aainstitutetext: Shing-Tung Yau Center of Southeast University, Nanjing
210096, Chinabbinstitutetext: School of Mathematics, Southeast University,
Nanjing 211189, China

#  Balanced Partial Entanglement and the Entanglement Wedge Cross Section

Qiang Wen<EMAIL_ADDRESS>
###### Abstract

In this article we define a new information theoretical quantity for any
bipartite mixed state $\rho_{AB}$. We call it the balanced partial
entanglement (BPE). The BPE is the partial entanglement entropy, which is an
integral of the entanglement contour in a subregion, that satisfies certain
balance requirements. The BPE depends on the purification hence is not
intrinsic. However, the BPE could be a useful way to classify the
purifications. We discuss the entropy relations satisfied by BPE and find they
are quite similar to those satisfied by the entanglement of purification. We
show that in holographic CFT2 the BPE equals to the area of the entanglement
wedge cross section (EWCS) divided by 4G. More interestingly, when we consider
the canonical purification the BPE is just half of the reflected entropy,
which also directly relate to the EWCS. The BPE can be considered as an
generalization of the reflected entropy for a generic purification of the
mixed state $\rho_{AB}$. We interpret the correspondence between the BPE and
EWCS using the holographic picture of the entanglement contour.

##  1 Introduction

Quantum entanglement has played a central role in the study of modern
theoretical physics, including quantum information theory, condensed matter
theory and quantum gravity. One of the most important entanglement measures is
the entanglement entropy, which captures the correlation between $A$ and $B$
for a bipartite system $A\cup B$ in a pure state. The study of entanglement
entropy gains huge amount of extra attention because of the Ryu-Takayanagi
(RT) Ryu:2006bv ; Ryu:2006ef  proposal, which reveals the deep connection
between spacetime geometry and quantum entanglement. In the context of the
AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj , let
us consider a static region $A$ in the boundary field theory and the minimal
surface $\mathcal{E}_{A}$ in the dual AdS bulk that anchored on the boundary
of $A$. The RT proposal relates the entanglement entropy of $A$ to the area of
$\mathcal{E}_{A}$ in Planck units, i.e.

$\displaystyle S_{A}=\frac{1}{4G}\text{Area}(\mathcal{E}_{A})\,.$ (1.1)

However, for bipartite (or multipartite) systems in a mixed state, the
entanglement entropy is not a good measure of entanglement as it mixes
classical and quantum correlations. New entanglement measures in quantum
information theory are proposed to take the place, for example, the mutual
information, the (logarithmic) entanglement negativity negativity1 ;
negativity2 ; negativity3 , the entanglement of purification (EoP) EoP , the
partial entanglement entropy (PEE) Vidal ; Wen:2018whg ; Kudler-Flam:2019oru ;
Wen:2019ubu ; Wen:2019iyq  etc.

On the other hand, from the perspective of holography there exists a geometric
quantity called the entanglement wedge cross section (EWCS) $\Sigma_{AB}$,
that may capture certain type of correlations between $A$ and $B$ for a mixed
state $\rho_{AB}$. For example, consider $AB\equiv A\cup B$ to be a subsystem
of the boundary of global AdS3, the reduced density matrix $\rho_{AB}$ has a
bulk dual named the entanglement wedge $\mathcal{W}_{AB}$ Czech:2012bh ;
Wall:2012uf ; Headrick:2014cta . The entanglement wedge is the causal
development of the homology surface $\mathcal{R}_{AB}$, which is a Cauchy
surface with the boundary being $\partial{\mathcal{R}_{AB}}=A\cup
B\cup\mathcal{E}_{AB}$. The EWCS $\Sigma_{AB}$ is then defined as the minimal
cross section of $\mathcal{R}_{AB}$ that separate $A$ from $B$. Since
$\Sigma_{AB}$ plays a special role in the bulk, it is very likely to represent
something special in quantum information theory.

So far, there are several proposals for the holographic dual of $\Sigma_{AB}$.
The first candidate is the entanglement of purification (EoP) EoP . It was
shown in HEoP1 ; HEoP2  that the EWCS and EoP satisfy the same entropy
relations in holographic theories. Also assuming a theory with a tensor
network description thus the surface/state correspondence Miyaji:2015yva  can
be realized, the calculation of $\Sigma_{AB}$ matches with the way we define
the EoP. However, it could be very hard to justify this proposal in more
generic cases due to the large optimization procedure in the definition of the
EoP. Another well-known candidate is half of the reflected entropy
Dutta:2019gen , which is defined on the canonical purification of $\rho_{AB}$.
The reflected entropy proposal can be confirmed under some mild assumptions.
There are also other proposals which claim that the EWCS is dual to, for
example the logarithmic negativity Kudler-Flam:2018qjo ; Kusuki:2019zsp , the
“odd entropy” Tamaoka:2018ned , the “differential purification”
Espindola:2018ozt  and so on. See also Agon:2018lwq ; Harper:2019lff ;
Bao:2019wcf ; Lin:2020yzf ; Du:2019emy ; Umemoto:2018jpc ; BabaeiVelni:2019pkw
; Ghodrati:2019hnn ; Ghodrati:2020vzm ; KumarBasak:2021lwm ; Basak:2020oaf ;
Chu:2019etd ; Khoeini-Moghaddam:2020ymm  for a incomplete list about recent
studies related to the EWCS. It is worth pointing out that the above proposals
are indeed in tension with each other (see for example Akers:2019gcv ), hence
a deeper understanding of the holographic picture for the EWCS is still in
need.

Recently a new entanglement measure called the partial entanglement entropy
(PEE) Vidal ; Wen:2018whg ; Kudler-Flam:2019oru ; Wen:2019ubu ; Wen:2019iyq
was proposed. For a given region $A$ and a subset $A_{i}$ of $A$, the PEE is
denoted as $s_{A}(A_{i})$. Physically it is assumed to capture the
contribution from $A_{i}$ to the entanglement entropy $S_{A}$. The key
property featured by the PEE is additivity, which is not possessed by any
other entanglement measures. The differential version of the PEE, named the
entanglement contour Vidal , is a function $f_{A}(\textbf{x})$ defined on $A$
that gives the contribution from the degrees of freedom at any point x in $A$
to $S_{A}$. In other words it is the density function of the entanglement
entropy $S_{A}$ that satisfies,

$\displaystyle S_{A}=\int_{A}f_{A}(\textbf{x})dx^{d}\,.$ (1.2)

Here $d$ gives the dimension of $A$. The PEE $s_{A}(A_{i})$ is then defined in
the following

$\displaystyle s_{A}(A_{i})=\int_{A_{i}}f_{A}(\textbf{x})dx^{d}\,.$ (1.3)

Note that the PEE $s_{A}(A_{i})$ only collect the contribution in the subset
$A_{i}$.

Let us assume that $\bar{A}$ is some system that purifies $A$. Since the PEE
$s_{A}(A_{i})$ in some sense captures the correlation between the subset
$A_{i}$ and $\bar{A}$, it should be invariant under the permutation between
$A_{i}$ and $\bar{A}$ Wen:2019iyq . In order to manifest this permutation
symmetry, we also write the PEE in the following way

$\displaystyle
s_{A}(A_{i})=\mathcal{I}(A_{i},\bar{A})=\mathcal{I}(\bar{A},A_{i})=s_{\bar{A}_{i}}(\bar{A})\equiv\mathcal{I}_{A_{i}\bar{A}}.$
(1.4)

Note that we should not mix the PEE $\mathcal{I}(A_{i},\bar{A})$ with the
mutual information $I(A_{i},\bar{A})$.

Unfortunately, the fundamental definition based on the reduced density matrix
for the PEE is still missing. According to its physical meaning, the PEE
should satisfy the following physical requirements 111 The requirements 1-6
are firstly given in Vidal , while the requirement 7 is recently given in
Wen:2019iyq :

  1. 1.

Additivity: if $A_{i}^{a}\cup A_{i}^{b}=A_{i}$ and $A_{i}^{a}\cap
A_{i}^{b}=\varnothing$, by definition we should have

$\displaystyle s_{A}(A_{i})$
$\displaystyle=s_{A}(A_{i}^{a})+s_{A}(A_{i}^{b})\,.$ (1.5)

  2. 2.

Invariance under local unitary transformations: $s_{A}(A_{i})$ should be
invariant under any local unitary transformations inside $A_{i}$ or $\bar{A}$.

  3. 3.

Symmetry: for any symmetry transformation $\mathcal{T}$ under which
$\mathcal{T}A=A^{\prime}$ and $\mathcal{T}A_{i}=A^{\prime}_{i}$, we have
$s_{A}(A_{i})=s_{A^{\prime}}(A^{\prime}_{i})$.

  4. 4.

Normalization: $S_{A}=s_{A}(A_{i})|_{A_{i}\to A}\,.$

  5. 5.

Positivity: $s_{A}(A_{i})\geq 0$.

  6. 6.

Upper bound: $s_{A}(A_{i})\leq S_{A_{i}}\,.$

  7. 7.

Symmetry under the permutation:
$\mathcal{I}(\bar{A},A_{i})=\mathcal{I}(A_{i},\bar{A})\,,$ which implies
$s_{A}(A_{i})=s_{\bar{A}_{i}}(\bar{A})$.

Recent explorations on entanglement contour or the PEE include Vidal:2002rm ;
Botero ; Vidal ; PhysRevB.92.115129 ; Coser:2017dtb ; Tonni:2017jom ;
Alba:2018ime ; Wen:2018whg ; Wen:2018mev ; Kudler-Flam:2019oru ; Wen:2019ubu ;
DiGiulio:2019lpb ; Han:2019scu ; Ageev:2019fjf ; Kudler-Flam:2019nhr ;
Wen:2019iyq ; Roy:2019gbi ; deBuruaga:2019xwv ; MacCormack:2020auw . People
propose formulas to construct the PEE (or entanglement contour) that satisfies
the above requirements. Each of the existed proposals are restricted to
special configurations. The first one is the Gaussian formula Botero ; Vidal ;
PhysRevB.92.115129 ; Coser:2017dtb ; Tonni:2017jom ; Alba:2018ime ;
DiGiulio:2019lpb ; Kudler-Flam:2019nhr  that applies to the Gaussian states in
free theories, where the density matrix can be completely characterized in
terms of the correlation matrix. The second proposal is a geometric
construction Wen:2018whg ; Wen:2018mev ; Han:2019scu  in holograhic theories,
which is inspired by a natural slicing of the entanglement wedge following the
boundary and bulk modular flows. The third one is the partial entanglement
entropy proposal Wen:2018whg ; Wen:2019ubu  that claims the PEE is given by an
additive linear combination of subset entanglement entropies. The fourth
proposal Wen:2019iyq  follows the construction of the extensive (or additive)
mutual information (EMI) Casini:2008wt  (see also Roy:2019gbi  for a related
construction), which tried to solve the above seven requirements in CFT.

