Boundary entropy of one-dimensional quantum systems at low temperature

The boundary beta function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta function, expressing it as the gradient of the boundary entropy s at fixed nonzero temperature. The gradient formula implies that s decreases under renormalization, except at critical points (where it stays constant). At a critical point, the number exp((s) is the "ground-state degeneracy," g, of Affleck and Ludwig, so we have proved their long-standing conjecture that g decreases under renormalization, from critical point to critical point. The gradient formula also implies that s decreases with temperature, except at critical points, where it is independent of temperature. It remains open whether the boundary entropy is always bounded below.     

Expectations and entropy inequalities for finite quantum systems

We prove that the relative entropy is decreasing under a trace-preserving expectation inB(K ¹), and we show the connection between this theorem and the strong subadditivity of the entropy. It is also proved that a linear, positive, trace-preserving map ofB(K) into itself such that 1 decreases the value of any convex trace function.     

Nonequilibrium entropy production for open quantum systems

We consider open quantum systems weakly coupled to a heat reservoir and driven by arbitrary time-dependent parameters. We derive exact microscopic expressions for the nonequilibrium entropy production and entropy production rate, valid arbitrarily far from equilibrium. By using the two-point energy measurement statistics for system and reservoir, we further obtain a quantum generalization of the integrated fluctuation theorem put forward by Seifert [Phys. Rev. Lett. 95, 040602 (2005)].     

Spacetime dynamical entropy of quantum systems

The definition of the dynamical entropy for single automorphisms of nuclear C ^*-algebras is extended to groups of several commuting automorphisms. This entropy of a Z ^v-action is shown to be nonzero only if all the corresponding Z -subactions (0<<v) have infinite entropy. As a simple consequence, the spacetime entropy of quantum lattice spin systems, and of one-dimensional continuous systems with physically reasonable quasifree states, vanishes.     

Generalized entropy production fluctuation theorems for quantum systems

Based on trajectory-dependent path probability formalism in state space, we derive generalized entropy production fluctuation relations for a quantum system in the presence of measurement and feedback. We have obtained these results for three different cases: (i) the system is evolving in isolation from its surroundings; (ii) the system being weakly coupled to a heat bath; and (iii) system in contact with reservoir using quantum Crooks fluctuation theorem. In Case (iii), we build on the treatment carried out by H T Quan and H Dong [arXiv/cond-mat:0812.4955], where a quantum trajectory has been defined as a sequence of alternating work and heat steps. The obtained entropy production fluctuation theorems (FTs) retain the same form as in the classical case. The inequality of second law of thermodynamics gets modified in the presence of information. These FTs are robust against intermediate measurements of any observable performed with respect to von Neumann projective measurements as well as weak or positive operator-valued measurements.     

Entanglement entropy in extended quantum systems

After a brief introduction to the concept of entanglement in quantum systems, I apply these ideas to many-body systems and show that the von Neumann entropy is an effective way of characterising the entanglement between the degrees of freedom in different regions of space. Close to a quantum phase transition it has universal features which serve as a diagnostic of such phenomena. In the second part I consider the unitary time evolution of such systems following a ‘quantum quench’ in which a parameter in the Hamiltonian is suddenly changed, and argue that finite regions should effectively thermalise at late times, after interesting transient effects.     

Control of temperature and entropy by frequent quantum measurements

We study deviations from thermal equilibrium between two-level systems (TLS) and a bath by frequent and brief quantum measurements of the TLS energy-states. The resulting entropy and temperature of both the system and the bath are found to be completely determined by the measurement rate, and unrelated to what is expected by standard thermodynamical rules that hold for Markovian baths. These anomalies allow for very fast control heating, cooling and state-purification (entropy reduction) of quantum systems much sooner than their thermal equilibration time.     

Decoherence and the rate of entropy production in chaotic quantum systems

We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.     

On the definition of entropy for non-equilibrium states

In this paper it is demonstrated how the mathematical theory of stability of motion can be applied to kinetic equations, describing irreversible processes in an isolated, homogeneous system. It turns out that functions having all the properties of entropy exist throughout the domain of definition of the kinetic equations. Since the kinetic equations depend only on variables defined outside equilibrium thermodynamics, it is possible to define entropy far beyond the range of validity of the thermodynamics of irreversible processes. It is shown that the commonly assumed properties of entropy are not sufficient, however, to single out just one entropy function.     

Measure of irreversibility and entropy production in open quantum systems

A new concept of a measure of irreversibility for quantum dynamics in open systems is introduced as a suitably regularized substitute for the common notion of entropy production, which, unfortunately, yields infinite values for so many irreversible processes of physical relevance.     

