Probability Representation Of Quantum Mechanics Research Articles

Even and Odd Cat States of Two and Three Qubits in the Probability Representation of Quantum Mechanics.

We derive the probability representation of even and odd cat states of two and three qubits. These states are even and odd superpositions of spin-1/2 eigenstates corresponding to two opposite directions along the z axis. The probability representation of even and odd cat states of an oscillating spin-1/2 particle is also discussed. The exact formulas for entangled probability distributions describing density matrices of all these states are obtained.

Probability Distributions Describing Qubit-State Superpositions.

We discuss qubit-state superpositions in the probability representation of quantum mechanics. We study probability distributions describing separable qubit states. We consider entangled states on the example of a system of two qubits (Bell states) using the corresponding superpositions of the wave functions associated with these states. We establish the connection with the properties and structure of entangled probability distributions.

Dynamics of System States in the Probability Representation of Quantum Mechanics.

A short description of the notion of states of quantum systems in terms of conventional probability distribution function is presented. The notion and the structure of entangled probability distributions are clarified. The evolution of even and odd Schrödinger cat states of the inverted oscillator is obtained in the center-of-mass tomographic probability description of the two-mode oscillator. Evolution equations describing the time dependence of probability distributions identified with quantum system states are discussed. The connection with the Schrödinger equation and the von Neumann equation is clarified.

Quantum Oscillator at Temperature T and the Evolution of a Charged-Particle State in the Electric Field in the Probability Representation of Quantum Mechanics

A short review constructing the probability representation of quantum mechanics is given, and examples of the probability distributions describing the states of quantum oscillator at temperature T and the evolution of quantum states of a charged particle moving in the electric field of an electrical capacitor are considered. Explicit forms of time-dependent integrals of motion, linear in the position and momentum, are used to obtain varying probability distributions describing the evolving states of the charged particle. Entropies corresponding to the probability distributions of initial coherent states of the charged particle are discussed. The relation of the Feynman path integral to the probability representation of quantum mechanics is established.

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

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

Entangled Qubit States and Linear Entropy in the Probability Representation of Quantum Mechanics.

The superposition states of two qubits including entangled Bell states are considered in the probability representation of quantum mechanics. The superposition principle formulated in terms of the nonlinear addition rule of the state density matrices is formulated as a nonlinear addition rule of the probability distributions describing the qubit states. The generalization of the entanglement properties to the case of superposition of two-mode oscillator states is discussed using the probability representation of quantum states.

Symplectic tomographic probability distribution of crystallized Schrödinger cat states

Within the framework of the probability representation of quantum mechanics, we study a superposition of generic Gaussian states associated to symmetries of a regular polygon of n sides; in other words, the cyclic groups (containing the rotational symmetries) and dihedral groups (containing the rotational and inversion symmetries). We obtain the Wigner functions and tomographic probability distributions (symplectic and optical tomograms) determining the density matrices of the states explicitly as the sums of Gaussian terms. The obtained Wigner functions demonstrate nonclassical behavior, i.e., contain negative values, while the tomograms show a series of maxima and minima different for each state, where the number of the critical points reflects the order of the group defining the states. We discuss general properties of such a generalization of normal probability distributions.

Quantum process in probability representation of quantum mechanics

In this work, the operator-sum representation of a quantum process is extended to the probability representation of quantum mechanics. It is shown that each process admitting the operator-sum representation is assigned a kernel, convolving of which with the initial tomogram set characterizing the system state gives the tomographic state of the transformed system. This kernel, in turn, is broken into the kernels of partial operations, each of them incorporating the symbol of the evolution operator related to the joint evolution of the system and an ancillary environment. Such a kernel decomposition for the projection to a certain basis state and a Gaussian-type projection is demonstrated as well as qubit flipping and amplitude damping processes.

Even and Odd Schrödinger Cat States in the Probability Representation of Quantum Mechanics

Even and Odd Schrödinger Cat States in the Probability Representation of Quantum Mechanics

Coherent States of a Free Particle with Varying Mass in the Probability Representation of Quantum Mechanics

Coherent States of a Free Particle with Varying Mass in the Probability Representation of Quantum Mechanics

Continuous Measurements in Probability Representation of Quantum Mechanics

The continuous quantum measurement within the probability representation of quantum mechanics is discussed. The partial classical propagator of the symplectic tomogram associated to a particular measurement outcome is introduced, for which the representation of a continuous measurement through the restricted path integral is applied. The classical propagator for the system undergoing a non-selective measurement is derived by summing these partial propagators over the entire outcome set. The elaborated approach is illustrated by considering non-selective position measurement of a quantum oscillator and a particle.

