# Inequalities for the holographic entanglement of purification in BCFT

Pan Li 1,2<EMAIL_ADDRESS>Yi Ling 1,2<EMAIL_ADDRESS>1Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
2 School of Physics, University of Chinese Academy of Sciences, Beijing
100049, China

###### Abstract

The entanglement wedge cross section in holographic picture provides a
geometrical description of the entanglement of purification for mixed states.
In this paper we study the inequalities for the entanglement of purification
in BCFT by holographic duality. In the presence of the boundary in CFT, the
entanglement exhibits abundant phase structures since the extremal surface may
end on the brane. We present a universal treatment which is applicable for all
the possible phases such that the inequalities for the holographic
entanglement of purification can be proved in an algebraic manner rather than
a diagrammatic manner. We show that the holographic entanglement of
purification in BCFT satisfies the same inequalities as in CFT.

##  I Introduction

Recent progress on gauge/gravity duality discloses the deep connections
between spacetime geometry and quantum entanglement, and sheds light on
solving the fundamental problems in black hole physics and quantum information
theory. In this direction the RT formula Ryu:2006bv , and the latter HRT
formula Hubeny:2007xt  play the central role in linking spacetime geometry to
quantum entanglement. Explicitly, they conjecture that the entanglement
entropy in quantum field theory defined on boundary of AdS spacetime can be
described by the extremal surface in the bulk, which is now also called
holographic entanglement entropy (HEE)Ryu:2006bv ; Hubeny:2007xt . The
holographic description of entanglement entropy provides an elegant
geometrical realization of complicated entanglement structures, and stimulates
further explorations on the relationship between the spacetime structure and
quantum entanglement, such the ER=EPR conjectureMaldacena:2013xja , the
quantum chaos Shenker:2013pqa  as well as the island paradigm recently
proposed for the black hole information loss paradoxAlmheiri:2019psf
Penington:2019npb .

With the power of RT and HRT formulae, some entanglement properties such as
the inequalities for entanglement entropy, which look very complicated and
hard to prove from the side of quantum field theory, can be transformed into
the tractable geometric problems on spacetime, which usually become more
transparent and have an intuitive interpretation. For instance, the
subadditivity and strong subadditivity of the entanglement entropy can be
easily and intuitively proved with the language of the minimal surface in a
geometric wayHeadrick:2007km . Moreover, it is found that HEE is subject to
some inequalities that generic quantum states do not satisfy, such as the
monogamy inequality for the mutual information (MMI)Hayden:2011ag  (at leading
order of $\frac{1}{G_{N}}$). These inequalities give us new insight to justify
which states in CFT may have classical bulk gravity duals. A lot of work in
this subject can be found in literature, for instance see Bao:2015bfa
HernandezCuenca:2019wgh .

Entanglement entropy is a good measure for the correlation between two
subsystems $A$ and $\bar{A}$ when the whole system $(A\cup\bar{A})$ is
described by a pure state $|\phi_{A\bar{A}}\rangle$. In this case the
entanglement entropy of $A$ is equal to that of $\bar{A}$:
$S(A)=tr(\rho_{A}ln\rho_{A})=S(\bar{A})=tr(\rho_{\bar{A}}ln\rho_{\bar{A}})$,
where $\rho_{A}$ or $\rho_{\bar{A}}$ is the reduced density matrix of the
subsystem $A$ or $\bar{A}$, respectively. However, when the union of two
subregions ($A\cup B\equiv AB$) is not described by a pure state, for instance
in a tripartite system with three subregions $A,B$ and $C$, then the
correlations between $A$ and $B$ can not be precisely captured by either the
entanglement entropy $S(A)$ or $S(B)$, and in general $S(A)$ is not equal to
$S(B)$ anymore. As a matter of fact, one has to face this problem if one
intends to understand the fine structure of correlations in an entangled many-
body system. In this circumstance, one traditional way is to introduce the
notion of mutual information, which is defined as $I(A:B)=S(A)+S(B)-S(AB)$, to
describe the correlations between $A$ and $B$. The other way is to purify the
subsystem $AB$ by embedding it into a larger system which is described by a
pure state, and then one finds the minimum of the entanglement entropy between
two subsystems including $A$ and $B$ respectively, among all the possible
purifications, which is called the entanglement of purification ($E_{P}$). In
a word, the entanglement of purification $E_{P}$ is proposed to measure the
correlations between $A$ and $B$ in mixed states. Nevertheless, the above
definition of $E_{P}$ involves in an extremal process, and the practical
operation is always cumbersome and complicated and thus evaluating $E_{P}$ is
a notoriously difficult task in field theoryHirai:2018jwy ; Caputa:2018xuf .
Thanks to holographic duality, inspired by the geometric description of
entanglement entropy, it is conjectured that the entanglement of purification
$E_{P}$ may also be evaluated by a geometric quantity, namely the entanglement
wedge cross section ($E_{W}$)Takayanagi:2017knl Nguyen:2017yqw . Given two
disjoint regions A and B on the boundary, one can firstly obtain the minimal
surface ($\Gamma_{AB}^{min}$) of $AB$ as the dual of entanglement entropy.
Then the bulk region surrounded by AB and $\Gamma_{AB}^{min}$ is the
entanglement wedge $M_{AB}$. $E_{W}$ is defined as the area of the minimal
surface which separates $M_{AB}$ into two bulk subregions associated to A and
B, respectively. Some evidences supporting this conjecture can be found in
Hirai:2018jwy Caputa:2018xuf , and further investigations on $E_{W}=E_{P}$ can
be found in Umemoto:2018jpc -Liu:2019qje . Alternatively, there are some other
quantities proposed in field theory as the avatars for $E_{W}$, which can be
found in Dutta:2019gen Kudler-Flam:2018qjo Tamaoka:2018ned .