The entropy of an ensemble of quantum mechanical systems

A mathematical result about a positive self-sdjoint operator of unit trace is proved whose physical interpretation is that when an observation is made on a mixed state of a quantum system leading to a change in its state, the entropy increases as it must.     

Time evolution for quantum systems at finite temperature

This paper investigates a new formalism to describe real time evolution of quantum systems at finite temperature. A time correlation function among subsystems will be derived which allows for a probabilistic interpretation. Our derivation is non-perturbative and fully quantized. Various numerical methods used to compute the needed path integrals in complex time were tested and their effectiveness was compared. For checking the formalism we used the harmonic oscillator where the numerical results could be compared with exact solutions. Interesting results were also obtained for a system that presents tunneling. A ring of coupled oscillators was treated in order to try to check self-consistency in the thermodynamic limit. The short time distribution seems to propagate causally in the relativistic case. Our formalism can be extended easily to field theories where it remains to be seen if relevant models will be computable.     

On the law of increasing entropy and the cause of the dynamics irreversibility of quantum systems

It has been shown that time reversal symmetry breaking in the dynamics of large systems originates from symmetry breaking in the occupation of the Hilbert space. The states ?, for which the entropy of the system (sub-system) increases, are automatically created in nature or can be prepared experimentally, in contrast to the respective complex-conjugate states (?*), for which the entropy decreases (although formally, according to the superposition principle, they can exist). It is indicated that, in the general case, the dynamics reversal of unknown states is impossible because the complex conjugation operator is antilinear. The complexity of reversal of the known state is exponential in the typical case of a large system. The formulated statements are illustrated by simple models.     

Diagonal entropy in many-body systems: Volume effect and quantum phase transitions

We investigate the diagonal entropy(DE) of the ground state for quantum many-body systems, including the XY model and the Ising model with next nearest neighbor interactions. We focus on the DE of a subsystem of L continuous spins. We show that the DE in many-body systems, regardless of integrability, can be represented as a volume term plus a logarithmic correction and a constant offset. Quantum phase transition points can be explicitly identified by the three coefficients thereof. Besides, by combining entanglement entropy and the relative entropy of quantum coherence, as two celebrated representatives of quantumness, we simply obtain the DE, which naturally has the potential to reveal the information of quantumness. More importantly, the DE is concerning only the diagonal form of the ground state reduced density matrix, making it feasible to measure in real experiments, and therefore it has immediate applications in demonstrating quantum supremacy on state-of-the-art quantum simulators.     

An entropy inequality for quantum measurements

It is proved that for an ideal quantum measurement the average entropy of the reduced states after the measurement is not greater than the entropy of the original state.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

A definition for time in quantum cosmology

A definition for the intrinsic time co-ordinate is proposed, using the phase of the wave function of the universe. This definition generalizes the notion of time co-ordinate which arises in the semiclassical cosmology. It also leads to acceptable results for the evaluation of expectation values of physical variables.     

Additivity of the Kerr effect in thin-film magnetic systems

A universal approach is developed for calculating magneto-optic coefficients in multimedia systems. Three configurations are described: (i) a two-media, one-boundary system, (ii) a film sandwiched between two media and (iii) a multilayer system, such as a superlattice or even a randomly stacked sequence. It is proven that in the thin-film limit, the Kerr effect for a multilayer system obeys an additivity law: it is equal to the algebraic sum of the Kerr signals of the individual magnetic films in the system. The prediction is verified experimentally and in numerical calculations on an Fe/Cu/Fe 3-layer stack grown on a Pd-substrate.     

Uncertainty relations with quantum memory for the Wehrl entropy

We prove two new fundamental uncertainty relations with quantum memory for the Wehrl entropy. The first relation applies to the bipartite memory scenario. It determines the minimum conditional Wehrl entropy among all the quantum states with a given conditional von Neumann entropy and proves that this minimum is asymptotically achieved by a suitable sequence of quantum Gaussian states. The second relation applies to the tripartite memory scenario. It determines the minimum of the sum of the Wehrl entropy of a quantum state conditioned on the first memory quantum system with the Wehrl entropy of the same state conditioned on the second memory quantum system and proves that also this minimum is asymptotically achieved by a suitable sequence of quantum Gaussian states. The Wehrl entropy of a quantum state is the Shannon differential entropy of the outcome of a heterodyne measurement performed on the state. The heterodyne measurement is one of the main measurements in quantum optics and lies at the basis of one of the most promising protocols for quantum key distribution. These fundamental entropic uncertainty relations will be a valuable tool in quantum information and will, for example, find application in security proofs of quantum key distribution protocols in the asymptotic regime and in entanglement witnessing in quantum optics.