Dephasing representation of quantum fidelity for general pure and mixed states

Abstract

General semiclassical expression for quantum fidelity (Loschmidt echo) of arbitrary pure and mixed states is derived. It expresses fidelity as an interference sum of dephasing trajectories weighed by the Wigner function of the initial state, and does not require that the initial state be localized in position or momentum. This general dephasing representation is special in that, counterintuitively, all of fidelity decay is due to dephasing and none due to the decay of classical overlaps. Surprising accuracy of the approximation is justified by invoking the shadowing theorem: twice--both for physical perturbations and for numerical errors. It is shown how the general expression reduces to the special forms for position and momentum states and for wave packets localized in position or momentum. The superiority of the general over the specialized forms is explained and supported by numerical tests for wave packets, non-local pure states, and for simple and random mixed states. The tests are done in non-universal regimes in mixed phase space where detailed features of fidelity are important. Although semiclassically motivated, present approach is valid for abstract systems with a finite Hilbert basis provided that the discrete Wigner transform is used. This makes the method applicable, via a phase space approach, e. g., to problems of quantum computation.

arXiv comment11 pages, 4 figures

References in corpus (23)

• Environment-independent decoherence rate in classically chaotic systems
• Sub-Planck spots of Schroedinger cats and quantum decoherence
• Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo
• Decoherence and the Loschmidt echo
• On general relation between quantum ergodicity and fidelity of quantum dynamics
• Sensitivity of Wave Field Evolution and Manifold Stability in Chaotic Systems
• Stability of quantum motion and correlation decay
• Quantum computers in phase space
• Quantum Dynamical Echoes in the Spin 'Diffusion' in Mesoscopic Systems
• A random matrix formulation of fidelity decay
• Semiclassical evaluation of quantum fidelity
• Decay of the Loschmidt Echo for quantum states with sub-Planck scale structures
• Theory of quantum Loschmidt echoes
• Universality of the Lyapunov regime for the Loschmidt echo
• Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions
• A Uniform Approximation for the Fidelity in Chaotic Systems
• Experimental verification of fidelity decay: From perturbative to Fermi Golden Rule regime
• Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics
• Quantum freeze of fidelity decay for a class of integrable dynamics
• Anomalous power law of quantum reversibility for classically regular dynamics
• Uniform semiclassical approach to fidelity decay in the deep Lyapunov regime
• Quantum freeze of fidelity decay for chaotic dynamics
• Uniform Semiclassical Approach to Fidelity Decay: from Weak to Strong Perturbation

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• Rotating Gaussian wave packets in weak external potentials
• Quantum non-Markovian behavior at the chaos border
• Non-monotonous short-time decay of the Loschmidt echo in quasi-one-dimensional systems
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