Probability Description and Entropy of Classical and Quantum Systems

Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered.     

Entangled states in the two-mode coherent fields interacting with a two-level atom

We investigate the entanglement properties of the two-mode coherent fields interacting with a two-level atom via the two-photon transition. We discuss the quantum entanglement between the two-mode coherent fields and the two-level atom by using the quantum reduced entropy and that between the two-mode coherent fields by using the quantum relative entropy. We also examine the influences of the initial states of the atom and the two-mode coherent fields on the quantum entanglement of the system. Our results show that three types of entangled states can be prepared via the two-mode coherent fields interacting with a two-level atom and choosing appropriately the initial-state parameters of the system.     

Preparation and control of entangled states in the two-mode coherent fields interacting with a moving atom via two-photon process

We investigate the preparation and the control of entangled states in a system with the two-mode coherent fields interacting with a moving two-level atom via the two-photon transition. We discuss entanglement properties between the two-mode coherent fields and a moving two-level atom by using the quantum reduced entropy, and those between the two-mode coherent fields by using the quantum relative entropy. In addition, we examine the influences of the atomic motion and field-mode structure parameter $p$ on the quantum entanglement of the system. Our results show that the period and the duration of the prepared maximal atom-field entangled states and the frequency of maximal two-mode field entangled states can be controlled, and that a sustained entangled state of the two-mode field, which is independent of atomic motion and the evolution time, can be obtained, by choosing appropriately the parameters of atomic motion, field-mode structure, initial state and interaction time of the system.     

Tomographic entropy and cosmology

The probability representation of quantum mechanics including propagators and tomograms of quantum states of the universe and its application to quantum gravity and cosmology are reviewed. The minisuperspaces modeled by oscillator, free pointlike particle and repulsive oscillator are considered. The notion of tomographic entropy and its properties are used to find some inequalities for the tomographic probability determining the quantum state of the universe. The sense of the inequality as a lower bound for the entropy is clarified.     

Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians

The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned.     

Tomographic characteristics of spin states

Spin states are studied in the tomographic-probability representation. The standard probability distribution of spin projection onto a direction in space is used instead of the spinor or the density matrix to identify the quantum state. The Shannon entropy and information are associated with the spin tomographic probability. A short review of the probability-theory notions is presented. Analysis of tomographic entropy and tomographic information for the Werner state is considered. The probability representation is used to describe a spin-3/2 particle and two qubits. The connection of tomographic entropy with the von Neumann entropy is discussed.     

New uncertainty relations for tomographic entropy: application to squeezed states and solitons

Using the tomographic probability distribution (symplectic tomogram) describing the quantum state (instead of the wave function or density matrix) and properties of recently introduced tomographic entropy associated with the probability distribution, the new uncertainty relation for the tomographic entropy is obtained. Examples of the entropic uncertainty relation for squeezed states and solitons of the Bose-Einstein condensate are considered.     

Entropy, Superposition and Dynamical Maps

Aspects of quantum entropy and relative quantum entropy are discussed in the Hilbert model. It is shown that finite values of the relative entropy of states implies a superposition relation between the states. The property is studied in case of tensor product of states and for state reductions. A “Schmidt-like” state, derived from the reduced states, is considered. It is shown that its entropy, relative to the product of the reduced states, is not smaller than the entropy of the reduced states. The main existing results concerning the changement of superposition and entropy under dynamical map are recalled in a uniform way. A class of possible dynamical maps, not necessarily linear, is proposed that do not decrease the entropy.     

A quantum steganography protocol based on <Emphasis Type="Italic">W</Emphasis> class entangled states

The condition of ɛ-security based on the quantum relative entropy is presented and a protocol that uses decoherence-free states is considered for the quantum steganography system.     

