Glauber Dynamics for the Quantum Ising Model in a Transverse Field on a Regular Tree

Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space.     

Probability representation and quantumness tests for qudits and two-mode light states

Using the tomographic-probability representation of spin states, the quantum behavior of qudits is examined. For a general j-qudit state, we propose an explicit formula for quantumness witness whose negative average value is incompatible with classical statistical model. The probability representations of quantum and classical (2j + 1)-level systems are compared within the framework of quantumness tests. In view of the Jordan–Schwinger map, the method is extended for checking the quantumness of two-mode light states.     

Logical foundation of quantum mechanics

The subject of this article is the reconstruction of quantum mechanics on the basis of a formal language of quantum mechanical propositions. During recent years, research in the foundations of the language of science has given rise to adialogic semantics that is adequate in the case of a formal language for quantum physics. The system ofsequential logic which is comprised by the language is more general than classical logic; it includes the classical system as a special case. Although the system of sequential logic can be founded without reference to the empirical content of quantum physical propositions, it establishes an essential part of the structure of the mathematical formalism used in quantum mechanics. It is the purpose of this paper to demonstrate the connection between the formal language of quantum physics and its representation by mathematical structures in a self-contained way.     

Femtosecond pump-probe fluorescence signals from classical trajectories: comparison with wave-packet calculations

A classical approach to simulate femtosecond pump-probe experiments is presented and compared to the quantum mechanical treatment. We restrict the study to gas-phase systems using the I₂ molecule as a numerical example. Thus, no relaxation processes are included. This allows for a direct comparison between purely quantum mechanical results and those obtained from classical trajectory calculations. The classical theory is derived from the phase-space representation of quantum mechanics. Various approximate quantum mechanical treatments are compared to their classical counterparts. Thereby it is demonstrated that the representation of the radial density as prepared in the pump-process is most crucial to obtain reliable signals within the classical approach. Received 28 March 2001     

Master Equation in Phase Space for a Spin in an Arbitrarily Directed Uniform External Field

The time evolution equation for the probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for the nonaxially symmetric problem of a quantum paramagnet subjected to a uniform magnetic field of arbitrary direction. This is accomplished by first rotating the coordinate system into one in which the polar axis is collinear with the field vector, then writing the reduced density matrix equation in the new coordinate system as an explicit inverse Wigner-Stratonovich transformation so that the phase space master equation may be derived just as in the axially symmetric case [Yu.P. Kalmykov et al., J. Stat. Phys. 131:969, 2008]. The properties of this equation, resembling the corresponding Fokker-Planck equation, are investigated. In particular, in the large spin limit, S→∞, the master equation becomes the classical Fokker-Planck equation describing the magnetization dynamics of a classical paramagnet in an arbitrarily directed uniform external field.     

The Phase Transition of the Quantum Ising Model is Sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called ‘random-parity’ representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.     

Quantum simulation of the t–J model

Computer simulation of a many-particle quantum system is bound to reach the inevitable limits of its ability as the system size increases. The primary reason for this is that the memory size used in a classical simulator grows polynomially whereas the Hilbert space of the quantum system does so exponentially. Replacing the classical simulator by a quantum simulator would be an effective method of surmounting this obstacle. The prevailing techniques for simulating quantum systems on a quantum computer have been developed for purposes of computing numerical algorithms designed to obtain approximate physical quantities of interest. The method suggested here requires no numerical algorithms; it is a direct isomorphic translation between a quantum simulator and the quantum system to be simulated. In the quantum simulator, physical parameters of the system, which are the fixed parameters of the simulated quantum system, are under the control of the experimenter. A method of simulating a model for high-temperature superconducting oxides, the t–J model, by optical control, as an example of such a quantum simulation, is presented.     

Quantum Image Edge Extraction Based on Improved Sobel Operator

Edge extraction is a basic task in image processing. This paper proposes a quantum image edge extraction algorithm based on improved sobel operator for the generalized quantum image representation (GQIR) to solve the real-time problem. The quantum image model of GQIR can store arbitrary quantum images with a size of H × W. Our scheme can calculate the gradients of image intensity of all the pixels simultaneously. Then, the concrete circuits of quantum image edge extraction algorithm are implemented by using a series of quantum operators which have been designed. Compared with existing quantum edge extraction algorithms, our scheme can achieve more accurate edge extraction, especially for diagonal edges. Finally, the complexity of the quantum circuits were been analyzed based on the basic quantum gates and give the simulation experiment results on classical computer.

     

Electromagnetic fields formed upon the interaction of ionizing radiation with matter

We propose a new approach to electromagnetic field calculations that involves the consecutive application of both quantum and classical electrodynamics methods. A current’s distribution is calculated via simulation modeling, using cross-section values obtained through quantum electrodynamics. The classical delayed potential of a current pulse of arbitrary form and duration, moving in a straight line at hyperlight velocity in the space-time representation, is calculated in an explicit form.     

