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Journal of Modern Optics

Entropy evolution of a displaced thermal state for diffusion noise and its quantum statistical properties

Yingyu Zhang, Yixing Wang

Published Oct 14, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.1080/09500340.2025.2572525

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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.
We investigate how displaced thermal states (DTSs) evolve in a laser channel. Remarkably, the initial DTS, an example of a mixed state, still remains mixed and thermal. At long times, they finally decay to a highly classical thermal field only related to the laser parameters κ and g . The normal ordering product of density operator of the DTS in the laser channel leads to obtaining the analytical time-evolution expressions of the photon number, Wigner function, and von Neumann entropy. Also, some interesting results are presented via numerically investigating these explicit time-dependent expressions.
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.
A displaced thermal state (DTS) is first introduced by Jeong et al. [Phys. Rev. Lett. 97, 100401 (2006)] and its superpositions can be used in transfer of nonclassicality. In this paper, the analytical evolution law of the DTS exposed to dissipative environments is obtained according to the Kraus operator for amplitude damping. The results manifestly show that, after undergoing the amplitude damping, the initial DTSs still keep mixed and thermal with the exponential decay due to amplitude damping. An appealing feature is that the entropy evolution in this process completely depends on the mixedness.