Quantum theory of irreversibility

A generalization of the Gibbs–von Neumann entropy is proposed based on the quantum BBGKY (Bogolyubov–Born–Green–Kirkwood–Yvon) hierarchy as the non-equilibrium entropy for an N$N$-body system. By using a generalization of the Liouville–von Neumann equation describing the evolution of a density superoperator, the entropy production for an isolated system is calculated, being non-zero in general. The existence of a non-zero entropy production allows us, following the procedure of non-equilibrium thermodynamics to introduce a master matrix for which a microscopic expression is obtained. After this, as a test of our theory the quantum Boltzmann equation is derived in terms of a transition superoperator related to this master matrix.     

A New Inequality for the von Neumann Entropy

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

The second law of thermodynamics under unitary evolution and external operations

The von Neumann entropy cannot represent the thermodynamic entropy of equilibrium pure states in isolated quantum systems. The diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, is a natural generalization of the von Neumann entropy and applicable to equilibrium pure states. We show that the diagonal entropy is consistent with the second law of thermodynamics upon arbitrary external unitary operations. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that quantum trajectories under unitary evolution are restricted by the Hamiltonian dynamics and that the external operation is performed without reference to the microscopic state of the system.     

The Saturation of Several Universal Inequalities in Information-Processing

In this paper, we characterize the saturation of four universal inequalities in quantum information theory, including a variant version of strong subadditivity inequality for von Neumann entropy, the coherent information inequality, the Holevo quantity, and average entropy inequalities. These results shed new light on quantum information inequalities.     

A Measure of Quantum Correlation Based on von Neumann Entropy and Positive Operator-Valued Measurement

Quantum correlations in composite quantum systems are at the origin of the most peculiar features of quantum mechanics such as the violation of Bells inequalities and non-locality. In quantum information theory, they are viewed as quantum resources used by quantum algorithms and communication protocols to outperform their classical analogs. In this paper, we define a new measure of quantum correlation based on von Neumann entropy and positive operator-valued measurement,which has clear physical meaning and we can prove that it satisfying many good property for a measure of quantumness.     

Entropy and superposition principle

Given a faithful normal state ? of a von Neumann algebra M, entropy and relative entropy for normal states of M are defined by Radon-Nikodyn derivatives of normal states with respect to ?. Most properties of entropy and relative entropy in finite quantum systems are shown to hold. It is also shown that the finiteness of relative entropy is related to the facial superposition principle in quantum theory [5].     

Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum

The entanglement entropy of a pure quantum state of a bipartite system A union or logical sumB is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. In one dimension, the entanglement of critical ground states diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find that the entanglement entropy of a standard class of z=2 conformal quantum critical points in two spatial dimensions, in addition to a nonuniversal "area law" contribution linear in the size of the AB boundary, generically has a universal logarithmically divergent correction, which is completely determined by the geometry of the partition and by the central charge of the field theory that describes the critical wave function.     

Replacing energy by von Neumann entropy in quantum phase transitions

We propose that quantum phase transitions are generally accompanied by non-analyticities of the von Neumann (entanglement) entropy. In particular, the entropy is non-analytic at the Anderson transition, where it exhibits unusual fractal scaling. We also examine two dissipative quantum systems of considerable interest to the study of decoherence and find that non-analyticities occur if and only if the system undergoes a quantum phase transition.     

A NEW MEASURE FOR THE JAYNES-CUMMINGS MODEL DYNAMICS DISTANCE BETWEEN DENSITY OPERATORS

We use the distance between density operators to study the dynamical evolution of the Jaynes-Cummings model with an additional Kerr medium, and to compare the result with the corresponding von Neumann entropy. We have shown that the distance between density operators can provide more detailed information about the dynamical behavior of the quantum system than von Neumann entropy.     

A NEW MEASURE FOR THE JAYNES-CUMMINGS MODEL DYNAMICS --- DISTANCE BETWEEN DENSITY OPERATORS

We use the distance between density operators to study the dynamical evolution of the Jaynes-Cummings model with an additional Kerr medium, and to compare the result with the corresponding von Neumann entropy. We have shown that the distance between density operators can provide more detailed information about the dynamical behavior of the quantum system than von Neumann entropy.     

Stochastic vacuum of quantum chromodynamics as an environment for color particles

The behavior of quarks is described within approaches used in quantum mechanics and related disciplines (quantum optics and quantum theory of information). The stochastic vacuum of quantum chromodynamics is treated as an environment (closed pool) for color particles (quarks). Their interaction results in a loss of information on the quark color state and consequently in the impossibility of observing it (the confinement of quarks). The processes are described using quantities of the quantum theory of information, such as von Neumann entropy, fidelity, and purity.     

