Ownerless island and partial entanglement entropy in island phases

In the context of partial entanglement entropy (PEE), we study the entanglement structure of the island phases realized in a holographic Weyl transformed CFT in two dimensions. The self-encoding property of the island phase changes the way we evaluate the PEE. With the contributions from islands taken into account, we give a generalized prescription to construct PEE and balanced partial entanglement entropy (BPE). Here the ownerless island region, which lies inside the island Is ( AB ) of A ∪ B but outside Is ( A ) ∪ Is ( B ) , plays a crucial role. Remarkably, we ﬁnd that under different assignments for the ownerless island, we get different BPEs, which exactly correspond to different saddles of the entanglement wedge cross-section (EWCS) in the entanglement wedge of A ∪ B . The assignments can be settled by choosing the one that minimizes the BPE. Furthermore, under this assignment we study the PEE and give a geometric picture for the PEE in holography, which is consistent with the geometric picture in the no-island phases.


Introduction
In the past few decades there have been significant progress in our understanding of the quantum information aspects of black holes, which is quite useful for our understanding of the black hole information paradox [1]. In the context of the AdS/CFT correspondence [2], the von Neumann entropy S A for a region A in the boundary CFT, was proposed to be dual to the area of a minimal surface E A in the bulk AdS geometry which is homologous to A [3,4], This relation between geometry and entanglement is the famous Ryu-Takayanagi (RT) formula. This formula is further refined to the quantum extremal surface (QES) formula [5,6] with the first order quantum correction from the bulk fields included. In [7,8] the QES formula is applied to calculate the entanglement entropy of the Hawking radiation for a evaporating black hole after Page time. The black hole is in a 2-dimensional JT gravity, which is coupled to a non-gravitational CFT bath at the boundary. The new insight is that, after the Page time, a new QES behind the horizon becomes dominant, and the region behind the QES is included in the entanglement wedge of the Hawking radiation. These new insights are further refined towards the so called island formula [9][10][11]. The proposal of island phase and entanglement islands not only opens a new window for us to understand the quantum information aspects of black holes, but also introduces novel matter phases and entanglement structures in quantum systems.
The island formula claims that, when we calculate the entanglement entropy of a region A in the bath, we should consider the possibility of including a region Is(A) inside gravity and calculate the entanglement entropy in the following way, Island f or mula I : S A = min ext Is(A) Area(∂ Is(A)) 4G N + S bulk (A ∪ Is(A)) .
More explicitly we consider all the possible island regions Is(A) and choose the one that minimizes the sum inside the brackets of (2). If the entanglement entropy calculated by (2) is smaller than the one calculated in the usual way without islands, then the configuration enters the island phase and (2) is the correct way to calculate the entanglement entropy. After the Page time, the JT gravity coupled with the CFT bath enters the island phase, which produces exactly the decreasing part of the Page curve. Furthermore, the island formula has been derived via gravitational path integrals where wormholes are allowed to join the black holes in different copies of the configurations in the replica trick [10,11]. Such a wormhole configuration turns out to be a new saddle when calculating the partition function in the replica manifold. Later it has been found that, the island phase can be realized in the AdS/BCFT setup [12] in a simple way, even without a black hole. See  for more references on recent developments on island formula in gravity and its applications.
Recently in [49], the island formula has been studied from a pure quantum information theoretic perspective. It has been demonstrated that when a quantum system (with or without gravity) is confined in a certain way, the Hilbert space of the system will reduce significantly and the state of a subset Is(A) is totally encoded in the state of another subset A. We call such confined systems self-encoded. Remarkably, in the self-encoded systems the entanglement entropy of A should be calculated by a formula very similar to (2), which we have called the Island formula I I [49], Island f or mula I I : S A = Area(∂ Is(A)) 4G N +S(A ∪ Is(A)) , |A〉 ⇒ |Is(A)〉 .
In Island formula I I, the first term proportional to the area of the boundary of Is(A) arises when Is(A) is settled in a gravitational background. The second termS(A ∪ Is(A)) indicates that the von Neumann entropy of the reduced density matrix is calculated by tracing out the degrees of freedom in the complement of A ∪ Is(A). As pointed out in [49], since the state of Is(A) is totally determined by the state of A, when we set boundary conditions for A to compute the elements of the reduced density matrix, we simultaneously set boundary conditions for Is(A), hence there is no room to trace out the degrees of freedom in Is(A) as they are non-independent with respect to A.
It has been further conjectured in [49] that, the two Island formulas I and I I are indeed the same, hence we can combine their advantages. On one hand, we can use the formula I to determine whether the system is in the island phase and find the corresponding island Is(A). On the other hand, we can use formula I I to generalize the island formula to non-gravitational systems and study their self-encoding properties. In [49], a holographic CFT 2 under a UV cutoff dependent Weyl transformation is proposed to sustain entanglement islands. We will introduce this holographic Weyl transformed CFT in more detail later in section 3.
