On entropy and information of quantum states

Shannon entropy and information are applied to study the properties of quantum states of a system in the probability representation of quantum mechanics. Examples of spin states and mixed Gaussian states of the two-mode system are considered. The relationship between the new entropy and the von Neumann entropy is reviewed. Two tomographic maps are considered within the framework of the star-product quantization. The explicit expression of tomographic entropy associated with photon-number tomogram of the two-mode state of photons is obtained in terms of Hermite polynomials of four variables. Based on a contribution to the International Conference “New Trends in Quantum Mechanics. Fundamental Aspects and Applications” (Palermo, Italy, November 2005).     

Probability-Representation Entropy for Spin-State Tomogram

The probability-representation entropy (tomographic entropy) of an arbitrary quantum state is introduced. Using the properties of the spin tomogram as the standard probability-distribution function, the tomographic entropy notion is discussed. The relation of tomographic entropy to Shannon entropy and von Neumann entropy is elucidated.     

Standard quantum mechanics featuring probabilities instead of wave functions

A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.     

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.     

MuSR method and tomographic-probability representation of spin states

The muon spin rotation/relaxation/resonance (MuSR) technique for studying matter structures is considered by means of a recently introduced probability representation of quantum spin states. A relation between experimental MuSR histograms and muon spin tomograms is established. The time evolution of muonium, anomalous muonium, and a muonium-like system is studied in the tomographic representation. The entanglement phenomenon of a bipartite muon–electron system is investigated, in view of the tomographic analogs of the Bell number and the positive partial transpose (PPT) criterion. Reconstruction of the muon–electron spin state as well as the total spin tomography of the composed system is discussed.     

Entropy and information characteristics of qubit states

Shannon entropy, Rényi entropy, and Tsallis entropy are discussed for the tomographic probability distributions of qubit states. Relative entropy and its properties are considered for the tomographic probability distribution describing the states of multi-spin systems. New inequalities for Hermite polynomials are obtained.     

Evolution Equation for a Joint Tomographic Probability Distribution of Spin-1 Particles

The nine-component positive vector optical tomographic probability portrait of quantum state of spin-1 particles containing full spatial and spin information about the state without redundancy is constructed. Also the suggested approach is expanded to symplectic tomography representation and to representations with quasidistributions like Wigner function, Husimi Q?function, and Glauber-Sudarshan P?function. The evolution equations for constructed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are found. The evolution equations are also obtained in special case of the quantum system of charged spin-1 particle in arbitrary electro-magnetic field, which are analogs of non-relativistic Proca equation in appropriate representations. The generalization of proposed approach to the cases of arbitrary spin is discussed. The possibility of formulation of quantum mechanics of the systems with spins in terms of joint probability distributions without the use of wave functions or density matrices is explicitly demonstrated.