Theoretical foundations of electron transport in mesoscopic systems, based on Landauer theory, Master equations or Onsager linear thermodynamics, are revisited to show that the noniteracting electrons model is insufficient to describe neither passive transport, nor generation of electromotive force (active transport). It is argued that 2-body electrostatic interactions creating double layers and surface charge distributions are crucial for the electron transport through a junction. Phenomenological modifications of the passive transport formulas based on the careful analysis of the fundamental notions of chemical, electrostatic, electrochemical, build-in potentials, band bending and bias voltage, are proposed. On the other hand active transport can be generated by a self-oscillating double layer (a pump) driven by an external heat, light or chemical energy source.

Nowadays, it is widely acknowledged that information can serve as a thermodynamic fuel capable of powering the extraction of useful work through measurements and feedback control. Moreover, it has been demonstrated that the work extractable from a correlated bipartite state can effectively distinguish entangled states from separable ones. In this context, we introduce an operational criterion denoted as [Formula: see text] to witness the presence of entanglement in a two-mode Gaussian state [Formula: see text], based on the amount of work that can be extracted from a thermal bath using a Szilard-like engine. In this setup, the state [Formula: see text] acts as the working medium of the engine, where modes [Formula: see text] and [Formula: see text] are generated during the first and second transitions of a nondegenerate three-level laser, and are coupled to a shared two-mode thermal bath. Our results reveal that the proposed criterion [Formula: see text] vanishes when [Formula: see text] is a product state. Furthermore, [Formula: see text] reaches its maximum value when [Formula: see text] is coupled to a vacuum bath, but it diminishes under the influence of thermal noise. Notably, the degree of entanglement detected by [Formula: see text] can be tuned via the atomic coherence of the laser. Finally, a comparison of [Formula: see text] with a reliable measure of entanglement such as logarithmic negativity, demonstrates perfect agreement in identifying the presence of entanglement in [Formula: see text].

The subgroups XU(2), YU(2), and ZU(2) of the unitary group U(2) allow to find twenty-four different decompositions of an arbitrary [Formula: see text] unitary matrix. Because the group YU(2) has not been considered before, 20 out of the 24 decompositions are new. Introducing this YU(2) group allows to highlight, within quantum circuit design, a symmetry similar to the threefold symmetry of the Pauli matrices of spin systems. Similar subgroups of the unitary group U([Formula: see text]) lead to various new decompositions of an arbitrary [Formula: see text] unitary matrix. Whereas for [Formula: see text] equal 2, this leads to the synthesis of single-qubit quantum gates, for [Formula: see text] equal a power of 2, this leads to new design methods for multiple-qubit circuits.

In this paper we demonstrate that any Markovian master equation defining a completely positive evolution for a quantum–classical hybrid state can always be written in terms of four basic coupling mechanisms. Each of them is characterized by a different “backaction” on each subsystem. On this basis, for each case, we find the conditions under which a diffusive limit is approached, that is, the time evolution can be approximated in terms of the first and second derivatives of the hybrid state with respect to a classical coordinate. In this limit, the restricted class of evolutions that guaranty the positivity of the hybrid state at all times (quantum Fokker-Planck master equations) emerges when the coupling mechanisms lead to infinitesimal (non-finite) changes in both the quantum and classical subsystems. A broader class of diffusive evolutions is obtained when positivity is only granted after a transient time or alternatively is granted after imposing an initial finite width on the state of the classical subsystem. A set of representative examples support these results.

By merging the Feynman-Vernon’s approach with the out-of-equilibrium Keldysh-Schwinger formalism, we construct the reduced generating functional through which all the time-dependent correlation functions of an open fermionic system can be directly derived by applying the appropriate functional derivatives. As a concrete example, we investigate the transverse Ising model, we derive the covariance matrix at the steady state of the system and we investigate its critical behaviour.

An analysis of the quantum measurement problem is presented which is a modest modification of the standard one often called the Copenhagen interpretation. The starting assumption is that QM is universal, and that all evolutions are unitary. We must also assume that the set of evolutions is restricted to allow the existence of stable structures for the equipment in our laboratory, including the measurement devices. The devices need some macroscopic properties to make the outcomes of the experiments stable data. There should not exist superdevices capable of retrieving the information needed to reverse these evolutions. As a result the quantum coherences complementary to the outcomes of experiments will typically be inaccessible and there is no need to remove them by decoherence. We need devices designed such that the outcome of an experiment is indicated by a pointer with the property that it has a definite but random final state. We then find that the outcomes of experiments can be considered factual data, and random in a classical sense. They can be considered to happen in a single world.

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the standard semi-circle random variable [Formula: see text], characterized by the fact that its probability distribution is the semi-circle law [Formula: see text] on [Formula: see text]. We prove that, in the identification of [Formula: see text] with the [Formula: see text]-mode interacting Fock space [Formula: see text], defined by the orthogonal polynomial gradation of [Formula: see text], [Formula: see text] is mapped into position operator and its canonically associated momentum operator [Formula: see text] into [Formula: see text] times the [Formula: see text]-Hilbert transform [Formula: see text] on [Formula: see text]. In the first part of the present paper, after briefly describing the simpler case of the [Formula: see text]-harmonic oscillator, we find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the semi-circle analogue of the translation group [Formula: see text] and of the semi-circle analogue of the free evolution [Formula: see text], respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of [Formula: see text] and [Formula: see text] on the [Formula: see text]-orthogonal polynomials is difficult, the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.

In phenomenological applications, time evolutions of Bloch-Redfield type are widely adopted for modelling open system dynamics, despite their nonpositive preserving character: this physical inconsistency, that in general shows up at small times, is usually cured by suitably restricting the space of allowed initial states. Nevertheless, additional problems may arise in relation to entanglement: specifically, we show that Redfield dynamics, though being a semigroup, can generate entanglement through a purely local action; moreover, this unphysical effect can persist on long time-scales.

A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to a GKSL master equation for the full density matrix.