An analytical relation between entropy production and quantum Lyapunov exponents for Gaussian bipartite systems

Abstract

Abstract. We study and compare the information loss of a large class of Gaussian bipartite systems. It includes the usual Caldeira-Leggett type model as well as Anosov models (parametric oscillators, the inverted oscillator environment, etc), which exhibit instability, one of the most important characteristics of chaotic systems. We establish a rigorous connection between the quantum Lyapunov exponents and coherence loss and show that in the case of unstable environments, coherence loss is completely determined by the upper quantum Lyapunov exponent, a behavior which is more universal than that of the Caldeira-Leggett type model.

Entanglement entropy converges to classical entropy around periodic orbits

“…preferred factorization into a degree of freedom and an environment is given, but note that we explicitly find that the entanglement entropy growth rate grows as the sum of the positive Lyapunov exponents, not just as the largest one. The original work of Zurek and Paz [6] had only one such positive exponent, and similarly with [12,18], though other studies have also found growth rates equal to the full sum [7,9]. We are also consistent with other earlier studies of entanglement entropy growth, but distinguished from them in that our system is not coupled to an external environment [6-8, 10, 14, 19, 21], our results are not perturbative [13] or numerical [34,35] and we do not work with a finite-dimensional Hilbert space [9,11] or invoke a random matrix or semi-classical approximation [15-17, 20, 36].…”

“…This is done for a closed system, and we will study the dependence on the choice of coarse-graining and initial state. The idea is to study the quantum counterparts of closed Hamiltonian chaotic dynamical systems with finitely many degrees of freedom, similarly to previous studies of closed systems in the quantum chaos and decoherence literature [9,12,18].…”

“…The rate at which the entanglement grows towards saturation depends, in general, on the details of these systems, although some general bounds exist [2][3][4][5]. Linear growth in time appears in many systems, for example, studies of decoherence and quantum chaos [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], and quenches of extended systems [22][23][24][25].When trying to apply these results in the context of black hole physics, we are usually confronted with two problems. First of all, the Hilbert space H is big.…”

Bridging entanglement dynamics and chaos in semiclassical systems

“…These authors conjectured that in a system coupled to an environment, the rate of entropy growth is equal to the sum of the positive Lyapunov exponents, the classical Kolmogorov-Sinai entropy rate [8][9][10]. A large body of numerical and analytical studies during the early 2000 [11][12][13][14][15][16][17][18][19][20][21][22][23][24] proved consistent with the Zurek-Paz surmise, and suggested further relationships between semiclassical entanglement dynamics and the chaoticity of the underlying trajectories. Related work focused on understanding the emergence of quantum irreversibility and decoherence through the dynamics of the purity and the Loschmidt echo [25,26].…”

“…( 44), this increase may be visualized as an enhancement of the projected volume spanned by the reduced quantum fluctuations within the subsystem's phase space, due to the progressive stretching of the global phase-space volume spanned by the quantum fluctuations. Similarly, the von Neumann entanglement entropy (19) can be computed as…”