Quantum operator entropies under unitary evolution

Abstract

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. 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 comprised 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. arXiv:1901.08956v4 [quant-ph]

Thermalization in a Quantum Harmonic Oscillator with Random Disorder

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“… From the density matrix ( 22 ) we can also calculate the von Neumann entropy (in bits) for the witness. The von Neumann entropy represents the number of bits of missing local information about the quantum state of witness m due to its entanglement with the device system, and through that, to other witnesses [ 21 ].…”

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“…This entropy is directly related to the length of λ (m) . The von Neumann entropy S m represents the number of bits of missing local information about the quantum state of witness m due to its entanglement with the device system, and through that, to other witnesses [17]. Figure 5(a) shows the time development of the coherence angles for the device with 8 witnesses, E int /γ = 5, and ϕ/ϕ 0 = 1/2.…”