Unitary and Nonunitary Evolution of Qubit States in Probability Representation of Quantum Mechanics

2019 ◽  
Vol 58 (6) ◽  
pp. 2054-2067 ◽  
Author(s):  
A. S. Avanesov ◽  
V. I. Manko
Keyword(s):  
Quantum Mechanics ◽  
Probability Representation ◽  
Qubit States

scholarly journals Superposition Principle and Born’s Rule in the Probability Representation of Quantum States

2019 ◽  
Vol 1 (2) ◽  
pp. 130-150 ◽  
Author(s):  
Igor Ya. Doskoch ◽  
Margarita A. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Probability Distribution ◽  
Particle System ◽  
Superposition Principle ◽  
Quantum States ◽  
Particle State ◽  
Probability Representation ◽  
Tomographic Probability ◽  
Born’S Rule ◽  
System States

The basic notion of physical system states is different in classical statistical mechanics and in quantum mechanics. In classical mechanics, the particle system state is determined by its position and momentum; in the case of fluctuations, due to the motion in environment, it is determined by the probability density in the particle phase space. In quantum mechanics, the particle state is determined either by the wave function (state vector in the Hilbert space) or by the density operator. Recently, the tomographic-probability representation of quantum states was proposed, where the quantum system states were identified with fair probability distributions (tomograms). In view of the probability-distribution formalism of quantum mechanics, we formulate the superposition principle of wave functions as interference of qubit states expressed in terms of the nonlinear addition rule for the probabilities identified with the states. Additionally, we formulate the probability given by Born’s rule in terms of symplectic tomographic probability distribution determining the photon states.

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 Probability representation of quantum mechanics and star product quantization

2019 ◽  
Vol 1348 ◽  
pp. 012101 ◽  
Author(s):  
V N Chernega ◽  
S N Belolipetskiy ◽  
O V Man’ko ◽  
V I Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Star Product ◽  
Probability Representation ◽  
Product Quantization

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.

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.

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