Though the above proposals have very different physical motivations, the PEE
calculated by different approaches are highly consistent with each other
Kudler-Flam:2019nhr ; Wen:2018whg ; Wen:2018mev ; Han:2019scu ; Wen:2019iyq .
This implies the PEE should be unique and well defined. So far, the uniqueness
of the PEE is only confirmed for Poincaré invariant theories Wen:2019iyq , by
showing that the above seven requirements in these theories have unique
solution. Due to the nice properties, the PEE is also useful to study the
entanglement structure in condensed matter theories 222The entanglement
contour gives a finer description for the entanglement structure. In condense
matter theories it can be used to discriminate between gapped systems and
gapless systems with a finite number of zero modes in $d=3$ Vidal . It has
been shown to be particularly useful to characterize the spreading of
entanglement when studying dynamical situations Vidal ; Kudler-Flam:2019oru ;
DiGiulio:2019lpb . Modular flows in two dimensions can be generated from the
PEE Wen:2019ubu . The entanglement contour is also a useful probe of slowly
scrambling and non-thermalizing dynamics for some interacting many-body
systems MacCormack:2020auw . Holographically the PEE Wen:2018whg ; Wen:2018mev
correspond to bulk geodesic chords which is a finer correspondence between
quantum entanglement and bulk geometry Wen:2018mev ; Abt:2018ywl . The new
concept of entanglement contour in quantum information will play an important
role in our understanding of the gauge/gravity duality and the entanglement
structure in quantum field theories (or many-body system)..

Since the entanglement contour is a finer description for the entanglement
structure, we expect that other entanglement measures could be extracted from
the contour function. In this paper, we study the correlations in mixed
bipartite states using PEE. For a mixed state $\rho_{AB}$, one can introduce
an auxiliary system $A^{\prime}B^{\prime}$ thus the combined system is in a
pure state $\left|\psi\right\rangle$, which is highly non-unique. The state
$\left|\psi\right\rangle$ is then called a purification of $\rho_{AB}$. In
section 2, we briefly review the PEE proposal and define a special PEE for any
purification. We call it the balanced partial entanglement (BPE), because the
partition of $A^{\prime}B^{\prime}$ should satisfy the following balance
requirement, $s_{AA^{\prime}}(A)=s_{BB^{\prime}}(B)$. In section 3, we study
aspects of the BPE for the case where $A$ and $B$ are adjacent. We calculate
the holographic BPE for the case of global AdS3, where $AB$ is a subsystem on
the boundary. We find the BPE gives the area of the EWCS. Interestingly we
find the crossing PEE $\mathcal{I}_{AB^{\prime}}=\frac{c}{6}\log 2$, which is
a constant independent from the length of $A$ and $B$. We discuss the entropy
relations satisfied by the BPE and find that, they are quite similar to those
satisfied by EoP. Also, we consider the minimal purification in the context of
the surface/state correspondence Miyaji:2015yva , and find the BPE differs
from the case of the global AdS3. In section 4, we discuss the cases where $A$
and $B$ are non-adjacent. We confirmed that the relation between the BPE and
$\Sigma_{AB}$ still holds. In section 5, we discuss the canonical purification
for generic $\rho_{AB}$ and show that, half of the reflected entropy is
identical to the BPE we defined. In section 6, we interpret the relation
between the BPE and the EWCS using the holographic picture for PEE, which are
the geodesic chords in the bulk that normal to the RT surfaces of relevant
regions. At last we give a discussion in section 7.

##  2 The balanced partial entanglement

###  2.1 Definition

Figure 1: The combined system $ABA^{\prime}B^{\prime}$ is in a pure state. The
red arrow represents the PEE $\mathcal{I}_{AB}$ between $A$ and $B$, while the
blue arrow represents $\mathcal{I}_{AB^{\prime}}$. $\mathcal{I}_{AB^{\prime}}$
and $\mathcal{I}_{BA^{\prime}}$ vary with the partition of the complement
$A^{\prime}B^{\prime}$. The balance requirement
$\mathcal{I}_{AB^{\prime}}=\mathcal{I}_{BA^{\prime}}$ can be satisfied by
adjusting the partition.

The balanced partial entanglement (BPE) is defined in the following: let
$\rho_{AB}$ be a density matrix on a bipartite system
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. We consider an auxiliary system
$A^{\prime}B^{\prime}$ that purifies $AB$, thus the whole system is in a pure
state $\left|\psi\right\rangle$ and
$\text{Tr}_{A^{\prime}B^{\prime}}\left|\psi\right\rangle\left\langle\psi\right|=\rho_{AB}$.
Let us partition the auxiliary system into $A^{\prime}$ and $B^{\prime}$
properly in the following way. Firstly we require the contribution from $A$ to
the entanglement entropy $S_{AA^{\prime}}$ equals to the contribution from $B$
to the entanglement entropy $S_{BB^{\prime}}$. We call this requirement the
balance requirement, i.e.

$\displaystyle\textit{balance requirement}:\qquad
s_{AA^{\prime}}(A)=s_{BB^{\prime}}(B)\,,\qquad
s_{AA^{\prime}}(A^{\prime})=s_{BB^{\prime}}(B^{\prime})\,.$ (2.6)

Since $S_{AA^{\prime}}=S_{BB^{\prime}}$, only one of the above requirements is
independent. In terms of the PEE we have,

$\displaystyle
s_{AA^{\prime}}(A)=\mathcal{I}_{AB}+\mathcal{I}_{AB^{\prime}}\,,\qquad
s_{BB^{\prime}}(B)=\mathcal{I}_{BA}+\mathcal{I}_{BA^{\prime}}\,.$ (2.7)

See Fig.1. The first term $\mathcal{I}_{AB}$ is supposed (though not proved in
general) to be intrinsic hence independent from the purification, because
unitary transformations acting outside $AB$ should not change the correlation
between $A$ and $B$. While the second term $\mathcal{I}_{AB^{\prime}}$ could
vary under different purifications. Since $\mathcal{I}_{AB}=\mathcal{I}_{BA}$,
the balance requirement can also be written as

$\displaystyle\mathcal{I}_{AB^{\prime}}=\mathcal{I}_{BA^{\prime}}\,.$ (2.8)

When $\mathcal{I}_{AB^{\prime}}$ (or $\mathcal{I}_{BA^{\prime}}$) satisfies
the balance requirement, we call it the crossing PEE of
$\left|\psi\right\rangle$, which can be used to classify the purifications.

The BPE is a natural quantity to consider for any purification
$\left|\psi\right\rangle$ of $\rho_{AB}$. However, in general the partition of
$A^{\prime}B^{\prime}$ that satisfies the balance requirement is not unique.
To clarify this ambiguity we propose the following minimal requirement.

  * •

Minimal requirement: among all the partitions that satisfy the balance
requirement we should choose the one such that $s_{AA^{\prime}}(A)$ reaches
its minimal value.

It is not hard to find a solution to this requirement since the purification
is already fixed. In most theories, the entanglement between any two local
degrees of freedom decreases with distance. When we set $B^{\prime}$ to be as
far from $A$ as we can and set $A^{\prime}$ to be as far from $B$ as we can,
we can simultaneously reduce $\mathcal{I}_{AB^{\prime}}$ and
$\mathcal{I}_{BA^{\prime}}$ while keeping the balance requirement satisfied.
Eventually we arrive at the following prescription for the natural and simple
partition: The whole system is partitioned into two parts $AA^{\prime}$ and
$BB^{\prime}$ thus separates $A$ from $B$. In some sense the region $A$ will
be surrounded by $A^{\prime}$ while $B$ is surrounded by $B^{\prime}$. Also,
there should be no embedding between the region $B^{\prime}$ and $A^{\prime}$.
Later we will partition the purifier $A^{\prime}B^{\prime}$ following this
prescription and not stress the minimal requirement any more.

One can understand our prescription via the examples in Fig.2. In the first
two figures $AB$ is a connected interval in a circle, while the partition of
the complement $A^{\prime}B^{\prime}$ is different. The circle is in a pure
state. We require these two configurations to have reflection symmetry between
the left and right hand side, hence the balance requirement can be satisfied
in both cases. It is obvious that the $s_{AA^{\prime}}(A)$ reaches its minimal
value in the second figure because, compare with the first figure, part of
$B^{\prime}$ get further from $A$ hence $\mathcal{I}_{AB^{\prime}}$ decreases.

Figure 2: All the three figures have reflection symmetry between the left and
right hand side, hence we can easily see that the balance requirement is
satisfied in all the figures. From the first figure to the second one, the
upper part of $A^{\prime}$ and $B^{\prime}$ exchange positions hence the upper
part of $B^{\prime}$ gets further from $A$ thus reduces the PEE between them.
The third figure shows the proper partition that satisfies our requirements
for the case where $AB$ is non-adjacent.

Then the BPE is defined as the partial entanglement entropy
$s_{AA^{\prime}}(A)$ under the partition of $A^{\prime}B^{\prime}$ satisfying
the above two requirements. We denote it as BPE$(A,B,\psi)$, i.e.

$\displaystyle\text{BPE}(A,B,\psi)=s_{AA^{\prime}}(A)|_{balance~{}requirements}\,.$
(2.9)

When the purification is specified we will omit the label $\psi$ thus write
BPE$(A,B)$.

