aainstitutetext: Institute for Theoretical Physics, University of Amsterdam,
1090 GL Amsterdam, The Netherlandsbbinstitutetext: Martin A. Fisher School of
Physics, Brandeis University, Waltham, MA 02453, USA

# Quantum bit threads

Andrew Rolph<EMAIL_ADDRESS>
###### Abstract

We give a bit thread prescription that is equivalent to the quantum extremal
surface prescription for holographic entanglement entropy. Our proposal is
inspired by considerations of bit threads in doubly holographic models, and
equivalence is established by proving a generalisation of the Riemannian max-
flow min-cut theorem. We explore our proposal’s properties and discuss ways in
which islands and spacetime are emergent phenomena from the quantum bit thread
perspective.

BRX-TH-6678

##  1 Introduction

The connection between quantum information and geometry is one of the deepest
aspects of quantum gravity to have emerged in recent decades, and this
connection is manifest in the Ryu-Takayanagi (RT) formula which relates von
Neumann entropies in holographic CFTs to minimal area surfaces in their bulk
duals Ryu_2006 :

$S(A)=\frac{1}{4G_{N}}\min_{m\sim A}\text{Area}(m).$ (1)

Bit threads Freedman2016  are a flow-based reformulation of holographic
entanglement entropy that was introduced in part to address some of the
conceptual issues arising from the surface-based RT prescription. Bit threads
replace the minimisation over boundary-anchored surfaces with maximisation of
boundary flux of flows into the bulk. In the original ‘classical’ bit thread
prescription, the von Neumann entropy of boundary region $A$ equals the
maximum flux of a divergenceless, norm-bounded vector field $v^{\mu}(x)$ out
of that boundary region:

$S(A)=\max_{v}\int_{A}n_{\mu}v^{\mu}$ (2)

with $v$ subject to the constraints that

$\nabla_{\mu}v^{\mu}=0\text{ and }|v|\leq\frac{1}{4G_{N}}.$ (3)

Equivalence of this prescription to RT follows from a generalisation of the
max flow-min cut theorem (MFMC) to continuous Riemannian manifolds
Headrick2017 . The classical RT surface, as a minimal area surface, acts a
bottleneck for the classical bit threads.

Both the RT formula and bit thread prescription (2) are only applicable to
time symmetric states and have corrections at finite $N$ and coupling
$\lambda$. At finite $\lambda$, when the effective bulk gravitational action
has higher curvature terms, the RT prescription receives Wald-like corrections
to the area functional Dong2013 ; Camps2013 ; Bhattacharyya2014 , and these
corrections are accounted for in the bit thread formulation with a spacetime-
dependent norm bound Harper2018 . To be accurate at finite $N$ the RT formula
is modified to the quantum extremal surface (QES) prescription Engelhardt2014
which includes a bulk entanglement entropy contribution:

$S(A)=\min_{m\sim
A}\left(\frac{\text{Area}(m)}{4G_{N}}+S_{\text{bulk}}(\sigma(m))\right).$ (4)

$\sigma(m)$ is the subregion of the bulk slice whose boundary is
$\partial\sigma(m)=A\cup m$.

Quantum corrections to the bit thread prescription were briefly considered in
the original paper Freedman2016 , where the authors qualitatively suggested
that the quantum correction be accounted for by allowing the bit threads
(integral curves of $v$) to start and end at points in the bulk111See also
Chen2018  for a discrete formulation of bit threads on MERA tensor networks
with sources and sinks.. Loosely speaking the physical idea is to allow bit
threads to tunnel between entangled bulk degrees of freedom, and this has a
hope of capturing the bulk entanglement entropy term given in (4) because
threads tunnelling across the RT surface allows for additional flux from $A$
roughly in proportion to the amount entanglement across that surface.

In this paper we find a quantum bit thread prescription: a flow-based
prescription for calculating entanglement entropy in holographic CFTs that is
accurate to all orders in $1/N$.222Regarding bulk entanglement entropy, we
neglect graviton fluctuations and potential issues of bulk Hilbert space
factorisation, as in the FLM and QES proposals. Also, to focus on the
generalisation to finite $N$, we neglect higher curvature corrections to the
bulk gravitational action and assume that the state is time-reflection
symmetric; a fully general prescription, which we leave for future work, would
not make these simplifying assumptions. This is in contrast to the original
bit thread prescription, which is accurate only to leading order in $1/N$.

The prescription is

$S(A)=\max_{v}\int_{A}n_{\mu}v^{\mu}$ (5)

with $v$ subject to the constraints

$|v|\leq\frac{1}{4G_{N}}\text{ and
}\forall(\sigma\in\Omega_{A}):\left(-\int_{\sigma}\nabla_{\mu}v^{\mu}(x)\leq
S_{bulk}(\sigma)\right)$ (6)

where $\Omega_{A}$ is the set of all bulk homology regions for $A$:

$\Omega_{A}:=\\{\sigma\subseteq\Sigma:\partial\sigma\supseteq A\\}.$ (7)

A homology region $\sigma\in\Omega_{A}$ can be thought of as a time slice of a
possible entanglement wedge for $A$; its boundary is $\partial\sigma=m\cup A$.
The prescription has replaced minimisation over surfaces with maximisation
over flows; there are no surfaces $m$ directly involved in the objective
function or the constraints. We prove that (5) is equivalent to the QES
prescription (4) using a technique from convex optimisation for mapping
maximisation problems to equivalent minimisation problems and vice versa.

The key difference between the quantum and classical bit thread prescriptions
is that the divergencelessness condition has been replaced. The new constraint
allows threads to end at bulk points, but bounds the total number that can end
in any given bulk homology region by the von Neumann entropy of the region.
Bit threads in flux-maximising flows have the appearance of jumping between
entangled bulk regions.

Quantum extremal surfaces and islands, which are not part of the bit thread
prescription, appear as properties of flux-maximising flows. Quantum extremal
surfaces are bottlenecks to the flow - just as RT surfaces are bottlenecks to
bit threads in the classical prescription. Islands are bulk regions where so
many bit threads reappear for a flux-maximising flow that the region’s
boundary is a novel disconnected bottleneck to the flow.

We also consider bit threads in doubly holographic models Almheiri2019b . The
benefit of this class of models is that the bulk entanglement entropy can be
computed using the classical RT formula in one higher dimension, so in these
models the quantum bit thread prescription in AdSd+1 in a sense follows
directly from the behaviour of classical bit threads along the boundary in the
highest dimensional picture.

Outline

In section 2 we gain some intuition for quantum bit threads by considering
classical bit threads in doubly holographic models, and we propose and comment
on a quantum bit thread prescription. In section 3 we prove the equivalence of
our quantum bit thread prescription to the quantum extremal surface
prescription, using tools from convex optimisation. In section 4 we conclude
with a discussion of the connection of quantum bit threads with ER=EPR,
emergent spacetime, and islands, and talk about possible future directions to
work towards. In appendix A we prove that quantum bit threads satisfy the
nesting property of flows, which is sufficient for a flow-based holographic
proof of strong subadditivity to all orders in $1/N$.

Note: As this work neared completion we learned of work by Cesar Agón and Juan
Pedraza Agon2021  which has partial overlap in scope, since they also study
modifications of the bit thread proposal to account for bulk quantum
corrections, and we have arranged to submit simultaneously to arXiv. Their
prescription is only accurate to next-to-leading order in $1/N$, not to all
orders, and also requires the position of the RT surface as input.