The conjecture of “$E_{W}=E_{P}$” was proposed partly based on the fact that
both $E_{W}$ and $E_{P}$ satisfy a set of identical inequalities. Some of
inequalities were previously proved for $E_{W}$ in the context of AdS/CFT
Takayanagi:2017knl , with the focus on some typical and specific diagrams (For
instance, each CFT subregion in inequalities is connected.). In this
diagrammatic proof, one need firstly determine the specific phase of the
minimal surface as well as the corresponding entanglement wedge, which usually
depends on the size and shape of the subregions and their relative positions.
The existence of many possible phases for the minimal surface will make the
diagrammatic proof tediously complicated and even become an obstacle. This
indeed happens when boundary subregions consist of many disconnected parts,
i.e.$A=A_{1}\cup A_{2}\cup...$, and in particular when we intend to prove
these inequalities in the context of AdS/BCFTTakayanagi:2011zk , where the
presence of the boundary of CFT brings lots of new entanglement phases from
the bulk point of view Chou:2020ffc . So in this paper we intend to develop an
algebraic proof for these inequalities in a quite general setup which applies
to the case that the subregions of CFT may have disconnected parts, and more
importantly to the case that the bulk spacetime may contain branes such as in
the context of AdS/BCFT. Our strategy is to define some general sets for the
candidates of the HEE and $E_{W}$. Then based on the structure of these sets
and by virtue of “wedge nesting” Czech:2012bh -Headrick:2013zda , we provide a
general treatment which is applicable for various entanglement phases so as to
avoid the complicated enumerations of all possible diagrams. With the similar
spirit the inequalities for HEE have been investigated in Hayden:2011ag ;
Bao:2015bfa ; Headrick:2013zda .

The paper is organized as follows. In next section we will construct the
general setup for the proof of $E_{W}$ inequalities, which largely relies on
the definitions of HEE and $E_{W}$, as well as the property of “wedge
nesting”. We will introduce a candidate set for HEE and a candidate set for
$E_{W}$ in both contexts of AdS/CFT and AdS/BCFT correspondence. In section 3,
we will apply the definition of those sets to accomplish the proof for the
inequalities of $E_{W}$ in an algebraic manner, which is applicable for
different entanglement phases. We compare this method with the diagrammatic
proof in the section of discussion.

##  II The general setup

We start with a general asymptotically AdS spacetime with CFT living on the
boundary. In the case of AdS/BCFTTakayanagi:2011zk , the bulk dual of BCFT
cantains brane Q. We will consider the simplest case where the brane $Q$ is
embedded into AdS spacetime perpendicular to the conformal boundary. We
consider a static time-reversal symmetric slice in the bulk. $M$ represents
the whole bulk spatial region, and $\partial M$ is the asymptotic AdS boundary
where the BCFT lives. In addition, we will prove the inequalities of
entanglement in the large N limit, which implies that only the leading term is
taken into account and the bulk fields satisfy the classical Einstein
equation.

Now, consider subregions on the boundary $A$, $B$, $C$ …. It is assumed that
they do not overlap with any non-zero size (but they can be adjacent).
Moreover, we stress that each of the subregion can be the union of many
disconnected parts, i.e. $A=A_{1}\cup A_{2}\cup...,\quad B=B_{1}\cup
B_{2}\cup...$. In addition, we remark that in this paper all elements linked
by the symbol $\cup$ do not overlap with one another. The entanglement wedge
in the bulk associated with the subregion $X$ on the boundary is denoted as
$M_{X}$.

###  II.1 HEE

In this subsection we will review the definition of holographic entanglement
entropy and then define the set consisting of the candidates for the minimal
surface.

####  II.1.1 HEE in AdS/CFT correspondence

Given a subregion $A$ on the boundary of AdS, $\Gamma_{A}$ is defined as a
surface in the bulk satisfying the following conditions:

  1. 1.

$\Gamma_{A}$ have the same boundary with $A:\quad\partial
A=\partial\Gamma_{A}$.

  2. 2.

$\Gamma_{A}$ is homologous with $A:\quad\exists R_{A}\subset M,\partial
R_{A}=A\cup\Gamma_{A}$.

We collect all these elements to define a set $\bm{\mathit{\Gamma_{A}}}$ for
candidates of the minimal surface

$\bm{\mathit{\Gamma_{A}}}=\left\\{1.\ \ \Gamma_{A}|\partial
A=\partial\Gamma_{A};2.\ \ \exists R_{A}\subset M,\partial
R_{A}=A\cup\Gamma_{A}\right\\}.$ (1)

By definition of Ryu:2006bv , we have

$S(A)=\underset{\Gamma_{A}}{min}\frac{\mathcal{A}(\Gamma_{A})}{4G_{N}},$ (2)

Figure 1:  The schematic demonstration of three different candidate surfaces
for the minimal surface, each of which has the same boundary with $A$ and
homologous with $A$. We pick the one that has the minimal area to be the
minimal surface, which is plotted in green. The corresponding entanglement
wedge $M_{A}$ is shaded in grey.

where $\mathcal{A}(\Gamma_{A})$ represents the area of $\Gamma_{A}$. We define
the element with the minimal area in set $\bm{\mathit{\Gamma_{A}}}$ as
$\Gamma^{min}_{A}$, and the above formula means that the area of
$\Gamma^{min}_{A}$ over $4G_{N}$ is interpreted as the entanglement entropy
between the subregion $A$ and its complementary. Correspondingly, the interior
region $M_{A}$ surrounded by $A$ and $\Gamma^{min}_{A}$ is called the
entanglement wedge of A (at a static slice). In Fig.(1) we plot three typical
configurations for candidates of the minimal surface.

