Entropy

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

Matyas I. Mechler, Margarita A. Man’ko, Vladimir I. Man’ko, Петер Адам

Published May 31, 2024• 8 citations

Abstract

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.

DOI: 10.3390/e26060485

Related papers

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

Vladimir N. Chernega, Olga V. Man’ko, Vladimir I. Man’ko

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.

EntropyApr 9, 202212 citations

Probability Distributions Describing Qubit-State Superpositions

Margarita A. Man’ko, Vladimir I. Man’ko

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.

EntropySep 22, 20236 citations

Dynamics of System States in the Probability Representation of Quantum Mechanics

Vladimir N. Chernega, Olga V. Man’ko

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.

EntropyMay 11, 202312 citations

Group SU(2) Irreducible Representations and Probability Distributions Describing the Density Matrices of Qubit States

Margarita A. Man’ko, Vladimir I. Man’ko

The general problem of describing quantum states not only by wave functions and density operators but also by probability distribution functions is discussed in this paper. The qubit-state density-matrix elements expressed in terms of probability distributions are connected with the irreducible representation of group SU(2). The transform of the probability distributions corresponding to the unitary transform of the qubit state of the spin-1/2 system is obtained and expressed in terms of matrix elements of the group SU(2). The possibility to extend the introduced formalism to other quantum states is suggested. For qubit systems, the notion of density operators algebra and the relation with the probability representation of quantum states are discussed. The Schrödinger equation for qubit states is obtained as an equation for probabilities.

PhysicsJul 3, 20260 citations

Evolving Probability Representations of Entangled Cat States in the Potentials of Harmonic and Inverted Oscillators

Matyas I. Mechler, Margarita A. Man’ko, Vladimir I. Man’ko +1 more

We determine the evolving probability representation of entangled cat states in the potential of either the harmonic oscillator or the inverted oscillator, assuming that the states are initially prepared in the potential of the harmonic oscillator. Such states have several applications in quantum information processing. The inverted quantum harmonic oscillator, where the potential energy corresponds to imaginary frequencies of the oscillator, can be applied in relation to cosmological problems. We also determine the evolving probability representation of cat states of an oscillating spin-1/2 particle of the inverted oscillator, in which the time evolution of the spin state is described by an arbitrary unitary operator. The properties of the determined entangled probability distributions are discussed.

Quantum ReportsMay 2, 20251 citation

Probability Representation of Quantum States

Olga V. Man’ko, Vladimir I. Man’ko

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.

EntropyApr 29, 202143 citations