##  2 Quantum bit threads

In this section we will give a flow-based bit thread prescription which is
equivalent to the quantum extremal surface (QES) prescription Engelhardt2014

$S(A)=\min_{m\sim
A}\left(\frac{\text{Area}(m)}{4G_{N}}+S_{\text{bulk}}(\sigma(m))\right)$ (8)

where $\sigma(m)$ is the subregion of the bulk slice whose boundary is
$\partial\sigma(m)=A\cup m$. We assume time reflection symmetry to separate
the challenges of creating covariant and quantum bit thread prescriptions.

The generalised entropy that features in the QES prescription has two sources
of UV divergences: from the bulk entanglement entropy and counterterms in the
gravitational action. We assume that these divergences cancel each other
rendering the generalised entropy finite and regulator independent
Solodukhin_1995 ; Susskind_1994 ; Solodukhin2011 ; Faulkner2013 .
$S_{\text{bulk}}$ is the finite piece of the bulk entanglement entropy, and
$G_{N}$ is the renormalised Newton’s constant, both of which are
renormalisation scheme-dependent.

###  2.1 Bit threads and double holography

Figure 1: Bit threads on a time slice of a doubly holographic model. The
geometry is asymptotically AdSd+2 with an end-of-the-world brane. The bit
threads are divergenceless; we assume the bulk is classical. For flux-
maximising flows, some flow lines go from $A$ to $\bar{A}$ by moving through
the AdSd+1 boundary while others move into the bulk. An observer restricted to
the AdSd+1 boundary sees threads appearing and disappearing on either side of
the QES (see figure 2). Figure 2: A flux-maximising quantum bit thread flow on
a time slice of AdSd+1. Bit threads can start and end at bulk points and some
appear to jump across the quantum extremal surface (green dashed line). The
number of threads that can appear and disappear is determined by von Neumann
entropy of bulk subregions, and in doubly holographic models this is given by
RT surface areas in the higher dimensional AdSd+2 geometry (see figure 1).

First, we take a small but instructive diversion to consider quantum bit
threads in doubly holographic models. In double holography we have three
equivalent descriptions Almheiri2019b : a CFTd, which has an (asymptotically)
AdSd+1 gravitational dual with a bulk matter CFTd+1, which in turn has an
(asymptotically) AdSd+2 gravitational dual. See Fig. 1. The advantage of such
models is that the bulk entanglement entropy in AdSd+1 can be calculated
holographically. We can use the classical Ryu-Takayanagi formula, if we assume
the quantum corrections in AdSd+2 to be subleading.

In doubly holographic models the quantum bit thread prescription in a sense
follows directly from the behaviour of classical bit threads along the
boundary branes in the highest dimensional picture. Classical bit threads
start at $A$ and can travel through, leave and join the boundary branes in the
AdSd+2 picture, though they cannot start or end on them Headrick2017 . In the
AdSd+1 picture, these look like bit threads starting and ending at bulk
points.

The bulk is classical in the AdSd+2 picture, so we use the classical bit
thread prescription given in (2). As Fig. 1 illustrates, the bit threads
separate into those that stay on the boundary and those that move into the
bulk. The one novelty is that what the $G_{N}$ appearing in the norm bound is
depends on whether the flow is through the bulk or along the boundary. The
flux of bit threads into the bulk from any subregion of the boundary is
bounded by the subregion’s RT surface.

How does this all look from the AdSd+1 perspective? As Fig. 2 depicts, some
bit threads appear to jump over the quantum extremal surface, and double
holography gives us an interesting interpretation of these threads as moving
into the higher dimensional bulk. The flux density from the boundary of AdSd+2
equals the divergence of $v$ on the AdSd+1 boundary. The bound on flux off
subregions of the AdSd+2 boundary translates to

$\left|\int_{\sigma}\nabla_{\mu}v^{\mu}\right|\leq S_{\text{bulk}}(\sigma)$
(9)

for all subregions $\sigma$ of the AdSd+1 time slice. Here, $\sigma$ need not
be a bulk homology region for $A$.

This property of flow configurations in doubly holographic models is very
suggestive of how to modify the bit thread prescription to account for quantum
corrections. We now turn to consider such modifications. In the discussion
section, we will revisit doubly holographic models when we discuss islands
from the quantum bit thread perspective.

###  2.2 Warm-up proposals

We do not claim there to be a unique quantum bit thread prescription. That
should not surprise us as even the original classical prescription is non-
unique; one can trivially generate a family of mathematically equivalent
prescriptions by redefining $v^{\mu}(x)\to f(x)\tilde{v}^{\mu}(x)$, which
superficially look different from the original prescription because
$\nabla_{\mu}\tilde{v}^{\mu}\neq 0$. The proposals we will discuss are not as
trivially equivalent as that, but the point stands.

Proposal I

To illustrate what makes some formulations more desirable than others, we
consider a short series of iterative improvements, starting from the crudest
way to account for quantum corrections in the bit thread formulation, which is
to add the entropy of the bulk homology region for the classical RT surface to
the final answer by hand

$S(A)=\max_{v}\left(\int_{A}n_{\mu}v^{\mu}\right)+S_{\text{bulk}}(\sigma(m_{RT}))\quad\text{
subject to }|v|\leq\frac{1}{4G_{N}}\text{ and }\nabla_{\mu}v^{\mu}=0.$ (10)

This is inaccurate beyond the leading order correction in $G_{N}$. There are
two other reasons why this prescription is lacking. Firstly, it requires
detailed foreknowledge about the bulk state and geometry, including where the
RT surface will be, to know what $S_{\text{bulk}}$ to add. As in the classical
prescription, the threads should have as little foreknowledge about the bulk
as possible. Secondly, the quantum correction is added by hand, rather than
determined dynamically by the threads themselves, like the area term is.

Proposal II

Proposal II is a flow prescription that is equivalent to the FLM surface
prescription given in Faulkner2013 . We can do better than proposal I by
modifying divergencelessness condition,

$S(A)=\max_{v}\left(\int_{A}n_{\mu}v^{\mu}\right)\quad\text{ subject to
}|v|\leq 1\text{ and }\nabla_{\mu}v^{\mu}(x)=-J(x)$ (11)

with source function $J$ to be determined. We assume that the maximal flow
saturates the norm bound on the QES, and then, by the divergence theorem

$\int_{A}n_{\mu}v^{\mu}=-\int_{\sigma(m)}\nabla_{\mu}v^{\mu}+\int_{m}n_{\mu}v^{\mu}$
(12)

(where the normal vector is outward pointing on $m$, and inward-pointing on
$A$) we see that to capture the bulk entropy term and match the QES
prescription we require

$\int_{\sigma(m_{QES})}J(x)=S_{\text{bulk}}(\sigma(m_{QES}))$ (13)

Thus $J$ is some kind of density function for the bulk von Neumann entropy.
Such density functions have been considered before, first in Chen2014 , and
dubbed entanglement contours. In a follow-up paper, we will explore
entanglement contours in detail Rolph2022 .

In previous work, bit thread flux density was interpreted as an entanglement
contour KudlerFlam2019a , which seems different from our proposal, which is
that the divergence of the flow field equals an entanglement contour, but the
connection is made in doubly holographic models where $\nabla_{\mu}v^{\mu}$ in
the AdSd+1 picture equals the flux density of bit threads off the boundary in
the AdSd+2 picture.

The prescription (11) is an improvement on (10) because the bulk entanglement
entropy is accounted for dynamically by the bit threads with only a simple and
local modification to the divergencelessness condition. It is still
unsatisfactory because it requires foreknowledge of where the Ryu-Takayanagi
surface will be, to know what source function $J$ to use, and is only accurate
to subleading order in $1/N$.