The third figure in Fig.2 shows the proper partition for the reflection
symmetric case where $AB$ is non-adjacent. For more generic cases with no
reflection symmetry, the boundary is partitioned similarly by two points and
the both of the region $A^{\prime}=A^{\prime}_{1}\cup A^{\prime}_{2}$ and
$B^{\prime}=B^{\prime}_{1}\cup B^{\prime}_{2}$ contain two disconnected pieces
which can be naturally set in pairs. For example in Fig.7, the two pairs are
$A_{1}^{\prime}\sim B_{1}^{\prime}$ and $A_{2}^{\prime}\sim B_{2}^{\prime}$.
Note that in the same sense $A\sim B$ are also a pair. The balance
requirements are indeed imposed on all the pairs,

$\displaystyle
s_{AA^{\prime}}(A_{1}^{\prime})=s_{BB^{\prime}}(B_{1}^{\prime})\,,\qquad
s_{AA^{\prime}}(A)=s_{BB^{\prime}}(B)\,,\qquad
s_{AA^{\prime}}(A_{2}^{\prime})=s_{BB^{\prime}}(B_{2}^{\prime})\,.$ (2.10)

Since $S_{AA^{\prime}}=S_{BB^{\prime}}$, only two of the above requirements
are independent, which are exactly the requirements that determine the two
partition points.

###  2.2 Review on the partial entanglement entropy proposal

In order to study the BPE, we need to calculate the PEE. The PEE proposal
Wen:2018whg ; Wen:2019ubu  may be the most powerful way to calculate the PEE
in two dimension theories. Since we heavily rely on this proposal, it is
necessary to give a brief review here. The proposal claims that, the PEE is
given by a linear combination of certain subset entanglement entropies. This
linear combination satisfies the key property of additivity. Furthermore, it
was shown to satisfy all the seven requirements using only the general
properties of entanglement entropy, thus can be applied to generic theories.
Especially in Poincaré invariant theories the PEE proposal has been shown to
be the unique solution to all the physical requirements Wen:2019iyq .

However, a definite order is required for all the degrees of freedom in $A$
for the satisfactory of the additivity. More explicitly, given a region $A$
and an arbitrary subset $\alpha$, when there is a definite order inside $A$,
in general it can be partitioned into $A=\alpha_{L}\cup\alpha\cup\alpha_{R}$
(see for example Fig.3). Here $\alpha_{L}$ ($\alpha_{R}$) is denoted as the
subset on the left (right) hand side of $\alpha$. In this configuration, the
PEE proposal claims that

$\displaystyle
s_{A}(\alpha)=\frac{1}{2}\left(S_{\alpha_{L}\cup\alpha}+S_{\alpha\cup\alpha_{R}}-S_{\alpha_{L}}-S_{\alpha_{R}}\right)\,.$
(2.11)

The order determines $\alpha_{L}$ and $\alpha_{R}$ unambiguously. In two-
dimensional theories the definite order always exist in the configurations
where the region $A$ (whether connected or disconnected) is embedded in a
larger one-dimensional chain or circle.

Figure 3:  The region $A$ is shown by the red interval. When an arbitrary
subset $\alpha$ is chosen, a natural decomposition of
$A=\alpha_{L}\cup\alpha\cup\alpha_{R}$ is determined. All the degrees of
freedom in $A$ lines in a definite order.

Note that, a definite order does not always exist in generic two-dimensional
systems. This has not been carefully discussed in previous studies. For
example in Fig.4, the pure state is settled on two disconnected circles. The
region $A$ is given by the two red half circles which are also disconnected.
When the subset $\alpha$ is chosen, $A=\alpha\cup\alpha_{1}\cup\alpha_{2}$ is
partitioned into three parts. In this case, the order between the three parts
is ambiguous. One can either take
$\alpha_{L}=\alpha_{1}\cup\alpha_{2}\,,\alpha_{R}=\varnothing$ or
$\alpha_{L}=\alpha_{1}\,,\alpha_{R}=\alpha_{2}$. These two choices represent
two different orders. Following (2.11), the two orders give different values
for $s_{A}(\alpha)$, thus the PEE become ambiguous. Also in the case where $A$
is a circle with no boundary, the order is also ambiguous. When the order is
not definite, then the proposal (2.11) become ambiguous. In these cases, we
should sue to other proposals to calculate the PEE.

Figure 4:  In this case the region $A$ is the red disconnected multi-interval.
Since they lines in different circles, there is no definite order inside $A$.
It is not clear whether $\alpha_{2}$ should be considered as the right-hand
side subset of left-hand subset of $\alpha$.

It could be quite useful to make a clarification about the configurations
where we can explicitly calculate the PEE or entanglement contour, thus study
the BPE.

  1. 1.

The entanglement contour for one dimensional regions in general theories with
a definite order can be calculated using the PEE proposal Wen:2018whg ;
Wen:2019ubu . The logic of the PEE proposal even works for disconnected
intervals with a definite order (see for example Kudler-Flam:2019nhr ).

  2. 2.

The entanglement contour for highly symmetric regions in higher dimensions,
which can be characterized by a single coordinate, can also be calculated by
the PEE proposal Han:2019scu . These are called the quasi-one-dimensional
configurations. For example the contour function for balls and annuli with
rotational symmetries, strips with translation symmetries. This also works for
general theories.

  3. 3.

In holographic theories, the entanglement contour for regions with local
modular Hamiltonian can be calculated by the geometric construction. This
works for the intervals (static or covariant) and balls in higher dimensions
Kudler-Flam:2019oru ; Han:2019scu . It is also valid for holographic theories
beyond AdS/CFT (see for example Wen:2018mev ).

  4. 4.

In Poincaré invariant CFTs with general dimensions, the PEE between any two
connected regions and the entanglement contour for any connected region can be
calculated by the general formula derived in Wen:2019iyq .

In this paper, we mainly use the PEE proposal and the geometric construction
to one-dimensional regions. We will also not discuss the Gaussian formula. We
will focus on systems in two-dimensional spacetime, especially those with a
geometric dual in the context of AdS/CFT. Because in these cases we have more
tools to calculate the PEE. The adjacent cases and non-adjacent cases will be
discussed separately.

##  3 Aspects of BPE when $A$ and $B$ are adjacent

In the previous section, we defined the balanced partial entanglement. Here we
claim that the BPE gives the area of the EWCS. In this section we explicitly
calculate the BPE in the case of the global AdS3 that duals to the vacuum
state of the boundary CFT2. We take $A$ and $B$ to be intervals on the AdS
boundary, then the boundary vacuum state is a purification of $\rho_{AB}$. We
firstly consider $A$ and $B$ to be adjacent and leave the non-adjacent case
for the next section. We will explicitly calculate the BPE$(A,B)$ and compare
it with the EWCS. Then we discuss the general entropy relations satisfied by
BPE$(A,B)$. Also BPE$(A,B)$ is calculated in the case of the minimal
purification in the context of the surface/state correspondence Miyaji:2015yva
.

Figure 5: The above circle is the boundary of the global AdS3 in a time slice,
while $AB$ is an interval in the circle. The complement $A^{\prime}\cup
B^{\prime}$ is partitioned by a point $P$. The balance requirements can be
satisfied by adjusting the position of $P$.

###  3.1 Holographic BPE for adjacent intervals

Let us consider the case in Fig.5, where the circle is the boundary of the
global AdS3. The balance requirement (2.6) will determine the position of the
point $P$ that partition the complement $A^{\prime}B^{\prime}$ into
$A^{\prime}$ and $B^{\prime}$. Note that, the requirement (2.6) does not refer
to any information from the bulk geometry.

Here we apply the PEE proposal to calculate the PEE in (2.6). The region here
we consider is $AA^{\prime}$ while the subset is $\alpha=A$. In this case if
we define $\alpha_{R}=A^{\prime}$, then $\alpha_{L}=\varnothing$. Following
(2.11) we have

$\displaystyle
s_{AA^{\prime}}(A)=\frac{1}{2}\left(S_{A}+S_{AA^{\prime}}-S_{A^{\prime}}\right)\,.$
(3.12)

Similarly we have

$\displaystyle
s_{BB^{\prime}}(B)=\frac{1}{2}\left(S_{B}+S_{BB^{\prime}}-S_{B^{\prime}}\right)\,.$
(3.13)

The balance requirement (2.6) then gives the following equation

$\displaystyle S_{A}+S_{AA^{\prime}}-S_{A^{\prime}}=$ $\displaystyle
S_{B}+S_{BB^{\prime}}-S_{B^{\prime}}\,,$ (3.14)
$\displaystyle\Rightarrow~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S_{A}-S_{B}=$
$\displaystyle S_{A^{\prime}}-S_{B^{\prime}}\,,$ (3.15)

which is enough to determine the point $P$.

In global AdS3, the entanglement entropy for an arbitrary interval of length
$l$ is given by

$\displaystyle
S=\frac{c}{3}\log\left(\frac{2}{\epsilon}\sin\frac{l}{2}\right)\,,$ (3.16)

where we have taken the length of the AdS boundary circle to be $2\pi$, the
AdS radius $\ell=1$ and $c=\frac{3}{2G}$. Let us use
$l_{A},\,l_{B},\,l_{A^{\prime}}$, and $l_{B^{\prime}}$ to denote the length of
all the relevant intervals. Obviously, we have

$\displaystyle l_{A}+l_{B}+l_{A^{\prime}}+l_{B^{\prime}}=$ $\displaystyle
2\pi\,,$ (3.17)

also (3.14) can be written as

$\displaystyle\frac{\sin(l_{A}/2)}{\sin(l_{B}/2)}=$
$\displaystyle\frac{\sin(l_{A^{\prime}}/2)}{\sin(l_{B^{\prime}}/2)}\,.$ (3.18)

The solution of the above two equations is given by

$\displaystyle l_{A^{\prime}}=$
$\displaystyle-2\cot^{-1}\left[\frac{1}{2}\csc\left(\frac{{l_{A}}}{2}\right)\left(\sin\left({l_{A}}+\frac{{l_{B}}}{2}\right)-3\sin\left(\frac{{l_{B}}}{2}\right)\right)\csc\left(\frac{{l_{A}}}{2}+\frac{{l_{B}}}{2}\right)\right]\,,$
(3.19) $\displaystyle l_{B^{\prime}}=$ $\displaystyle~{}2\pi-
l_{A}-l_{B}-l_{A^{\prime}}\,.$ (3.20)

Plugging the solution into the PEE (3.12), we get the BPE,

$\displaystyle\text{BPE}(A,B)=\frac{c}{6}\log\left[\frac{4}{\epsilon}\frac{\sin(l_{A}/2)\sin(l_{B}/2)}{\sin((l_{A}+l_{B})/2)}\right]\,.$
(3.21)

One can check that, the $\text{BPE}(A,B)$ exactly gives the area of the EWCS
calculated in HEoP1 ; HEoP2 .