####  II.1.2 HEE in AdS/BCFT correspondence

The boundary of CFT may be implemented by embedding branes into the bulk
spacetime. We consider the simplest case where branes are set to be
perpendicular to the boundary of AdS such that the backreaction of the branes
can be ignored. In this situation, the possible configurations of the minimal
surface become slightly different, which may end on the branes. Therefore, we
define the candidate set for the minimal surface for BCFT asMiao:2017gyt
Chang:2018pnb :

$\bm{\mathit{\Gamma_{A}}}=\left\\{\Gamma_{A}|1.\ \
\partial\Gamma_{A}|_{\partial M}=\partial A,\ \
\partial\Gamma_{A}|_{Q}=\partial Q_{A};\ \ 2.\ \ \exists R_{A}\subset
M,\partial R_{A}=A\cup Q_{A}\cup\Gamma_{A}\right\\},$ (3)

and

$S(A)=\underset{\Gamma_{A}}{min}\frac{\mathcal{A}(\Gamma_{A})}{G_{N}},$ (4)

where $Q$ denotes branes in AdS, and $Q_{A}$ represents the part of the
boundary of $R_{A}$ which locates in the brane $Q$. It is noticed that the
area of $Q_{A}$ is not included as part of $S(A)$. The different phases of the
minimal surface in AdS/BCFT are illustrated in Fig. 2 and we name these phases
following Chou:2020ffc .

Figure 2:  Three possible phases for the minimal surface and the corresponding
entanglement wedge when the subregion is located at different positions
relative to the brane $Q$ in boundary CFT Chou:2020ffc .

###  II.2 $E_{W}$ in AdS/BCFT correspondence

In this subsection, we would like to extend the notion of entanglement wedge
cross section $E_{W}$ originally proposed in Takayanagi:2017knl Nguyen:2017yqw
in AdS/CFT to the context of AdS/BCFT 111The same extension for $E_{W}$ in
AdS/BCFT appears in Li:2021dmf ; Ling:2021vxe , where $E_{W}$ is conjectured
to be dual to the reflected entropy rather than the entanglement of
purification, and the effects of islands and quantum corrections due to bulk
fields are taken into account as well.. One new feature of $E_{W}$ in AdS/BCFT
duality is that $E_{W}$ may end on the brane Q. Here we will introduce a set
for candidates of the minimal entanglement wedge cross section.

In comparison with the holographic description of entanglement entropy, the
key difference in the holographic description of $E_{P}$ is that it always
involves in at least two subregions $A$ and $B$, and we need to divide the
entanglement wedge $M_{AB}$ into two parts to define the cross section, which
looks more complicated. However, we remark that it is very worthy introducing
such sets to collect all the candidates of $\Sigma_{A:B}^{min}$. Once such
sets are constructed, then the proof of inequalities will become very concise
and elegant, as we will demonstrate in next sections. For clearness, we divide
the procedure of figuring out $E_{W}(A:B)$ into following steps.

Figure 3:  Three typical phases of S(AB), which depend on relative positions
of $A$, $B$ and the brane Q.

  1. 1.

Given subregions $A$ and $B$ on the boundary CFT, one can directly figure out
the minimal surface $\Gamma_{AB}^{min}$. Then the corresponding entanglement
wedge of $AB$ in the bulk is denoted as $M_{AB}$, which is defined by

$\partial M_{AB}=A\cup B\cup\Gamma_{AB}^{min}\cup Q_{AB},$ (5)

as we demonstrate in Fig.(3).

  2. 2.

Division: We divide $\Gamma_{AB}^{min}$ and $Q_{AB}$ into two individual parts
respectively

$\displaystyle\Gamma_{AB}^{min}=\Gamma_{AB}^{A}\cup\Gamma_{AB}^{B},$ (6)
$\displaystyle Q_{AB}=Q_{AB}^{A}\cup Q_{AB}^{B}.$ (7)

  3. 3.

Recombination: We define

$\displaystyle\widetilde{A}=A\cup\Gamma_{AB}^{A}\cup Q_{AB}^{A},$ (8)
$\displaystyle\widetilde{B}=B\cup\Gamma_{AB}^{B}\cup Q_{AB}^{B},$ (9)

such that the contour of the entanglement wedge $M_{AB}$ is composed of these
two parts, namely $\partial M_{AB}=\widetilde{A}\cup\widetilde{B}$.

  4. 4.

Minimization: Usually the procedure of the minimization has the following
steps. Firstly, one specifies $\widetilde{A}$ and $\widetilde{B}$, and then
finds the minimal surface $\Sigma^{min}_{A:B}(\widetilde{A}$) homologous to
$\widetilde{A}$, as illustrated in the left plot of Fig.(4). Secondly, varying
$\widetilde{A}$ and $\widetilde{B}$, one finds the corresponding
$\Sigma^{min}_{A:B}(\widetilde{A}$) again, as illustrated in the right plot of
Fig.(4). Finally, one figures out the minimal cross section
$\Sigma_{A:B}^{min}$ among $\Sigma^{min}_{A:B}(\widetilde{A}$) for all the
possible $\widetilde{A}$, and identifies its area as the entanglement of
purification Takayanagi:2017knl , namely

$E_{P}(A:B)=E_{W}(A:B)={min}\frac{\mathcal{A}\left(\Sigma_{A:B}^{min}(\widetilde{A})\right)}{4G_{N}}.$
(10)

We may rephrase the above procedure of minimization by defining a set
$\bm{\mathit{\Sigma^{min}_{A:B}}}$, which collect all
$\Sigma_{A:B}^{min}(\widetilde{A})$ as the candidates for the minimal
entanglement wedge cross section

$\begin{split}\bm{\mathit{\Sigma^{min}_{A:B}}}=\\{\Sigma_{A:B}^{min}(\widetilde{A})|\
\ &1.\ \partial\Sigma_{A:B}=\partial\widetilde{A};\quad 2.\ \exists
R_{A}\subset M_{AB},\partial R_{A}=\Sigma_{A:B}\cup\widetilde{A};\\\ &3.\ \
\Sigma_{A:B}^{min}(\widetilde{A})\text{ is the minimal surface among
}\Sigma_{A:B}\text{ with }\widetilde{A}\text{ specified}\\}.\end{split}$ (11)

As a result, the element with the minimal area in the above set corresponds to
$E_{P}(A:B)$.