###  2.3 Quantum bit thread proposal

Proposal III

The last prescription we consider requires no foreknowledge and is accurate to
all orders in $G_{N}$. It replaces the divergencelessness condition with a
constraint on the flux into all bulk homology regions for $A$.

The prescription is

$S(A)=\max_{v}\int_{A}n_{\mu}v^{\mu}$ (14)

with $v$ subject to the constraints

$|v|\leq\frac{1}{4G_{N}}\text{ and
}\forall(\sigma\in\Omega_{A}):\left(-\int_{\sigma}\nabla_{\mu}v^{\mu}(x)\leq
S_{bulk}(\sigma)\right)$ (15)

and where $\Omega_{A}$ is the set of all bulk homology regions for $A$:

$\Omega_{A}:=\\{\sigma\subseteq\Sigma:\partial\sigma\supseteq A\\}.$ (16)

The prescription needs neither more nor less information about the bulk slice
than the QES prescription, as expected. It needs the bulk metric and the
renormalised von Neumann entropies of all bulk homology regions for $A$.

Neither of the constraints are regulator-independent, as both the renormalised
Newton’s constant $G_{N}$ and the renormalised bulk von Neumann entropy
$S_{\text{bulk}}$ in (14) are regulator-dependent. Nevertheless, the
constraints together are regulator-independent - one way to argue this is
through its equivalence to the QES prescription, which is regulator-
independent Susskind_1994 ; Cooperman2014 .

How is this prescription equivalent to the QES prescription? Consider an
arbitrary surface $m$ homologous to $A$, and its associated bulk region
$\sigma(m)$. The constraints and the divergence theorem (12) bound the flux
out of $A$

$\int_{A}n_{\mu}v^{\mu}\leq\frac{\text{Area}(m)}{4G_{N}}+S_{\text{bulk}}(\sigma(m))$
(17)

for all $m\sim A$. We can try to saturate this inequality and so turn it into
an equality by minimising the right-hand side over surfaces $m$ homologous to
$A$, which minimises on $m=m_{QES}$, and maximising the left-hand side with
respect to $v$. The inequality is saturated if we assume that there exists a
flow field configuration $v$ that:

  1. 1.

Satisfies the constraints of our prescription (14).

  2. 2.

Has $v^{\mu}=n^{\mu}/4G_{N}$ on the QES, with $n^{\mu}$ the unit normal.

  3. 3.

Saturates the divergence bound when $m=m_{QES}$:

$-\int_{\sigma(m_{QES})}\nabla_{\mu}v^{\mu}=S_{\text{bulk}}(\sigma(m_{QES}))$
(18)

This existence assumption is reasonable, though the precise argument for why
is somewhat involved, so to avoid interrupting the flow we will come back to
it at the end of the section. With the assumption, (17) becomes

$\max_{v}\int_{A}n_{\mu}v^{\mu}=\min_{m\sim
A}\left(\frac{\text{Area}(m)}{4G_{N}}+S_{\text{bulk}}(\sigma(m))\right)$ (19)

which establishes the equivalence of our quantum bit thread prescription to
the QES prescription. We will give a rigorous proof of the equivalence to the
QES prescription, which does not rely on existence assumptions, in section 3.

####  2.3.1 Properties

We have modified the classical bit thread proposal by replacing the
$\nabla_{\mu}v^{\mu}=0$ condition with a constraint that allows bit threads to
start and end at bulk points, but that limit the number of threads that can
end in all bulk homology regions. Our prescription is non-local, in that it
constrains the flow over codimension-0 (with respect to the bulk time slice)
regions; this is not unexpected, and it may be it cannot be improved upon,
given the non-local nature of bulk entanglement.

As expected, the classical bit thread prescription is recovered in the
$N\to\infty$ limit. First, we rescale $v$ by $4G_{N}$, which gives the
prescription

$S(A)=\frac{1}{4G_{N}}\max_{v}\int_{A}v$ (20)

with $v$ subject to the constraints

$|v|\leq 1\text{ and
}\forall(\sigma\in\Omega_{A}):\int_{\sigma}\nabla_{\mu}v^{\mu}(x)\geq-4G_{N}S_{bulk}(\sigma).$
(21)

There exist states and regions for which $S_{bulk}(\sigma)$ is order
$1/G_{N}$, but with reasonable assumptions and for finite energy states the
(renormalised) entropy of any sufficiently small bulk region is $O(1)$, and
for these states and regions we have $\lim_{G_{N}\to 0}(G_{N}S_{bulk})=0$.
Next, for the divergence constraint to hold for all $\sigma$ requires
$\nabla_{\mu}v^{\mu}(x)\geq 0$ for all $x$. If it were not then the constraint
would be violated for the $\sigma$ equal to $\Sigma$ minus the neighbourhood
of $x$ where $\nabla_{\mu}v^{\mu}<0$. Finally, this constraint
$\nabla_{\mu}v^{\mu}\geq 0$ can be further tightened, without affecting the
optimum, to $\nabla_{\mu}v^{\mu}(x)=0$ \- which is the divergencelessness
constraint of the original bit thread prescription - because positive sources
for $v$ in the bulk cannot possibly increase the flux from $A$.

Another consequence of the constraint: if the global bulk state is pure then
when applied to $\sigma=\Sigma\in\Omega_{A}$ our divergence constraint implies
that

$\int_{\Sigma}\nabla_{\mu}v^{\mu}(x)\geq 0.$ (22)

For pure states, the flux out of bulk slice is thus non-negative; there cannot
be more bit threads that end at points in the bulk than start.

(a)

(b)

Figure 3: Flux-maximising flows for two different states on the same bulk
slice, which illustrates the quantum extremal surface phase transition from
the bit thread perspective. The QES, depicted by the green dashed line, is
always the bottleneck to the flow. In (a) there is little entanglement across
the QES, and the QES is $l_{P}$ away from the minimal-area classical RT
surface. In (b) the bulk entanglement has increased until a new bottleneck
appears in the bulk region where the flow is constrained to reappear, which
generally will not be close to the classical RT surface.

Even when the QES and the classical RT surface are not close together (in
Planck units), our quantum bit thread prescription (14) is still correct and
equivalent to the QES prescription, as we will prove in section 3. Such
situations have been of recent interest in the context of the black hole
information paradox Almheiri2019a ; Penington2019 . The QES and classical RT
surfaces are often perturbatively close to each other, due to the bulk
entanglement entropy term having a $G_{N}$ suppression relative to the area
term, but there are cases when the bulk entanglement across a candidate QES is
sufficiently large to overcome the relative $G_{N}$ suppression. As the bulk
entanglement across the QES increases, there may come a point where its
position jumps discontinuously. Fig. 3 shows what is happening from the
quantum bit thread perspective. As the bulk entanglement across the minimal
area RT surface increases, more and more threads are allowed to jump across
that bottleneck, but the threads eventually become maximally packed in some
new region where they are forced to reappear due to the divergence constraint,
and a new bottleneck emerges at the quantum extremal surface. As the bulk
entanglement increases, there is a continuous change in the flow vector field,
in contrast to the discontinuous jump of the extremal surface, and there is
still a bottleneck to the flow, but it is no longer the minimal area surface.
The QES is always the bottleneck to the flow.

