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Triangle Geometry for Qutrit States in the Probability Representation

Vladimir N. Chernega, Olga V. Man’ko, Vladimir I. Man’ko

arXiv:1804.04886Journal of Russian Laser ResearchSep 1, 201735 citations

Geometric aspects of quantum spin states

Michael Aizenman, Bruno Nachtergaele

arXiv:cond-mat/9310009Communications in Mathematical PhysicsJul 1, 1994152 citations

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

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

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.

Open Systems & Information DynamicsSep 1, 20192 citations

From quantum carpets to quantum suprematism—the probability representation of qudit states and hidden correlations

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

Abstract We develop the tomographic-probability picture where quantum states of qudits (spin- j systems, N -level atoms) are described by means of sets of fair classical probability distributions associated with properties of sets of dichotomic classical random variables. We formulate the superposition principle of pure quantum states as nonlinear addition rule of standard probabilities. The invertible map of the density matrices of qudit states onto the probability distributions is illustrated by their geometric representation in terms of triangles and squares called Malevich’s squares. We discuss an analogy of the squares with the quantum carpet picture. New entropic inequalities obtained due to the introduced map of the density matrices onto Malevich’s squares are studied. The correlations in systems without subsystems (hidden correlations) are discussed.Abstract We develop the tomographic-probability picture where quantum states of qudits (spin- j systems, N -level atoms) are described by means of sets of fair classical probability distributions associated with properties of sets of dichotomic classical random variables. We formulate the superposition principle of pure quantum states as nonlinear addition rule of standard probabilities. The invertible map of the density matrices of qudit states onto the probability distributions is illustrated by their geometric representation in terms of triangles and squares called Malevich’s squares. We discuss an analogy of the squares with the quantum carpet picture. New entropic inequalities obtained due to the introduced map of the density matrices onto Malevich’s squares are studied. The correlations in systems without subsystems (hidden correlations) are discussed.

Physica ScriptaJun 26, 201823 citations

Entangled Qubit States and Linear Entropy in the Probability Representation of Quantum Mechanics

Vladimir N. Chernega, Olga V. Man’ko, Vladimir I. Man’ko

The superposition states of two qubits including entangled Bell states are considered in the probability representation of quantum mechanics. The superposition principle formulated in terms of the nonlinear addition rule of the state density matrices is formulated as a nonlinear addition rule of the probability distributions describing the qubit states. The generalization of the entanglement properties to the case of superposition of two-mode oscillator states is discussed using the probability representation of quantum states.The superposition states of two qubits including entangled Bell states are considered in the probability representation of quantum mechanics. The superposition principle formulated in terms of the nonlinear addition rule of the state density matrices is formulated as a nonlinear addition rule of the probability distributions describing the qubit states. The generalization of the entanglement properties to the case of superposition of two-mode oscillator states is discussed using the probability representation of quantum states.

EntropyApr 9, 202212 citations

The quantum geometry of spin and statistics

Robert Oeckl

arXiv:hep-th/0008072Journal of Geometry and PhysicsSep 1, 200114 citations

Triangle Geometry of Spin States and Nonlinear Superposition of Probabilities Describing These States†

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