Bit threads and holographic entanglement of purification

Abstract

The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a "bit thread" formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for S(AB) in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.

The connection between holographic entanglement and complexity of purification

“…Therefore, we conjecture that both EoP and CoP could be written in terms of these flows as well. In fact, just recently, after our work, in [66] the bit thread formalism for EoP has been worked out. Note that, EoP could be interpreted as the number of threads that pass through the minimal wedge cross section, the surface Γ in any construction.…”

Notes on entanglement wedge cross sections

“…This inequality has been proven in holography [4]. Inequalities for several boundary regions for entanglement of purification are also known (see, e.g., [6]), but such analogous inequalities for E W have not been exhaustingly tested (see, however, [7]). In all of the cases we study in this paper we found that the inequality (1.1) is satisfied.…”

“…Let us remark that in [9][10][11] it was suggested that in holographic framework the mutual information, multiplied by a constant, could be a dual to the entanglement negativity E. This was revised by a suggestion [14] that the entanglement wedge cross section E W , supplemented by the backreaction of the hanging minimal (cosmological) surfaces to the geometry, is actually a gravity dual of entanglement negativity. While lots of important work [4,6,7, has been carried out and most obvious inequalities tested, we feel that it is premature to identify E W one way or the other. In particular, as we will discuss in the following, E W can be discontinuous and non-monotonous, unlike E P or E in gauge theories at finite N .…”

Towards a bit threads derivation of holographic entanglement of purification

“…purifications that saturate this bound comprise a class of optimal purifications or completions of the state in question. The E P = E W conjecture has been further extended, studied, and generalized in recent work [14][15][16][17][18][19][20][21][22].…”