Balanced Partial Entanglement and the Entanglement Wedge Cross Section

Abstract

In this article we define a new information theoretical quantity for any bipartite mixed state $ρ_{AB}$. We call it the \textit{balanced partial entanglement} (BPE). The BPE is the partial entanglement entropy, which is an integral of the entanglement contour in a subregion, that satisfies certain balance requirements. The BPE depends on the purification hence is not intrinsic. However, the BPE could be a useful way to classify the purifications. We discuss the entropy relations satisfied by BPE and find they are quite similar to those satisfied by the entanglement of purification. We show that in holographic CFT$_2$ the BPE equals to the area of the entanglement wedge cross section (EWCS) divided by 4G. More interestingly when we consider the canonical purification, the BPE is just half of the reflected entropy, which is also directly related to the EWCS. The BPE can be considered as an generalization of the reflected entropy for a generic purification of the mixed state $ρ_{AB}$. We interpret the correspondence between the BPE and EWCS using the holographic picture of the entanglement contour.