A note on classical and quantum unimodular gravity

We discuss unimodular gravity at a classical level, and in terms of its extension into the UV through an appropriate path integral representation. Classically, unimodular gravity is locally a gauge fixed version of general relativity (GR), and as such it yields identical dynamics and physical predictions. We clarify this and explain why there is no sense in which it can “bring a new perspective” to the cosmological constant problem. The quantum equivalence between unimodular gravity and GR is more of a subtle question, but we present an argument that suggests one can always maintain the equivalence up to arbitrarily high momenta. As a corollary to this, we argue, whenever inequivalence is seen at the quantum level, that just means we have defined two different quantum theories that happen to share a classical limit. We also present a number of alternative formulations for a covariant unimodular action, some of which have not appeared, to our knowledge, in the literature before.     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

The Pauli Principle and the Restricted Primitive Model

The restricted primitive model is an electrically neutral, classical model consisting of hard spheres charged either +q or –q. We show that, by appropriately selecting the diameter of the hard spheres, the pressure when q=0 can be made equal to that for a fluid of Maxwell–Boltzmann point ions and an ideal Fermi gas of electrons. We compare the series expansion of these classical and quantum systems and find that, except for intermediate de Broglie density and moderate to strong electrical interaction strength, the restricted primitive model gives a reasonable representation of the pressure of the corresponding quantum system. Much of the current interest, however, has been focused on the above, excepted region.     

Two-state system coupled to a boson mode: Quantum dynamics and classical approximations

The relation between classical and quantum mechanical integrability is investigated for a boson mode coupled to a two-level system. Different semi-classical approximations of this system are considered which are obtained by (i) factorization of expectation values of the two-state variable and the boson, (ii) making a WKB-type approximation, (iii) replacing the boson by a classical field of constant amplitude and fixed frequency and (iv) putting the boson into a self-consistent coherent state. The results vary considerably and include cases of non-integrable and integrable classical dynamics. Quantum mechanically the system is found to satisfy a criterion of quantum mechanical integrability, which we formulate, but the separated Hamiltonian of the boson alone does not have a well-defined classical limit. Numerical results for the energy spectrum and expectation values are obtained, which show a high degree of regularity but also display overlapping avoided crossings usually associated with non-integrable Hamiltonians. The exact dynamics of the occupation probabilities of the two levels is also analysed numerically. The dependence of quantum mechanical recurrence effects (in quantum optics known as revivals) on coupling strength, frequency detuning and initial conditions is studied. The revivals are found to disappear in the case of strong coupling. The Fourier spectra of the dynamical expectation values are also calculated     

The noncommutative values of quantum observables

We discuss the notion of representing the values of physical quantities by the real numbers, and its limits to describe the nature to be understood in the relation to our appreciation that the quantum theory is a better theory of natural phenomena than its classical analog. Getting from the algebra of physical observables to their values for a fixed state is, at least for classical physics, really a homomorphic map from the algebra into the real number algebra. The limitation of the latter to represent the values of quantum observables with noncommutative algebraic relation is obvious. We introduce and discuss the idea of the noncommutative values of quantum observables and its feasibility, arguing that at least in terms of the representation of such a value as an infinite set of complex numbers, the idea makes reasonable sense theoretically as well as practically.     

Multiplication rule of similarity transformations in the coherent state representation and its mapping onto quantum optical generalized ABCD law

We find that classical similarity transformations in the coherent state representation projects onto the similarity transformation operators (STO), these operators constitute a loyal representation of symplectic group. Remarkably, the multiplication rule of the STOs naturally leads to the quantum optical generalized ABCD law, which is the quantum mechanical correspondence of the classical optical ABCD law. Throughout the whole derivation, the technique of integration within an ordered product (IWOP) of operators is employed.     

Quantum and classical correlations for a two-qubit X structure density matrix

We derive explicit expressions for quantum discord and classical correlation for an X structure density matrix. Based on the characteristics of the expressions, the quantum discord and the classical correlation are easily obtained and compared under different initial conditions using a novel analytical method. We explain the relationships among quantum discord, classical correlation, and entanglement, and further find that the quantum discord is not always larger than the entanglement measured by concurrence in a general two-qubit X state. The new method, which is different from previous approaches, has certain guiding significance for analysing quantum discord and classical correlation of a two-qubit X state, such as a mixed state.     

Interplay between classical and quantum signals: partial trace and measurement channels

We continue the study of similarities between quantum information theory and theory of classical Gaussian signals. The possibility of using quantum entropy for classical Gaussian signals was explored a long time ago. Recently we demonstrated that some basic quantum channels can be represented as linear transforms of classical Gaussian signals. Here we consider bipartite quantum systems and show that an important quantum channel given by the partial trace operation has a simple classical representation, namely, a coordinate projection of a classical “prequantum signal.” We also consider the classical signal realization of quantum channels corresponding to state transforms in the process of measurement. The latter induces a difficult interpretational problem — the output signal corresponding to one system depends on a measurement that has been done on the second system. This situation might be interpreted as a sign of quantum nonlocality, action at a distance. Although we do not exclude such a possibility, i.e., that, in complete accordance with Bell, the creation of a realistic prequantum model is impossible without action at a distance, we found another interpretation of this situation that is not related to quantum nonlocality.     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

Integral Representation for the Eigenfunctions of a Quantum Periodic Toda Chain

An integral representation for the eigenfunctions of a quantum periodic Toda chain is constructed for the N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for the N=2,3 and 4-particle chain. Our method of solving the problem combines the ideas of Gutzwiller and the R-matrix approach of Sklyanin with the classical results in the theory of Whittaker functions. In particular, we calculate Sklyanin's invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining a natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.