Bit thread decomposition of holographic
entanglement
Bogdan Stoica
Martin A. Fisher School of Physics
Brandeis University

25 June 2018

S. X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica, M. Walter
Bogdan Stoica

Holographic entanglement decomposition

25 June 2018, OISTMotivation: Gravity and entanglement

Recent developments at the intersection of gravity, field theory
and quantum information suggest that the connection between
gravity and entanglement is essential for understanding how gravitational degrees of freedom emerge from quantum ones.

How does geometry arise from information? What mathematicalresults can be uncovered along the way? What other ingredientsare missing?

Bogdan Stoica

Holographic entanglement decomposition

25 June 2018, OIST(Holographic) entanglement entropy
In quantum systems, entanglement entropy is a measure of thecorrelation of spatial region A with its complement Ac . Hard tocompute in field theory!
For quantum field theory states with smooth geometric duals,
entanglement entropy has a holographic prescription as the bulk
minimal area m(A) homologous to the boundary region A,m(A)S(A) =. [Ryu-Takayanagi]
4GNAcAm(A)Bogdan Stoica

Holographic entanglement decomposition

25 June 2018, OIST(Qu)Bit threads
A local prescription for entanglement entropy?
Imagine threads of entanglement through the bulk, connecting any
two Planck-area regions on the boundary. These are bit threads.
Divergenceless bulk vector flows of bounded norm.
Entanglement entropy between A and Ac : Maximum number ofthreads passing through the bulk from A to Ac .
The bottleneck constraining the maximum number of flows between A and Ac is the bulk minimal area m(A) ([Ryu-Takayanagi]
recovered).
[Freedman-Headrick]Bogdan Stoica

Holographic entanglement decomposition[Headrick–Hubeny]
25 June 2018, OISTEntanglement from (Qu)Bit threads
Key insight
Properties of bit thread flows ⇔ Properties of HEEB
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Bogdan Stoica

Holographic entanglement decomposition

25 June 2018, OISTConcurrent flows theorem
Holographic entropic inequalities follow from theorems on flows.

TheoremAny bulk with n boundary regions Ai , there exist n(n − 1)/2 pairwise
flows vij , i < j, between regions Ai and Aj , such that:
The flows for region Ai realize its entanglement entropy,S(Ai ) =XZ
j6=iAivij .

Flows vij can be concurrently realized on the manifold.

Corollary
The monogamy of holographic mutual information holds.
Bogdan Stoica

Holographic entanglement decomposition

25 June 2018, OISTEntanglement decomposition
Decomposition theorem
The entanglement of three boundary regions can be decomposed into
the entanglement of three Bell pairs.

Decomposition conjectureThe entanglement of four boundary regions can be decomposed into
the entanglement of six Bell pairs, plus a remainder four-region perfecttensor entanglement.
The conjecture implies the monogamy of mutual information: Thesix Bell pairs have vanishing contribution to MMI, the four-region
perfect tensor contributes positively.
Bogdan Stoica

Holographic entanglement decomposition

25 June 2018, OISTGraph decomposition conjecture+

Bogdan Stoica=+++

Holographic entanglement decomposition

25 June 2018, OISTThank you!

Bogdan Stoica

Holographic entanglement decomposition

25 June 2018, OIST