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Keywords = symplectic tomography

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11 pages, 257 KB  
Article
Entangled Probability Distributions for Center-of-Mass Tomography
by Ivan V. Dudinets, Margarita A. Man’ko and Vladimir I. Man’ko
Physics 2024, 6(3), 1035-1045; https://doi.org/10.3390/physics6030064 - 13 Aug 2024
Cited by 3 | Viewed by 2037
Abstract
We review the formalism of center-of-mass tomograms, which allows us to describe quantum states in terms of probability distribution functions. We introduce the concept of separable and entangled probability distributions for center-of-mass tomography. We obtain the time evolution of center-of-mass tomograms of entangled [...] Read more.
We review the formalism of center-of-mass tomograms, which allows us to describe quantum states in terms of probability distribution functions. We introduce the concept of separable and entangled probability distributions for center-of-mass tomography. We obtain the time evolution of center-of-mass tomograms of entangled states of the inverted oscillator. Full article
19 pages, 337 KB  
Article
Even and Odd Cat States of Two and Three Qubits in the Probability Representation of Quantum Mechanics
by Matyas Mechler, Margarita A. Man’ko, Vladimir I. Man’ko and Peter Adam
Entropy 2024, 26(6), 485; https://doi.org/10.3390/e26060485 - 31 May 2024
Cited by 8 | Viewed by 1653
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 [...] Read more.
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. Full article
(This article belongs to the Section Quantum Information)
17 pages, 319 KB  
Article
Dynamics of System States in the Probability Representation of Quantum Mechanics
by Vladimir N. Chernega and Olga V. Man’ko
Entropy 2023, 25(5), 785; https://doi.org/10.3390/e25050785 - 11 May 2023
Cited by 10 | Viewed by 2283
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
17 pages, 322 KB  
Article
Symplectic Polar Duality, Quantum Blobs, and Generalized Gaussians
by Maurice de Gosson and Charlyne de Gosson
Symmetry 2022, 14(9), 1890; https://doi.org/10.3390/sym14091890 - 9 Sep 2022
Cited by 4 | Viewed by 2196
Abstract
We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of [...] Read more.
We apply the notion of polar duality from convex geometry to the study of quantum covariance ellipsoids in symplectic phase space. We consider in particular the case of “quantum blobs” introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle in its strong Robertson–Schrödinger form. We show that these phase space units can be characterized by a simple condition of reflexivity using polar duality, thus improving previous results. We apply these geometric constructions to the characterization of pure Gaussian states in terms of partial information on the covariance ellipsoid, which allows us to formulate statements related to symplectic tomography. Full article
(This article belongs to the Topic Quantum Information and Quantum Computing)
13 pages, 294 KB  
Article
Symplectic Radon Transform and the Metaplectic Representation
by Maurice A. de Gosson
Entropy 2022, 24(6), 761; https://doi.org/10.3390/e24060761 - 28 May 2022
Cited by 9 | Viewed by 2427
Abstract
We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform [...] Read more.
We study the symplectic Radon transform from the point of view of the metaplectic representation of the symplectic group and its action on the Lagrangian Grassmannian. We give rigorous proofs in the general setting of multi-dimensional quantum systems. We interpret the Radon transform of a quantum state as a generalized marginal distribution for its Wigner transform; the inverse Radon transform thus appears as a “demarginalization process” for the Wigner distribution. Full article
(This article belongs to the Special Issue Quantum Mechanics and Its Foundations II)
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