  5. 5.

Extension: for our purpose we may remove condition 3 in
$\bm{\mathit{\Sigma^{min}_{A:B}}}$ to define a larger set

$\bm{\mathit{\Sigma_{A:B}}}=\\{\Sigma_{A:B}|1.\ \
\partial\Sigma_{A:B}=\partial\widetilde{A};\quad 2.\ \ \exists R_{A}\subset
M_{AB},\partial R_{A}=\Sigma_{A:B}\cup\widetilde{A}\\}.$ (12)

Figure 4:  The process demonstration of figuring out the minimal cross section
$\Sigma_{A:B}^{min}$ for CFT, which is identified as the one with the minimal
area among $\Sigma^{min}_{A:B}(\widetilde{A})$ for all the possible
$\widetilde{A}$. Figure 5:  The process demonstration of figuring out the
minimal cross section $\Sigma^{min}_{A:B}$ for BCFT, where there are more
possible phases for $\Sigma_{A:B}^{min}(\widetilde{A})$. We just show two
typical phases here.

In fact, after the extension, all surfaces homologous to all possible
$\widetilde{A}$ are collected in this set (In contrast,
$\bm{\mathit{\Sigma^{min}_{A:B}}}$ just collects the cross sections with the
minimal area which are homologous to any possible $\widetilde{A}$), as
illustrated in Fig.(4) and (5). Roughly speaking, we could understand this set
as the collection of all the possible cross sections which divide the
entanglement wedge $M_{AB}$ into two parts and successfully separate $A$ and
$B$. Moreover, by definition we know that the element with the smallest area
in $\bm{\mathit{\Sigma^{min}_{A:B}}}$ must be the element with the smallest
area in $\bm{\mathit{\Sigma_{A:B}}}$ as well. It means that the entanglement
of purification $E_{P}$ can also be identified as the area of the minimal
cross section over 4$G_{N}$ in set $\bm{\mathit{\Sigma_{A:B}}}$, namely

$E_{P}(A:B)=E_{W}(A:B)={min}\frac{\mathcal{A}\left(\Sigma_{A:B}\right)}{4G_{N}}.$
(13)

From now on, we restrict $\Sigma_{A:B}^{min}$ to represent the minimal-area
element in $\bm{\mathit{\Sigma_{A:B}}}$. In other words,
$\mathcal{A}\left(\Sigma_{A:B}^{min}\right)=E_{W}(A:B)$, where for
convenience, we have also simply set $4G_{N}=1$.

###  II.3 Wedge nesting

To prove some inequalities of $E_{P}$ in an algebraic way, it is essential to
employ the properties called “wedge nesting”, which was investigated in
Czech:2012bh ; Wall:2012uf ; Headrick:2014cta ; Headrick:2013zda . Basically,
“wedge nesting” contains the following two statements about the relation
between boundary regions and the corresponding entanglement wedge

    1. (a)

$A\cap B=\emptyset\Rightarrow$ $M_{A}\cap M_{B}=\emptyset$.

    2. (b)

$A\subset B\Rightarrow M_{A}\subset M_{B}$.

These two properties play a key role in the proof of some inequalities for
$E_{P}$, as we show in the next section.

##  III The inequalities of $E_{P}$ in AdS/BCFT duality

Now with all the ingredients at hand, we intend to prove the following five
inequalities of $E_{W}$ in the context of AdS/BCFT.

$\displaystyle\min\left\\{S(A),S(B)\right\\}\geq E_{W}(A:B),$ (14)
$\displaystyle E_{W}(A:B)\geq\frac{1}{2}I(A:B),$ (15) $\displaystyle
E_{W}(A:BC)\geq E_{W}(A:B),$ (16) $\displaystyle
E_{W}(A:BC)\geq\frac{1}{2}I(A:B)+\frac{1}{2}I(A:C),$ (17) $\displaystyle
E_{W}(AB:CD)\geq E_{W}(A:C)+E_{W}(B:D).$ (18)

Previously such inequalities have been considered for AdS/CFT in
Takayanagi:2017knl . In addition, some of inequalities for holographic
entanglement entropy in the context of AdS/BCFT was presented inChou:2020ffc .

In principle, we can prove the above inequalities as the way performed in
Chou:2020ffc . Classifying and listing all possible entanglement phases
appearing in those inequalities, visualizing each phase with an individual
diagram, and then showing the proof in all possible diagrams. But as we
mentioned before, the boundary effect of BCFT brings many new entanglement
phases in the dual bulk. Even just for the proof of the inequalities for HEE
in BCFT, there are a large number of diagrams to show Chou:2020ffc , not to
mention the phases of entanglement wedge cross sections which are constructed
based on the phases of entanglement entropy, and thus become more complicated.
(seeing Fig.(6) for an example.) Therefore, with the help of the candidate
sets $\bm{\mathit{\Gamma_{A}}}$, $\bm{\mathit{\Sigma_{A:B}}}$, here we develop
an algebraic way to prove the inequalities of $E_{W}$, which does not depend
on the entanglement phases.

Figure 6:  The phase transition of $E_{W}(A:B)$ when changing the length of A,
B and their relative positions with brane Q.