The above calculation shows that the BPE captures the specific correlation
between $A$ and $B$, which is represented by the area of the EWCS. Since the
BPE can be defined in general quantum system, the BPE could be considered as a
generalization of the EWCS to non-holographic systems.

More interestingly we find that the crossing PEE $\mathcal{I}_{AB^{\prime}}$
in this case is a constant independent from $l_{A}$ and $l_{B}$. Using the PEE
proposal and the solution (3.19), we find

$\displaystyle\mathcal{I}_{AB^{\prime}}=s_{A^{\prime}AB}(A)=$
$\displaystyle\frac{1}{2}\left(S_{A^{\prime}A}+S_{AB}-S_{A^{\prime}}-S_{B}\right)$
(3.22) $\displaystyle=$
$\displaystyle\frac{c}{6}\log\left[\frac{\sin\left((l_{A}+l_{A^{\prime}})/2\right)\sin\left((l_{A}+l_{B})/2\right)}{\sin\left(l_{A^{\prime}}/2\right)\sin\left(l_{B}/2\right)}\right]$
(3.23) $\displaystyle=$ $\displaystyle\frac{c}{6}\log 2\,.$ (3.24)

This is surprising that the crossing PEE is independent from the partition of
the pure state, hence can be used to characterize or classify the
purifications. Later we will show that the crossing PEE for the canonical
purification is also $\frac{c}{6}\log 2$. However, there is no evidence that
the crossing PEE is invariant under all the purifications.

###  3.2 Entropy relations for BPE

The PEEs which can be written as a linear combination of the entanglement
entropies $S_{A},~{}S_{B}$ and $S_{AB}$ are of course purification
independent. For example,

$\displaystyle\mathcal{I}(A,B)$ $\displaystyle=\frac{1}{2}I(A,B)\,,$ (3.25)
$\displaystyle\mathcal{I}(A,A^{\prime}B^{\prime})$
$\displaystyle=\frac{1}{2}\left(S_{A}+S_{AB}-S_{B}\right)\,,$ (3.26)
$\displaystyle\mathcal{I}(B,A^{\prime}B^{\prime})$
$\displaystyle=\frac{1}{2}\left(S_{B}+S_{AB}-S_{A}\right)\,.$ (3.27)

Note that the relation (3.25) between $\mathcal{I}(A,B)$ and the mutual
information $I(A,B)$ only holds for the adjacent cases. The balance
requirements give one more purification independent quantity

$\displaystyle\left(\mathcal{I}_{AA^{\prime}}-\mathcal{I}_{BB^{\prime}}\right)=S_{A}-S_{B}.$
(3.28)

Unfortunately the crossing PEE, as well as BPE$(A,B)$, is not purification
independent. BPE$(A,B)$ is only invariant under the unitary transformations
which keep $\mathcal{I}_{AB^{\prime}}$ fixed. These include

  * •

the local unitary transformations on $A^{\prime}$ and $B^{\prime}$
respectively,

  * •

the unitary transformations that adding or removing correlations between
$A^{\prime}$ and $B^{\prime}$,

  * •

the unitary transformations that adding or removing correlation between $A$
and $A^{\prime}$ while adding or removing the same amount of correlation
between $B$ and $B^{\prime}$, i.e. keeping
$\mathcal{I}_{AA^{\prime}}-\mathcal{I}_{BB^{\prime}}$ fixed.

Note that, the correlation here means the PEE.

Now we discuss the general entropy relations satisfied by BPE$(A,B)$. The
upper bound of the PEE indicates $s_{AA^{\prime}}(A)\leq S_{A}$ and
$s_{BB^{\prime}}(B)\leq S_{B}$. Imposing the balance requirement, we directly
get

$\displaystyle property~{}1:~{}\text{BPE}(A,B)\leq\text{min}(S_{A},S_{B})\,.$
(3.29)

The above relation can be satisfied in general cases.

When the balance requirements are satisfied,
BPE$(A,B)=\mathcal{I}_{AB}+\mathcal{I}_{AB^{\prime}}$. In the adjacent cases
we also have (3.25). Then the positivity of the PEE directly gives the
following relation

$\displaystyle property~{}2:~{}\text{BPE}(A,B)\geq\frac{1}{2}I(A,B).$ (3.30)

Note that in the non-adjacent cases the above relation cannot be proved in the
same way. In terms of the PEEs, the entanglement entropy satisfies the
following decomposition

$\displaystyle
S_{A}=\mathcal{I}_{AB}+\mathcal{I}_{AA^{\prime}}+\mathcal{I}_{AB^{\prime}}\,,\qquad
S_{B}=\mathcal{I}_{AB}+\mathcal{I}_{BA^{\prime}}+\mathcal{I}_{BB^{\prime}}\,.$
(3.31)

In the cases where $AB$ is a connected region, we also have

$\displaystyle S_{AB}=$
$\displaystyle\mathcal{I}_{AA^{\prime}}+\mathcal{I}_{AB^{\prime}}+\mathcal{I}_{BB^{\prime}}+\mathcal{I}_{BA^{\prime}}\,.$
(3.32)

One can easily see that the above decompositions directly gives

$\displaystyle I(A,B)=S_{A}+S_{B}-S_{AB}=2\mathcal{I}(A,B).$ (3.33)

However the decomposition (3.32) is not accurate for the non-adjacent
cases333Note that this evaluation of entanglement entropy using PEE is very
subtle for disconnected regions. For example, previous studies
Berthiere:2019lks ; Roy:2019gbi ; Wen:2019iyq  showed that naively taking an
uniform cutoff for all the endpoints of multi-intervals in CFT2 will give the
results of Refs. Casini:2005rm ; Casini:2004bw ; Calabrese:2004eu ;
Hubeny:2007re , which is only justified for free fermions while incorrect in
more general theories. So the relation $I(A,B)=2\mathcal{I}(A,B)$ only holds
when $AB$ is a connected region., thus the comparison between
$\text{BPE}(A,B)$ and $\frac{1}{2}I(A,B)$ is not clear.

The properties (3.29) and (3.30) directly lead to other interesting entropy
relations. For example the polygamy inequality for a the system $ABC$ in a
pure state,

$\displaystyle
property~{}3:~{}\text{BPE}(A,B)+\text{BPE}(A,C)\geq\text{BPE}(A,BC)\,.$ (3.34)

and the saturation of the upper bound when the Araki-Lieb inequality is
saturated

$\displaystyle
property~{}4:~{}|S_{A}-S_{B}|=S_{AB}~{}~{}\Rightarrow~{}~{}\text{BPE}(A,B)=\min(S_{A},S_{B})\,.$
(3.35)

The monotonicity of the BPE$(A,B)$ can also be justified using the additivity
and positivity of the PEE. We assume that the partition point $P$ satisfies
the balance requirement. Let us consider a region $C$ which is inside
$B^{\prime}$ and adjacent to $B$. We define the complement of $C$ inside
$B^{\prime}$ to be $B^{\prime\prime}$ thus $B^{\prime}=C\cup
B^{\prime\prime}$. According to the additivity and positivity we have

$\displaystyle s_{AA^{\prime}}(A)=s_{BB^{\prime}}(B)<s_{BB^{\prime}}(BC)\,.$
(3.36)

Then we combine $B$ and $C$, and consider BPE$(A,BC)$. Since $B$ expands to
$BC$, the balance requirement is now $s_{AA^{\prime}}(A)=s_{BB^{\prime}}(BC)$.
We need to adjust the position of $P$ to go back to balance. In other words we
should increase $s_{AA^{\prime}}(A)$ and reduce $s_{BB^{\prime}}(BC)$ by
adjusting $P$. Since PEE is additive, we can write

$\displaystyle s_{AA^{\prime}}(A)$
$\displaystyle=\mathcal{I}(A,BC)+\mathcal{I}(A,B^{\prime\prime})$ (3.37)
$\displaystyle s_{BB^{\prime}}(BC)$
$\displaystyle=\mathcal{I}(BC,A)+\mathcal{I}(BC,A^{\prime})$ (3.38)

Our goal can be easily achieved by moving $P$ towards $A$, thus
$B^{\prime\prime}$ expands while $A^{\prime}$ shrinks. Due to the positivity
and additivity, this procedure increases $\mathcal{I}(A,B^{\prime\prime})$ and
reduces $\mathcal{I}(BC,A^{\prime})$. After $P$ is properly settled down such
that $s_{AA^{\prime}}(A)=s_{BB^{\prime}}(BC)$, the PEE
$s_{AA^{\prime}}(A)=\text{BPE}(A,BC)$ is bigger than its previous value
$\text{BPE}(A,B)$. In other words, we get the monotonicity of BPE,

$\displaystyle property~{}5:~{}\text{BPE}(A,BC)\geq\text{BPE}(A,B)\,.$ (3.39)

In the cases where $A$ and $B$ are non-adjacent, the partition point is more
than one. A similar argument also lead to the monotonicity using the
positivity and additivity of the PEE.

In summary our arguments for properties 1 and 5 also applies for non-adjacent
cases. The argument for property 2 applies to the adjacent cases in a generic
theory. The properties 3 and 4 follows from properties 1 and 2. The properties
1-5 are all satisfied by the EoP.

