[Paper Review] The Multiparticle Quantum Arnol'd Cat: a test case for the decoherence approach to quantum chaos

Giorgio Mantica|arXiv (Cornell University)|Jul 6, 2010

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper proposes a multi-particle extension of the quantum Arnol'd cat map to test whether decoherence can restore classical chaos in quantum systems. Using the Alicki-Fannes quantum dynamical entropy as an information-theoretic measure, numerical results show that increasing the number of scattering particles enhances quantum-classical correspondence in information production, suggesting decoherence may justify the correspondence principle for chaotic systems under controlled dynamical conditions.

ABSTRACT

A multi-particle extension of the Arnol'd Cat Hamiltonian system is defined and examined. We propose to compute its Alicki-Fannes quantum dynamical entropy, to validate (or disprove) the validity of the decoherence approach to quantum chaos. A first set of numerical experiments is presented and discussed.

Motivation & Objective

• To investigate whether decoherence can restore chaotic signatures in quantum systems that otherwise fail to exhibit classical chaos under unitary evolution.
• To develop a fully dynamical, quantal model of decoherence that avoids phenomenological approximations used in prior work.
• To use information-theoretic measures—specifically Alicki-Fannes quantum dynamical entropy—to assess the degree of quantum-classical correspondence in the presence of decoherence.
• To determine whether the time span of quantum-classical agreement in information production scales favorably with system parameters like particle number and mass.
• To explore whether consistent histories emerge in the quantum model under controlled decoherence, mirroring classical dynamical cylinder measures.

Proposed method

• Define a multi-particle extension of the classical Arnol'd cat map as a kicked particle system on a torus with multiple small scattering particles.
• Construct a fully quantal scattering model for small particles on the torus, solving the associated scattering matrix problem to model decoherence dynamically.
• Compute the finite-time Shannon entropy required for the Alicki-Fannes (A-F) quantum dynamical entropy using numerical diagonalization of the time-evolution operator.
• Use the A-F entropy as a proxy for dynamical complexity, comparing its growth with classical Shannon entropies for the same partition of phase space.
• Perform large-scale numerical simulations on high-performance clusters to compute A-F entropies for varying numbers of small particles (I), masses (M, m), and system sizes.
• Analyze scaling behavior of quantum entropy with respect to the number of scattering particles and system parameters to assess correspondence with classical dynamics.

Experimental results

Research questions

• RQ1 Does the inclusion of multiple scattering particles in a quantum Arnol'd cat system lead to a longer time window of quantum-classical correspondence in information production?
• RQ2 Can the Alicki-Fannes quantum dynamical entropy reproduce classical Kolmogorov-Sinai entropy growth rates under controlled decoherence?
• RQ3 Does increasing the number of small particles (I) lead to a power-law increase in the duration of quantum-classical agreement, rather than logarithmic scaling?
• RQ4 Are there parameter sequences (M, I) that grow polynomially with the maximal time J_max for which quantum and classical entropies coincide within a fixed precision ε?
• RQ5 Do individual quantum histories become consistent with classical dynamical cylinders under the same decoherence conditions that yield agreement in global information production?

Key findings

• Increasing the number of small scattering particles (I) from 1 to 3 extends the time span during which quantum A-F entropy matches classical Shannon entropy, even at fixed system mass M.
• For M = 2^8 h, the quantum A-F entropy closely approximates the classical entropy up to J = 6, indicating strong quantum-classical correspondence under these conditions.
• The increase in the duration of quantum-classical agreement with I is consistent with a power-law dependence rather than a logarithmic one, suggesting favorable scaling.
• The computational cost of calculating A-F entropies scales with the Hilbert space dimension N, which grows exponentially with I, requiring thousands of CPU hours on large clusters.
• Preliminary results indicate that the A-F entropy can detect the emergence of classical-like information production in a fully dynamical, quantal model of decoherence.
• The results support the hypothesis that decoherence can justify the correspondence principle for chaotic systems when information-theoretic measures are used, though full validation requires further parameter-space exploration.