Entanglement in the Bimodal Jaynes–Cummings Model with the Two-Mode Squeezed Vacuum State

In this paper, we study the interaction between the two-level atom and a bimodal cavity field, namely, two-mode Jaynes–Cummings model when the atom and the modes are initially in the atomic superposition state and two-mode squeezed vacuum state, respectively. For this system we investigate the atomic inversion, linear entropy and atomic Wehrl entropy. We show that there is a connection between all these quantities. Also we prove that the atomic Wehrl entropy exhibits behaviors similar to those of the linear entropy and the von Neumann entropy. Moreover, we show that the bipartite exhibits periodical disentanglement and derive the explicit forms of the states of the atom and the modes at these values of the interaction times.     

Entanglement properties in a system of a pairwise entangled state

Based on the quantum information theory, this paper has investigated the entanglement properties of a system which is composed of the two entangled two-level atoms interacting with the two-mode entangled coherent fields. The influences of the strength of light field and the two parameters of entanglement between the two-mode fields on the field entropy and on the negative eigenvalues of partial transposition of density matrix are discussed by using numerical calculations. The result shows that the entanglement properties in a system of a pairwise entangled states can be controlled by appropriately choosing the two parameters of entanglement between the two-mode entangled coherent fields and the strength of two light fields respectively.     

多模简并多光子Tavis-Cummings模型中的量子纠缠特性

利用冯·纽曼约化熵理论研究了多模相干态光场与两等同二能级原子简并多光子相互作用系统量子纠缠演化特性,得到了多模光场量子纠缠的解析表达式,并给出了双模光场与两原子相互作用时量子纠缠的数值计算结果.结果表明:量子纠缠随着光子简并度的增大而增强;随着初始平均光子数的增加,量子纠缠演化的周期性变得越来越明显;当场与原子远离共振时,量子纠缠随着频率失谐量的增大而减弱;当失谐量足够大时,场与原子几乎总是处于纠缠状态.这些结论对于纠缠态或纯态的制备及获取光学系统中的量子信息研究中有一定参考价值.     

双模压缩真空态与运动原子相互作用系统的场熵演化特性

利用全量子理论,研究了双模压缩真空场与运动原子相互作用系统的场熵演化特性,讨论了不同初始状态态下原子的运动速度、模场结构对场熵的影响。结果表明,场熵通过时间因子sin2(εpgt/2)度与场模结构参数,原子运动和场模结构引起了附加的“熵振荡”。     

双模场与原子相互作用中的量子纠缠和内禀退相干

通过求解系统的Milburn方程，研究了两能级原子与双模SU(1,1)相干态光场发生相互作用系统中，原子与场的纠缠及双模SU(1,1)相干态场的模间纠缠随时间的演化问题，讨论了内禀退相干、双模光子数差等对纠缠度的影响.结果表明，存在内禀退相干时，随着时间的演化，场-原子纠缠逐渐减小到一个确定值，而模间纠缠逐渐增大到一个确定值，两者演化的最终值只取决于双模光子数差和平均光子数，而与内禀退相干因子无关. 关键词： Milburn理论 SU(1 1)相干态 量子约化熵 量子相对熵     

Polarization States of Light and Their Quantum Tomography

The notion and main features of polarization states of light are discussed within the framework of classical and quantum optics. This notion is shown to be correctly defined for arbitrary light beams only within quantum optics by using the P-quasispin formalism developed earlier. Polarization states of quantum light are shown to be fully described by a polarization density operator (PDO) obtained via reducing the total field density operator. Theoretical foundations are given for quantum tomography of polarization states of light fields considered as a way of measuring PDO. Herewith, the main attention is paid to a method where proper polarization tomographic observables (PDO “measurers”) are used. The method is shown to be adequately formulated by means of quasi-spectral tomographic expansions of PDO in special operator bases (given by finite sums of partially orthogonal projectors), which determine probability distributions of tomographic observables as expansion coefficients. Matrix versions of such “tomographic” PDO representations are obtained. In particular, projections of these expansions on quasiclassical operator bases, determining polarization quasiprobability functions, are given. An example of experimental implementation of polarization tomography of unpolarized light (biphoton radiation with hidden polarization) is analyzed.     