###  3.3 BPE in the minimal purification and the entanglement of purification

Then we consider another purification, which is closely related to the EoP.
The EoP is defined in the following: assuming a bipartite system $AB$ is in a
mixed state $\rho_{AB}$. Let
$\left|\psi\right\rangle\in\mathcal{H}_{AA^{\prime}}\otimes\mathcal{H}_{BB^{\prime}}$
be a purification of $\rho_{AB}$. The EoP $E_{p}(A,B)$ EoP  is defined by:

$\displaystyle
E_{p}(A,B)=\mathop{\text{min}}\limits_{\phi,A^{\prime}}S_{AA^{\prime}}\,,$
(3.40)

where minimization is over all the purifications and all the partitions of
$A^{\prime}B^{\prime}$. The EoP is an intrinsic entanglement measure
independent from purifications.

It was proposed HEoP1 ; HEoP2  that in the context of AdS/CFT the EoP
$E_{p}(A,B)$ is dual to the area of the EWCS $\Sigma_{AB}$ in the following
way

$\displaystyle E_{p}(A,B)=\frac{Area(\Sigma_{AB})}{4G}$ (3.41)

Also the entropy relations satisfied by $\Sigma_{AB}$ are the same with those
satisfied by EoP. Though it is hard to prove this duality, in the context of
the surface/state correspondence Miyaji:2015yva  the calculation of the area
of $\Sigma_{AB}$ perfectly agrees with the definition of the EoP HEoP1 ;
Guo:2019azy . The surface/state correspondence proposes much more general
correspondence between bulk codimension-2 convex Miyaji:2015yva  spacelike
surfaces and quantum states, based on the tensor network description of the
AdS/CFT correspondence. In the case of AdS3/CFT2 the main points of the
surface/state correspondence are in the following.

  * •

Convex curves that homologous to a point correspond to pure states (for
example circles in the bulk with no black hole inside), otherwise the curves
correspond to mixed states (like intervals in the bulk or circles that
surrounding a black hole).

  * •

Any two surfaces $\sigma_{1}$ and $\sigma_{2}$ that are connected by a smooth
deformation preserving convexity, are related by an unitary transformation.

  * •

The entanglement entropy for any convex curves $\sigma$ are calculated by the
area of the minimal surfaces that homologous $\sigma$, which is a straight
forward generalization of the RT formula.

In the case of global AdS3, the boundary $ABA^{\prime}B^{\prime}$ is a circle
in a pure state. Let us consider the simple case of Fig.5. In the context of
the surface/state correspondence it is convenient to perform unitary
transformations on $A^{\prime}B^{\prime}$ by deforming of the curve
$A^{\prime}B^{\prime}$ while keeping the endpoints fixed. Among all the
deformations and partitions, $S_{AA^{\prime}}$ arrives at its minimal value
when $A^{\prime}B^{\prime}$ is deformed to the RT surface $\mathcal{E}_{AB}$
and $A^{\prime}B^{\prime}$ is properly partitioned thus
$\mathcal{E}_{AA^{\prime}}$ is normal to
$\mathcal{E}_{A^{\prime}B^{\prime}}=A^{\prime}B^{\prime}$. It is easy to find
that $\mathcal{E}_{AA^{\prime}}$ coincides with $\Sigma_{AB}$, (see Fig.6).
Hence the calculation of the length of $\Sigma_{AB}$ can be achieved via the
minimization of $S_{AA^{\prime}}$ among all the purifications and partitions.

Figure 6: Here $A^{\prime}B^{\prime}$ is deformed to be the RT surface of
$AB$, it is also the RT surface of itself. $S_{AA^{\prime}}$ reaches its
minimal value when $\mathcal{E}_{AA^{\prime}}$ coincide with $\Sigma_{AB}$,
which is the purple line.

Then let us consider the configuration of Fig.6 without specifying the
partition of $A^{\prime}B^{\prime}$. We can evaluate the BPE$(A,B)$ using the
PEE proposal and balance requirements. Since $A^{\prime}B^{\prime}$ is a
minimal surface, using the generalized RT formula for the bulk convex curves
we have

$\displaystyle S_{A^{\prime}}=\frac{l_{A^{\prime}}}{4G}\,,\qquad
S_{B^{\prime}}=\frac{l_{B^{\prime}}}{4G}\,.$ (3.42)

Since $\mathcal{E}_{AB}=A^{\prime}\cup B^{\prime}$ we have

$\displaystyle S_{A^{\prime}}+S_{B^{\prime}}=S_{AB}\,.$ (3.43)

The balance requirement requires that

$\displaystyle S_{A^{\prime}}-S_{B^{\prime}}=S_{A}-S_{B}$ (3.44)

Solving the above two equations we have

$\displaystyle
S_{A^{\prime}}=\frac{c}{6}\log\left[\frac{2\sin\frac{l_{A}}{2}\sin\frac{l_{A}+l_{B}}{2}}{\epsilon\sin\frac{l_{B}}{2}}\right]\,,\qquad
S_{B^{\prime}}=\frac{c}{6}\log\left[\frac{2\sin\frac{l_{B}}{2}\sin\frac{l_{A}+l_{B}}{2}}{\epsilon\sin\frac{l_{A}}{2}}\right]\,.$
(3.45)

The above solution determines the position of the partition point $P$. It is
interesting that the point $P$ is exactly where $\Sigma_{AB}$ intersecting
with $\mathcal{E}_{AB}$. One may wonder that here the BPE$(A,B)$ may also give
the length of $\Sigma_{AB}$. This is obviously not true because

$\displaystyle S_{AA^{\prime}}>s_{AA^{\prime}}(A)\,.$ (3.46)

Also according to the generalized RT formula and (3.21) we have

$\displaystyle
S_{AA^{\prime}}=\frac{Area(\Sigma_{AB})}{4G}=\frac{c}{6}\log\left[\frac{4}{\epsilon}\frac{\sin(l_{A}/2)\sin(l_{B}/2)}{\sin((l_{A}+l_{B})/2)}\right]\,,$
(3.47)

while the BPE$(A,B)$ is given by

$\displaystyle\text{BPE}(A,B)=$ $\displaystyle
s_{AA^{\prime}}(A)=\frac{1}{2}\left(S_{A}+S_{AA^{\prime}}-S_{A^{\prime}}\right)$
(3.48) $\displaystyle=$
$\displaystyle\frac{c}{6}\log\left[\frac{2\sin\frac{l_{A}}{2}\sin\frac{l_{B}}{2}}{\epsilon\sin\frac{l_{A}+l_{B}}{2}}\right]+\frac{c}{12}\log
2$ (3.49) $\displaystyle=$ $\displaystyle\mathcal{I}_{AB}+\frac{c}{12}\log
2\,.$ (3.50)

Here we used the solution (3.45). Also one can easily read that the crossing
PEE is just given by

$\displaystyle\mathcal{I}_{AB^{\prime}}=\frac{c}{12}\log 2\,,$ (3.51)

which is again a constant independent from $l_{A}$ and $l_{B}$. However, it is
different from the constant $\frac{c}{6}\log 2$ for the case of the global
AdS3. In other words, in the context of the surface/state correspondence the
unitary transformation, that evolve the boundary vacuum state of the global
AdS3 to the “minimal” purification shown in Fig.6, changes the crossing PEE
$\mathcal{I}_{AB^{\prime}}$ hence changes the BPE$(A,B)$.

##  4 BPE for non-adjacent intervals

Figure 7: The region $AB$ is disconnected two-interval with connected
entanglement wedge. The partition of the combined system
$ABA^{\prime}B^{\prime}$ follows our prescription to minimize the
$s_{AA^{\prime}}(A)$ and is determined by the position of two points $P_{1}$
and $P_{2}$ under the balance requirements,.

Then we consider the cases where $A$ and $B$ are non-adjacent intervals on the
AdS boundary and the entanglement wedge $\mathcal{W}_{AB}$ is connected. For
example, see Fig.7. In this case the compliment $A^{\prime}B^{\prime}$ of $AB$
on the boundary is also disconnected. It is partitioned by two points $P_{1}$
and $P_{2}$ into four regions that are classified into two pairs. Let us
denote, for example, the region partitioned by $P_{1}$ as $A^{\prime}_{1}\cup
B^{\prime}_{1}$. As we have mentioned previously, the position of $P_{1}$ and
$P_{2}$ can be determined by the balance requirements (2.10). Here we rewrite
the two independent requirements in the following

$\displaystyle
s_{AA^{\prime}}(A^{\prime}_{1})=s_{BB^{\prime}}(B^{\prime}_{1})\,,\qquad
s_{AA^{\prime}}(A^{\prime}_{2})=s_{BB^{\prime}}(B^{\prime}_{2})\,.$ (4.52)

Note that $s_{AA^{\prime}}(A)=s_{BB^{\prime}}(B)$ follows from
$S_{AA^{\prime}}=S_{BB^{\prime}}$.