Von Neumann entropy in a Rashba-Dresselhaus nanodot; dynamical electronic spin-orbit entanglement

The main purpose of the present article is to report the characteristics of von Neumann entropy, thereby, the electronic hybrid entanglement, in the heterojunction of two semiconductors, with due attention to the Rashba and Dresselhaus spin-orbit interactions. To this end, we cast the von Neumann entropy in terms of spin polarization and compute its time evolution; with a vast span of applications. It is assumed that gate potentials are applied to the heterojunction, providing a two dimensional parabolic confining potential (forming an isotropic nanodot at the junction), as well as means of controlling the spin-orbit couplings. The spin degeneracy is also removed, even at electronic zero momentum, by the presence of an external magnetic field which, in turn, leads to the appearance of Landau states. We then proceed by computing the time evolution of the corresponding von Neumann entropy from a separable (spin-polarized) initial state. The von Neumann entropy, as we show, indicates that electronic hybrid entanglement does occur between spin and two-dimensional Landau levels. Our results also show that von Neumann entropy, as well as the degree of spin-orbit entanglement, periodically collapses and revives. The characteristics of such behavior; period, amplitude, etc., are shown to be determined from the controllable external agents. Moreover, it is demonstrated that the phenomenon of collapse-revivals’ in the behavior of von Neumann entropy, equivalently, electronic hybrid entanglement, is accompanied by plateaus (of great importance in quantum computation schemes) whose durations are, again, controlled by the external elements. Along these lines, we also make a comparison between effects of the two spin-orbit couplings on the entanglement (von Neumann entropy) characteristics. The finer details of the electronic hybrid entanglement, which may be easily verified through spin polarization measurements, are also accreted and discussed. The novel results of the present article, with potent applications in the field of quantum information processing, provide a deeper understanding of the electronic von Neumann entropy and hybrid entanglement that occurs in two-dimensional nanodots.     

Quantum phase transition and entanglement in Li atom system

By use of the exact diagonalization method, the quantum phase transition and entanglement in a 6-Li atom system are studied. It is found that entanglement appears before the quantum phase transition and disappears after it in this exactly solvable quantum system. The present results show that the von Neumann entropy, as a measure of entanglement, may reveal the quantum phase transition in this model.     

量子条件振幅算子性质的研究

分析量子条件振幅算子的性质，该算子起一个类似于在经典信息理论中的条件概率的作用.论证表示一个量子双组元系统的条件算子的频谱在局域幺正变换下是不变的，并且表明它的不可分性.证明一个可分态的条件振幅算子不能有一个超过1的本征值.得出一个在von Neumann条件熵的非负性基础上的相关的可分性条件. 关键词： 条件概率 条件振幅算子 von Neumann条件熵 可分性条件     

Quantum games entropy

We propose the study of quantum games from the point of view of quantum information theory and statistical mechanics. Every game can be described by a density operator, the von Neumann entropy and the quantum replicator dynamics. There exists a strong relationship between game theories, information theories and statistical physics. The density operator and entropy are the bonds between these theories. The analysis we propose is based on the properties of entropy, the amount of information that a player can obtain about his opponent and a maximum or minimum entropy criterion. The natural trend of a physical system is to its maximum entropy state. The minimum entropy state is a characteristic of a manipulated system, i.e., externally controlled or imposed. There exist tacit rules inside a system that do not need to be specified or clarified and search the system equilibrium based on the collective welfare principle. The other rules are imposed over the system when one or many of its members violate this principle and maximize its individual welfare at the expense of the group.     

Intrinsic Entropy of Squeezed Quantum Fields and Nonequilibrium Quantum Dynamics of Cosmological Perturbations

Density contrasts in the universe are governed by scalar cosmological perturbations which, when expressed in terms of gauge-invariant variables, contain a classical component from scalar metric perturbations and a quantum component from inflaton field fluctuations. It has long been known that the effect of cosmological expansion on a quantum field amounts to squeezing. Thus, the entropy of cosmological perturbations can be studied by treating them in the framework of squeezed quantum systems. Entropy of a free quantum field is a seemingly simple yet subtle issue. In this paper, different from previous treatments, we tackle this issue with a fully developed nonequilibrium quantum field theory formalism for such systems. We compute the covariance matrix elements of the parametric quantum field and solve for the evolution of the density matrix elements and the Wigner functions, and, from them, derive the von Neumann entropy. We then show explicitly why the entropy for the squeezed yet closed system is zero, but is proportional to the particle number produced upon coarse-graining out the correlation between the particle pairs. We also construct the bridge between our quantum field-theoretic results and those using the probability distribution of classical stochastic fields by earlier authors, preserving some important quantum properties, such as entanglement and coherence, of the quantum field.     

On the mixing-enhancing dynamical maps and mixing-enhancing evolution of a quantum system

The mixing-enhancing (in the sense of Uhlmann) dynamical maps and dynamical evolution is studied. We give a necessary and sufficient condition for a dynamical map (and dynamical evolution) of a quantum system to be mixing-enhancing. In the case of a finite- dimensional Hilbert space this condition is equivalent to the condition that the dynamical map (dynamical evolution) preserve the most mixed state and the von Neumann entropy be non- decreasing. It is proved that, in contrast with the finite-dimensional case, increasing of the von Neumann entropy under a dynamical map (for any initial state) does not imply that the dynamical map is mixing-enhancing. We also give a necessary and sufficient condition for an infinitesimal generator of a norm-continuous dynamical semigroup to be mixing-enhancing.     

Decoherence and dynamical entropy generation in quantum field theory

We formulate a novel approach to decoherence based on neglecting observationally inaccessible correlators. We apply our formalism to a renormalised interacting quantum field theoretical model. Using out-of-equilibrium field theory techniques we show that the Gaussian von Neumann entropy for a pure quantum state increases to the interacting thermal entropy. This quantifies decoherence and thus measures how classical our pure state has become. The decoherence rate is equal to the single particle decay rate in our model. We also compare our approach to existing approaches to decoherence in a simple quantum mechanical model. We show that the entropy following from the perturbative master equation suffers from physically unacceptable secular growth.