scholarly journals Quantum process in probability representation of quantum mechanics

2022 ◽  
Author(s):  
Yan Przhiyalkovskiy
Keyword(s):  
Quantum Mechanics ◽  
Evolution Operator ◽  
System State ◽  
Probability Representation ◽  
Quantum Process ◽  
Amplitude Damping ◽  
Gaussian Type ◽  
Joint Evolution

Abstract In this work, the operator-sum representation of a quantum process is extended to the probability representation of quantum mechanics. It is shown that each process admitting the operator-sum representation is assigned a kernel, convolving of which with the initial tomogram set characterizing the system state gives the tomographic state of the transformed system. This kernel, in turn, is broken into the kernels of partial operations, each of them incorporating the symbol of the evolution operator related to the joint evolution of the system and an ancillary environment. Such a kernel decomposition for the projection to a certain basis state and a Gaussian-type projection is demonstrated as well as qubit flipping and amplitude damping processes.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Quantum Process ◽  
Hidden Variable ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classic probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classic probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic hidden variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. Uncovering this variable confirms the possibility that it could be the stochastic archetype of quantum probability.

scholarly journals A search for a stochastic archetype of quantum probability

2021 ◽  
Author(s):  
Tim C Jenkins
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Probability Density ◽  
Random Variables ◽  
Quantum Probability ◽  
Interference Term ◽  
Mathematical Foundation ◽  
Classical Probability ◽  
Quantum Process ◽  
Probability Densities

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years Man’ko and co-authors have successfully reconciled quantum and classical probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely that mathematically the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. This hidden generic variable appears to be such an archetype.

scholarly journals Superposition Principle and Born’s Rule in the Probability Representation of Quantum States

2019 ◽  
Vol 1 (2) ◽  
pp. 130-150 ◽  
Author(s):  
Igor Ya. Doskoch ◽  
Margarita A. Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Probability Distribution ◽  
Particle System ◽  
Superposition Principle ◽  
Quantum States ◽  
Particle State ◽  
Probability Representation ◽  
Tomographic Probability ◽  
Born’S Rule ◽  
System States

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.

Superposition of causal orders for quantum discrimination of quantum processes

2019 ◽  
Vol 17 (07) ◽  
pp. 1950055
Author(s):  
Seid Koudia ◽  
Abdelhakim Gharbi
Keyword(s):  
Quantum Mechanics ◽  
Memory Effects ◽  
Causal Order ◽  
Quantum Processes ◽  
Quantum Process ◽  
Order Case ◽  
Ultimate Bound ◽  
Process Discrimination

We address the superposition of causal orders in the quantum switch as a convenient framework for quantum process discrimination in the presence of noise in qubit systems, using Bayes strategy. We show that, for different kinds of qubit noises, the indefinite causal order between the unitary to be discriminated and noise gives enhancement compared to the definite causal order case without reaching the ultimate bound of discrimination in general. Whereas, for entanglement breaking channels, the enhancement is significant, where the quantum switch allows for the attainability of the ultimate bound for discrimination posed by quantum mechanics. Memory effects escorting the superposition of causal orders are discussed, where we point out that processes describing an indefinite causal order, violate the notion of Markov locality. Accordingly, a suggestion for the simulation of indefinite causal orders in more generic scenarios beyond the quantum switch is given.

scholarly journals Probability representation of quantum mechanics and star product quantization

2019 ◽  
Vol 1348 ◽  
pp. 012101 ◽  
Author(s):  
V N Chernega ◽  
S N Belolipetskiy ◽  
O V Man’ko ◽  
V I Man’ko
Keyword(s):  
Quantum Mechanics ◽  
Star Product ◽  
Probability Representation ◽  
Product Quantization
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