We address the problem of synthesizing a generalized Gaussian noise with exponent 1/2 by means of a nonlinear memoryless transformation applied to a uniform noise. We show that this transformation is expressable in terms of a special function known under the name of the Lambert W function. We review the main methods for numerical evaluation of the relevant branch of the (multivalued) Lambert W function with controlled accuracy and complement them with an original rational function approximation. Based on these methods, synthesis of the generalized Gaussian noise can be performed with arbitrary accuracy. We construct a simple and fast evaluation algorithm with prescribed accuracy, which is especially suited for Monte Carlo simulation requiring large numbers of realizations of the generalized Gaussian noise.

A deterministic periodic signal plus a stationary random noise is applied to a static nonlinearity taking the form of a monovariable arbitrary function on real numbers. The property of noise-enhanced signal transmission through stochastic resonance is studied for this class of static nonlinear systems. A theory is developed that provides expressions for the output autocorrelation function, power spectral density, signal-to-noise ratio, and input-output phase shift, in the presence of a periodic input, a noise distribution, and a static nonlinearity, all three being arbitrary. Both white and colored input noises are successively considered. For white input noise, exact expressions are derived in a discrete-time framework directly confrontable to simulations or experiments. The theory is applied to describe stochastic resonance in various examples of static nonlinear systems, for instance, a diode nonlinearity. In addition, confrontations with experiments and simulations are given that support the theory. In particular, interesting effects are reported such as a signal-to-noise ratio larger at the output than at the input or stochastic resonance at zero frequency. Finally, the validity of the theory is extended to dynamic nonlinear systems that can be decomposed into a static nonlinearity followed by an arbitrary dynamic linear system.

A generic qubit unitary operator affected by depolarizing noise is duplicated and inserted in a quantum switch process, realizing a superposition of causal orders. The characterization of the resulting switched quantum channel is worked out for its action on the joint state of the probe-control qubit pair. The switched channel is then specifically investigated for the important metrological task of phase estimation on the noisy unitary operator, with the performance assessed by the Fisher information, classical or quantum. A comparison is made with conventional techniques of estimation where the noisy unitary is directly probed in a one-stage or two-stage cascade with definite order or several uses of them with two or more qubits. In the switched channel with indefinite order, specific properties are reported, meaningful for estimation and not present with conventional techniques. It is shown that the control qubit, although it never directly interacts with the unitary, can nevertheless be measured alone for effective estimation, while discarding the probe qubit that interacts with the unitary. Also, measurement of the control qubit maintains the possibility of efficient estimation in difficult conditions where conventional estimation becomes less efficient, as for instance with ill-configured input probes, or in blind situations when the axis of the unitary is unknown. Effective estimation by measuring the control qubit remains possible even when the input probe tends to align with the axis of the unitary, or with a fully depolarized input probe, while in these conditions conventional estimation becomes inoperative. Measurement of the probe qubit of the switched channel is also analyzed and shown to add useful capabilities for phase estimation. The results contribute to the ongoing identification and analysis of the properties and capabilities of switched quantum channels with indefinite order for information processing and uncover new possibilities for quantum estimation and qubit metrology.

A nonlinear system is considered where an aperiodic binary input signal is added to an arbitrarily distributed noise and compared to a fixed threshold to determine the binary output signal. Noise enhancement of the transmission of the aperiodic signal via stochastic resonance is demonstrated and studied in this nonlinear information channel. The characterization developed goes up to the calculation of the information capacity of the channel, defined as the maximal achievable input-output transinformation occurring when the statistics of the input signal is matched to the noise. It is then demonstrated that a regime exists where the information capacity of the channel can be increased by means of an increase of the noise, up to an optimal noise level where the capacity resonates at a maximum value. The influence on this resonance of the noise distribution is also studied.