Properties of Quantizer and Dequantizer Operators for Qudit States and Parametric Down-Conversion

We review the method of quantizers and dequantizers to construct an invertible map of the density operators onto functions including probability distributions and discuss in detail examples of qubit and qutrit states. The biphoton states existing in the process of parametric down-conversion are studied in the probability representation of quantum mechanics.

Correction to: Quantized-Energy Equation for N-Level Atom in the Probability Representation of Quantum Mechanics

In section Acknowledgments, the Russian Science Foundation Grant No. 1871-10091 should read No. 1971-10091.

Quantized-Energy Equation for N-Level Atom in the Probability Representation of Quantum Mechanics

For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr¨odinger equation for qudit (spin-j system, N-level atom) with the state vector |ψ〉 in a new form of the linear eigenvalue equation for the matrix ℋ = (H ⊗ 1N) and the probability eigenvector |p〉 identified with quantum states in the probability representation of quantum mechanics. We discuss the possibility to experimentally detect the difference between the system states described by the solutions, corresponding to the Schrodinger equation with Hermitian and non-Hermitian Hamiltonians, by measuring the probabilities of artificial spin-1/2 projections m = ± 1/2, sets of which are identified with qudit states. We show that different symmetries of systems, including 𝒫𝒯 -symmetry and broken 𝒫𝒯 -symmetry, are determined by a set of N complex eigenvalues of the Hamiltonian matrix H.

PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

PT-symmetric qubit-system states are considered in the probability representation of quantum mechanics. The new energy eigenvalue equation for probability distributions identified with qubit and qutrit states is presented in an explicit form. A possibility to test PT-symmetry and its violation by measuring the probabilities of spin projections for qubits in three perpendicular directions is discussed.

Schrödinger Equation for Energy Levels as a Linear Equation for Probability Distributions Identified with Quantum States

In this paper, we show that the energy spectrum of a qudit system (spin-j, N-level atom) with Hamiltonian N×N matrix H is determined by the equation for eigenvalues of the N2×N2 matrix ℋ and its eigenvectors | p⟩ with components, which are probability distributions identified with quantum states in the probability representation of quantum mechanics. We derive this equation based on the conventional Schrodinger equation for the state vector | ψ⟩. We discuss the example of qubit in detail and study new entropic characteristics of the energy levels.

Probability Representation of Quantum Mechanics and the Quantizer—Dequantizer Formalism

A review of the approach where the states of quantum systems are identified with fair probability distributions is presented. The quantizer–dequantizer operators used to construct the invertible map of the density operators onto the probability distributions are applied to obtain the kinetic equations for probability distributions identified with the quantum system states. For qubit states, the von Neumann evolution equation for the density operator is explicitly given in the form of kinetic equation for the probability distribution. Simplest tomographic probability distributions describing the states of multimode quantum oscillator are constructed.

SU(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics

In view of the probabilistic quantizer–dequantizer operators introduced, the qubit states (spin-1/2 particle states, two-level atom states) realizing the irreducible representation of the S U ( 2 ) symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1/2 particle projections m = ± 1 / 2 onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schrödinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer–dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg–Weyl symmetry properties.

Observables, interference phenomenon and Born’s rule in the probability representation of quantum mechanics

The description of quantum states by probability distributions of classical-like random variables associated with observables is presented. An invertible map of the wave functions and density matrices onto the probability distributions is constructed. The relation of the probability distributions to quasidistributions like the Wigner function is discussed. The interference phenomenon and superposition principle of pure quantum states are given in the form of nonlinear addition of the probabilities identified with the quantum states. The probability given by Born’s rule is expressed as a function of the probabilities describing the system states. The suggested probability representation of quantum mechanics is presented using examples of harmonic oscillators and qubits.

Probability Representation of Quantum Mechanics Where System States Are Identified with Probability Distributions

The probability representation of quantum mechanics where the system states are identified with fair probability distributions is reviewed for systems with continuous variables (the example of the oscillator) and discrete variables (the example of the qubit). The relation for the evolution of the probability distributions which determine quantum states with the Feynman path integral is found. The time-dependent phase of the wave function is related to the time-dependent probability distribution which determines the density matrix. The formal classical-like random variables associated with quantum observables for qubit systems are considered, and the connection of the statistics of the quantum observables with the classical statistics of the random variables is discussed.