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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.

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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 probecontrol 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. Especially, 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.