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Modern Physics Letters A

Time evolution of entropy variation for coherent state in quantum diffusion channel

Zhi-Long Wan, Hong-Yi Fan, Xiao-Lei Yin, H Yuan

Published Oct 30, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732325500907

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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 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.
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.
In classical information theory, the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov–Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes–Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy were proposed [10,[Formula: see text]3,[Formula: see text]8,[Formula: see text]39,[Formula: see text]15,[Formula: see text]44,[Formula: see text]9,[Formula: see text]27,[Formula: see text]28,[Formula: see text]2,[Formula: see text]19,[Formula: see text]45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S-mixing entropy by Ohya [26,[Formula: see text]27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S-mixing entropy. In this paper, based on research into the dynamical entropy and mean mutual entropy of quantum systems, this study investigates the characteristics of quantum channels that exhibit entanglement, a property unique to quantum systems, and investigates the behaviour of quantum entropy for compound quantum channels by changing the perspective from information transmission between input and output to a new perspective of information transmission from the initial state to the final state.
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.
The evolution of a pure coherent state into a chaotic state is described very well by a master equation, as is validated via an examination of the coherent state's evolution during the diffusion process, fully utilizing the technique of integration within an ordered product (IWOP) of operators. The same equation also describes a limitation that maintains the coherence in a weak diffusion process, i.e., when the dissipation is very weak and the initial average photon number is large. This equation is dρ/ d t = −κ[ a + a ρ − a + ρ a − a ρ a + + ρ aa + ]. The physical difference between this diffusion equation and the better-known amplitude damping master equation is pointed out.