The divergence constraint in our quantum bit thread prescription is similar to
the property of flows in doubly holographic models we found earlier, which was

$\forall\sigma\subseteq\Sigma:\qquad\left|\int_{\sigma}\nabla_{\mu}v^{\mu}\right|\leq
S_{\text{bulk}}(\sigma).$ (23)

Doubly holographic models are a subset of all holographic models, so, for
consistency, their flows, which satisfy (23), must also satisfy the
constraints of our quantum bit thread prescription given in (14), which they
do.

The property (23) may or may not be too tight a constraint to impose in non-
doubly-holographic models. It is stronger than the constraint in (14) in two
ways: it applies to all bulk subregions, not just bulk homology regions for
$A$, and the absolute value is taken. Taking the absolute value is certainly
too strong; $S_{bulk}$ is a renormalised entropy, it can be negative, and the
constraint (23) can not be satisfied for any $v$ when it is.

####  2.3.2 Quantum bit threads near the QES

Now we come back to justify the assumption that was key to establishing
equivalence of prescriptions, that there exists a flow field $v$ that
satisfies the conditions given in (18), one of which was that it saturates the
norm bound with $v^{\mu}=n^{\mu}/4G_{N}$ on the QES. There is cause to
question this assumption because in the classical prescription (with
$\nabla_{\mu}v^{\mu}=0$) it is not possible to saturate the norm bound
$|v|\leq 1/4G_{N}$ on any surface homologous to $A$ other than the minimal
area one without violating the norm bound elsewhere. In the quantum
prescription, however, due to the non-zero divergence of $v$, threads can
start and end anywhere in the bulk, and this makes it possible for the norm
bound to be saturated on a non-minimal area surface, such as the QES, without
the norm bound being violated immediately off the surface.

Since we assumed that the bit threads of a flux maximising flow are maximally
packed on the QES, the most sensible place to check whether the norm bound
will be violated is in the neighbourhood of the QES. From the definition of
the QES, shape deformations of its generalised entropy vanish to first order,
which implies that

$\begin{split}0&=\frac{\delta}{\delta
m(x)}\left(\frac{\text{Area}(m)}{4G_{N}}+S_{\text{bulk}}(\sigma(m))\right)\\\
&=\frac{K(x)}{4G_{N}}+\frac{\delta}{\delta
m(x)}S_{\text{bulk}}(\sigma(m))\end{split}$ (24)

on the QES, with $K$ the trace of the extrinsic curvature. Suppose now also
that the divergence bound is saturated not only for $m=m_{QES}$ as in (18),
but for first order shape deformations away from the QES too, then

$\nabla_{\mu}v^{\mu}(x)=-\frac{\delta}{\delta m(x)}S_{\text{bulk}}(\sigma(m))$
(25)

on the QES. By assumption $|v|=1/4G_{N}$ on the QES and what we now check is
whether our supposition (25) is sufficient to ensure that the norm bound
$|v|\leq 1/4G_{N}$ is not violated at first order in directions perpendicular
to the QES; this requires $v^{\mu}\partial_{\mu}|v|=0$. This equation was
called the ‘linear obstruction equation’ in Harper2018  and was shown to be
equivalent to

$\nabla_{\mu}v^{\mu}(x)=\frac{K(x)}{4G_{N}}$ (26)

on any surface on which $v^{\mu}=n^{\mu}/4G_{N}$ (which for us is the QES).
Now if (26) were incompatible with the constraints of our prescription then we
would be in trouble, but (24) and (25) do together imply (26), which shows
that the supposition (25) (which is within our prescription’s constraints) is
sufficient.

##  3 Proof of equivalence to the QES prescription

At the end of the last section, we did a local analysis in the neighbourhood
of the QES and found strong evidence for the existence of flux-maximising
flows that make the quantum bit thread and QES prescriptions equivalent. In
this section, with a completely different approach, we prove the
equivalence333To a physicist’s level of rigour.. We will use tools from convex
optimisation, which were also used by the authors of Headrick2017  to prove
the equivalence of the classical Ryu-Takayanagi and bit thread prescriptions.

The key tool is the application of Lagrangian duality, the basic idea of which
is to

  1. 1.

Take a constrained optimisation problem, the ‘primal’.

  2. 2.

Introduce Lagrange multiplier terms to enforce the constraints.

  3. 3.

Optimise over the original variables, leaving us with a new optimisation
problem called the ‘dual’.

For us, the primal problem is the quantum bit thread prescription and the
target dual problem is the quantum extremal surface prescription.

The steps to finding the Lagrangian dual of a constrained optimisation problem
which we’ve enumerated are mathematically straightforward, and we do not need
to go into the general theory of Lagrangian duality and convex optimisation,
but readers who would like to learn more about the mathematical background can
read the review in section 2 of Headrick2017  and the references therein.

The only non-trivial result from convex optimisation we need is Slater’s
condition, which is a sufficient condition for strong duality to hold. Strong
duality means that the optima of the primal and dual problems are equal.
Without strong duality, the primal and dual optimisation problems are not
equivalent, which to this section’s purpose would be fatal.

###  3.1 From the quantum bit thread prescription to a dual optimisation
problem

Lagrangian dualisation begins with a constrained optimisation problem, the
primal. Our bit thread proposal meets the definition of a special class of
optimisation problems, called concave optimisation problems, because both the
objective function we are maximising over

$\int_{A}v$ (27)

and the constraints we are imposing

$\frac{1}{4G_{N}}-|v|\geq 0$ (28)

and

$\forall(\sigma\in\Omega_{A}):-\int_{\sigma}\nabla_{\mu}v^{\mu}\leq
S_{\text{bulk}}(\sigma)$ (29)

are concave functions of $v$. Recall that $\Omega_{A}$ is the set of bulk
homology regions for $A$, and $S_{bulk}$ is the renormalised bulk von Neumann
entropy of the reduced state. In our simplified notation, $\int_{A}v$ is short
for $\int_{A}\sqrt{h}n_{\mu}v^{\mu}$ with $n$ the inward pointing unit normal
to the bulk Cauchy slice, and we suppress the argument of $v$ and the measure
on $\Sigma$.

Slater’s condition is satisfied by the quantum bit thread prescription. This
is important because if Slater’s condition is satisfied for a convex or
concave optimisation problem then strong duality holds. Slater’s condition
requires there to exist a strictly feasible point in the domain of the primal
problem. A strictly feasible point is a point that satisfies the constraints,
including strictly satisfying all the non-linear constraints. For us (28) is
the only non-linear constraint that needs to be strictly satisfied, and $v=0$
is a strictly feasible point, so strong duality holds. This establishes that
our quantum bit thread proposal is equivalent to whatever optimisation
problems we can turn it into using our Lagrangian dualisation procedure444As
in Headrick2017  we assume that what is true about convex optimisation
problems with a finite number of constraints is also true when there are an
infinite number of constraints..