The general idea is following: the geometric correspondence of entropy
inequalities in holography are essentially about the area of different
surfaces. It is noticed that all inequalities we intend to prove can be
translated into the following form:
$\mathcal{A}(X)\geq\mathcal{A}(\Gamma^{min}_{A})$ or
$\mathcal{A}(Y)\geq\mathcal{A}(\Sigma^{min}_{B:C})$. (A, B and C are just
arbitrarily chosen.) Recall that $\Gamma^{min}_{A}$ and $\Sigma^{min}_{B:C}$
are the minimal surfaces in $\bm{\mathit{\Gamma_{A}}}$ or
$\bm{\mathit{\Sigma_{B:C}}}$. If one can prove that the surface $X$ or $Y$ on
the left-hand side belongs to the set $\bm{\mathit{\Gamma_{A}}}$ or
$\bm{\mathit{\Sigma_{B:C}}}$, then above inequalities will be satisfied,
according to the definitions of $\bm{\mathit{\Gamma_{A}}}$ and
$\bm{\mathit{\Sigma_{B:C}}}$. In this way one great advantage is that one does
not need to figure out the specific phases of surfaces $X$ or $Y$, on the
contrary, one just needs to show they belong to $\bm{\mathit{\Gamma_{A}}}$ or
$\bm{\mathit{\Sigma_{B:C}}}$, respectively.

###  III.1 $\min\left\\{S(A),S(B)\right\\}\geq E_{W}(A:B)$

This inequality indicates that both holographic entanglement entropy $S(A)$
and $S(B)$ must be larger than or equal to the entanglement of purification
between subregion $A$ and $B$. It can be proved as follows:

Obviously, by definition, the quantity $E_{W}(A:B)$ on the right-hand side in
this inequality corresponds the surface $\Sigma_{A:B}^{min}$ (the minimal-area
element in $\bm{\mathit{\Sigma_{A:B}}}$), therefore following the above
strategy we just need to prove that the surface associated with any quantity
on the left-hand side is an element belonging to the set
$\bm{\mathit{\Sigma_{A:B}}}$ as well. Without loss of generality, let us
consider the entanglement entropy of region $A$. Since the corresponding
minimal surface of $S(A)$ is $\Gamma_{A}^{min}$, we just need to prove
$\Gamma_{A}^{min}\in\bm{\mathit{\Sigma_{A:B}}}$. Firstly, based on the
definition of $\Gamma_{A}^{min}$, we know that

$\displaystyle(\partial\Gamma_{A}^{min})|_{\partial M}$
$\displaystyle=\partial A,$ (19)
$\displaystyle(\partial\Gamma_{A}^{min})|_{Q}$ $\displaystyle=\partial Q_{A},$
(20) $\displaystyle\quad\partial M_{A}$
$\displaystyle=\Gamma_{A}^{min}\cup\left(A\cup Q_{A}\right).$ (21)

Secondly, employing “wedge nesting” we know $M_{A}\subset M_{AB}$, so let
$\widetilde{A}=\left(A\cup Q_{A}\right)$, then we get

$\displaystyle\partial\Gamma_{A}^{min}$ $\displaystyle=\partial\widetilde{A},$
(22) $\displaystyle M_{A}\subset M_{AB},\quad\partial M_{A}$
$\displaystyle=\Gamma_{A}^{min}\cup\widetilde{A}.$ (23)

Two conditions in (12) for the elements in $\bm{\mathit{\Sigma_{A:B}}}$ are
satisfied. Therefore, $\Gamma_{A}^{min}\in\bm{\mathit{\Sigma_{A:B}}}$, and one
must have $S(A)\geq E_{W}(A:B)$, following the definition in (12). Or
intuitively, we may state that $\Gamma_{A}^{min}$ is a cross section which
divides the entanglement wedge $M_{AB}$ into two parts and separates $A$ and
$B$, by definition its area must not be smaller than the minimal cross section
of $M_{AB}$, which is juts defined as $E_{W}(A:B)$. For the same reason one
can get $S(B)\geq E_{W}(A:B)$, thus the above inequality is proved.

###  III.2 $E_{W}(A:B)\geq\frac{1}{2}I(A:B)$

This inequality states that $E_{W}(A:B)$ must not be smaller than one half of
the holographic mutual information between $A$ and $B$. Since the mutual
information is constructed by the entanglement entropy, explicitly
$I(A,B)=S(A)+S(B)-S(AB)$, we may change the inequality into the form

$2E_{W}(A:B)+S(AB)\geq S(A)+S(B).$ (24)

Two examples of diagrammatic proof are demonstrated in Fig.(8) and (8) for CFT
and BCFT respectively, where one typical phase of $E_{W}(A:B)$ is plotted to
demonstrate this inequality geometrically. Nevertheless, from Fig.(3) and
Fig.(6) one knows that there are various possible phases for $S(AB)$ and
$E_{W}(A:B)$. If one insists to prove this inequality in a diagrammatic
manner, then she/he has to enumerate all the possible phases case by case,
that would be very cumbersome. Therefore, we present the following proof in an
algebraic manner.

Figure 7:  The diagrammatic demonstration of inequality (24) in AdS/CFT. Since
the union of the green line and the red dotted line is homologous to $A$,
while the blue line is the minimal surface homologous to $A$, we have
$\mathcal{A}(\Gamma_{AB}^{A})+E_{W}(A:B)\geq S(A)$. Similarly,
$\mathcal{A}(\Gamma_{AB}^{B})+E_{W}(A:B)\geq S(B)$. Combining these two
inequalities one can get $2E_{W}(A:B)+S(AB)\geq S(A)+S(B)$.

Figure 8:  The diagrammatic demonstration of inequality (24) in AdS/BCFT.

