On the entropy of systems undergoing time evolution

Hiroki Majima, Akira Suzuki

Chunjian Zhao, Zhiguo Lü, Hang Zheng

The ground state and the spectral structure of lower-lying excited states of a dissipative two-level system coupled to a sub-Ohmic bath (s = 1/2) with nonzero bias have been studied using the unitary transformation method. By calculating the ground-state entanglement entropy, the ground-state average of σ(z)(G), and the static susceptibility of the two-level system, we explore the nature of the transition (crossover) between the delocalized and localized state of the two-level system. Furthermore, we calculate the time-dependent expectation (σ(z)(t)) and the time evolution of the entanglement entropy to show that, when the system undergoes a transition (crossover) from the delocalized to the localized state, the time evolution of the two-level system changes from coherent to decoherent dynamics.

Gisela A. Lassner, Gisela A. Lassner

Craig S. Lent

For a quantum state undergoing unitary Schrödinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator [R. S. Ingarden, Quantum information theory, Rep. Math. Phys. 10, 43 (1976)RMHPBE0034-487710.1016/0034-4877(76)90005-7]. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system composed of a set of highly interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.

Craig S. Lent

Zhi-Long Wan, Hong-Yi Fan, Xiao-Lei Yin +1 more

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.