The next step of the Lagrangian dualisation procedure is to add Lagrange
multiplier terms for each constraint. The divergence constraint given by (29)
applies to every element of $\Omega_{A}$, and the norm bound (28) applies at
every point in the bulk time slice $\Sigma$. Adding Langrange multiplier terms
for these constraints gives

$\sup_{v}\inf_{\mu,\phi}\left[\int_{A}v+\int_{\Sigma}\phi(x)\left(\frac{1}{4G_{N}}-|v|\right)+\int_{\Omega_{A}}d\mu(\sigma)\left(\left(\int_{\Sigma}\chi(\sigma,x)\nabla_{\mu}v^{\mu}\right)+S_{\text{bulk}}(\sigma)\right)\right]$
(30)

where $\mu$ is a (non-negative) measure on $\Omega_{A}$, $\phi$ a non-
negative555Lagrange multipliers usually enforce equality constraints, while we
have inequality constraints, so $\mu$ and $\phi$ are a generalisation called
Karush–Kuhn–Tucker multipliers, which is also why $\mu$ and $\phi$ need to be
non-negative. scalar function on $\Sigma$, and $\chi(\sigma,x)$ the
characteristic function for $\sigma\subseteq\Sigma$ defined by

$\chi(\sigma,x):=\begin{cases}1\qquad\text{ for }x\in\sigma\\\ 0\qquad\text{
for }x\in\Sigma\backslash\sigma.\end{cases}$ (31)

Minimising with respect to the Lagrange multipliers $\mu$ and $\phi$ would
return us to the primal problem. Instead, we maximise with respect to the
original variable $v$, after integrating the $\nabla_{\mu}v^{\mu}$ term by
parts, which gives666We have assumed that the orders of integration can be
reversed.

$\begin{split}\inf_{\mu,\phi}\sup_{v}&\left[\left(1-\int_{\Omega_{A}}d\mu(\sigma)\right)\int_{A}v+\int_{\Sigma}\left(\phi(x)\left(\frac{1}{4G_{N}}-|v|\right)-v^{\mu}\partial_{\mu}\left(\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x)\right)\right)\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int_{\Omega_{A}}d\mu(\sigma)S_{\text{bulk}}(\sigma)\right]\end{split}$
(32)

The supremum of this with respect to $v$ is $+\infty$, unless both

$\int_{\Omega_{A}}d\mu(\sigma)=1$ (33)

and

$\phi(x)\geq\left|\partial_{\mu}\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x)\right|.$
(34)

Since we are minimising with respect $\mu$ and $\phi$, we certainly want to
avoid the region of the domain of the optimisation problem (32) where either
(33) or (34) are not satisfied and the optimum is $+\infty$. With that in
mind, (32) is equivalent to

$\inf_{\mu,\phi}\left[\frac{1}{4G_{N}}\int_{\Sigma}\phi(x)+\int_{\Omega_{A}}d\mu(\sigma)S_{\text{bulk}}(\sigma)\right]$
(35)

as long as we impose (33) and (34) as constraints. The constraint (33) tells
us that $\mu$ is a probability measure on $\Omega_{A}$. The minimisation of
(35) with respect to $\phi$ is trivial; it saturates the inequality (34). This
leaves us with a Lagrangian dual to our quantum bit thread prescription, which
is to find

$\begin{split}\inf_{\mu}&\left[\frac{1}{4G_{N}}\int_{\Sigma}\left|\partial_{\mu}\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x)\right|+\int_{\Omega_{A}}d\mu(\sigma)S_{\text{bulk}}(\sigma)\right]\\\
&\qquad\qquad\qquad\text{ subject to
}\int_{\Omega_{A}}d\mu(\sigma)=1\end{split}$ (36)

What remains to show is that this dual problem is equivalent to the QES
prescription. We have two ways of showing this.

###  3.2 From the dual optimisation problem to the QES prescription

Argument 1

Our first argument starts by noting that if in (36) we could take the
$\int_{\Omega_{A}}d\mu(\sigma)$ outside of the norm, then our optimisation
problem would be

$\begin{split}&\inf_{\mu}\left[\int_{\Omega_{A}}d\mu(\sigma)\left(\frac{1}{4G_{N}}\int_{\Sigma}\left|\partial_{\mu}\chi(\sigma,x)\right|+S_{\text{bulk}}(\sigma)\right)\right]\\\
=&\inf_{\mu}\left[\int_{\Omega_{A}}d\mu(\sigma)\left(\frac{\text{Area}(m)}{4G_{N}}+S_{\text{bulk}}(\sigma)\right)\right]\\\
\end{split}$ (37)

This is a minimisation of generalised entropy over probability measures on the
set of bulk homology regions for $A$, which is equivalent to the QES
prescription and what we wanted. The first equality uses that the normal
derivative of the characteristic function $\chi(\sigma,x)$ is a surface delta
function on $m$, with $\partial\sigma=m\cup A$.

What stops us from taking $\int_{\Omega_{A}}d\mu(\sigma)$ outside the norm?
The first term in (36) can be thought of as the norm of the integral of such
surface delta functions over those $m$ on which $\mu(\sigma)$ has support.
$\mu$ can have support on surfaces whose contributions to
$\int_{\Sigma}\left|\partial_{\mu}\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x)\right|$
partially cancel against each other inside the norm. For this cancellation to
happen for a pair of bulk regions $\sigma_{1}$ and $\sigma_{2}$, with
boundaries $\partial\sigma_{1}=m_{1}\cup A$ and $\partial\sigma_{2}=m_{2}\cup
A$, the surfaces $m_{1}$ and $m_{2}$ must have some overlap with oppositely
oriented surface normals, an example of which is shown in Fig. 4. To spell out
the issue, for the $m_{1}$ and $m_{2}$ shown in Fig. 4 the following is not
true:

$\int_{\Sigma}|\partial_{\mu}\chi(\sigma_{1},x)+\partial_{\mu}\chi(\sigma_{2},x)|=\int_{\Sigma}\left(|\partial_{\mu}\chi(\sigma_{1},x)|+|\partial_{\mu}\chi(\sigma_{2},x)|\right).$
(38)

Figure 4: $m_{1}$ and $m_{2}$ are two surfaces homologous to $A$ whose
intersection is such that their contributions to the first term in (36)
partially cancel. By cutting and pasting we can define new surfaces $m_{3}$
and $m_{4}$ with lower total bulk entropy and whose contributions to the first
term in (37) do not cancel.

Now, suppose we are given a measure $\mu$ that has support on a pair of bulk
regions that are problematic in the sense we have been discussing; they do not
satisfy (38). Since we are minimising over probability measures on
$\Omega_{A}$, it would be sufficient to show that we can define a new measure
$\mu^{\prime}$ from $\mu$, which does not increase the value of the objective
function we are minimising over, and does not have support on both
$\sigma_{1}$ and $\sigma_{2}$.

Take such a probability measure $\mu$ with support on $\sigma_{1}$ and
$\sigma_{2}$, and assume without loss of generality that $d\mu(\sigma_{1})\leq
d\mu(\sigma_{2})$. Define new bulk homology regions that are also in
$\Omega_{A}$

$\begin{split}\sigma_{3}&:=\sigma_{1}\cup\sigma_{2}\\\
\sigma_{4}&:=\sigma_{1}\cap\sigma_{2}\end{split}$ (39)

and a new probability measure $\mu^{\prime}$ which is identical to $\mu$
except we move support from $\sigma_{1}$ and $\sigma_{2}$ to $\sigma_{3}$ and
$\sigma_{4}$:

$\begin{split}d\mu^{\prime}(\sigma_{1})&=0\\\
d\mu^{\prime}(\sigma_{2})&=d\mu(\sigma_{2})-d\mu(\sigma_{1})\\\
d\mu^{\prime}(\sigma_{3})&=d\mu(\sigma_{3})+d\mu(\sigma_{1})\\\
d\mu^{\prime}(\sigma_{4})&=d\mu(\sigma_{4})+d\mu(\sigma_{1}).\\\ \end{split}$
(40)

Changing $\mu\mapsto\mu^{\prime}$ cannot increase the objective function in
(36); the first term is unchanged, and the entropy term cannot increase
because of the strong subadditivity of bulk entropies

$\begin{split}S_{\text{bulk}}(\sigma_{1})+S_{\text{bulk}}(\sigma_{2})&\geq
S_{\text{bulk}}(\sigma_{1}\cap\sigma_{2})+S_{\text{bulk}}(\sigma_{1}\cup\sigma_{2})\\\
&=S_{\text{bulk}}(\sigma_{3})+S_{\text{bulk}}(\sigma_{4}).\end{split}$ (41)

For an arbitrary measure $\mu$ we imagine repeating this process, tracing a
path through the space of probability measures, until we are left with a
measure with support only on surfaces which do not cancel one another inside
the norm of (36), and whose evaluation in the objective function of the
Lagrangian dual can not be greater than that of the original measure. This
establishes that (36) and (37) are equivalent, and so our quantum bit thread
prescription and the QES prescription are equivalent.

