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.     

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.     

Tomographic representation of minisuperspace quantum cosmology and noether symmetries

The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select “observable” universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.     

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.     

Tomographic probability representation for quantum fermion fields

Tomographic probability representation is introduced for fermion fields. The states of the fermions are mapped onto the probability distribution of discrete random variables (spin projections). The operators acting on the fermion states are described by fermionic tomographic symbols. The product of the operators acting on the fermion states is mapped onto the star-product of the fermionic symbols. The kernel of the star-product is obtained. The antisymmetry of the fermion states is formulated as a specific symmetry property of the tomographic joint probability distribution associated with the states.     

Quantum correlations and tomographic representation

We review the probabilistic representation of quantum mechanics within which states are described by the probability distribution rather than by the wavefunction and density matrix. Uncertainty relations have been obtained in the form of integral inequalities containing measurable optical tomograms of quantum states. Formulas for the transition probabilities and purity parameter have been derived in terms of the tomographic probability distributions. Inequalities for Shannon and Rényi entropies associated with quantum tomograms have been obtained. A scheme of the star product of tomograms has been developed.     

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.     

The driven-oscillator evolution in the tomographic-probability representation

We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system??s phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic propagator for this equation is studied.     

Calculation of Quantum Probability in O（2,2） String Cosmology with a Dilaton Potential

The quantum properties of O（2,2） string cosmology with a dilaton potential are studied in this paper. The cosmological solutions are obtained on three-dlmensional space-time. Moreover, the quantum probability of transition between two duality universe is calculated through a Wheeler-De Witt approach.     

Calculation of Quantum Probability in O(2,2) String Cosmology with a Dilaton Potential

The quantum properties of O(2,2) string cosmology with a dilaton potential are studied in this paper. The cosmological solutions are obtained on three-dimensional space-time. Moreover, the quantum probability of transition between two duality universe is calculated through a Wheeler-De Witt approach.     

Squeezed and Correlated States of Parametric Oscillator and Free Particle in the Probability Representation of Quantum Mechanics

Journal of Russian Laser Research - The states of quantum oscillator with time-dependent frequency are described by the tomographic probability distributions. The integrals of motion, being linear...     

Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups IGL(n,R) and GL(n,R) respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties.     

Mixtures and pure states in relativistic Schrödinger theory

In relativistic Schrödinger theory, a physical system can be described by a wave function (? “pure state”) or by an intensity matrix (? “mixture”). Since the space-time evolution of the system is described by anon-Hermitian Hamiltonian, transmutations of mixtures into pure states (and vice versa) would be formally possible. Nevertheless, the transition of a mixture into a pure state is dynamically forbidden, whereas the pure states are unstable and decay into mixtures. This effect is demonstrated by considering the Klein-Gordon-Higgs equations over an expanding Robertson-Walker universe.     

General Bell-CHSH type and entropic inequalities based on quantum tomograms

Review of Bell-CHSH type and entropic inequalities in composite quantum correlated systems in the probability representation of states is presented. The upper bounds for some new Bell-CHSH type inequalities within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-CHSH type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states. Joint tomographic entropies of multiqubit systems are studied. Limitations on inequalities for tomographic entropies are obtained. A negative result of possible connection between the violation of entropic and Bell-CHSH type inequalities in multi-partite states is reported.     

Interaction of electromagnetic field with the OH radical: A non-perturbative approach

A non-perturbative approach for the study of the interaction of a hydroxyl (OH) radical with infra-red radiation is presented. The dressed states and vibrational transition probability of OH radical are defined by a quasi-energy approach (non-perturbative).     

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.     

Generalized Qubit Portrait of the Qutrit-State Density Matrix

We obtain new inequalities for tomographic probability distributions and density matrices of qutrit states by generalization of the qubit-portrait method. We propose an approach based on the quditportrait method of obtaining new entropic inequalities. Our approach can be applied to the case of arbitrary nonnegative hermitian matrices, including the density matrices of multipartite qudit states.     

A Possible Operational Motivation for the Orthocomplementation in Quantum Structures

In the foundations of quantum mechanics Gleason’s theorem dictates the uniqueness of the state transition probability via the inner product of the corresponding state vectors in Hilbert space, independent of which measurement context induces this transition. We argue that the state transition probability should not be regarded as a secondary concept which can be derived from the structure on the set of states and properties, but instead should be regarded as a primitive concept for which measurement context is crucial. Accordingly, we adopt an operational approach to quantum mechanics in which a physical entity is defined by the structure of its set of states, set of properties and the possible (measurement) contexts which can be applied to this entity. We put forward some elementary definitions to derive an operational theory from this State–COntext–Property (SCOP) formalism. We show that if the SCOP satisfies a Gleason-like condition, namely that the state transition probability is independent of which measurement context induces the change of state, then the lattice of properties is orthocomplemented, which is one of the ‘quantum axioms’ used in the Piron–Solèr representation theorem for quantum systems. In this sense we obtain a possible physical meaning for the orthocomplementation widely used in quantum structures.