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arXiv:1205.0945 · Journal of Physics A Mathematical and Theoretical

Connecting the von Neumann and Rényi entropies for fermions

Mark Fannes, Nicholas Van Ryn

Published Sep 5, 2012• 9 citations
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Abstract

We explore the relation between the von Neumann entropy and the Rényi entropies of integer orders for shift-invariant quasi-free fermionic lattice systems. We investigate the approximation of the von Neumann entropy by a combination of integer-order Rényi entropies and give an estimate for the quality of such an approximation.
DOI
10.1088/1751-8113/45/38/385003

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The main aim of this thesis is to study quantum dynamics, with a focus on irreversible, dissipative dynamics, by collecting repeated observations and arranging them to form a process. This leads to quantum states of extended models with some simplifying features. The models explored here are quantum spins on Fermionic lattices and chains of quantum spins. We focus in particular on the randomising properties of the dynamics. The thesis begins by exploring the relation between the average von Neumann and Renyi entropies of integer orders for shift-invariant quasi-free Fermionic lattice systems. This then leads to an investigation into approximating the von Neumann entropy in terms of a combination of integer-order Renyi entropies and an estimate for the quality of such an approximation is given. Later, a rather general technique is introduced to model a quantum system’s dynamics on a semi-infinite half-chain of quantum spins. Here, bounds are given for the average entropy of the model system as well as a general scheme for computing the average Renyi entropy of integer order of such models. Finally, the half-chain model is made explicit with the example of a simple qubit evolving under a thermalising dissipative dynamics. With this example, the integer-order Renyi entropies are explicitly calculated. By optimising over all possible choices of models, a model-independent expression for the average Renyi entropies is derived.
We study the entanglement entropy of connected bipartitions in free-fermion gases of N particles in arbitrary dimension d . We show that the von Neumann and Rényi entanglement entropies grow asymptotically as N ( d −1)/ d ln N , with a prefactor that is analytically computed using the Widom conjecture both for periodic and open boundary conditions. The logarithmic correction to the power-law behavior is related to the area-law violation in lattice free fermions. These asymptotic large- N behaviors are checked against exact numerical calculations for N -particle systems.