Argument 2777We credit Matt Headrick with the basic idea behind this argument.

Strong subadditivity of the bulk entropies is also key to the second way of
arguing equivalence of our Lagrangian dual problem to the QES prescription. We
start from an entropy inequality that follows from the strong subadditivity
relation being repeatedly applied to the von Neumann entropies of $N$ regions
Casini2012 :

$\sum_{i=1}^{N}S(X_{i})\geq S(\cup_{i}X_{i})+S(\cup_{\\{ij\\}}X_{i}\cap
X_{j})+...+S(\cap_{i}X_{i}).$ (42)

We apply this relation to an arbitrary set of $N$ bulk homology regions
$\sigma_{i}$ (with as usual $\partial\sigma_{i}=m_{i}\cup A$). The argument of
the $n$th term on the right-hand side of (42) is the bulk region $r(n/N)$,
which is the union of all points which are in at least a fraction $n/N$ of
those bulk regions $\sigma_{i}$,

$r(n/N):=\\{x\in\Sigma:\psi(x)\geq n/N\\}$ (43)

with $\psi(x)$ the fraction of $\sigma_{i}$ that contain the point $x$,

$\psi(x):=\frac{1}{N}\sum_{i=1}^{N}\chi(\sigma_{i},x)$ (44)

With these definitions (42) becomes

$\sum_{i=1}^{N}S_{\text{bulk}}(\sigma_{i}))\geq\sum_{i=1}^{N}S_{\text{bulk}}(r(i/N))$
(45)

which in the $N\to\infty$ limit (after dividing both sides by $N$) becomes

$\int_{\Omega_{A}}d\mu(\sigma)S_{\text{bulk}}(\sigma)\geq\int_{0}^{1}dp\,S_{\text{bulk}}(r(p))$
(46)

with $\mu$ again an arbitrary probability measure on $\Omega_{A}$. This is the
first relation we need to prove equivalence of (36) to the QES prescription.

In the $N\to\infty$ limit, $\psi(x)$ as defined in (44) becomes the expression
that appears in the first term of (36):

$\psi(x)=\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x).$ (47)

From the fact that $\mu$ is a probability measure, $\psi$ satisfies
$0\leq\psi(x)\leq 1$ and $\psi(x)|_{\partial\Sigma}=\chi(A,x)$, where
$\chi(A,x)$ is the characteristic function of $A\subseteq\partial\Sigma$,
whose domain is $\partial\Sigma$ and equals 1 for $x\in A$ and 0 for
$x\in\partial\Sigma\backslash A$.

From the codimension-0 subregions we defined earlier

$r(p):=\\{x\in\Sigma:\psi(x)\geq p\\},$ (48)

where $p$ is now a continuous variable in $[0,1]$, we define the associated
codimension-1 level sets $m(p):=\partial r(p)\backslash A$. This allows us to
rewrite the first term in the objective function of (36) as the average area
of the level sets $m(p)$ (recalling that the normal derivative of a
characteristic function is a surface delta function):

$\begin{split}\int_{\Sigma}\left|\partial_{\mu}\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x)\right|&=\int_{\Sigma}|\partial_{\mu}\psi(x)|\\\
&=\int_{0}^{1}dp\text{ Area}(m(p))\end{split}$ (49)

This is the second relation we need, which in combination with (46), allows us
to write the objective function of our Lagrangian dual optimisation problem as

$\begin{split}\int_{0}^{1}dp\left(\frac{\text{Area}(m(p))}{4G_{N}}+S_{\text{bulk}}(r(p))\right).\end{split}$
(50)

$r(p)$ with its associated level set $m(p)$ is determined by $\mu$ through
(47) and (48). Minimising this function with respect to $\mu$ is equivalent to
the QES prescription; the function is minimised when $\mu$ only has support on
the entanglement wedge time slice, when

$\begin{split}\psi(x)&=\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x)\\\
&=\chi(\sigma(m_{QES}),x).\end{split}$ (51)

This completes our proof that the quantum bit thread prescription is
equivalent to the quantum extremal surface prescription.

##  4 Discussion and future directions

Figure 5: It from qubit threads? The figure depicts three pairs of holographic
CFTs with differing degrees of entanglement between them, showing how quantum
bit threads are a mechanism (at least pictorially) for bulk spacetime
emergence. The red lines depict a flux-maximising flow configuration, the
dashed red lines connect entangled bulk degrees of freedom, and the fuzzy red
region represents the macroscopic wormhole.

Quantum bit threads give an interesting perspective on ER = EPR Maldacena2013
and the emergence of spacetime from entanglement. Consider the set-up shown in
Fig. 5, whose three subfigures depicts unentangled, weakly entangled, and
strongly entangled pairs of holographic CFTs. Our prescription tells us that
the number of threads that can jump from one bulk to the other is limited by
the bulk entanglement entropy. In the first pair of CFTs, the state is
unentangled between them, so no threads can jump. In the second pair, the CFTs
are weakly entangled, and we suppose that there are a few EPR pairs shared
between the two bulk spacetimes which allows an order $\mathcal{O}(1)$ number
of threads to jump across. Rather than thinking of the quantum bit threads as
jumping discontinuously from one bulk to the other, one could alternatively
imagine that each quantum bit thread, loosely speaking, travels between bulks
through its own wormhole of Planck scale cross-section. In the last pair of
CFTs we keep adding EPR pairs until we eventually form an entangled pair of
black holes joined by a classical wormhole. If we take this microscopic
wormhole idea seriously, then the natural interpretation of what has happened
is that the $\mathcal{O}(G_{N}^{-1})$ microscopic wormholes have coalesced
into a single macroscopic wormhole. This speculative interpretation, that bit
threads are in some sense Planck scale wormholes, and that bulk spacetimes are
built from them, is a direction it would be interesting to explore further.

(a) AdSd+2 picture.

(b) AdSd+1 picture.

Figure 6: Quantum bit threads in the island phase of a doubly holographic
model. $A$ and $\bar{A}$ are a pair of CFTd’s, and the lines with arrows are
representative of a flux-maximising flow from $A$ to $\bar{A}$. From the
AdSd+1 perspective, the quantum bit threads start from $A$, disappear at some
bulk point, reappear elsewhere, then end on $\bar{A}$. Some reach $\bar{A}$ by
reappearing in the island region $\mathcal{I}$, whose boundary
$\partial\mathcal{I}$ is an emergent bottleneck for quantum bit threads in the
island phase.

