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Chinese Physics B

Entropy evolution law of a general linear state for diffusion noise

Yingyu Zhang, Yixing Wang

Published May 7, 2025• 1 citation
Publisher

Abstract

Abstract Based on the Kraus operator-sum representation of the analytical solution of the diffusion equation, we obtain the evolution of a general linear state in the diffusion channel. Also, we study the quantum statistical properties of the initial general linear state and its von-Neumann entropy evolution in the diffusion channel, especially find that the entropy evolution is influenced by the diffusion noise and the thermal parameter but without the displacement.

DOI
10.1088/1674-1056/add4f2

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Abstract Based on the Kraus operator-sum representation of the analytical solution of the diffusion equation, we obtain the evolution of a general linear state in the diffusion channel. Also, we study the quantum statistical properties of the initial general linear state and its von-Neumann entropy evolution in the diffusion channel, especially find that the entropy evolution is influenced by the diffusion noise and the thermal parameter but without the displacement.
Using the correspondence of the operators between the real and fictitious modes, we find the Kraus operator-sum form of the analytical solution of the diffusion equation and obtain the evolution of a displaced thermal state in the diffusion channel. Also, we investigate the quantum statistical properties of the initial displaced thermal state and its von Neumann entropy evolution in the diffusion channel, especially find that the entropy evolution is affected by diffusion and mixedness, but without the displacement. These results provide theoretical basis for quantifying the evolved entropy in quantum information.
Abstract Using the operator correspondence of the real and fictious modes in the thermo entangled state representation, we solve the quantum master equation describing the diffusion channel and obtain the Kraus operator-sum representation of its analytical solution. we find that the pure coherent states evolve into the new mixed thermal superposed states in the diffusion channel. Also, we investigate the statistical properties of the initial coherent states and their entropy evolutions in the diffusion channel, and find that the entropy evolutions are only related to the decay time and without the amplitudes of the initial coherent states.
In quantum statistics theory, very few systems’ quantum entropy can be evaluated, since the quantum entropy’s formula by von-Neumann is [Formula: see text] where [Formula: see text] is density operator, [Formula: see text] is the Boltzmann constant, and the natural logarithm [Formula: see text] is quite difficult to treat. In this work, we tackle how to derive the entropy evolution of an initial coherent state undergoing a quantum diffusion process. We employ the method of integration within ordered product (IWOP) of operators to derive the final mixed state, which is a displaced chaotic state, and then deduce its entropy.
Using the integration within ordered products, we obtain the analytical density-operator evolution of the general quadratic state ρ 0 in a laser channel, which always maintains the same form as the initial density operator ρ 0. Introducing the differential representation of the von-Neumann entropy and two important resemble formulas, we analytically and numerically investigate the entropy evolution of the state ρ 0 in a laser channel, which is determined by not only the competition of the gain and loss of the laser channel, but also the coefficients α, β of the initial quadratic state ρ 0.
By virtue of the properties of thermal entanglement, we succeed to obtain the solution to the representative phase diffusion master equation and give the operator sum representation of density operator. After analyzing the time evolution process of density operator at different initial state, we find out density operators keep invariable with the initial state being Fock state or thermal state. But the system always maintains coherence with phase diffusion when the initial state is coherent state.