Entanglement contour

Abstract

In the context of characterizing the structure of quantum entanglement in many-body systems, we introduce the entanglement contour, a tool to identify which real-space degrees of freedom contribute, and how much, to the entanglement of a region A with the rest of the system B. The entanglement contour provides a complementary, more refined approach to characterizing entanglement than just considering the entanglement entropy between A and B, with several concrete advantages. We illustrate this in the context of ground states and quantum quenches in fermionic quadratic systems. For instance, in a quantum critical system in D = 1 spatial dimensions, the entanglement contour allows us to determine the central charge of the underlying conformal field theory from just a single partition of the system into regions A and B, (using the entanglement entropy for the same task requires considering several partitions). In D ≥ 2 dimensions, the entanglement contour can distinguish between gapped and gapless phases that obey a same boundary law for entanglement entropy. During a local or global quantum quench, the time-dependent contour provides a detailed account of the dynamics of entanglement, including propagating entanglement waves, which offers a microscopic explanation of the behavior of the entanglement entropy as a function of time.

Entanglement Renormalization and Integral Geometry

“…For pure AdS 3 , the holographic entanglement contour for I ℓ→∞ is 5) wherez is regarded as the UV cutoff. In the UV limitz → 0, the entanglement contour s(|x −x|;z = 0) ∝ 1/|x −x|, this agrees with the result in [30]. Moreover, it is easy to see the aforementioned fact that Θ(ℓ ≥ ,x;z) = 2s(|x −x| := ℓ ≥ ;z).…”

“…By this way, we recognize the kinematic measure can be understood as entanglement contour, i.e., an additive measure of entanglement density proposed in [30]. Furthermore, by adopting the SS-duality, we can see geometrically how the long-distance entanglement contour renormalizes under the change of RG scale.…”

“…At this moment, it is interesting to compare the entanglement contour holographically obtained here (in the UV limit) with the one for the gapless system on the 1D lattice solved numerically in [30]. For pure AdS 3 , the holographic entanglement contour for I ℓ→∞ is 5) wherez is regarded as the UV cutoff.…”

“…This kind of the quantity is recently proposed in [30] and is called "entanglement contour" which should be positive and summed to the entanglement entropy (see also [40,41] for a similar concept). Formally, entanglement contour s A (i) ≥ 0 is defined as contribution to the entanglement entropy…”

A contour for the entanglement entropies in harmonic lattices

“…As for the harmonic lattices, a proposal which satisfies (6) is based on the results obtained by Botero and Reznik [4] and it has been studied in [27]. Following [26], throughout this manuscript the function s Many functions satisfying (6) and (7) could be constructed and a complete list of properties characterising the contour function for the entanglement entropies in a unique way is not available. In [26], the authors made a step in this direction by proposing three further reqirements beside (6) and (7) involving the contour functions for the entanglement entropies.…”