Invariant Correlation Entropy and Complexity of Quantum States

Abstract

We define correlational (von Neumann) entropy for an individual quantum state of a system whose time-independent hamiltonian contains random parameters and is treated as a member of a statistical ensemble. This entropy is representation independent and can be calculated as a trace functional of the density matrix which describes the system in its interaction with the noise source. We analyze perturbation theory in order to show the evolution from the pure state to the mixed one. Exactly solvable examples illustrate the use of correlational entropy as a measure of the degree of complexity in comparison with other available suggestions such as basis-dependent information entropy. It is shown in particular that a harmonic oscillator in a uniform field of random strength comes to a quasithermal equilibrium; we discuss the relation between effective temperature and canonical equilibrium temperature. The notion of correlational entropy is applied to a realistic numerical caculation in the framework of the nuclear shell model. In this system, which reveals generic signatures of quantum chaos, correlational entropy and information entropy calculated in the mean field basis display similar qualitative behavior.

34 pages, REVTeX, 2 ps-figures. Submitted to Phys. Rev. E

References in corpus (5)

• Chaos and Quantum Thermalization
• Chaos {\sl vs} Thermalization in the Nuclear Shell Model
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• Quantum Chaos, Delocalization, and Entanglement in Disordered Heisenberg Models
• Nuclear pairing: new perspectives
• Thermodynamics of pairing in mesoscopic systems
• Entanglement and Tensor Product Decomposition for Two Fermions
• Invariant correlational entropy as a signature of quantum phase transitions in nuclei
• Application of information entropy to nuclei
• The dependence of information entropy of uniform Fermi systems on correlations and thermal effects
• Coherent and chaotic properties of nuclear pairing
• Exploring quantum dynamics in an open many-body system: Transition to superradiance
• High-lying single-particle modes, chaos, correlational entropy, and doubling phase transition
• Collision dynamics and entanglement generation of two initially independent and indistinguishable boson pairs in one-dimensional harmonic confinement
• Statistical measure of complexity and correlated behavior of Fermi systems
• Quantum Dynamics Against a Noisy Background
• Information entropy and nucleon correlations in nuclei
• Low energy properties of non-perturbative quantum systems: a space reduction approach
• Complexity of two-level systems