Here, we focus on the so called entanglement wedge cross-section (EWCS) in the gravitational dual of a boundary state in island phase and its dual quantum information quantity. The EWCS of the entanglement wedge of two non-overlapping regions AB ≡ A ∪ B is a natural measure for the mixed state correlation between A and B. Since the measure for mixed state correlations is also not well studied in quantum information theory, the study of EWCS is also quite interesting from the quantum information perspective. In [50,51], it was proposed that the quantum information quantity corresponding to the EWCS is the entanglement of purification, since it satisfies a similar set of inequalities as the EWCS. Since then, a series of quantum information quantities have been proposed to be the dual of the EWCS, which includes the entanglement negativity [52][53][54], the reflected entropy [55], the "odd entropy" [56], the "differential purification" [57], the entanglement distillation [58,59]. See [60][61][62][63][64][65][66][67] for more explorations along these lines. Nevertheless, most of these quantities are defined in terms of an optimization problem, which makes them extremely difficult to calculate and the evidence for their correspondence to the EWCS is not enough. The reflected entropy is defined as the entanglement entropy under a special canonical purification, hence calculable in general quantum systems. Moreover, the correspondence between the EWCS and the reflected entropy in island phases was explicitly studied in [68,69] In this work we will, in particular, study the balanced partial entanglement entropy (BPE) [70][71][72], which has also been proposed to be dual to the EWCS. For a purification A 1 B 1 AB of the mixed state ρ AB , the BPE is a special partial entanglement entropy (PEE) s AA 1 (A) satisfying certain balanced conditions. The BPE is easy to calculate, as we have several powerful prescriptions to construct the PEE [73][74][75] in two dimensions. Moreover, unlike the reflected entropy and entanglement of purification which are defined on some special purifications, the BPE can be defined in generic purifications and is claimed to be purification independent. The purification independence and the correspondence to the EWCS for the BPE have been tested in global and Poincaré AdS 3 [71], holographic CFT 2 with an arbitrary Weyl transformation [71], holographic CFT 2 with gravitational anomalies [72] and BMS 3 symmetric field theories dual to 3-dimensional asymptotically flat spacetimes [71]. In particular, BPE can be regarded as a generalization of the reflected entropy, as BPE reduces to the reflected entropy for the particular case of canonical purification.
The main task of this paper is to study the BPE for the Weyl transformed CFT 2 in island phase and match it with the EWCS. This is a highly non-trivial task. Firstly, the phase structure of the EWCS is more complicated than the entanglement entropy (or RT surfaces), and we need to reproduce the phase structure from the BPE, which is purely evaluated from field theory side without any reference to the geometric picture. Secondly, in island phase when we calculate the entanglement entropy of a certain region A, it may involve other degrees of freedom outside A there is an entanglement island Is(A). This essentially change the way we calculate the entanglement entropy and PEE, hence we need to generalized the way we construct the PEE and BPE to the scenarios with entanglement islands. We find the generalization involves the assignment of the contribution from the ownerless island regions. For two non-overlapping regions A and B, when the entanglement island of AB is larger than the union of the islands of A and B, i.e.
then the region Is(AB)/(Is(A) ∪ Is(B)) inside Is(AB) but outside Is(A) ∪ Is(B) is called the ownerless island regions. The ownerless island regions are closely related to the so-called reflected islands [68]. The key to calculate the BPE(A, B) correctly, is to assign the contributions from the ownerless island regions to the right PEE. We will see that, different assignments for the ownerless island regions correspond to different balance point for the BPE, as well as different saddle point for the EWCS. We should choose the balance point that gives the minimal BPE or the EWCS saddle point with the minimal area. Eventually we get the matching between the BPE and EWCS. This paper is organized as follows. In Sec.2, we will briefly introduce the partial entanglement entropy and the balanced partial entanglement entropy for usual quantum systems without islands. Then in section 3, we introduce the new concept of ownerless island regions and why it changes the way we evaluate the PEE and BPE in island phase. After taking into account the contribution from the entanglement islands and ownerless islands, we generalize the way we construct the PEE and BPE in the no-island phases to the scenarios with entanglement islands. In Sec.4 we give a classification for the EWCS in island phases in two dimensions. We also provide a naive calculation for BPE following the standard construction of the PEE and BPE in no-island phase, and find that it does not match with the EWCS. In Sec.5, with the generalized version of the ALC proposal and generalized balance requirements, we calculate the BPE for various configurations in island phase. We find that under different assignments of the ownerless island regions we get different BPEs, which correspond to different saddles of the EWCS. The minimal BPE matches exactly with the minimal EWCS. In section 6, under the assignments for the ownerless island that gives the minimal BPE, we evaluate the contributions s AB (A) and s AB (B) for various configurations and find consistency with the geometric picture. At last, in section 7 we give a summary of our results, a discussion on the physical significance of our results and provide an outlook for the future directions.

