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International Journal of Quantum Information

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

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

Published Jan 31, 2020• 11 citations
Publisher

Abstract

The description of quantum states by probability distributions of classical-like random variables associated with observables is presented. An invertible map of the wave functions and density matrices onto the probability distributions is constructed. The relation of the probability distributions to quasidistributions like the Wigner function is discussed. The interference phenomenon and superposition principle of pure quantum states are given in the form of nonlinear addition of the probabilities identified with the quantum states. The probability given by Born’s rule is expressed as a function of the probabilities describing the system states. The suggested probability representation of quantum mechanics is presented using examples of harmonic oscillators and qubits.

DOI
10.1142/s0219749919410211

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The description of quantum states by probability distributions of classical-like random variables associated with observables is presented. An invertible map of the wave functions and density matrices onto the probability distributions is constructed. The relation of the probability distributions to quasidistributions like the Wigner function is discussed. The interference phenomenon and superposition principle of pure quantum states are given in the form of nonlinear addition of the probabilities identified with the quantum states. The probability given by Born’s rule is expressed as a function of the probabilities describing the system states. The suggested probability representation of quantum mechanics is presented using examples of harmonic oscillators and qubits.
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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.
The probability representation of quantum mechanics where the system states are identified with fair probability distributions is reviewed for systems with continuous variables (the example of the oscillator) and discrete variables (the example of the qubit). The relation for the evolution of the probability distributions which determine quantum states with the Feynman path integral is found. The time-dependent phase of the wave function is related to the time-dependent probability distribution which determines the density matrix. The formal classical-like random variables associated with quantum observables for qubit systems are considered, and the connection of the statistics of the quantum observables with the classical statistics of the random variables is discussed.
Review of probability representation of quantum mechanics where the system states are identified with fair probability distributions is presented. Examples of qubits and qutrits are given in detail. The explicit expressions of the qubit density matrices and qubit state vectors in terms of probabilities of dichotomic random variables are discussed. The wave function of a qubit state, and its phase which is not contained in the density matrix of the state, are expressed in terms of the dichotomic probabilities. The generalisation of the probability representation to the case of qudit state is considered. The Schrödinger–like equation for spectra of Hermitian and non Hermitian Hamiltonians is written in the new form of equation for the dichotomic probabilities.
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.