Fidelity serves as a benchmark for the relieability in quantum information processes, and has recently atracted much interest as a measure of the susceptibility of dynamics to perturbations. A rich variety of regimes for fidelity decay have emerged. The purpose of the present review is to describe these regimes, to give the theory that supports them, and to show some important applications and experiments. While we mention several approaches we use time correlation functions as a backbone for the discussion. Vanicek's uniform approach to semiclassics and random matrix theory provides an important alternative or complementary aspects. Other methods will be mentioned as we go along. Recent experiments in micro-wave cavities and in elastodynamic systems as well as suggestions for experiments in quantum optics shall be discussed.Comment: Review article with some original results in integrable systems and random matrix models; 133 pages, 53 figure

We propose to study echo dynamics in a random matrix framework, where we assume that the perturbation is time independent, random and orthogonally invariant. This allows to use a basis in which the unperturbed Hamiltonian is diagonal and its properties are thus largely determined by its spectral statistics. We concentrate on the effect of spectral correlations usually associated to chaos and disregard secular variations in spectral density. We obtain analytic results for the fidelity decay in the linear response regime. To extend the domain of validity, we heuristically exponentiate the linear response result. The resulting expressions, exact in the perturbative limit, are accurate approximations in the transition region between the "Fermi golden rule" and the perturbative regimes, as examplarily verified for a deterministic chaotic system. To sense the effect of spectral stiffness, we apply our model also to the extreme cases of random spectra and equidistant spectra. In our analytical approximations as well as in extensive Monte Carlo calculations, we find that fidelity decay is fastest for random spectra and slowest for equidistant ones, while the classical ensembles lie in between. We conclude that spectral stiffness systematically enhances fidelity.

The concept of fidelity decay is discussed from the point of view of the scattering matrix, and the scattering fidelity is introduced as the parametric cross-correlation of a given S-matrix element, taken in the time domain, normalized by the corresponding autocorrelation function. We show that for chaotic systems, this quantity represents the usual fidelity amplitude, if appropriate ensemble and/or energy averages are taken. We present a microwave experiment where the scattering fidelity is measured for an ensemble of chaotic systems. The results are in excellent agreement with random matrix theory for the standard fidelity amplitude. The only parameter, namely the perturbation strength could be determined independently from level dynamics of the system, thus providing a parameter free agreement between theory and experiment.

In a complex scattering system with few open channels, say a quantum dot with leads, the correlation properties of the poles of the scattering matrix are most directly related to the internal dynamics of the system. We may ask how to extract these properties from an analysis of cross sections. In general this is very difficult, if we leave the domain of isolated resonances. We propose to consider the cross correlation function of two different elastic or total cross sections. For these we can show numerically and to some extent also analytically a significant dependence on the correlations between the scattering poles. The difference between uncorrelated and strongly correlated poles is clearly visible, even for strongly overlapping resonances. PACS number(s): 05.45.Mt, 03.65.Nk

The fidelity decay in a microwave billiard is considered, where the coupling to an attached antenna is varied. The resulting quantity, coupling fidelity, is experimentally studied for three different terminators of the varied antenna: a hard-wall reflection, an open wall reflection, and a 50 Ω load, corresponding to a totally open channel. The model description in terms of an effective Hamiltonian with a complex coupling constant is given. Quantitative agreement is found with the theory obtained from a modified VWZ approach [J. J. M. Verbaarschot, Phys. Rep. 129, 367 (1985)].

We study decoherence of two non-interacting qubits. The environment and its interaction with the qubits are modelled by random matrices. Decoherence, measured in terms of purity, is calculated in linear response approximation. Monte Carlo simulations illustrate the validity of this approximation and of its extension by exponentiation. The results up to this point are also used to study one qubit decoherence. Purity decay of entangled and product states are qualitatively similar though for the latter case it is slower. Numerical studies for a Bell pair as initial state reveal a one to one correspondence between its decoherence and its internal entanglement decay. For strong and intermediate coupling to the environment this correspondence agrees with the one for Werner states. In the limit of a large environment the evolution induces a unital channel in the two qubits, providing a partial explanation for the relation above.

The scattering matrix was measured for a flat microwave cavity with classically chaotic dynamics. The system can be perturbed by small changes of the geometry. We define the "scattering fidelity" in terms of parametric correlation functions of scattering matrix elements. In chaotic systems and for weak coupling the scattering fidelity approaches the fidelity of the closed system. Without free parameters the experimental results agree with random matrix theory in a wide range of perturbation strengths, reaching from the perturbative to the Fermi golden rule regime. The stability of quantum motion has been a topic of increasing interest in recent years. In Ref.[1], Peres proposed to consider the time evolution of wave packets governed by two slightly different Hamiltonians. Starting from the same initial state, their overlap provides a natural measure for the stability of the quantum evolution. As "fidelity" and "quantum Loschmidt echo", this quantity has since been investigated extensively (see Ref.[2] and references therein). Nowadays, it has become a standard benchmark for the reliability of quantum information processing [3]. Following Ref.[2], one may define fidelity as F (t) = |f (t)| 2 and fidelity amplitude aswhere the unitary operators U ′ (t) and U (t) describe the perturbed and unperturbed time evolution of an arbitrary initial state ψ(0). Depending on the strength of the perturbation one can discern three regimes. In the perturbative regime, where time-independent perturbation theory can be applied, the decay of the fidelity is Gaussian. For larger perturbations a cross-over to exponential decay is observed, with a decay constant obtained from Fermi's golden rule [4,5]. For very strong perturbations the decay constant saturates at the classical Lyapunov exponent [6].Since the first spin-echo experiment by Hahn [7], echo experiments have been performed with many different quantum and classical wave systems (e.g. Ref. [8,9]). However, wave functions are usually not accessible to experiments, and only some reduced information is available, such as the nuclear induction averaged over the probe in a magnetic resonance experiment [10,11], or the transmission between two antennas in a microwave or ultrasound experiment [12,13,14].Here, we report on the experimental measurement of fidelity decay in a flat electromagnetic cavity, using the equivalence of Helmholtz and stationary Schrödinger equation [15]. Instead of following the evolution of wave packets, we measure stationary spectra of scattering matrix elements, separately, for the perturbed and the unperturbed system. Then, for a given scattering matrix element, we compute the Fourier transform of the crosscorrelation function between the two spectra. After an appropriate normalization, this defines the scattering fidelity amplitude. Averaging this quantity over a large number of uniformly distributed antennas with small transmission yields the standard fidelity amplitude. Yet for integrable systems, this may still lead to system specific results. For chao...

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