Partial entanglement entropy
The entanglement contour is a concept in quantum information conjectured in [76], which is a function s A (x) that describes the contribution from each site x inside a subsystem A to the entanglement entropy of A. This can be regarded as a density function of entanglement entropy inside A, that not only depends on the site x but also on the region A. By definition the entanglement contour function should satisfy where dσ x is the infinitesimal area element of A. Later a systematic study on the partial entanglement entropy (PEE) has been carried out in [73][74][75]77]. The PEE is defined as the contribution from a subset α of A to the entanglement entropy S A , which can be expressed as The expression s A (α) displays the information about the contribution from the subregions, hence is called the contribution representation of the PEE. Later it was found that the PEE can be interpreted as an additive two-body correlation, and it is usually more convenient to express it in the following form whereĀ is the complement of A andĀ ∪ A makes a pure state. We call the notation on the left hand side the two-body-correlation representation of the PEE, while the notation on the right hand side the contribution representation of the PEE. These two representations are equivalent to each other in the usual quantum systems without islands. The PEE should satisfy a set of physical requirements [75,76] including those satisfied by the mutual information I(A, B) 1 and an additional one, the additivity property. For non-overlapping regions A, B and C, the physical requirements for the PEE are classified in the following: For more details about the well (or uniquely) defined scope of the PEE and the ways to construct the PEEs in different situations, the readers may consult [73][74][75][77][78][79][80]. These details are also summarized in the background introduction sections of [71,72]. Here we only introduce one particular proposal to construct the PEE in generic two-dimensional theories with all the degrees of freedom settled in a unique order (for example settled on a line or a circle), which we call the additive linear combination (ALC) proposal [73,75,77].
• The ALC proposal: Consider a region A which is partitioned in the following way, A = α L ∪α∪α R , where α is some subregion inside A and α L (α R ) denotes the regions left (right) to it. The proposal claims that: The Additivity and Permutation symmetry properties of the PEE indicate that, any PEE I(A, B) can be evaluated by the summation of all the two-point PEEs I(x, y) with x and y located inside A and B respectively [75], i.e.
Note that the two-point PEE is an intrinsic entanglement structure of the system, in the sense that it is independent of the choice of the regions A and B.