By definition, the geometric correspondence of these quantities are given by

$\displaystyle S(AB)=\mathcal{A}\left(\Gamma_{AB}^{min}\right),$ (25)
$\displaystyle E_{W}(A:B)=\mathcal{A}(\Sigma_{A:B}^{min}).$ (26)

It is noticed that $\Sigma_{A:B}^{min}$ divides $\Gamma_{AB}^{min}$ into two
parts $\Gamma_{AB}^{min}=\Gamma_{AB}^{A}\cup\Gamma_{AB}^{B}$. Note the
quantity on the left-hand side in (24) can be recombined as

$\mathcal{A}(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min})+\mathcal{A}(\Gamma_{AB}^{B}+\Sigma_{A:B}^{min}).$
(27)

We demonstrate this procedure in Fig.(8) for CFT case, while in Fig.(8) for
BCFT case. Now, our key observation is that
$\left(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min}\right)$ actually becomes a surface
which belongs to the set $\bm{\mathit{\Gamma_{A}}}$, namely the collection of
candidates for the minimal surface associated with subregion $A$ (See the
example in Fig.(8) and (8)). That is,
$\left(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min}\right)\in\bm{\mathit{\Gamma_{A}}}$,
since the conditions in the definition of $\bm{\mathit{\Gamma_{A}}}$ in (3)
are satisfied by $\left(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min}\right)$. We check
these conditions explicitly as follows.

For condition 1. By the definition of $\Sigma_{AB}^{min}$, we have

$\partial\Sigma_{AB}^{min}=\partial\widetilde{A}=\partial\left(A\cup\Gamma_{AB}^{A}\cup
Q_{AB}^{A}\right).$ (28)

This is equivalent to

$\displaystyle\partial\left(\Sigma_{A:B}^{min}\cup{\Gamma}_{AB}^{A}\right)|_{\partial
M}$ $\displaystyle=\partial A,$ (29)
$\displaystyle\partial\left(\Sigma_{A:B}^{min}\cup{\Gamma}_{AB}^{A}\right)|_{Q}$
$\displaystyle=\partial Q_{AB}^{A}.$ (30)

So $\left(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min}\right)$ satisfies condition 1 of
$\bm{\mathit{\Gamma_{A}}}$ in (3), and for the same reason we know
$\left(\Gamma_{AB}^{B}+\Sigma_{A:B}^{min}\right)$ also satisfies condition 1
of $\bm{\mathit{\Gamma_{B}}}$.

For condition 2 in (3). By the definition of $\Sigma_{A:B}^{min}$, there
exists a corresponding $R_{A}\subset M_{AB}$, which satisfies

$\displaystyle\partial R_{A}=\Sigma_{A:B}^{min}\cup\widetilde{A}.$ (31)

From above formulae we can get

$R_{A}\subset M\ \ \ \ \ \text{and }\ \ \partial
R_{A}=\left(\Sigma^{min}_{A:B}\cup\Gamma_{AB}^{A}\right)\cup A\cup
Q_{AB}^{A}.$ (32)

Therefore, condition 2 in (3) is also satisfied. We conclude that
$(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min})$ belongs to $\bm{\mathit{\Gamma_{A}}}$
indeed. As a result, we must have

$\mathcal{A}(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min})\geq\mathcal{A}(\Gamma_{A}^{min}).$
(33)

Similarly, we also have

$\mathcal{A}(\Gamma_{AB}^{B}+\Sigma_{A:B}^{min})\geq\mathcal{A}(\Gamma_{B}^{min}).$
(34)

Finally, combining these two inequalities we can prove the initial inequality
on the relation between $E_{W}$ and the holographic mutual information

$\mathcal{A}(\Gamma_{AB}^{A}+\Sigma_{A:B}^{min})+\mathcal{A}(\Gamma_{AB}^{B}+\Sigma_{A:B}^{min})=2E_{W}(A:B)+S(AB)\geq
S(A)+S(B).$ (35)

###  III.3 $E_{W}(A:BC)\geq E_{W}(A:B)$

Figure 9:  A diagrammatic demonstration of the proof for inequality
$E_{W}(A:B)\leq E_{W}(A:BC)$, which is geometrically described by the fact
that the area of the orange line ($\Sigma^{min}_{A:B}$) is always less than
that of the purple solid line ($\Sigma^{\prime}_{A:B}$), which is just a part
of $\Sigma^{min}_{A:BC}$, as illustrated in the right plot. Here for the
clearness a simple configuration is plotted, while it should be stressed that
the procedures of our proof are quite general and applicable for other
configurations which may be very complicated.

Figure 10:  A diagrammatic demonstration of the proof for inequality
$E_{W}(A:BC)\geq E_{W}(A:B)$ in AdS/BCFT, which is geometrically described by
the fact that the area of the orange line ($\Sigma^{min}_{A:B}$) is always
less than that of the purple line ($\Sigma^{\prime}_{A:B}$). Notice that there
is a phase transition for $E_{W}$ from $\Sigma^{min}_{A:B}$ (which ends on the
$Q$ brane) to $\Sigma^{min}_{A:BC}$ (which ends on the minimal surface
$\Gamma^{min}_{ABC}$).

This inequality means that enlarging any one of region A and B does not
decrease the correlations between these two regions. This inequality can be
intuitively understood as follows. The minimal cross section of $E_{W}(A:BC)$
separates the entanglement wedge of ABC, namely $M_{ABC}$, into two parts
$R_{A}$ and $R_{BC}$ which include A and BC, respectively. At the same time it
must separate $M_{AB}$ into $R^{\prime}_{A}$ and $R^{\prime}_{B}$ as well, as
illustrated in Fig.(10) and Fig.(10), but $E_{W}(A:B)$ is the minimal area to
separate $M_{AB}$ into any possible $R^{\prime}_{A}$ and $R^{\prime}_{B}$, so
we should have $E_{W}(A:BC)\geq E_{W}(A:B)$.