Entanglement and localization of a two-mode Bose-Einstein condensate

A simple second quantization model is used to describe a two-mode Bose-Einstein condensate (BEC), which can be written in terms of the generators of a SU(2) algebra with three parameters. We study the behavior of the entanglement entropy and localization of the system in the parameter space of the model. The phase transitions in the parameter space are determined by means of the coherent state formalism and the catastrophe theory, which besides let us get the best variational state that reproduces the ground state energy. This semiclassical method let us organize the energy spectrum in regions where there are crossings and anticrossings. The ground state of the two-mode BEC, depending on the values of the interaction strengths, is dominated by a single Dicke state, a spin collective coherent state, or a superposition of two spin collective coherent states. The entanglement entropy is determined for two recently proposed partitions of the two-mode BEC that are called separation by boxes and separation by modes of the atoms. The entanglement entropy in the boxes partition is strongly correlated to the properties of localization in phase space of the model, which is given by the evaluation of the second moment of the Husimi function. To compare the fitness of the trial wavefunction its overlap with the exact quantum solution is evaluated. The entanglement entropy for both partitions, the overlap and localization properties of the system get singular values along the separatrix of the two-mode BEC, which indicates the phase transitions which remain in the thermodynamical limit, in the parameter space.     

好腔中的Bell态原子与纠缠相干态光场相互作用系统的保真度

考虑原子间偶极相互作用,求出好腔中的Bell态原子与纠缠相干态光场相互作用系统的保真度。结果表明,对于理想腔,若原子初始时刻处于相干保持态,系统保真度始终等于1;若原子初始时刻处于其余Bell态之一,腔场初态的平均光子数很小,系统保真度在0～1之间作周期性振荡,随着腔场初态的平均光子数的增加,系统保真度的振荡频率增大,振幅减小。对于好腔,若原子初始时刻处于相干保持态,系统保真度呈指数单调衰减;若原子初始时刻处于其余Bell态之一,系统保真度呈指数振荡衰减,且随着腔场初态的平均光子数的增加,系统保真度的振荡频率增大,振幅减小。     

好腔中的Bell态原子与纠缠相干态光场相互作用系统的保真度

考虑原子间偶极相互作用,求出好腔中的Bell态原子与纠缠相干态光场相互作用系统的保真度.结果表明,对于理想腔,若原子初始时刻处于相干保持态,系统保真度始终等于1;若原子初始时刻处于其余Bell态之一.腔场初态的平均光子数很小,系统保真度在0～1之间作周期性振荡,随着腔场初态的平均光子数的增加,系统保真度的振荡频率增大,振幅减小.对于好腔,若原子初始时刻处于相干保持态,系统保真度呈指数单调衰减;若原子初始时刻处于其余Bell态之一,系统保真度呈指数振荡衰减,且随着腔场初态的平均光子数的增加,系统保真度的振荡频率增大,振幅减小.     

Probabilistic Teleportation of an Arbitrary Two-mode <Emphasis Type="Italic">N</Emphasis>-photon Entangled State in Cavity QED

We propose a scheme for teleportation of an arbitrary two-mode N-photon entangled states in cavity QED. The scheme is based on the resonant interaction between Λ-type atoms and two-mode cavity fields. In contrast to all the theoretical schemes proposed previously in cavity QED for teleportation of two-mode cavity field states, in the present scheme, the established entanglement for the quantum channel is the type of the multi-dimensional entanglement between the symmetric multi-atom Dicke states and two-mode N-photon states. Therefore, the scheme extends the scope of the theoretical study of the teleportation.     

Information Entropy Squeezing for a Atom in Mode-Mode Competition System

The entropy squeezing properties for a two-level atom interacting with a two-mode field via two different competing transitions are investigated from a quantum information point of view. The influences of the initial state of the system and the relative coupling strength between the atom and the field on the atomic information entropy squeezing are discussed. Our results show that the squeezed direction and the frequency of the information entropy squeezing can be controlled by choosing the phase of the atom dipole and the relative competing strength of atom-field, respectively. We find that, under the same condition, no atomic variance squeezing is predicted from the HUR while optimal entropy squeezing is obtained from the EUR, so the quantum information entropy is a remarkable precision measure for the atomic squeezing in the considered system.