Let us denote the length of the intervals to be

$\displaystyle l_{A}=2a,\quad l_{B}=2b,\quad l_{A_{1}^{\prime}}=2a_{1},\quad
l_{B_{1}^{\prime}}=2b_{1}\quad l_{A_{2}^{\prime}}=2a_{2},\quad
l_{B_{2}^{\prime}}=2b_{2}.$ (4.53)

Given the length and position of $A$ and $B$, the length and position of
$A_{1}^{\prime}B_{1}^{\prime}$ and $A_{2}^{\prime}B_{2}^{\prime}$ are also
determined. Assuming $a_{1}+b_{1}=\alpha$ we have
$a_{2}+b_{2}=2\pi-2a-2b-2\alpha$. Since $\alpha$ is known to us, there remains
only two undetermined parameters. Let us take them to be $a_{1}$ and $a_{2}$
thus

$\displaystyle b_{1}=\alpha-a_{1}\,,\qquad b_{2}=\pi-\alpha-a-b-a_{2}\,.$
(4.54)

The balance requirements (4.52) give that

$\displaystyle
S_{A_{1}^{\prime}}-S_{B_{1}^{\prime}}=S_{AA_{2}^{\prime}}-S_{BB_{2}^{\prime}}\,,\qquad
S_{A_{2}^{\prime}}-S_{B_{2}^{\prime}}=S_{AA_{1}^{\prime}}-S_{BB_{1}^{\prime}}\,.$
(4.55)

Using the holographic result for entanglement entropy (3.16) and the relations
(4.54), the above equations can be rewritten as,

$\displaystyle\frac{\sin[a_{1}]}{\sin[\alpha-a_{1}]}=$
$\displaystyle\frac{\sin[a+a_{2}]}{\sin[\alpha+a+a_{2}]}\,,$ (4.56)
$\displaystyle\frac{\sin[a_{2}]}{\sin[\alpha+a+b+a_{2}]}=$
$\displaystyle\frac{\sin[a+a_{1}]}{\sin[a+a_{2}]}\,,$ (4.57)

which uniquely determine the partition points of $A^{\prime}B^{\prime}$. The
solutions are in the following,

$\displaystyle a_{1}=\cos^{-1}$
$\displaystyle\left[\frac{\sin(a-\alpha+a_{2})+3\sin(a+\alpha+a_{2})}{\sqrt{2}\sqrt{-2\cos(2(a+\alpha+a_{2}))-2\cos(2(a+a_{2}))+\cos(2\alpha)+3}}\right]\,,$
(4.58) $\displaystyle a+2a_{2}=\tan^{-1}$
$\displaystyle\Big{[}\sin(a-b)(\sin(a)\cos(\eta)-\sin(b))-2\lambda\sin(a)\sin(\eta),$
(4.59)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\sin(a)\sin(\eta)\sin(a-b)-2\lambda\sin(a)\cos(\eta)+2\lambda\sin(b)\Big{]}\,,$
(4.60)

where

$\displaystyle\lambda=$
$\displaystyle\sqrt{\sin(a)\sin(b)\sin(a+\alpha)\sin(\alpha+b)}\,,$ (4.61)
$\displaystyle\eta=$ $\displaystyle a+b+2\alpha\,.$ (4.62)

Following the PEE proposal we have

$\displaystyle s_{AA^{\prime}}(A)$
$\displaystyle=\frac{1}{2}\left(S_{AA_{1}^{\prime}}+S_{AA_{2}^{\prime}}-S_{A_{1}^{\prime}}-S_{A_{2}^{\prime}}\right)$
(4.63)
$\displaystyle=\frac{c}{6}\log\left[\frac{\sin[a+a_{1}]\sin[a+a_{2}]}{\sin[a_{1}]\sin[a_{2}]}\right]\,.$
(4.64)

The above PEE gives the BPE$(A,B)$ when we plug in the solutions
(4.58)-(4.59). At last, we find

$\displaystyle\text{BPE}(A,B)$
$\displaystyle=\frac{c}{6}\log\left[2\frac{\lambda+\sin(a)\sin(b)}{\sin(\alpha)\sin(a+\alpha+b)}+1\right]$
(4.65)

Though the solution to the balance requirements is a bit complicated, the BPE
has a simple expression. One can check that it exactly matches with the length
of $\Sigma_{AB}$ previously calculated in HEoP2 . Compare with the calculation
of HEoP2 , our requirements are simple and donot have to refer to any
information from the bulk geometry. Later we will explain why this matching
appears using the holographic picture for entanglement contour.

The BPE also satisfies certain entropy relations when $A$ and $B$ are non-
adjacent. The property 1 holds in general. The property 2 is not easy to prove
for disconnected $AB$ in a generic purification. This may due to the
disadvantage that our understanding of the entanglement contour for
disconnected regions is not clear Wen:2019iyq . However, for holographic
cases, because the mutual information is monogamous Hayden:2011ag , it was
proved in Wen:2019ubu  that the PEE satisfies the following inequality,

$\displaystyle
s_{AA^{\prime}}(A)\geq\frac{1}{2}I(A,BB^{\prime})\geq\frac{1}{2}I(A,B)\,.$
(4.66)

Since the BPE is also a PEE we have $\text{BPE}(A,B)\geq\frac{1}{2}I(A,B)$.

For non-adjacent cases, so far we donot have the proof for property 2 for non-
holographic theories. We want to point out that, the monogamy of the mutual
information is not a necessary condition for the property 2. So it is still
possible to prove it in general cases. We leave this point for future study.
Since the property 3 and 4 follow from property 2, they are also only
justified for holographic theories. While the property 5 of monotonicity can
be understood for generic configurations using the similar arguments in the
previous section.

##  5 The canonical purification and the reflected entropy

The canonical purification discussed in Dutta:2019gen  is another example
where we can explicitly study the BPE. In Dutta:2019gen  a new quantity named
the reflected entropy was defined and its holographic relation to the EWCS was
established. In this section, we will show that the reflected entropy is
indeed identical to the BPE for canonical purifications. Consequently the
relation between the BPE and $\Sigma_{AB}$ in the canonical purification
follows directly. Note that the reflected entropy is only defined in the
canonical purification cases, while the BPE can be defined for a generic
purification. The BPE can be considered as a generalization of the reflected
entropy for generic purifications.

Let us firstly give a brief review on the canonical purification and the
reflected entropy. Consider a bipartite system $A\cup B$ with the Hilbert
space $\mathcal{H}_{AB}$ and the orthonormal bases
$\\{\left|\psi_{i}\right\rangle\\}$. The system is in a mixed state

$\displaystyle\rho_{AB}=\sum\limits_{i}p_{i}\left|\psi_{i}\right\rangle\left\langle\psi_{i}\right|\,.$
(5.67)

Then we introduce a system $A^{\prime}\cup B^{\prime}$ with the same copy of
the Hilbert space, and the partition is just a reflection of the partition of
$AB$. The canonical purification is given by the following pure state for the
combined system $ABA^{\prime}B^{\prime}$,

$\displaystyle\left|\sqrt{\rho_{AB}}\right\rangle=\sum\limits_{i}\sqrt{p_{i}}\left|\psi_{i}\right\rangle_{AB}\left|\psi_{i}\right\rangle^{*}_{A^{\prime}B^{\prime}}\,,$
(5.68)

where $\\{\left|\psi_{i}\right\rangle^{*}\\}$ is another orthonormal basis of
$\mathcal{H}_{AB}$. Now the mixed state is the reduced density matrix
$\rho_{AB}=\text{Tr}_{A^{\prime}B^{\prime}}\left|\sqrt{\rho_{AB}}\right\rangle\left\langle\sqrt{\rho_{AB}}\right|$.
The thermo-field double state is a simple case of the canonical purification.
The reflected entropy is then defined as the von Neumann entropy (or
entanglement entropy) for $AA^{\prime}$,

$\displaystyle
S_{R}(A:B)=S_{AA^{\prime}}=-\text{Tr}\rho_{AA^{\prime}}\log\rho_{AA^{\prime}}\,,$
(5.69)

where
$\rho_{AA^{\prime}}=\text{Tr}_{BB^{\prime}}\left|\sqrt{\rho_{AB}}\right\rangle\left\langle\sqrt{\rho_{AB}}\right|$.

Figure 8: The homology surfaces $\mathcal{R}_{AB}$ and
$\mathcal{R}_{A^{\prime}B^{\prime}}$ are glued at the RT surface
$\mathcal{E}_{AB}$, which is the blue line. This configuration
$\mathcal{RR^{\prime}}_{AB}$ is the gravity dual of the canonical purification
of $\rho_{AB}$ and has the reflection symmetry at $\mathcal{E}_{AB}$. The
solid purple line is the $\Sigma_{AB}$ while the dashed purple line is its
image under reflection.

It is important that, for holographic systems the canonical purification has a
bulk geometric description. For example the eternal black hole
Maldacena:2001kr  describes the thermo-field double state. For more generic
cases where $AB$ is a subsystem of a holographic boundary state, the geometric
dual for the canonical purification is the manifold glued from the homology
surface $\mathcal{R}_{AB}$ and its CPT conjugate
$\mathcal{R}_{A^{\prime}B^{\prime}}$ along the RT surfaces $\mathcal{E}_{AB}$
and $\mathcal{E}_{A^{\prime}B^{\prime}}$. For example see Fig.8. This glued
manifold, denoted as $\mathcal{RR^{\prime}}_{AB}$, is the gravity dual of the
canonical purification $\left|\rho_{AB}\right\rangle$. By construction it has
reflection symmetry at the RT surface $\mathcal{E}_{AB}$. This holographic
construction is proposed in Dutta:2019gen  using the Engelhardt-Wall procedure
Engelhardt:2018kcs ; Engelhardt:2017aux . The entanglement entropy
$S_{AA^{\prime}}$ is then holographically calculated by the minimal surface
$\mathcal{E}_{AA^{\prime}}$ in $\mathcal{RR^{\prime}}_{AB}$ that anchored on
the boundary of $AA^{\prime}$.

