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

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 131
Author(s):  
Peter Adam ◽  
Vladimir A. Andreev ◽  
Margarita A. Man’ko ◽  
Vladimir I. Man’ko ◽  
Matyas Mechler
Keyword(s):  
Quantum Mechanics ◽  
Probability Distributions ◽  
Probability Representation ◽  
Density Operators ◽  
Parametric Down Conversion ◽  
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.

Interference of Quantum States and Superposition Principle in Probability Representation of Quantum Mechanics

2019 ◽  
Vol 26 (03) ◽  
pp. 1950016 ◽  
Author(s):  
Margarita A. Man’ko ◽  
Vladimir I. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Probability Distributions ◽  
Superposition Principle ◽  
State Density ◽  
Quantum States ◽  
Density Matrices ◽  
Probability Representation ◽  
Density Operators ◽  
Pure States

The superposition of pure quantum states explicitly expressed in terms of a nonlinear addition rule of state density operators is reviewed. The probability representation of density matrices of qudit states is used to formulate the interference of the states as a combination of the probability distributions describing pure states. The formalism of quantizer–dequantizer operators is developed. Examples of spin-1/2 states and f-oscillator systems are considered.

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

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1099 ◽  
Author(s):  
Peter Adam ◽  
Vladimir A. Andreev ◽  
Margarita A. Man’ko ◽  
Vladimir I. Man’ko ◽  
Matyas Mechler
Keyword(s):  
Probability Distributions ◽  
Star Product ◽  
Kinetic Equations ◽  
Random Variables ◽  
Continuous Variables ◽  
Probability Representation ◽  
Density Operators ◽  
Symmetry Properties ◽  
Weyl Symmetry ◽  
Qubit States

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.

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

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1702 ◽  
Author(s):  
Vladimir N. Chernega ◽  
Margarita A. Man’ko ◽  
Vladimir I. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Explicit Form ◽  
Probability Distributions ◽  
Energy Eigenvalue ◽  
Pt Symmetry ◽  
Eigenvalue Equation ◽  
Probability Representation ◽  
Qubit System ◽  
New Energy ◽  
System States

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.

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

2020 ◽  
Vol 18 (01) ◽  
pp. 1941021 ◽  
Author(s):  
Margarita A. Man’ko ◽  
Vladimir I. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Probability Distributions ◽  
Superposition Principle ◽  
Quantum States ◽  
Wave Functions ◽  
Harmonic Oscillators ◽  
Probability Representation ◽  
Interference Phenomenon ◽  
Born’S Rule ◽  
System States

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.

scholarly journals God plays coins or superposition principle for quantum states in probability representation of quantum mechanics

2019 ◽  
Vol 220 ◽  
pp. 01008
Author(s):  
Olga Man’ko ◽  
Vladimir Chernega
Keyword(s):  
Quantum Mechanics ◽  
Probability Distributions ◽  
Superposition Principle ◽  
Quantum States ◽  
Probability Representation

The superposition principle of quantum states is expressed as the addition rule of probability distributions which are identified with these quantum states in new representation of quantum mechanics. Example of two spin-1/2 states is considered explicitly.

scholarly journals Probability Representation of Quantum States

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 549
Author(s):  
Olga V. Man’ko ◽  
Vladimir I. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Open Systems ◽  
Probability Distributions ◽  
Free Particle ◽  
Classical Mechanics ◽  
Quantum Tomography ◽  
Quantum States ◽  
Space Representation ◽  
Continuous Variables ◽  
Probability Representation

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.

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

2020 ◽  
Vol 2 (1) ◽  
pp. 64-79 ◽  
Author(s):  
Vladimir Chernega ◽  
Olga Man'ko ◽  
Vladimir Man'ko
Keyword(s):  
Quantum Mechanics ◽  
Probability Distributions ◽  
Random Variables ◽  
Time Dependent ◽  
Continuous Variables ◽  
Probability Representation ◽  
Classical Statistics ◽  
Quantum Observables ◽  
System States ◽  
Dependent Probability

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.

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