. Classical and Quantum Gravity 22, 3295.CrossRefADSMathSciNetMATHGoogle Scholar
  14. Giovannetti, V., Lloyd, S., and Maccone, L. (2004). Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330.CrossRefADSGoogle Scholar
  15. Gambini, R. and Porto, R. A. (2001). Relational time in generally covariant quantum systems: four models. Physical Review D 63, 105014.CrossRefADSMathSciNetGoogle Scholar
  16. Gambini, R., Porto, R., and Pullin, J. (2004). A relational solution to the problem of time in quantum mechanics and quantum gravity induces a fundamental mechanism for decoherence. New Journal of Physies 6, 45.CrossRefADSGoogle Scholar
  17. Gambini, R., Porto, R., and Pullin, J. (2005). Fundamental decoherence in quantum gravity. Brazilian Journal of Physics 35, 266.CrossRefADSGoogle Scholar
  18. Goldstone, J., Salam, A., and Weinberg, S. (1962). Broken symmetries. Physical Review 127, 965.CrossRefADSMathSciNetMATHGoogle Scholar
  19. Haag, R. (1992). Local Quantum Physics: Fields, Particles, Algebras. Springer.Google Scholar
  20. Hartle, J. B., Laflamme, R., and Marolf, D. (1995). Conservation laws in the quantum mechanics of closed systems. Physical Review D 51, 7007.CrossRefADSMathSciNetGoogle Scholar
  21. Kempe, J., Bacon, D., Lidar, D. A., and Whaley, K. B. (2001). Theory of decoherence-free fault-tolerant universal quantum compuation. Physical Review A 63, 42307.CrossRefADSGoogle Scholar
  22. Kershaw, D., and Woo, C. H. (1974). Experimental test for the charge superselection rule. Physical Review Letters 33(15).Google Scholar
  23. Kribs, D., Laflamme, R., and Poulin, D. (2005). A unified and generalized approach to quantum error correction. Physical Review Letters 94, 180501.CrossRefADSGoogle Scholar
  24. Knill, E., Laflamme, R., and Viola, L. (2002). Theory of quantum error correction for general noise. Physical Review Letters 84, 2525–2528.CrossRefADSMathSciNetGoogle Scholar
  25. Kraus, K. (1983). States, Effects and Operations. Fundamental Notions of Quantum Theory, Academic Press, Berlin.MATHGoogle Scholar
  26. Kogut, J. B., and Susskind, L. (1975). Kogut, J. B. and Susskind, L. (1975). Hamiltonian formulation of Wilson’s lattice gauge theories. Physical Review D 11, 395.CrossRefADSGoogle Scholar
  27. Leggett, A. J. (2000). Topics in the theory of the ultracold dilute alkali gases. In C. M. Savage and M. -P. Das, (eds.), Bose-Einstein Condensation. p.1.Google Scholar
  28. Lindblad, G. (1999). A general no-cloning theorem. Letter in Mathematical Physics 47, 189–196.CrossRefMathSciNetMATHGoogle Scholar
  29. Lloyd, S. (2005). The computational universe: quantum gravity from quantum computation.Google Scholar
  30. Major, S. (1999). A spin network primer. American Journal of Physics 67, 972.CrossRefADSMathSciNetGoogle Scholar
  31. Marolf, D. (2000). Group averaging and refined algebraic quantization: Where are we now?Google Scholar
  32. Mazzucchi, S. (2000). On the observables describing a quantum reference frame. arxiv.org:quant-ph/0006060Google Scholar
  33. Milburn, G. J. (2003). Lorentz invariant intrinsic decoherence.Google Scholar
  34. Milburn, G. J. and Poulin, D. (2005). Relational time for systems of oscillators. International Journal of Quantum Info.Google Scholar
  35. Mølmer, K. (1997). Optical coherence: a convenient fiction. Physical Review A 55, 3195.CrossRefADSGoogle Scholar
  36. Nielsen, M. A., and Poulin, D. (2005). Algebraic and information-theoretic conditions for operator quantum error-correction.Google Scholar
  37. Ollivier, H., Poulin, D., and Zurek, W. H. (2004). Objective properties from subjective quantum states: environment as a witness. Physical Review Letters 93, 220401.CrossRefADSGoogle Scholar
  38. Pegg, D. T. (1991). Time in a quantum mechanical world. Journal of Physics 24, 3031.Google Scholar
  39. Penrose, R. (1971). Angular momentum: an approach to combinatorial space-time. In T. Bastin, editor. Quantum Theory and Beyond, Cambridge University Press, Cambridge, UK. p. 151.Google Scholar
  40. Poulin, D. (2004). Emergence of a classical world from within quantum theory. PhD thesis, University of Waterloo, Providence, RI (Rhode Island), USA.Google Scholar
  41. Page, D. N. and Wootters, W. K. (1983). Evolution without evolution: dynamics described by stationary observables. Physical Review D 27, 2885.CrossRefADSGoogle Scholar
  42. Rovelli, C. (1991). Quantum reference systems. Classical and Quantum Gravity 8, 317.CrossRefADSMathSciNetGoogle Scholar
  43. Rovelli, C. (1996). Relational quantum mechanics. International Journal of Theoretical Physics 35, 1637.CrossRefADSMathSciNetMATHGoogle Scholar
  44. Rovelli, C. (2004). Quantum Gravity, Cambridge University Press, Cambridge, UK.Google Scholar
  45. Rovelli, C. and Smolin, L. (1995). Spin networks and quantum gravity. Physical Review D 52, 5743.CrossRefADSMathSciNetGoogle Scholar
  46. Sakurai, J. (1994). Modern quantum mechanics, Addison-Welay,Ontario, Canada.Google Scholar
  47. Toller, M. (1997). Quantum reference frames and quantum transformations. Il Nuovo Cimento 112, 1013.MathSciNetGoogle Scholar
  48. Unruh, W. G. and Wald, R. M. (1989). Time and the interpretation of canon ical quantum gravity. Physical Review D 40, 2598.CrossRefADSMathSciNetGoogle Scholar
  49. von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press, New Jersey.MATHGoogle Scholar
  50. Wheeler, J. A. (1991). Information, physics, quantum: The search for links. In W. H. Zurek, (ed.) Complexity, Entropy and the Physics of Information, Addison-Wesley, Ontario, Canada.Google Scholar
  51. Wigner, E. P. (1957). Relativistic invariance and quantum phenomena. Reviews of Modern Physics 29, 255.CrossRefADSMathSciNetMATHGoogle Scholar
  52. Zanardi, P. (2001). Stabilizing quantum information. Physical Review A 63, 12301.CrossRefADSMathSciNetGoogle Scholar
  53. Zurek, W. H. (2003). Decoherence, einselection and the quantum origins of the classical. Reviews of Modern Physics 75, 715–775.CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.School of Physical SciencesThe University of QueenslandQueenslandAustralia

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