Balanced partial entanglement
Compared to the entanglement entropy, the entanglement contour or the PEE is a finer description for the entanglement structure of a quantum system. Then it is possible to extract other quantum information quantities from the PEE. In this paper we focus on the so-called balanced partial entanglement entropy (BPE), which is a special PEE that satisfies certain balance conditions, and can be considered as a generalization of the reflected entropy in generic purifications of a mixed state. The BPE was proposed in [70] and is claimed to be dual to the EWCS. Furthermore in [71], it was proposed that the BPE captures exactly the reflected entropy in a mixed state and is purification independent. These proposals have passed various tests in covariant scenarios [72], holographic CFT 2 with gravitational anomalies [72], CFT 2 with different purifications [71] and 3-dimensional flat holography [71, 81] 2 . For a bipartite system H A ⊗H B in a mixed state ρ AB , one can introduce an auxiliary system A 1 B 1 to purify AB such that the whole system ABA 1 B 1 is in a pure state |ψ〉, and Tr A 1 B 1 |ψ〉 〈ψ| = ρ AB . The way we purify ρ AB is highly nonunique. The balanced requirements that defines the BPE are in the following: 1. Balance requirement: Among all possible configurations for the partition of A 1 B 1 , we should choose the one satisfying the following condition When A and B are adjacent, (10) is enough to determine the partition of A 1 B 1 , or equivalently, the balance point.
However, when A and B are non-adjacent (see the right panel of Fig.1), the complement AB is disconnected and we need two partition points (see Fig.1) to divide AB. In this case the balance requirements are generalized to two conditions or Since is automatically satisfied if the above two conditions are satisfied.
Furthermore, in [71] it was observed that I(A,

Island phase in holographic Wely transformed CFT
The island formula is often realized in the context of the AdS/BCFT correspondence [12]. Combined with the braneworld holography [88][89][90], the island formula emerges in an effective 2d description of AdS/BCFT [91][92][93][94]. The PEE has also been explored in this context; for example, the authors of [95,96] have studied the entanglement contour for the Hawking radiation. Nevertheless, these explorations are based on the straightforward application of the ALC proposal for PEE in the island phase. In [97], the contribution from the island Is(A) for certain region A in the Hawking radiation was also discussed.
In this paper, we would like to study the PEE aspects of the island phase, which is realized in the Weyl transformed holographic CFT 2 setup proposed in [49]. Let us start from the vacuum state of a holographic CFT 2 on a Euclidean flat space with the metric ds 2 = 1 δ 2 dτ 2 + d x 2 3 . Here δ is an infinitesimal constant representing the UV cutoff of the boundary CFT. One may apply a Weyl transformation to the metric, 3 Here the overall factor 1 δ 2 is inspired by AdS/CFT, where the metric is precisely the boundary metric of the dual AdS 3 geometry ds 2 = ℓ 2 z 2 (−d t 2 + d x 2 + dz 2 ) with the radius coordinate settled to be z = δ.
which effectively changes the cutoff scale following δ ⇒ e −ϕ(x) δ, where ϕ(x) is usually negative. The entanglement entropy of a generic interval A = [a, b] in the CFT after the Weyl transformation picks up additional contributions from the scalar field ϕ(x) as follows [71,98,99] This formula can be achieved by performing the Weyl transformation on the two-point function of the twist operators. Then we perform a UV cutoff dependent Weyl transformation for the x < 0 region such that the metric in this region is proportional to the AdS 2 metric, where κ is an undetermined constant. After the Weyl transformation the metric at x < 0 becomes Such a special Weyl transformation changes the CFT essentially, and the cutoff scale at x < 0 is no longer related to δ, rather it is characterized by the coordinate x. Let us apply the Island formula I to the system. We consider A to be the semi-infinite region x > a. The entanglement entropy calculated by the island formula is given by 4 where we take the region I = (−∞, −a ′ ] as a possible choice for Is(A). One can easily check that, when a = a ′ the above expression attains its minimal value: which is always smaller than the entanglement entropy computed through the usual techniques. Similarly when we consider A to be an interval [a, b] inside the region x > 0 and [−b ′ , −a ′ ] is any possible choice of Is(A), the island formula will give 5 which has the saddle point 4 Note that, the area term does not appear since the Weyl transformed CFT is not gravitational. 5 Here we need to assume that the entanglement entropy for two disjoint intervals in the holographic Weyl transformed CFT exhibit similar phase transitions as the RT formula [100,101], under certain sparseness conditions on the spectrum and OPE coefficients of bulk and boundary operators and large c limit. We leave this for future investigation.
As was found in [49], the above result is smaller than the usually computed entanglement entropy In other words, for any interval [a, b] with a/b < r * , the interval admits an island and the entanglement entropy should be calculated through the island formula.
In [49], the subjection of the term ϕ(x) in the entanglement entropy formula (16) is interpreted as putting a cutoff sphere in the AdS 3 background with radius |φ(x)| and center (δ, x). For the specific Weyl transformation (17), the common tangent line of all such cutoff spheres plays exactly the role of the end of world (EoW) brane in the AdS/BCFT correspondence. The RT surfaces are allowed to be anchored on the tangent line in the sense that they are cut off there.
In the context of AdS/BCFT, we can either consider the regions with no islands or regions In the cases with islands, the entanglement entropies are given by the two-point functions of twist operators settled at reflection symmetric sites. These are also well-defined in the holographic Weyl transformed CFT 2 , as the RT surfaces can also anchor at the common tangent vertically. Moreover, in holographic Weyl transformed CFT 2 , the Weyl transformed two-point functions for twist operators go beyond those configurations, i.e. they are also computable for sites that are not reflection-symmetrically settled. For example, we can set the two twist operators at −b and a with a > 0, b > 0 and a = b. The physical interpretation for these two-point functions are not clear so far. It is quite tempting to interpret these two-point functions as the entanglement entropies for the intervals [−b, a] with no reflection symmetry. Nevertheless, in the following, we argue that this interpretation is not correct, and we will provide the correct interpretation in Sec.3.3.
For b < a, although the two-point function is still computable, according to the island formula I I, the region (0, a] determines the state of its island region [−a, 0). When we set boundary conditions for (0, a], we simultaneously set boundary conditions on [−a, 0), which means we can only trace the degrees of freedom outside [−a, a], which is a larger region covering [−b, a]. This implies that the entanglement entropy for [−b, a] with b < a is not well defined. Similar problems arise for the region However, the above problem does not arise in the cases with b > a, where the region [−b, 0) covers the island [−a, 0). Nevertheless, in these cases the entanglement entropy for [−b, a] is also not well defined. The reason comes from the self-encoding property of the system, which makes the situation different from the normal systems where the degrees of freedom at different sites are independent of each other. When we compute the reduced density matrix on

