Entanglement wedge cross sections require tripartite entanglement
Chris Akers, Pratik Rath
Abstract
A bstract We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well- supported conjectures about the entanglement wedge cross section surface EW . If EW is related to either the CFT’s reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at $$ O\left(\frac{1}{G_N}\right). $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mfrac> </mml:mfenced> <mml:mo>.</mml:mo> </mml:math> We prove that this implies holographic CFT states must have $$ O\left(\frac{1}{G_N}\right). $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mfrac> </mml:mfenced> <mml:mo>.</mml:mo> </mml:math> amounts of tripartite entanglement. This proof involves a new Fannes-type inequality for the reflected entropy, which itself has many interesting applications.