In this work, we show that the dissipation in a many-body system under an arbitrary nonequilibrium process is related to the R\'enyi divergences between two states along the forward and reversed dynamics under a very general family of initial conditions. This relation generalizes the links between dissipated work and R\'enyi divergences to quantum systems with conserved quantities whose equilibrium state is described by the generalized Gibbs ensemble. The relation is applicable for quantum systems with conserved quantities and can be applied to protocols driving the system between integrable and chaotic regimes. We demonstrate our ideas by considering the one-dimensional transverse quantum Ising model and the Jaynes-Cummings model which are driven out of equilibrium.
We show that the quantum Jarzynski equality generalizes to
PT-symmetric quantum mechanics with unbroken
PT symmetry. In the regime of broken
PT symmetry, the Jarzynski equality does not hold as also the
CPT norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system---two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken
PT symmetry is thermodynamically inhibited as the irreversible work diverges at the critical point.
In this work, we show that a universal quantum work relation for a quantum system driven arbitrarily far from equilibrium extends to a parity-time- (PT-) symmetric quantum system with unbroken PT symmetry, which is a consequence of microscopic reversibility. The quantum Jarzynski equality, linear response theory, and Onsager reciprocal relations for the PT-symmetric quantum system are recovered as special cases of the universal quantum work relation in a PT-symmetric quantum system. In the regime of broken PT symmetry, the universal quantum work relation does not hold because the norm is not preserved during the dynamics.
The concept of classical [Formula: see text]-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz’ quasi-entropies (standard [Formula: see text]-divergences), Matsumoto’s maximal [Formula: see text]-divergences, measured [Formula: see text]-divergences, and sandwiched and [Formula: see text]-[Formula: see text]-Rényi divergences. In this paper, we give a systematic overview of the various concepts of quantum [Formula: see text]-divergences, with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum [Formula: see text]-divergence by a quantum operation. In particular, we compare the standard and the maximal [Formula: see text]-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum [Formula: see text]-divergences are strictly different for non-commuting operators unless [Formula: see text] is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard [Formula: see text]-divergences. We also study the monotonicity of the [Formula: see text]-[Formula: see text]-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters [Formula: see text] where monotonicity holds, and where the preservation of the [Formula: see text]-[Formula: see text]-Rényi divergence implies the reversibility of the quantum operation.
The effect of \mathcal{PT} 𝒫 𝒯 -symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT} 𝒫 𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT} 𝒫 𝒯 -symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT} 𝒫 𝒯 -symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.
The concept of classical
f-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard
f-divergences), Matsumoto's maximal
f-divergences, measured
f-divergences, and sandwiched and
α-
z-R\'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum
f-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum
f-divergence by a quantum operation. In particular, we compare the standard and the maximal
f-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum
f-divergences are strictly different for non-commuting operators unless
f is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard
f-divergences. We also study the monotonicity of the
α-
z-R\'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\'enyi divergence invariant, and determine domains of the parameters
α,z where monotonicity holds, and where the preservation of the
α-
z-R\'enyi divergence implies the reversibility of the quantum operation.