The PEE structure of the island phase and the ownerless island
Previously we have introduced two representations for the PEE, i.e. the contribution representation and the two-body-correlation representation. The contribution representation needs the input of the region A and its subset α, while the two-body-correlation representation is an intrinsic structure of the state which does not rely on the choices of the regions and their subsets. As we have shown that, given a region A and one of its subsets α, we can generate all the contributions s A (α) from I(x, y) by integration (or summation). So we can claim that the two representations are equivalent to each other.
In island phase, the two-body-correlation structure is still an intrinsic structure of the state, while the contribution representation changes a lot in island phase. The reason is that, in island phase when we talk about certain region A, it also involves other degrees of freedom (which is in the island region Is(A)) outside A when it admits an island. In this case, not only the degrees of freedom inside A, but also those in its island contribute to S A . The self-encoding property of the state indicates that the degrees of freedom at different sites are no longer independent of each other, and essentially change the way we evaluate the entanglement entropy of a region A, as well as the contribution s A (α) from the subset α. Later in this paper, our discussion will be conducted mainly in the two-body-correlation representation.
More explicitly, let us consider a region A which admits island. The entanglement entropy S A is calculated by the island formula S A =S A∪Is(A) , which is the von Neumann entropy of the reduced density matrixρ A∪Is(A) computed by tracing out the degrees of freedom outside A ∪ Is(A). This strongly indicates that, we should collect all the two-point PEEs I(x, y) with x and y located in A ∪ Is(A) and its complement separately, i.e.
A direct consequence is that, the normalization requirement S A = s A (A) = I(A,Ā) in the non-island phase breaks down as the island also contributes to S A .

• In other words, the PEE between Is(A) and C should contribute to S A since the state of Is(A) is totally determined by A and should be understood as a window through which A can entangle with A ∪ Is(A).
One can also write the above equation as which is quite different from the normalization property S A = I(A,Ā) in non-island phase. In the above formula (25), the region A can be either connected or disconnected 6 .
• We conclude that, in island phase the relation s A (α) = I(α,Ā) between the two representations as well as the ALC proposal no longer holds. When we compute the contribution s A (α) from the two-body-correlation I(x, y), we should take into account the island configuration carefully.
Now we discuss the entanglement entropy for the union of two non-overlapping regions AB ≡ A∪B and the contribution s AB (A) and s AB (B) in terms of the PEEs in two-body-correlation representation. These configurations have more complicated structures in the presence of islands. In the following, we classify the island configurations of these configurations into three classes and explicitly discuss how the island structure affects the contributions.