It was shown in Dutta:2019gen  that $\mathcal{E}_{AA^{\prime}}$ is closely
related to $\Sigma_{AB}$. In general $\Sigma_{AB}$ can be determined by the
following two properties. Firstly it should be a geodesic chord in
$\mathcal{R}_{AB}$ that separates $A$ from $B$. Secondly it should be anchored
on the RT surface $\mathcal{E}_{AB}$ vertically, which is required by the
reflection symmetry. For example, in Fig.8 where $A$ and $B$ are adjacent,
$\Sigma_{AB}$ is the geodesic chord that emanates from the joint point of $A$
and $B$ and end on $\mathcal{E}_{AB}$ vertically, which is shown by the solid
purple line. The dashed purple line is the image of $\Sigma_{AB}$ under
reflection. The RT surface $\mathcal{E}_{AA^{\prime}}$ is just formed by
$\Sigma_{AB}$ and its image. This directly gives that

$\displaystyle Area(\mathcal{E}_{AA^{\prime}})=2Area(\Sigma_{AB})\,.$ (5.70)

The reflected entropy is then related to $\Sigma_{AB}$ in the following way

$\displaystyle\frac{1}{2}S_{R}(A,B)=\frac{Area(\mathcal{E}_{AA^{\prime}})}{8G}=\frac{Area(\Sigma_{AB})}{4G}$
(5.71)

Now let us calculate the BPE$(A,B)$. We see that in Fig.8 the combined system
$ABA^{\prime}B^{\prime}$ forms a circle and $AA^{\prime}$ is a connected
interval with a definite order. In this case we can calculate the PEE
$s_{AA^{\prime}}(A)$ using the PEE proposal (2.11),

$\displaystyle
s_{AA^{\prime}}(A)=\frac{1}{2}(S_{A}+S_{AA^{\prime}}-S_{A^{\prime}})=\frac{1}{2}S_{AA^{\prime}}.$
(5.72)

We used $S_{A}=S_{A^{\prime}}$ in the above equation, which follows from the
reflection symmetry. However the definite order for $AA^{\prime}$ is not
guaranteed for generic configurations. Fortunately there exists a generic way
to calculate $s_{AA^{\prime}}(A)$ in canonical purification cases using the
reflection symmetry. The reflection symmetry indicates that the contribution
from $A$ and $A^{\prime}$ to $S_{AA^{\prime}}$ are equal. Regarding the
normalization property that
$s_{AA^{\prime}}(A)+s_{AA^{\prime}}(A^{\prime})=S_{AA^{\prime}}$, in general
we have

$\displaystyle
s_{AA^{\prime}}(A)=s_{AA^{\prime}}(A^{\prime})=\frac{1}{2}S_{AA^{\prime}}=\frac{1}{2}S_{R}(A,B).$
(5.73)

Similarly we have

$\displaystyle
s_{BB^{\prime}}(A)=s_{BB^{\prime}}(A^{\prime})=\frac{1}{2}S_{BB^{\prime}}.$
(5.74)

Since $S_{AA^{\prime}}=S_{BB^{\prime}}$, we straightforwardly find

$\displaystyle s_{AA^{\prime}}(A)=s_{BB^{\prime}}(A^{\prime})\,,$ (5.75)

which is exactly the balance requirement. In summary in the canonical
purifications for any $\rho_{AB}$ where the partition of
$A^{\prime}B^{\prime}$ is a reflection of the partition of $AB$, we have
$s_{AA^{\prime}}(A)=\text{BPE}(A,B)$. This further more indicates that the
BPE$(A,B)$ is directly related to the EWCS,

$\displaystyle\text{BPE}(A,B)=\frac{1}{2}S_{R}(A,B)=\frac{Area(\Sigma_{AB})}{4G}\,.$
(5.76)

Then we consider the case where $AB$ is a disconnected but has a connected
entanglement wedge, which is shown in the left figure of Fig.9. In this case
$AA^{\prime}$ has no boundary hence $\mathcal{E}_{AA^{\prime}}$ is determined
totally by the homology constraint, which turns out to be the minimal circle
that warps on the bulk wormhole geometry. It is easy to see the relation
(5.70) also holds. Note that, in this case the order in $AA^{\prime}$ is
ambiguous hence the PEE proposal does not apply. Using the reflection
symmetry, we can easily find that
$s_{AA^{\prime}}(A)=s_{BB^{\prime}}(A^{\prime})=\frac{1}{2}S_{R}(A,B)$, hence
(5.76) follows. In the right figure of Fig.9 where $\mathcal{E}_{AB}$ is
disconnected, one can also find the relation (5.76) holds using similar
arguments.

Figure 9: In the left figure $A$ and $B$ are non-adjacent and
$\mathcal{W}_{AB}$ is connected. In the right figure there is a black hole in
the bulk and $\mathcal{E}_{AB}$ is disconnected. Here the PEE proposal does
not apply because $AA^{\prime}$ is a circle.

The canonical purification is different from case of global AdS3. It is
interesting to find that in both of the two purifications the BPE$(A,B)$ gives
the area of $\Sigma_{AB}$, which indicates that the crossing PEE are the same
for these two purifications. One can easily check this for the case of Fig.8,
where the crossing PEE can be calculated by the PEE proposal,

$\displaystyle\mathcal{I}_{AB^{\prime}}=\mathcal{I}_{A^{\prime}B}=$
$\displaystyle\frac{1}{2}(S_{AB}+S_{AA^{\prime}}-S_{B}-S_{A^{\prime}})$ (5.77)
$\displaystyle=$
$\displaystyle\frac{c}{6}\log\left(\frac{\sin[(l_{A}+l_{B})\pi/L]}{\sin[l_{A}\pi/L]\sin[l_{B}\pi/L]}\frac{2\sin[l_{A}\pi/L]\sin[l_{B}\pi/L]}{\sin[(l_{A}+l_{B})\pi/L]}\right)$
(5.78) $\displaystyle=$ $\displaystyle\frac{c}{6}\log 2\,.$ (5.79)

In the above equation we used the relation $S_{A^{\prime}}=S_{A}$. Also
$S_{AA^{\prime}}$ is calculated by twice of the length of $\Sigma_{AB}$, which
is given by (3.21). Again we arrive at the constant $\frac{c}{6}\log 2$ which
is the exactly the one we got for global AdS3.

##  6 Interpretation for the correspondence between the BPE and the
entanglement wedge cross section

In this section we show that the EWCS can be interpreted as certain types of
PEE following the holographic picture for entanglement contour proposed in
Wen:2018whg ; Wen:2019ubu . We find that, the PEE that correspond to
$\Sigma_{AB}$ satisfies the balance requirement thus is a BPE. This justifies
our previous claim that the BPE gives $\Sigma_{AB}$ when $\rho_{AB}$ has a
geometric dual.

###  6.1 Brief review on holographic entanglement contour

In Wen:2018whg , the author gave a holographic picture for the entanglement
contour for a single interval in the context of AdS3/CFT2. Following the bulk
and boundary modular flows, it was shown in Wen:2018whg  that the entanglement
wedge $\mathcal{W}_{A}$ has a natural slicing using the modular slices (see
the left figure in Fig.10). A modular slice is the orbit of a boundary modular
flow line in the bulk444The modular flow that exactly settled at the boundary
has no orbit in the bulk, because the boundary modular flow equals to the bulk
modular flow. Here the boundary modular flow is not exactly at but infinitely
close to the boundary Wen:2018whg ; Wen:2019ubu .. This slicing gives a one-
to-one correspondence between the points in the interval $A$ and the points in
its RT surface $\mathcal{E}_{A}$. More explicitly the correspondence means the
contribution to $S_{A}$ from any point in $A$ is represented by its partner
point on $\mathcal{E}_{A}$. In all the cases where both of the above geometric
construction and the PEE proposal applies, the two proposal give the same
results. This consistency even goes beyond the AdS/CFT correspondence
Wen:2018mev .

When the interval $A$ is static, the point-to-point correspondence have a
simple description using geodesics normal to $\mathcal{E}_{A}$. It was shown
in Han:2019scu  that, any point on $\mathcal{E}_{A}$ can be connected to its
partner point on $A$ via a static geodesic that is normal to $\mathcal{E}_{A}$
(see the right figure in Fig.10). These normal geodesics are where the modular
slices intersect with the static homology surface $\mathcal{R}_{A}$.

Figure 10: The left figure shows the slicing of the entanglement wedge with
modular slices (the blue surface), while the left figure shows a time slice of
the entanglement wedge and the fine correspondence between the points in the
interval $A$ and its RT surface $\mathcal{E}_{A}$ using the dashed purple
geodesics normal to $\mathcal{E}_{A}$.

In the same sense this correspondence induces a correspondence between the
geodesic chords $\mathcal{E}_{i}$ on $\mathcal{E}_{A}$ and the PEE of certain
subset $A_{i}$ in $A$,

$\displaystyle s_{A}(A_{i})=\frac{Length\left(\mathcal{E}_{i}\right)}{4G}\,.$
(6.80)

For example see Fig.11. The above relation gives a finer correspondence
between quantum entanglement and bulk geometry than the RT formula.

Figure 11: The purple dashed lines are the static geodesics that normal to
$\mathcal{E}_{A}$. They are the lines which set the subsets of $A$ and the
subsets of $\mathcal{E}_{A}$ in pairs in the sense of (6.80). The points in
$A_{i}$ correspond to the points on $\mathcal{E}_{i}$.

###  6.2 The holographic BPE and the entanglement wedge cross section

The geodesics normal to the RT surfaces play an important role in the fine
correspondence. As we know the EWCS $\Sigma_{AB}$ is also a geodesic chord
normal to the RT surface $\mathcal{E}_{AB}$, $\Sigma_{AB}$ can be interpreted
as the gravity dual of certain PEE following (6.80). In other words, we can
relate the length of $\Sigma_{AB}$ to a PEE, and furthermore to a linear
combination of the entanglement entropies of relevant boundary intervals
following the PEE proposal (2.11). The prescription to determine this PEE is
to extend $\Sigma_{AB}$ to a RT surface of some boundary region. As a portion
of this RT surface, $\Sigma_{AB}$ will correspond to a PEE in this region.

