Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time
Open accessAbstract
The environment – external or internal degrees of freedom coupled to the system – can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time t ℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale t ℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.
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Simple quantum-classical correspondence breaks down for chaotic systems because of rapid spreading of the packet; even an initial minimum-uncertainty packet spreads over macroscopic regions of phase space within experimentally accessible timescales.
“However, such simple quantum-classical correspondence breaks down for chaotic systems, because of the rapid spreading of the packet: even an initial minimum-uncertainty packet spreads over macroscopic regions of phase space within experimentallyaccessible timescales (Zurek 1998).”
From De Broglie-Bohm Pilot-Wave Theory: Many Worlds in Denial? (2008) · cites this paper as Zurek 1998
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References 7 · Cited by 150
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- [14]14. Zurek, W. H., pp. 145-149 in Frontiers of Nonequilibrium Statistical Mechanics , G. T. Moore and M. O. Scully, eds. (Plenum, New York, 1986).
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- [39]39. Zurek, W. H., 'Information transfer in quantum measurements', pp. 87-116 in Quantum Optics, Experimental Gravity, and the Measurement Theory , P. Meystre and M. O. Scully, eds. (Plenum, New York, 1983).
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- [52]52. Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.-O., and Zeh, H. D., Decoherence and the Appearance of a Classical World in Quantum Theory , (Springer, Berlin, 1996).
- Authors
- Zurek, W. H.
- Affiliations
- Los Alamos National Laboratory (US)
- Corresponding author
- Zurek, Wojciech H.
- ORCID
- Zurek, Wojciech H.
- Venue
- Physica Scripta
- Publisher
- IOP Publishing
- Published
- 1998
- Volume
- T76
- Issue
- 1
- Pages
- 186
- ISSN
- 0031-8949 (ISSN-L)
- OpenAlex
- W2140431882
- Cited by
- 150
- References
- 7
- Citations by year
- 2026: 3 · 2025: 9 · 2024: 4 · 2023: 6 · 2022: 6 · 2021: 9 · 2020: 4 · 2019: 7 · 2018: 4 · 2017: 6 · 2016: 2 · 2015: 7 · 2014: 2 · 2013: 5 · 2012: 4
- FWCI
- 5.14
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physica Scripta (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Keywords
- decoherence, quantum chaos, einselection, quantum-classical correspondence, arrow of time, Lyapunov exponents, von Neumann entropy, entropy production, Schrödinger cat states, solar system dynamics, environmental monitoring, quantum measurement
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