Class 1
In the first class, the region AB, as well as A and B, does not admit an island. In this case, since all the regions do not involve degrees of freedom outside AB, s AB (A) and s AB (B) can be calculated by the ALC proposal for PEE in the non-island phase,

Class 2
In the second class, all the three regions AB, A and B admit islands and Let us denote C as the complement of AB ∪ Is(AB). In this case, the entanglement entropy S AB = I(AB ∪Is(AB), C) has contribution I(Is(AB), C) from Is(AB). Since Is(AB) is just the union of Is(A) and Is(B), this island contribution can be decomposed into I(Is(A), C) + I(Is(B), C), where the two terms should be assigned to s AB (A) and s AB (B) respectively. More explicitly we have These PEEs can be calculated by writing them as a linear combination ofS α . For example, According to the formula (25), it is evident that the contribution s AB (A) and s AB (B) can still be written as the linear combination of entanglement entropies (including island contribution) following the ALC proposal.

Class 3
In the third class, the region AB admits an island Is(AB), but Is(A) ∪ Is(B) does not cover the full Is(AB), i.e.
Here we do not require the two subregions A and B to admit individual islands. If A (or B) does not admit an island, then Is(A) = (or Is(B) = ) in (31). In this case, there are degrees of freedom that belong to Is(AB) but outside Is(A) ∪ Is(B). We call these degrees of freedom the ownerless island region and denote them as Io(AB), i.e.
It is also possible that neither A nor B admits islands and hence the total island Is(AB) is ownerless. Let us again denote the complement of AB ∪ Is(AB) as It is clear that the ownerless island region contributes to the entanglement entropy S AB , since However, it is not clear whether we should assign this contribution I(Io(AB), C) to s AB (A), s AB (B) or to both of them. The assignment for the contribution from the ownerless island region has not been discussed before. Let us divide Io(AB) into two parts where Io(A) is assumed to contribute to s AB (A), while Io(B) is assumed to contribute to s AB (B), i.e.
One can check that, if we naively apply the ALC proposal, then the contribution from the ownerless island regions will be missing which results in a wrong answer. So in these cases the ALC proposal no longer holds and we should treat the ownerless island regions carefully. Later we will see that different assignments will give us different BPEs which correspond to different saddles of the EWCS. Furthermore, when we have multiple balance points, we should choose the one that gives the minimal BPE, which helps us further determine the decomposition of the ownerless islands. Eventually we will see that the region Io(A) ∪ Is(A) (or Io(B) ∪ Is(B)) will coincide with the so-called reflected entropy island of A (or B) defined in [68]. For simplicity we define which can be understood as a generalized version of islands when calculating contributions from subsets with the ownerless island regions taken into account. The contributions in (36) then can be expressed as