Figure 12: Here the $\Sigma_{AB}$ is extended to a geodesic that anchored on
the boundary, which is just the RT surface $\mathcal{E}_{AA^{\prime}}$. Then
we consider the PEE in $AA^{\prime}$ and use $\mathcal{E}_{AB}$, which is the
blue line normal to $\mathcal{E}_{AA^{\prime}}$, to

Firstly, let us consider the cases in Fig.12 in the context of AdS3/CFT2. The
upper figures are Poincaré AdS3 and the lower figures are global AdS3. Firstly
we determine the RT surface that contains $\Sigma_{AB}$. This can be easily
done by extending $\Sigma_{AB}$ to a geodesic anchored on the boundary, which
is a RT surface of certain boundary interval. For the left figures in Fig.12,
$AB$ is connected and one of the endpoints of $\Sigma_{AB}$ is settled on the
boundary. The extension of $\Sigma_{AB}$ will also intersect with the boundary
on another $P$. The point $P$ partition the complement of $AB$ into two parts,
which we denote as $A^{\prime}$ and $B^{\prime}$. The extension of
$\Sigma_{AB}$ is just the RT surface $\mathcal{E}_{AA^{\prime}}$ (or
$\mathcal{E}_{BB^{\prime}}$) of the interval $A^{\prime}A$ (or $BB^{\prime}$).
Secondly, since the two RT surfaces $\mathcal{E}_{AB}$ and
$\mathcal{E}_{A^{\prime}A}$ are normal to each other, $\mathcal{E}_{AB}$ can
be considered to play the role of the dashed purple line in Fig.12. According
to the fine correspondence (6.80) we learn that $\Sigma_{AB}$ corresponds to
the $s_{AA^{\prime}}(A)$,

$\displaystyle\frac{Area(\Sigma_{AB})}{4G}=s_{A^{\prime}A}(A)\,.$ (6.81)

Note that $\mathcal{E}_{AA^{\prime}}$ is also the RT surface
$\mathcal{E}_{BB^{\prime}}$ of $BB^{\prime}$. Using the fine correspondence
between the points on $\mathcal{E}_{BB^{\prime}}$ and $BB^{\prime}$, we also
have

$\displaystyle\frac{Area(\Sigma_{AB})}{4G}=s_{BB^{\prime}}(B)\,.$ (6.82)

Then we directly find that the balance requirement is satisfied by the
partition induced by the extension of $\Sigma_{AB}$,

$\displaystyle s_{A^{\prime}A}(A)=s_{BB^{\prime}}(B)=\text{BPE}(A,B)\,.$
(6.83)

One can also explicitly check that the partition of $A^{\prime}B^{\prime}$
induced by the balance requirement and the extension of $\Sigma_{AB}$ are
exactly the same. This confirms our previous observation that

$\displaystyle\text{BPE}(A,B)=\frac{Area(\Sigma_{AB})}{4G}\,.$ (6.84)

Similarly, for the non-adjacent $AB$ with connected $\mathcal{W}_{AB}$ (see
the figures on the right hand side), the extension of $\Sigma_{AB}$ is a
geodesic anchored on the boundary at two points $P_{1}$ and $P_{2}$. We see
that $P_{1,2}$ partition the boundary into two regions $AA^{\prime}$ and
$BB^{\prime}$ and the extended $\Sigma_{AB}$ is the RT surface
$\mathcal{E}_{AA^{\prime}}$ of $AA^{\prime}$. In this case
$A^{\prime}=A^{\prime}_{1}\cup A^{\prime}_{2}$ and
$B^{\prime}=B^{\prime}_{1}\cup B^{\prime}_{2}$ are disconnected. Since
$\mathcal{E}_{AA^{\prime}}$ is normal to $\mathcal{E}_{AB}$, according to the
fine correspondence (6.80) and the PEE proposal we have

$\displaystyle\frac{Area(\Sigma_{AB})}{4G}$ $\displaystyle=s_{AA^{\prime}}(A)$
(6.85)

Also note that $\mathcal{E}_{AA^{\prime}}=\mathcal{E}_{BB^{\prime}}$, the fine
correspondence between $BB^{\prime}$ and $\mathcal{E}_{BB^{\prime}}$ indicates
that

$\displaystyle\frac{Area(\Sigma_{AB})}{4G}$
$\displaystyle=s_{BB^{\prime}}(B)\,,$ (6.86)

hence, again we arrive at (6.84).

Then we discuss the case where the boundary is in a mixed state. For example a
thermal state that duals to a BTZ black hole. For simplicity we take $AB$ to
be the whole boundary. Here we consider the thermal field double state
Maldacena:2001kr  which is a canonical purification discussed in Dutta:2019gen
. The gravity dual of the thermofield double state is the eternal black hole
Maldacena:2001kr . The two copies of CFT on each boundary are entangled thus
purifies each other. The Hilbert space factorizes into the left and right
subspaces $\mathcal{H}=\mathcal{H}_{L}\times\mathcal{H}_{R}$. The thermo-field
double state is in the following:

$\displaystyle\left|\Psi\right\rangle=\frac{1}{Z}\sum\limits_{n}e^{\frac{-\beta
E_{n}}{2}}\left|E_{n}\right\rangle_{L}\left|E_{n}\right\rangle_{R}.$ (6.87)

where $E_{n}$ is the energy eigenvalue of the energy eigenstates
$\left|E_{n}\right\rangle$.

Figure 13: A time slice of the eternal black hole. The blue circle is the
horizon, while the purple lines are $\Sigma_{AB}$ and its extension.

The two figures in Fig.13 draw the left and right hand side of the eternal
black hole in a time slice. They are glued together at the horizon. Assuming
that $B$ is larger than $A$. When $A$ is small enough the $\Sigma_{AB}$ is
just the RT surface $\mathcal{E}_{A}$. As $A$ becomes larger there will be a
phase transition for $\Sigma_{AB}$ from $\mathcal{E}_{A}$ to the two geodesic
chords emanating from the endpoints of $A$ and intersecting with the horizon
vertically (see the two solid purple lines in Fig.13). Let us consider the
later case and use the previous prescription to determine the partition of
$A^{\prime}B^{\prime}$. Since the two bulk sides are glued together at the
horizon, the extended $\Sigma_{AB}$ will enter the right bulk side through the
horizon and eventually intersect with the right boundary. See the dashed
purple line in Fig.13. The intersection points are the partition points that
divide the right boundary into $A^{\prime}\cup B^{\prime}$. The partition
determined by the extension of $\Sigma_{AB}$ is exactly a reflection of the
partition of $AB$, which is just same as the canonical purification cases.
Following our discussion for the canonical purifications, we have

$\displaystyle\text{BPE}(A,B)=\frac{Area(\Sigma_{AB})}{4G}\,.$ (6.88)

When $A$ is small, $\Sigma_{AB}=\mathcal{E}_{A}$ which cannot be extended to
the right bulk side. We may need to use the reflection image of $\Sigma_{AB}$
to partition $A^{\prime}B^{\prime}$, which is identical to the canonical
purifications cases.

For the cases where $AB$ does not cover the entire boundary, the reflection
symmetry between $AB$ and $A^{\prime}B^{\prime}$ no longer exist, which
differs from the canonical purifications. Since there are subtleties for
evaluating the PEEs, we will not discuss for these cases further.

##  7 Discussion

The entanglement contour is a finer and more comprehensive description for the
entanglement structure of a quantum system. It has the key property of
additivity, which makes it different from all the other known entanglement
measures. It is natural to expect that, other entanglement measures can be
extracted from the entanglement contour. In this paper, we consider a special
PEE satisfying the balance requirements for any purification of a bipartite
mixed state $\rho_{AB}$. We call it the balanced partial entanglement
BPE$(A,B,\psi)$ for the purification $\left|\psi\right\rangle$, which is
omitted when the purification is specified. We find that, for canonical
purifications the BPE$(A,B)$ is identical to half of the reflected entropy.
While for the holographic purification on the AdS boundary, the BPE$(A,B)$
gives the area of the EWCS divided by $4G$. These results show that, the BPE
unifies the quantum information interpretation for the EWCS in both the
canonical purification and purifications on the AdS boundaries. Since the BPE
can be defined in general quantum systems, in some sense it generalizes the
concept of the reflected entropy to generic purifications, and generalize the
EWCS to purifications with no geometric description.

Again, note that the partition that satisfies the balanced requirements is not
unique. We eliminate this ambiguity by imposing the minimal requirement, which
can be satisfied by our prescription to partition the purifier introduced in
section 2. For continuous systems, the balance requirements can always be
satisfied by continuously adjusting the partition. However, for discrete
systems, especially few-body systems, the continuous adjusting no longer exist
and the number of partitions is finite. In these cases, there is no obvious
reason for the existence of a partition that satisfies the balanced
requirements. We hope to clarify this point in the future. The study of the
BPE in few-body systems is feasible and will be interesting. Calculations in
section 3 and 4 can also be generalized to lattice models on one-dimensional
chains or circles, if the entanglement entropies for single intervals can be
calculated.

Also the study of the BPE can be extended to higher dimensions at least for
several highly symmetric configurations where the PEE can be explicitly
calculated. It will be very interesting to test the relation between the BPE
and the EWCS in higher dimensions. We can also explore the relation between
the BPE and other entanglement measures like EoP, entanglement negativity and
odd entropy in non-holographic systems.

Though the BPE depends on purification, it is independent from a large class
of unitary transformations on the complement $A^{\prime}B^{\prime}$. The
purifications with the same crossing PEE $\mathcal{I}_{AB^{\prime}}$ gives the
same BPE. Interestingly we show that, in both of the canonical purifications
and the holographic purification on the boundary of global AdS3, the crossing
PEE equals to $\frac{c}{6}\log 2$, which is a constant independent from the
partition of the pure state. While for the “minimal” purification in the
context of the surface/state correspondence, the crossing PEE equals to
$\frac{c}{12}\log 2$. It seems that the crossing PEE is a constant that
characterize the purifications, hence could be a useful tool to classify
purifications or quantum states. Is there any bounds for the crossing PEE and
how can they be saturated? Is the crossing PEE a useful tool to distinguish
between holographic and non-holographic states? The physical meaning of these
constants deserves further investigation.

The study of BPE bases on our understanding of the PEE or entanglement
contour. However, our understanding of the entanglement contour or the PEE is
still on a primitive stage. The fundamental definition for the PEE based on
density matrix is still not clear and the proposals are not enough to
calculate the PEE for even a generic quantum system in two dimensions. It is
also important to point out that the BPE is not sensitive to the phase
transition between connected and disconnected entanglement wedge. More
explicitly the BPE$(A,B)$ does not vanish when the entanglement wedge of $AB$
become disconnected. A naive explanation is that the subset entanglement
entropies in the PEE proposal are all connected regions, which are insensitive
to the phase transition. This confusion may be understand if we have a deeper
understanding of the entanglement contour for disconnected regions.

## Acknowledgments

The author would like to thank Tatsuma Nishioka, Tadashi Takayanagi and Huajia
Wang for helpful discussions. Especially I would like to thank Muxin Han for
early collaboration. I thank the Yukawa Institute for Theoretical Physics at
Kyoto University. Discussions during the workshop YITP-T-19-03 “Quantum
Information and String Theory 2019” were useful to complete this work. This
work is supported by the NSFC Grant No.11805109 and the “Zhishan” Scholars
Programs of Southeast University.

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