Abstract
We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv:1104.1004, for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.
Highlights
Quantum systems that are spatially separated can share information that cannot be accounted for by the relativistic laws of classical physics
As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain
Understanding entanglement in bipartite systems is of fundamental importance in quantum information, but at the same time it is often fraught with technical difficulties
Summary
Quantum systems that are spatially separated can share information that cannot be accounted for by the relativistic laws of classical physics. In the case of disjoint intervals in a spin chain there is the extra complication due to the fact that in the fermionic space the operators between blocks contribute to the entropy, because the Jordan–Wigner transformation is not local This problem was tackled using CFT by Fagotti and Calabrese [17]. In the model (1.3), the Fermi operators in between blocks do not appear in the computation of the reduced density of states; the approach adopted in [26] applies This simplification allows a rigorous computation of the asymptotic behaviour of the entanglement entropy as m → ∞ while at the same time preserving the physical phenomenon that we want to study. Our approach is based on the Riemann–Hilbert method, which has the additional advantage of being mathematically rigorous
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