Links between Dissipation and R\'{e}nyi Divergences in PT\mathcal{PT}-Symmetric Quantum Mechanics

Thermodynamics and information theory have been intimately related since the times of Maxwell and Boltzmann. Recently it was shown that the dissipated work in an arbitrary non-equilibrium process is related to the R\'{e}nyi divergences between two states along the forward and reversed dynamics. Here we show that the relation between dissipated work and Renyi divergences generalizes to $\mathcal{PT}$-symmetric quantum mechanics with unbroken $\mathcal{PT}$ symmetry. In the regime of broken $\mathcal{PT}$ symmetry, the relation between dissipated work and Renyi divergences does not hold as the norm is not preserved during the dynamics. This finding is illustrated for an experimentally relevant system of two-coupled cavities.Comment: 7 page

Linking Phase Transitions and Quantum Entanglement at Arbitrary Temperature

In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derived a set of universal functional relations between the matrix elements of two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first order phase transitions are signaled by the matrix elements of reduced density matrix while the second order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near second order phase transition point, we show that the first derivative of the reduced density matrix elements present universal scaling behaviors. Finally we establish a theorem which connects the phase transitions and entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.Comment: 8 pages,2 figure

Fluctuation Relations for Heat Exchange in the Generalized Gibbs Ensemble

In this work, we investigate the heat exchange between two quantum systems whose initial equilibrium states are described by the generalized Gibbs ensemble. First, we generalize the fluctuation relations for heat exchange discovered by Jarzynski and W\'ojcik to quantum systems prepared in the equilibrium states described by the generalized Gibbs ensemble at different generalized temperatures. Second, we extend the connections between heat exchange and R\'enyi divergences to quantum systems with very general initial conditions.These relations are applicable for quantum systems with conserved quantities and are universally valid for quantum systems in the integrable and chaotic regimes.Comment: 7 page

Insights into Phase Transitions and Entanglement from Density Functional Theory

Density functional theory has made great success in solid state physics, quantum chemistry and in computational material sciences. In this work we show that density functional theory could shed light on phase transitions and entanglement at finite temperatures. Specifically, we show that the equilibrium state of an interacting quantum many-body system which is in thermal equilibrium with a heat bath at a fixed temperature is a universal functional of the first derivatives of the free energy with respect to temperature and other control parameters respectively. This insight from density functional theory enables us to express the average value of any physical observable and any entanglement measure as a universal functional of the first derivatives of the free energy with respect to temperature and other control parameters. Since phase transitions are marked by the nonanalytic behavior of free energy with respect to control parameters, the physical quantities and entanglement measures may present nonanalytic behavior at critical point inherited from their dependence on the first derivative of free energy. We use an experimentally realizable model to demonstrate the idea. These results give new insights for phase transitions and provide new profound connections between entanglement and phase transition in interacting quantum many-body physics.Comment: 10 pages, 5 figure

Quantum Work Relations and Response Theory in PT\mathcal{PT}-Symmetric Quantum Systems

In this work, we show that a universal quantum work relation for a quantum system driven arbitrarily far from equilibrium extend to $\mathcal{PT}$-symmetric quantum system with unbroken $\mathcal{PT}$ symmetry, which is a consequence of microscopic reversibility. The quantum Jarzynski equality, linear response theory and Onsager reciprocal relations for the $\mathcal{PT}$-symmetric quantum system are recovered as special cases of the universal quantum work relation in $\mathcal{PT}$-symmetric quantum system. In the regime of broken $\mathcal{PT}$ symmetry, the universal quantum work relation does not hold as the norm is not preserved during the dynamics.Comment: 6 page

Dissipation in the Generalized Gibbs Ensemble

In this work, we show that the dissipation in a many-body system under an arbitrary non-equilibrium process is related to the R\'{e}nyi divergences between two states along the forward and reversed dynamics under very general family of initial conditions. This relation generalizes the links between dissipated work and Renyi 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 which is driven out of equilibrium by the instantaneous switching of the transverse magnetic field.Comment: 6 pages. arXiv admin note: text overlap with arXiv:1710.0605

Relations between Heat Exchange and R\'{e}nyi Divergences

In this work, we establish an exact relation which connects the heat exchange between two systems initialized in their thermodynamic equilibrium states at different temperatures and the R\'{e}nyi divergences between the initial thermodynamic equilibrium state and the final non-equilibrium state of the total system. The relation tells us that the various moments of the heat statistics are determined by the Renyi divergences between the initial equilibrium state and the final non-equilibrium state of the global system. In particular the average heat exchange is quantified by the relative entropy between the initial equilibrium state and the final non-equilibrium state of the global system. The relation is applicable to both finite classical systems and finite quantum systems.Comment: 5 page

Relations between Dissipated Work and R\'enyi Divergences

In this paper, we establish a general relation which directly links the dissipated work done on a system driven arbitrarily far from equilibrium, a fundamental quantity in thermodynamics, and the R\'{e}nyi divergences, a fundamental concept in information theory. Specifically, we find that the generating function of the dissipated work under an arbitrary time-dependent driving process is related to the R\'{e}nyi divergences between a non-equilibrium state in the driven process and a non-equilibrium state in its time reversed process. This relation is a consequence of time reversal symmetry in driven process and is universally applicable to both finite classical system and finite quantum system, arbitrarily far from equilibrium.Comment: 5 pages and 2 figure

Fidelity Susceptibility in the Quantum Rabi Model

Quantum criticality usually occurs in many-body systems. Recently it was shown that the quantum Rabi model, which describes a two-level atom coupled to a single model cavity field, presents quantum phase transitions from a normal phase to a superradiate phase when the ratio between the frequency of the two level atom and the frequency of the cavity field extends to infinity. In this work, we study quantum phase transitions in the quantum Rabi model from the fidelity susceptibility perspective. We found that the fidelity susceptibility and the generalized adiabatic susceptibility present universal finite size scaling behaviors near the quantum critical point of the Rabi model if the ratio between frequency of the two level atom and frequency of the cavity field is finite. From the finite size scaling analysis of the fidelity susceptibility, we found that the adiabatic dimension of the fidelity susceptibility and the generalized adiabatic susceptibility of fourth order in the Rabi model are $4/3$ and $2$, respectively. Meanwhile the correlation length critical exponent and the dynamical critical exponent in the quantum critical point of the Rabi model are found to be $3/2$ and $1/3$ respectively. Since the fidelity susceptibility and the generalized adiabatic susceptibility are the moments of the quantum noise spectrum which is directly measurable by experiments in linear response regime, the scaling behavior of the fidelity susceptibility in the Rabi model could be tested experimentally. The simple structure of the quantum Rabi model paves the way for experimentally observing the universal scaling behavior of the fidelity susceptibility at a quantum phase transition.Comment: 6 pages, 6 Figure

Wave Function and Pair Distribution Function of a Dilute Bose Gas

The wave function of a dilute hard sphere Bose gas at low temperatures is discussed, emphasizing the formation of pairs. The pair distribution function is calculated for two values of $\sqrt{\rho a^3}$.Comment: 2 pages, 1 figur