Local measures of entanglement in black holes and CFTs

Abstract

We study the structure and dynamics of entanglement in CFTs and black holes. We use a local entanglement measure, the entanglement contour, which is a spatial density function for von Neumann entropy with some additional properties. The entanglement contour can be calculated in many 1+1d condensed matter systems and simple models of black hole evaporation. We calculate the entanglement contour of a state excited by a splitting quench, and find universal results for the entanglement contours of low energy non-equilibrium states in 2d CFTs. We also calculate the contour of a non-gravitational bath coupled to an extremal AdS_22 black hole, and find that the contour only has finite support within the bath, due to an island phase transition. The particular entanglement contour proposal we use quantifies how well the bath’s state can be reconstructed from its marginals, through its connection to conditional mutual information, and the vanishing contour is a reflection of the protection of bulk island regions against erasures of the boundary state.

Balanced partial entanglement and mixed state correlations

“…The entanglement contour has been studied extensively in the context of the evolution of entanglement [47,53,58,63,64]. PEE is rather a quasi-local measure of entanglement and gained significant importance in the context of holography [27,55] as it provides a finer description between quantum entanglement of the boundary theory and the geometry of the bulk [55].…”

“…This balanced PEE can be shown to be applicable for generic purification and naturally incorporates the reflected entropy [22] as a specific case. More details on PEE, especially a first law-like version of the entanglement contour and its role in recently proposed island proposal can be found in [63,65,66].…”

“…The vanishing of the crossing PEE may have a special physical meaning related to the Markov recovery process. This follows from the observation that the PEE, hence also the BPE and the crossing PEE, can be expressed in terms of the conditional mutual information (CMI) [63]. The CMI for a three-party system is defined as [73]…”

Covariant entanglement wedge cross-section, balanced partial entanglement and gravitational anomalies

“…The entanglement contour [49], which is the derivative version of the PEE, has been explored in the Gaussian state of several free lattice models [49][50][51][52][53][54][55][56] and is used to characterize the density distribution of the entanglement entropy S A for the region A. It has been studied to capture evolution of the entanglement distribution [49,55,57,58], in holographic theories [21,28,[59][60][61] and quantum information theories [59,60,62]. In the context of holography, the entanglement contour has a geometric interpretation, which is a fine corresponce between the points on the boundary interval A and the points on the RT surface E A [21,28].…”

The PEE aspects of entanglement islands from bit threads

“…Nevertheless, it is interesting to investigate the entanglement details between the different parts of the system more specifically in such a semi-classical picture. However, previous work usually focused on the entanglement entropy of a subregion of the system (although see [76,77]). In this paper, our research interest is to further study the partial entanglement entropies (PEEs) of the various parts in the subsystem in the context involving island in this semi-classical picture.…”