Fig. 6 shows a flux-maximising flow configuration in the island phase of a
doubly holographic set up888We expect that the comments we make here about
islands and quantum bit threads are true for more than just doubly holographic
models. We use these models because the behaviour of flux-maximising quantum
bit thread flows follows immediately from the behaviour of classical bit
threads in the highest dimensional picture, which are easier to work with.. To
capture the Area($\partial\mathcal{I}$) term of the island formula, in the
AdSd+2 picture some bit threads in the flux-maximising flow must rejoin the
boundary in the island region and travel through $\partial\mathcal{I}$. From
the quantum bit thread perspective in AdSd+1, islands are regions of the bulk
where, due to the entanglement structure of the bulk state, the bit threads of
flux-maximising flows are constrained to reappear; so many that a new
bottleneck emerges. The island phase transition is not discontinuous for bit
threads like it is for the surfaces involved. The island is disjoint from the
rest of $A$’s entanglement wedge, but they are connected by the quantum bit
threads that jump from one to the other; another manifestation of ER=EPR. In
some sense islands are an emergent phenomenon that materialise when too many
quantum bit threads try to reappear in a bulk region.

We found a prescription for bit threads that is accurate to all orders in
$1/N$, to leading order in the ’t Hooft coupling $\lambda$, and valid for time
reflection symmetric states. A significant goal of the bit thread program is
to find a prescription that is as widely applicable as current surface-based
holographic entanglement entropy constructions. There are covariant bit thread
prescriptions, valid at leading order in $1/N$ and for non-time reflection
symmetric states Headrick2022 , so it is natural to look for a covariant
quantum bit thread prescription. This is important because both time
dependence and quantum corrections are necessary to apply holographic
entanglement entropy to evaporating black holes. Regarding finite $\lambda$
corrections, classical bit threads have been generalised to bulk actions with
higher curvature gravitational terms, such as Gauss-Bonnet, see Harper2018 ,
and there are no obvious obstacles to directly combining finite $\lambda$ and
finite $N$ corrections to bit threads.

We can ask whether there are other quantum bit thread prescriptions than the
ones we have discussed. We do not know whether it is possible to have a
prescription that modifies the divergence constraint in a local way, say to
$\nabla_{\mu}v^{\mu}(x)=J(x)$, without needing an oracle to tell us where the
entangling surface will be as in proposal II.

We have proven equivalence of one particular proposal to the QES prescription,
but not uniqueness, nor, as we have mentioned, should we expect uniqueness.
One step in this direction is to ask the extent to which one could loosen or
tighten constraints in the prescription we have given. Our stance has been to
place constraints on $v$ that are sufficiently strong so that the flux through
$A$ does not exceed $S(A)$, but no stronger than that. One such tightening of
constraints would be to turn the divergence inequality constraint into an
equality, but this is too strong; it would require (25) to be satisfied at
each $x$ for all surfaces $m$ that pass through that point, which is generally
impossible.

###### Acknowledgements.

I would like to thank Matt Headrick and Qiang Wen for useful discussions and
comments on an earlier draft. I would also like to thank Gurbir Arora, Ben
Freivogel, Harsha Hampapura and Juan Pedraza for useful discussions. This
research was supported by the Stichting Nederlandse Wetenschappelijk Onderzoek
Instituten (NWO-I) through ‘Scanning New Horizons’, DoE grant DE-SC0009987 and
the Simons Foundation.

##  Appendix A Nesting and entropy inequalities

We want to prove that quantum bit threads satisfy the nesting property. This
is because there are flow-based proofs of the subadditivity and strong
subadditivity entropy inequalities which require the nesting property to hold
Freedman2016 . Since the Araki-Lieb inequality and positivity of von Neumann
entropy follow by subadditivity, a proof that quantum bit threads satisfy the
nesting property is an indirect proof of an important set of entropy
inequalities.

The nesting property for flows says that for any pair of boundary regions $A$
and $B$ there exists a flow $v$ which simultaneously maximises the flux
through both $A$ and $AB$999$AB$ is short-hand for $A\cup B$.. We want to show
that there exists a $v$, which satisfies the quantum bit thread prescription’s
constraints, such that

$\int_{AB}v=S(AB)\text{ and }\int_{A}v=S(A)$ (52)

where $S(A)$ is the von Neumann entropy of boundary region $A$, to all orders
in $1/N$, as calculated by either the QES or quantum bit thread prescriptions.

To prove that the nesting property holds, it is sufficient to show that

$\sup_{v}\left(\int_{A}v+\int_{AB}v\right)$ (53)

equals $S(A)+S(AB)$.

The flow $v$ is subject to the norm bound

$|v|\leq\frac{1}{4G_{N}}$ (54)

and a constraint that makes $v$ a valid quantum bit thread flow for both $A$
and $AB$

$\forall(\sigma\in\Omega_{A}\cup\Omega_{AB}):\qquad-\int_{\sigma}\nabla_{\mu}v^{\mu}\leq
S_{bulk}(\sigma)$ (55)

$\Omega_{A}$ is the set of bulk homology regions for boundary subregion $A$:
I’m not sure about my definition of $\Omega_{A}$. Is there a difference
between $m\sim A$ and $m\sim Arel\partial A$?

$\Omega_{A}:=\\{\sigma\subseteq\Sigma\>:\>\partial\sigma\supseteq A\\}$ (56)

and $\Omega_{AB}$ the corresponding set of bulk homology regions for $AB$.

The supremum (53) cannot be greater than $S(A)+S(AB)$ because the supremum of
the sum cannot be greater than the sum of the suprema, so if we can also show
that it cannot be less than $S(A)+S(AB)$ then it must be equal. This is a
proof strategy that we are adapting from Headrick2017 .

To (53) we add Karush-Kuhn-Tucker multiplier terms with non-negative
multipliers for the inequality constraints:

$\begin{split}\sup_{v}\inf_{\phi,\mu,\mu^{\prime}}\left[\int_{A}v+\int_{AB}v+\int_{\Sigma}\phi(x)\left(\frac{1}{4G_{N}}-|v|\right)+\int_{\Omega_{A}}d\mu(\sigma)\left((\int_{\Sigma}\chi(\sigma,x)\nabla\cdot
v(x))+S_{bulk}(\sigma)\right)\right.\\\
+\left.\int_{\Omega_{AB}}d\mu^{\prime}(\sigma)\left((\int_{\Sigma}\chi(\sigma,x)\nabla\cdot
v(x))+S_{bulk}(\sigma)\right)\right]\end{split}$ (57)

and then we integrate by parts the terms with derivatives of $v$

$\begin{split}\sup_{v}\inf_{\phi,\mu,\mu^{\prime}}\left[\left(2-\int_{\Omega_{A}}d\mu-\int_{\Omega_{AB}}d\mu^{\prime}\right)\int_{A}v+\left(1-\int_{\Omega_{AB}}d\mu^{\prime}\right)\int_{B}v-\int_{\Sigma}v\cdot\partial\psi(x)+\int_{\Sigma}\phi(x)\left(\frac{1}{4G_{N}}-|v|\right)\right.\\\
+\left.\int_{\Omega_{A}}d\mu(\sigma)S_{bulk}(\sigma)+\int_{\Omega_{AB}}d\mu^{\prime}(\sigma)S_{bulk}(\sigma)\right]\end{split}$
(58)

We have used that, by its definition, $\chi(\sigma,x)=1$ for $x\in A$ and $0$
for $x\in(\partial\Sigma\backslash A)$ for all $\sigma\in\Omega_{A}$. The
$\mu^{\prime}$ in the above equations should not be confused with that defined
in section 3.2; at this stage $\mu$ and $\mu^{\prime}$ are independent
measures on different sets of bulk homology regions. We have also introduced a
new quantity $\psi$ defined by

$\psi(x):=\int_{\Omega_{A}}d\mu(\sigma)\chi(\sigma,x)+\int_{\Omega_{AB}}d\mu^{\prime}(\sigma)\chi(\sigma,x)$
(59)

We are free to switch the order in which the supremum and infimum are taken in
(58) because Slater’s conditions are satisfied and so strong duality holds -
the constraint (55) is linear in $v$, and the non-linear constraint (54) has
the strictly feasible point $v=0$.

