A uniform approximation for the fidelity in chaotic systems

Cerruti, Nicholas R.; Tomsovic, Steven

doi:10.1088/0305-4470/36/12/334

Cited by 61 publications

(66 citation statements)

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“…The asymptotic rate of entanglement production in chaotic systems depends on the strength of the interaction between the two particles, and is explicitely given by a classical time-correlator. We note that, as is the case for the Loschmidt Echo [16,17], this regime is also adequately captured by an approach …”

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confidence: 65%

Semiclassical Time Evolution of the Reduced Density Matrix and Dynamically Assisted Generation of Entanglement for Bipartite Quantum Systems

Jacquod

2004

Phys. Rev. Lett.

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Two particles, initially in a product state, become entangled when they come together and start to interact. Using semiclassical methods, we calculate the time evolution of the corresponding reduced density matrix ρ1, obtained by integrating out the degrees of freedom of one of the particles. To quantify the generation of entanglement, we calculate the purity P(t) = Tr[ρ1 (t) 2 ]. We find that entanglement generation sensitively depends (i) on the interaction potential, especially on its strength and range, and (ii) on the nature of the underlying classical dynamics. Under general statistical assumptions, and for short-ranged interaction potentials, we find that P(t) decays exponentially fast if the two particles are required to interact in a chaotic environment, whereas it decays only algebraically in a regular system. In the chaotic case, the decay rate is given by the golden rule spreading of one-particle states due to the two-particle coupling, but cannot exceed the system's Lyapunov exponent. When two systems (. . . ) enter into temporary interaction (. . . ), and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. This is how entanglement was characterized by Schrödinger almost seventy years ago [1]. Entanglement is arguably the most puzzling property of multipartite quantum systems, and often leads to counterintuitive predictions due to, in Einstein's words, spooky action at a distance. Entanglement has received a renewed, intense interest in recent years in the context of quantum information theory [2].In the spirit of Schrödinger's above formulation, one is naturally led to ask the following question: "What determines the rate of entanglement production in a dynamical system ?". Is this rate mostly determined by the interaction between two, initially unentangled particles, or does it depend on the underlying classical dynamics ? This is the question we address in this paper. Previous attempts to answer this question have mostly focused on numerical investigations [3,4,5,6,7,8], with claims that entanglement is favored by classical chaos, both in the rate it is generated [3,4,7] and in the maximal amount it can reach [5]. In particular, strong numerical evidences have been given by Miller and Sarkar for an entanglement production rate given by the system's Lyapunov exponents [4]. These findings have however been recently challenged by Tanaka et al. [6], whose numerical findings show no increase of the entanglement production rate upon increase of the Lyapunov exponents in the strongly chaotic but weakly coupled regime, in agreement with their analytical calculations relating the rate of entanglement production to classical time correlators. Ref.[6] is seemingly in a paradoxical disagreement with the almost identical analytical approach of Ref. [7], where entanglement production was found to be faster in chaotic systems than in regular ones [9]. This controversy t...

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confidence: 65%

Semiclassical Time Evolution of the Reduced Density Matrix and Dynamically Assisted Generation of Entanglement for Bipartite Quantum Systems

Jacquod

2004

Phys. Rev. Lett.

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“…Random matrix models for breaking fundamental or dynamical symmetries have been introduced for a variety of cases since the work of [37] for breaking time reversal invariance. Examples include ensembles to describe parity violation [68], parametric statistical correlations [69], modeling transport barriers [25,26,70], and the fidelity [71]. The structure imposed by the particular symmetry is incorporated into the unperturbed ensemble (zero th -order piece) and a tunable-strength ensemble is added that violates that structure.…”