To prove the above inequality in an algebraic method, our strategy is to show
that one portion of the left-hand side of the inequality corresponds to the
area of surface $\Sigma^{\prime}_{A:B}$ which belongs to
$\bm{\mathit{\Sigma_{A:B}}}$, while the right-hand side is the area of the
minimal surface in $\bm{\mathit{\Sigma_{A:B}}}$. Then one obtains the
relations between their areas as
$E_{W}(A:BC)\geq\mathcal{A}(\Sigma^{\prime}_{A:B})\geq E_{W}(A:B)$. (See
Fig.(10)(10).)

Now we need to construct such $\Sigma^{\prime}_{A:B}$. In fact,
$\Sigma^{\prime}_{A:B}$ may be defined as “cut $\Sigma_{A:BC}^{min}$”, which
is the part of $\Sigma_{A:BC}^{min}$ in the interior of $M_{AB}$, and given by

$\Sigma^{\prime}_{A:B}=M_{AB}\cap\Sigma_{A:BC}^{min}.$ (36)

Next we just need to show that $\Sigma^{\prime}_{A:B}$ belongs to
$\bm{\mathit{\Sigma_{A:B}}}$. For this purpose we just check that
$\Sigma^{\prime}_{A:B}$ satisfies two conditions in the definition of
$\bm{\mathit{\Sigma_{A:B}}}$.

Firstly, by the definition of $\Sigma_{A:BC}^{min}$, there exists a bulk
region $R_{A}$ which satisfies

$\displaystyle R_{A}\subset M_{ABC},$ (37) $\displaystyle\partial
R_{A}=\widetilde{A}\cup\Sigma_{A:BC}^{min}.$ (38)

Then we define “cut $R_{A}$” to be the part of $R_{A}$ in $M_{AB}$ as

$R^{\prime}_{A}=R_{A}\cap M_{AB}.$ (39)

Now we intend to argue that for $\Sigma^{\prime}_{A:B}$, $R^{\prime}_{A}$ is
exactly the bulk region such that one has $\partial
R^{\prime}_{A}=\Sigma^{\prime}_{A:B}\cup\widetilde{A^{\prime}}$. (Notice that
here $\widetilde{A^{\prime}}$ is different from the above $\widetilde{A}$. The
former is associated with $\Sigma_{A:B}^{min}$, while the latter is associated
with $\Sigma_{A:BC}^{min}$.) The existence of such a $R^{\prime}_{A}$ will
lead to $\Sigma^{\prime}_{A:B}\in\bm{\mathit{\Sigma_{A:B}}}$. To prove this
explicitly, we consider the boundary of “cut $R_{A}$”

$\partial R^{\prime}_{A}=\partial\left(M_{AB}\cap R_{A}\right).$ (40)

Now we intend to figure out what one would obtain on the right-hand side of
this equation. It is noticed that in general the boundary of the intersection
of two regions $X_{A}$ and $X_{B}$ is the union of the boundary of $X_{A}$ in
the interior of $X_{B}$, the boundary of $X_{B}$ in the interior of $X_{A}$
and the common boundary of two regions $X_{A}$ and $X_{B}$, as illustrated in
Fig.(11).

Figure 11:  The boundary of $X_{A}\cap X_{B}$ consists of three parts: one
part is the boundary of $X_{A}$ in the interior of $X_{B}$, and the second
part is the boundary of $X_{B}$ in the interior of $X_{A}$, while the third
part is the common boundary of $X_{A}$ and $R_{B}$.

Similarly, in our case we find the boundary of $R^{\prime}_{A}$ contains the
following three parts. Firstly, from the wedge nesting property $M_{AB}\subset
M_{ABC}$ and the definition of $R_{A}$, one can get the boundary of $R_{A}$ in
the interior of $M_{AB}$ is just $\Sigma^{\prime}_{A:B}$; secondly, according
to the definition of $\Gamma^{A}_{AB}$, the boundary of $M_{AB}$ in the
interior $R_{A}$ must be one choice of $\Gamma^{A}_{AB}$; thirdly, it is
obviously seen that region $A$ is the unique common boundary of $M_{AB}$ and
$R_{A}$. As a result, we have the following expressions

$\displaystyle\partial R^{\prime}_{A}$ $\displaystyle=\partial\left(M_{AB}\cap
R_{A}\right)$ (41) $\displaystyle=\Sigma^{\prime}_{A:B}\cup\Gamma_{AB}^{A}\cup
A$ (42) $\displaystyle=\Sigma^{\prime}_{A:B}\cup\widetilde{A^{\prime}}.\ $
(43)

From the above results, we know there exists a bulk region
$R^{\prime}_{A}\subset M_{AB}$, such that $\partial
R^{\prime}_{A}=\Sigma^{\prime}_{A:B}\cup\widetilde{A^{\prime}}$. In addition,
since $R^{\prime}_{A}$ is a closed region, we have
$\partial\Sigma^{\prime}_{A:B}=\partial\widetilde{A^{\prime}}$. Therefore, we
conclude that $\Sigma^{\prime}_{A:B}$ belongs to $\bm{\mathit{\Sigma_{A:B}}}$,
then the inequality in this subsection so far is proved.

