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

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

doi:10.1142/s0219749919410211

Cited by 11 publications

(3 citation statements)

References 39 publications

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“…We point out that discussed examples of studying parametric systems can be used to apply the results associated with the behavior of physical systems like photons in cavities with time-dependent locations of boundaries to dynamical Casimir effect (see [49]) and its analog in superconducting circuits [50,51]. One can discuss nonunitary evolution of systems, which have no subsystems, using hidden correlations [52], which are present in noncomposite systems. In this appendix, the expressions of the covariance matrix and the mean values of the Gaussian system given in Sec.…”

Section: Discussionmentioning

confidence: 99%

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

López-Saldívar

Man’ko

2020

Entropy

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In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian H ^ . We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schrödinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.

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Section: Discussionmentioning

confidence: 99%

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

López-Saldívar

Man’ko

2020

Entropy

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“…If the operators Â have the properties of density operators, i.e., ρ † = ρ, Tr ρ = 1, and all the eigenvalues of the operator Â are nonnegative, and the dequantizer operators Û X have the same properties, we arrive at the probability representation of operators Â, in view of Born's rule [26][27][28]. The vector X can contain parameters of random variables and condition parameters for the case of the probability representation of the density operator ρ.…”

Section: Hilbert Spaces and Quantizer-dequantizer Operator Formalismmentioning

confidence: 99%

Probability Distributions Describing Qubit-State Superpositions

Man’ko,

Man’ko

2023

Entropy

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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.

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“…The quantum system dynamics is closely related with the symmetry of systems. We consider systems with Hermitian Hamiltonian dynamics and unitary evolution [38]. It is worth noting that there exist the models with non-Hermitian dynamics [39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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.

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

Cited by 11 publications

References 39 publications

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

Probability Distributions Describing Qubit-State Superpositions

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

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