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confidence: 99%

Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

et al. 2018

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We derive universal entanglement entropy and Schmidt eigenvalue behaviors for the eigenstates of two quantum chaotic systems coupled with a weak interaction. The progression from a lack of entanglement in the noninteracting limit to the entanglement expected of fully randomized states in the opposite limit is governed by the single scaling transition parameter, Λ. The behaviors apply equally well to few-and many-body systems, e.g. interacting particles in quantum dots, spin chains, coupled quantum maps, and Floquet systems as long as their subsystems are quantum chaotic, and not localized in some manner. To calculate the generalized moments of the Schmidt eigenvalues in the perturbative regime, a regularized theory is applied, whose leading order behaviors depend on √ Λ. The marginal case of the 1/2 moment, which is related to the distance to closest maximally entangled state, is an exception having a √ Λ ln Λ leading order and a logarithmic dependence on subsystem size. A recursive embedding of the regularized perturbation theory gives a simple exponential behavior for the von Neumann entropy and the Havrda-Charvát-Tsallis entropies for increasing interaction strength, demonstrating a universal transition to nearly maximal entanglement. Moreover, the full probability densities of the Schmidt eigenvalues, i.e. the entanglement spectrum, show a transition from power laws and Lévy distribution in the weakly interacting regime to random matrix results for the strongly interacting regime. The predicted behaviors are tested on a pair of weakly interacting kicked rotors, which follow the universal behaviors extremely well.

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“…In many systems the rate of the exponential increases as the square of the perturbation strength [8] (see [14] for an exceptional case) until saturating at the underlying classical systems' Lyapunov exponent [7,9] or at the bandwidth of the Hamiltonian [8]. The crossover between the various regimes [15,16,17] and the fidelity saturation level [18,19] have also been explored. Quantum fidelity decay simulations have also been carried out in weakly chaotic systems [20], and at the edge of quantum chaos [21].…”

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Quantum fidelity decay in quasi-integrable systems

Weinstein

Hellberg

2005

Phys. Rev. E

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We show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type. Manifestations of chaos and complexity in the quantum realm have been widely explored in connection with the correspondence principle between classical and quantum mechanics [1]. An example is a system's response to small perturbations of its Hamiltonian. Peres [2,3] conjectured that this response serves as an indicator of chaos applicable to both the classical and quantum realms. That is, in both realms the behavior of fidelity between a state evolved under perturbed and unperturbed dynamics depends on whether or not the dynamics is chaotic.Peres' conjecture found an experimental venue in nuclear magnetic resonance (NMR) polarization echoes. In these experiments the initial state of the system is evolved forward under its internal dipolar Hamiltonian and then inverted by a sequence of radio-frequency pulses [4]. The inverted Hamiltonian, however, will be not be an exact reversal of the internal Hamiltonian due to pulse imperfections and interactions with the environment. These perturbations reduce the subsequent echo amplitude which is the measure of fidelity.The polarization echo as a means of studying dynamical irreversibility was applied in Ref. [5], where it was noted that the echo decay behavior as a function of time can be exponential or Gaussian, depending on the molecule under investigation. The connection between these results and the exponential fidelity decay predicted for systems exhibiting quantum chaos [3] was made in Ref. [6].Encouraged by these experimental investigations, Jalabert and Pastawski [7] applied semi-classical analysis to the evolution of what they termed the Loschmidt echo, or fidelity decay. Their analysis showed that for chaotic systems, when the perturbation is strong enough such * To whom correspondence should be addressed; Electronic address: weinstei@dave.nrl.navy.mil † Electronic address: hellberg@dave.nrl.navy.mil that perturbation theory fails, the fidelity decay is comprised of two exponentially decaying terms. The first of these terms is dominant for small errors and can be described by the Fermi golden rule [8,9]. The second term is dominant for strong errors, independent of perturbation strength...

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A uniform approximation for the fidelity in chaotic systems

Cited by 61 publications

References 43 publications

Semiclassical Time Evolution of the Reduced Density Matrix and Dynamically Assisted Generation of Entanglement for Bipartite Quantum Systems

Semiclassical Time Evolution of the Reduced Density Matrix and Dynamically Assisted Generation of Entanglement for Bipartite Quantum Systems

Eigenstate entanglement between quantum chaotic subsystems: Universal transitions and power laws in the entanglement spectrum

Quantum fidelity decay in quasi-integrable systems

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