Let us then switch the order: take the supremum over $v$, and then the infimum
over $\mu$, $\mu^{\prime}$ and $\phi$. The supremum over $v$ on
$\partial\Sigma$ is infinite unless the coefficients of the boundary $v$ flux
terms in (58) are zero, so a finite infimum of the supremum must have $\mu$
and $\mu^{\prime}$ which are probability measures on $\Omega_{A}$ and
$\Omega_{AB}$ respectively:

$\int_{\Omega_{A}}d\mu=1;\qquad\int_{\Omega_{AB}}d\mu^{\prime}=1.$ (60)

This in turn implies that $0\leq\psi(x)\leq 2$ for $x\in\Sigma$, and that on
the boundary of $\Sigma$ we have $\psi(x)=2$ on $A$, $\psi(x)=1$ on $B$, and
$\psi(x)=0$ on $\partial\Sigma\backslash(AB)$.

Just as in (34) the supremum over $v$ in the interior of the bulk slice is
infinite unless $\phi(x)\geq|\partial\psi(x)|$, and taking the infimum with
respect to $\phi$ saturates this inequality. This leaves us with

$\inf_{\mu,\mu^{\prime}}\left[\frac{\int_{\Sigma}|\partial\psi(x)|}{4G_{N}}+\int_{\Omega_{A}}d\mu\,S_{\text{bulk}}(\sigma)+\int_{\Omega_{AB}}d\mu^{\prime}\,S_{\text{bulk}}(\sigma)\right]$
(61)

Next, we use a simple extension of the result (46) which is

$\int_{\Omega_{A}}d\mu
S_{\text{bulk}}(\sigma)+\int_{\Omega_{AB}}d\mu^{\prime}S_{\text{bulk}}(\sigma)\geq\int_{0}^{2}dpS_{\text{bulk}}(r(p))$
(62)

We have defined

$r(p):=\\{x\in\Sigma\,|\,\psi(x)\geq p\\}$ (63)

which has the properties

$\begin{split}r(p\leq 0)&=\Sigma\\\ r(0<p\leq 1)\cap\partial\Sigma&=A\cup B\\\
r(1<p\leq 2)\cap\partial\Sigma&=A\\\ r(2<p)&=\varnothing.\end{split}$ (64)

Equation (62) is proven the same way as (46): start from (42), except with
$2N$ bulk regions split evenly between $\Omega_{A}$ and $\Omega_{AB}$.

We also need

$\int_{\Sigma}|\partial\psi(x)|=\int dp\,\text{Area}(m(p))$ (65)

where $m(p):=(\partial r(p)\backslash\partial\Sigma)$. Note that
$m(p)=\varnothing$ for $p\leq 0$ and $p>2$, so the right hand side of (65)
only gets a contribution from $0<p\leq 2$. What if $m(p)$ reaches the boundary
somewhere other than $A$ or $B$?

Plugging (65) and (62) into (61) we arrive at the result that the supremum
(53) is lower bounded by

$\inf_{\mu,\mu^{\prime}}\left[\int_{0}^{2}dp\left(\frac{\text{Area}(m(p))}{4G_{N}}+S_{bulk}(r(p))\right)\right].$
(66)

Equation (66) is equal to the QES result for $S(AB)+S(A)$ because $m(p)$ is
homologous to $AB$ for $0<p\leq 1$, and to $A$ for $1<p\leq 2$; the infimum
has $\mu$ equal to a delta-function measure which picks out the bulk homology
region for the QES homologous to $A$, and similarly for $\mu^{\prime}$ with
respect to $AB$.

This completes the proof of the nesting property.

## References

  * (1) S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de Sitter space/conformal field theory correspondence, Physical Review Letters 96 (May, 2006).
  * (2) M. Freedman and M. Headrick, Bit threads and holographic entanglement, Comm. Math. Phys. 352, 407 (2017) (Apr., 2016) [arXiv:1604.00354].
  * (3) M. Headrick and V. E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35: 10 (2018) (Oct., 2017) [arXiv:1710.09516].
  * (4) X. Dong, Holographic entanglement entropy for general higher derivative gravity, JHEP 01 (2014) 044, [arXiv:1310.5713].
  * (5) J. Camps, Generalized entropy and higher derivative gravity, Journal of High Energy Physics 2014 (Oct., 2013) [arXiv:1310.6659].
  * (6) A. Bhattacharyya and M. Sharma, On entanglement entropy functionals in higher derivative gravity theories, JHEP 1410 (2014) 130 (May, 2014) [arXiv:1405.3511].
  * (7) J. Harper, M. Headrick, and A. Rolph, Bit threads in higher curvature gravity, J. High Energ. Phys. (2018) 2018: 168 (July, 2018) [arXiv:1807.04294].
  * (8) N. Engelhardt and A. C. Wall, Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 (Aug., 2014) [arXiv:1408.3203].
  * (9) C.-B. Chen, F.-W. Shu, and M.-H. Wu, Quantum bit threads of MERA tensor network in large $c$ limit, Chinese Physics C 44 (Apr., 2018) 075102, [arXiv:1804.00441].
  * (10) A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, Journal of High Energy Physics 2020 (Aug., 2019) [arXiv:1908.10996].
  * (11) C. A. Agón and J. F. Pedraza, Quantum bit threads and holographic entanglement, Journal of High Energy Physics 2022 (May, 2021) [arXiv:2105.08063].
  * (12) S. N. Solodukhin, Conical singularity and quantum corrections to the entropy of a black hole, Physical Review D 51 (Jan, 1995) 609–617.
  * (13) L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Physical Review D 50 (Aug, 1994) 2700–2711.
  * (14) S. N. Solodukhin, Entanglement entropy of black holes, Living Rev. Relativity 14, (2011), 8 (Apr., 2011) [arXiv:1104.3712].
  * (15) T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum corrections to holographic entanglement entropy, arXiv:1307.2892.
  * (16) Y. Chen and G. Vidal, Entanglement contour, J. Stat. Mech. (2014) P10011 (June, 2014) [arXiv:1406.1471].
  * (17) A. Rolph, Local measures of entanglement in black holes and CFTs, SciPost Phys. 12 (2022), no. 3 079, [arXiv:2107.11385].
  * (18) J. Kudler-Flam, I. MacCormack, and S. Ryu, Holographic entanglement contour, bit threads, and the entanglement tsunami, J. Phys. A: Math. Theor. 52 (2019) 325401 (Feb., 2019) [arXiv:1902.04654].
  * (19) J. H. Cooperman and M. A. Luty, Renormalization of entanglement entropy and the gravitational effective action, JHEP 12 (2014) 045, [arXiv:1302.1878].
  * (20) A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, Journal of High Energy Physics 2019 (May, 2019) [arXiv:1905.08762].
  * (21) G. Penington, Entanglement wedge reconstruction and the information paradox, Journal of High Energy Physics 2020 (May, 2019) [arXiv:1905.08255].
  * (22) H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Physical Review D 85 (Feb., 2012) 125016, [arXiv:1202.5650].
  * (23) J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781–811, [arXiv:1306.0533].
  * (24) M. Headrick and V. E. Hubeny, Covariant bit threads, arXiv:2208.10507.