Crossing versus locking: Bit threads and continuum multiflows

Abstract

Bit threads are curves in holographic spacetimes that manifest boundary entanglement, and are represented mathematically by continuum analogues of network flows or multiflows. Subject to a density bound, the maximum number of threads connecting a boundary region to its complement computes the Ryu-Takayanagi entropy. When considering several regions at the same time, for example in proving entropy inequalities, there are various inequivalent density bounds that can be imposed. We investigate for which choices of bound a given set of boundary regions can be "locked", in other words can have their entropies computed by a single thread configuration. We show that under the most stringent bound, which requires the threads to be locally parallel, non-crossing regions can in general be locked, but crossing regions cannot, where two regions are said to cross if they partially overlap and do not cover the entire boundary. We also show that, under a certain less stringent density bound, a crossing pair can be locked, and conjecture that any set of regions not containing a pairwise crossing triple can be locked, analogously to the situation for networks.

Deriving the PEE proposal from the Locking bit thread configuration

“…More specifically, we explicitly identify the PEE as the flux of the component f low in a locking bit thread configuration [17,41]. In other words, we identify the entanglement contour as the component flow in a multiflow that describes the locking thread configuration.…”

“…The motivation of our research arises from the recent study on the locking bit thread configurations under the different precise definitions of thread density bound and involving various types of sets of the boundary subregions [41]. The terminology "lock" is borrowed from the network flow theory.…”

“…Naively, one may want to use the same definition of thread density as in the previous and still consider the locking thread configuration constructed by a ν v multiflow. Indeed, it has been proved that there exists an almost the most powerful locking theorem for the ν v multiflow as follows [41]:…”

“…Happily, there is a ready solution to this problem. It has been shown in [41] that if we choose another reasonable definition of the density of bit threads, i.e., defining the thread density as the number per unit area intersecting a small disk, maximized over the orientation of the disk, then for the corresponding so-called ν a multiflow, we have a more powerful locking theorem as follows:…”

“…A few comments: one may worry that we change the definition of thread density in this section, however, as point out in [41], changing the density definition of bit threads from the ν v version to the ν a version does not affect the really essential characteristics of bit threads, i.e., as long as we still require the thread density to be less than 1 everywhere in the bulk, the maximum flux of threads through any surface is still equal to the area of the minimal bulk surface homologous to it, thus we can still reproduce the desired RT formula.…”