###  III.4 $E_{W}(A:BC)\geq\frac{1}{2}I(A:B)+\frac{1}{2}I(A:C)$

Having proved the former inequalities, we find the proof of this inequality
becomes pretty easy. Firstly, we just apply the known inequality (15)

$E_{W}(A:B)\geq\frac{1}{2}I(A:B).$ (44)

Replacing $B$ with $BC$ in this inequality, we get

$E_{W}(A:BC)\geq\frac{1}{2}I(A:BC).$ (45)

Then with the use of the holographic MMI inequalityHayden:2011ag Chou:2020ffc

$I(A:BC)\geq I(A:B)+I(A:C),$ (46)

we prove this inequality

$E_{W}(A:BC)\geq\frac{1}{2}I(A:B)+\frac{1}{2}I(A:C).$ (47)

###  III.5 $E_{W}(AB:CD)\geq E_{W}(A:C)+E_{W}(B:D)$

In this inequality we notice that two terms on the right-hand side are nothing
but the areas of minimal-area elements in $\bm{\mathit{\Sigma_{A:C}}}$ and
$\bm{\mathit{\Sigma_{B:D}}}$, namely $\mathcal{A}(\Sigma_{A:C}^{min})$ and
$\mathcal{A}(\Sigma_{B:D}^{min})$, while the term on the left-hand side is
just $\mathcal{A}(\Sigma_{AB:CD}^{min})$. So our strategy is to prove that
there exist two non-overlapping surfaces $X$ and $Y$ contained in
$\Sigma_{AB:CD}^{min}$. Moreover, $X$ and $Y$ belong to
$\bm{\mathit{\Sigma_{A:C}}}$ and $\bm{\mathit{\Sigma_{B:D}}}$, respectively.
Then we can get the following relations

$\displaystyle
E_{W}(AB:CD)=\mathcal{A}(\Sigma_{AB:CD}^{min})\geq\mathcal{A}(X)+\mathcal{A}(Y)\geq
E_{W}(A:C)+E_{W}(B:D).$ (48)

So next our target is to show the existence of such $X$ and $Y$ contained in
$\Sigma_{AB:CD}^{min}$.

Firstly, exploiting “wedge nesting” we can get

$AC\cap BD=\emptyset\Rightarrow M_{AC}\cap M_{BD}=\emptyset.$ (49)

As in (36), we may define two “cut $\Sigma_{AB:CD}^{min}$”

$\displaystyle\Sigma^{\prime}_{A:C}=\Sigma^{min}_{AB:CD}\cap M_{AC},$ (50)
$\displaystyle\Sigma^{\prime}_{B:D}=\Sigma^{min}_{AB:CD}\cap M_{BD}.$ (51)

Because

$M_{AC}\cap M_{BD}=\emptyset,$ (52)

we have

$\Sigma^{\prime}_{A:C}\cap\Sigma^{\prime}_{B:D}=\emptyset.$ (53)

It actually means that

$\Sigma^{min}_{AB:CD}=\Sigma^{\prime}_{A:C}\cup\Sigma^{\prime}_{B:D}\cup(\text{the
remaining surfaces}).$ (54)

Next we need to prove

$\displaystyle\Sigma^{\prime}_{A:C}\in\bm{\mathit{\Sigma_{A:C}}},$ (55)
$\displaystyle\Sigma^{\prime}_{B:D}\in\bm{\mathit{\Sigma_{B:D}}}.$ (56)

Because $M_{AC}$ does not overlap with $M_{BD}$, one can prove above two
formulae independently. Moreover, based on the definition of
$\Sigma^{\prime}_{A:C}$ and $\Sigma^{\prime}_{B:D}$, one can prove each of
them belongs to set $\bm{\mathit{\Sigma_{A:C}}}$ or
$\bm{\mathit{\Sigma_{B:D}}}$ in a parallel way as we performed in subsection
(III.3). Then finally we have

$E_{W}(AB:CD)\geq\mathcal{A}(\Sigma^{\prime}_{A:C})+\mathcal{A}(\Sigma^{\prime}_{B:D})\geq
E_{W}(A:C)+E_{W}(B:D).$ (57)

We remark that this inequality is just satisfied by quantum states with
holographic dual, not applicable to arbitrary quantum statesBao:2018gck . So
it can be used to justify whether a quantum state in CFT has a smooth bulk
dual.

##  IV Conclusion and Discussion

In this paper we have investigated the inequalities for the entanglement of
purification in the context of AdS/BCFT duality. We have constructed the
candidate sets for HEE and $E_{W}$ in general cases, and developed an
algebraic method to show that $E_{W}$ satisfies the same inequalities as in
AdS/CFT. We have only considered the case that the bulk is pure AdS vacuum. As
a matter of fact, one can show that in AdS black hole background (which is
dual to BCFT at finite temperature) these inequalities should still hold. It
is quite interesting to compare the algebraic method and the diagrammatic
method in the proof of these inequalities. The diagrammatic method is applied
naturally in the geometric description of entanglement entropy, and the
relevant inequalities can be understood and then proved intuitively in this
way. However, when the system consists of multi partitions and the subregion
is the union of many disconnected parts, then the corresponding diagram would
become complicated. In particular, when the CFT has a boundary as in the
context of AdS/BCFT, the entanglement exhibits abundant phase structure, and
the diagrammatic method suffers from the enumeration of all the possible
diagrams involved in inequalities. In this situation, the algebraic method is
very suitable for proving these inequalities because the process of proof does
not depend on the specific phases of the entanglement. The price that one has
to pay is that the proof might become abstract. Therefore, in this paper we
have also illustrated some typical diagrams to assist us to understand the
process of proof.

Our work in this paper may be pushed forward in following directions in
future. Firstly, one may check the inequalities of $E_{W}$ in time-dependent
bulk gravity, which could improve our understanding on the dynamical
properties of $E_{W}$. Secondly, it is very desirable to apply the algebraic
method proposed in this work to explore more inequalities of $E_{W}$,
especially holographic $E_{W}$ inequalities such as MMI, which might impose
new constraints on holographic states. Finally, it is also very interesting to
consider the inequalities beyond the leading order of the holographic
entanglement. For instance, one may take into account the bulk corrections to
the holographic entanglement as discussed in Akers:2019lzs Akers:2021lms , and
then check whether the behaviours of those inequalities would be changed.

## Acknowledgments

We are very grateful to Yuxuan Liu and Zhuo-Yu Xian for helpful discussions.
This work is supported in part by the Natural Science Foundation of China
under Grant No. 11875053 and 12035016. It is also supported by Beijing Natural
Science Foundation under Grant No. 1222031.

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