Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems

We consider even and odd coherent states (Schrödinger cat states) in the probability representation of quantum mechanics. The probability representation of the cat states is explicitly given using the formalism of quantizer and dequantizer operators, that provides the existence of an invertible map of operators acting in a Hilbert space onto functions called symbols of the operators. We employ a special set of quantizers and dequantizers to construct Wigner functions of the Schrödinger cat states and obtain the relation of the Wigner functions and the probability distributions by means of the Radon integral transform. The notion of entangled classical probability distributions is introduced in probability theory.

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“…In view of the relation of the wave function of cat states ( 9) to their tomograms (17) given in terms of Gaussian integrals of the form…”

Section: Johann Karl August Radon (1887 -1956) President Of the Austr...mentioning

confidence: 99%

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

Ádám

Man’ko

2022

J Russ Laser Res

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“…In the case of one-qubit state, the relationship between the probabilities and mean values is very simple, but for N-qubit states, it is more complicated; for details, see [52][53][54][55].…”

Section: O(3) Transforms Of Probabilities and Spin-projection Mean Va...mentioning

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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“…The star product formalism of symbols for N -dimensional systems is described in detail in [51]. For qubit states, the set of quantizers and dequantizers for the spin tomogram was considered, e.g., in [52] and a detailed analysis of the spin Wigner functions and probability distributions is given in [53][54][55]. Using this formalism the relations between tomograms and Wigner functions for one and two qubits have been determined [51,56,57].…”

Section: Introductionmentioning

confidence: 99%

Star Product Formalism for Probability and Mean Value Representations of Qudits

Ádám¹,

Andreev²,

Man’ko³

et al. 2019

Preprint

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The quantizer-dequantizer formalism is developed for mean value and probability representation of qubits and qutrits. We derive the star-product kernels providing the possibility to derive explicit expressions of the associative product of the symbols of the density operators and quantum observables for qubits. We discuss an extension of the quantizer-dequantizer formalism associated with the probability and observable mean-value descriptions of quantum states for qudits.

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Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems

Cited by 9 publications

References 53 publications

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

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

Star Product Formalism for Probability and Mean Value Representations of Qudits

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