The generalized ALC proposal and generalized balance requirements
In the previous subsection we have shown that for AB without any island, the contributions are still given by the ALC proposal. When AB admits an island and no ownerless island appears, then the ALC proposal also applies with the entanglement entropies in the linear combination calculated by the island formula. Nevertheless, in class 3 the situation is different since Ir(A) is not the island of A. It is impossible to write the PEE I(AIr(A), C) in terms of the entanglement entropies of subsets.
In order to explicitly compute the PEE, we need to find a way to write the PEE in terms of other quantities that are computable.
As we have discussed, in the Weyl transformed CFT we can compute the two-point correlation function of twist operators, when the two points are settled with reflection symmetry the correlation function computes the entanglement entropy with islands. While when there is no reflection symmetry the physical meaning of the two point function is not clear due to the self-encoding property of the system. Here we propose that these two-point functions indeed give the PEE between the region enclosed by the two points and the complement of this region. More explicitly, let us consider the twist operators settled at a and b (b > a) and denote the connected region [a, b] as γ. Then, we propose the following: In non-island phases, the above equation holds as both of the I(γ,γ) and the two-point function give the entanglement entropy of the region γ. While in island phase, the PEE I(γ,γ) can be classified into the following three classes.
• When a > 0 and γ does not admit island, the two point function gives the entanglement of γ as in the non-island phase, • When a = −b and b > 0, the two point function gives the entanglement entropy for the region A = [0, b], where Is(A) = [−b, 0) and γ = A ∪ Is(A).
• When a = −b and a < 0, I(γ,γ) is not the entanglement entropy of any region. By definition it is just the integration (or summation) of two point PEEs Although in the third class I(γ,γ) is not an entanglement entropy, it is also computable following (40). This is crucial since all the PEE I(A, B) between any two non-overlapping regions, can be written as a linear combination of the special type of PEE between a region and its complement, i.e. I(γ,γ). We will see that, this linear combination of I(γ,γ) has the same structure as the ALC proposal if we also denote I(γ,γ) as Later we will also encounter cases with disconnected γ = where we have used the basic proposal 1 in the second line. The above equation indicates that the two basic proposals are not consistent unless I(E, F) = 0. This is possible since E and F are separated by the region A and its generalized island Ir(A), which indicates that the entanglement wedge for E ∪ F is disconnected. Later we will give a demonstration in support of this statement. Therefore, the basic proposal 1 given in (40), is the only conjecture we made in this paper to compute PEE.
This is a generalization of the formula (30) with the island Is(A) and Is(B) replaced by the reflected islands Ir(A) and Ir(B). Interestingly, the above linear combination have the same structure as the ALC proposal (26). The only difference is that, every region appearing in the linear combination has a generalized island companion, For the non-adjacent case, we need to consider the scenarios where A is sandwiched by two intervals A 1 and A 2 and compute the contribution where C is the complement of A 1 Ir(A 1 )AIr(A)A 2 Ir(A 2 ). According to the additivity property, we can write the contribution in the following way which can further be written as It is evident that the above formula coincides with the ALC proposal (8) with the replacement (48). We call the new formulas (47) and (51) the generalized ALC proposal in the island phase. It reduces to (8) when Ir(γ) = , and reduces to (30) when there is no ownerless islands, i.e. Ir(γ) = Is(γ).
Since the way we compute the contribution is generalized for island phase, the balance requirements should also be modified to a new version in terms of PEEs. For adjacent cases, the generalized balanced requirement is given by For Disjoint cases, the two balanced requirements are generalized to be

Classification for entanglement wedge cross-sections
In this section, we turn to the BPE in island phase and check its correspondence with the EWCS. Before we explicitly solve the balance requirements and compute the BPE, in this section we give a classification for the EWCS of the entanglement wedge of A∪ B, where the intervals A and B are defined as When b 2 = b 3 the two intervals are adjacent, while when b 2 < b 3 they are disjoint. For both of the adjacent and disjoint cases, we classify the EWCS in the following.

Phase A1 and D1: Configurations when
does not admit an island. In this phase, the entanglement wedge of AB is the same as the one in non-island phase, as well as the EWCS Σ AB (see Fig.2). Also the area of the EWCS has been studied in [50] and are given by, Phase-D1 :  In these configurations the RT surface E AB homologous to [b 1 , b 4 ] becomes disconnected 7 (see Fig.3), and is decomposed in two pieces where, for example, RT (b 1 ) denotes the piece that emanates from x = b 1 and lands at the bulk (cutoff) brane. Note that there are three saddle points for the area of the entanglement wedge cross-section, as one of the endpoints of Σ AB can anchor on three choices: RT (b 1 ), the brane, and RT (b 2 ). One should choose the EWCS with the minimal area. Nevertheless, any of the three choices can be the minimal one if we adjust the four parameters b i properly. So these configurations can further be classified into three different phases for which the areas of the EWCS have been studied in [50,102,103] and these are given by: 1. Σ AB is anchored on RT(b 1 ), the RT surface connecting b 1 and the brane: Phase D2a : 2. Σ AB is anchored on the brane: Phase D2b :

A naive calculation of BPE with ownerless islands
Now we show that for the configurations with ownerless islands, if we insist on applying the ALC proposal to calculate the PEE, then the resulting BPE does not match with the EWCS. To be specific, let us consider a typical configuration (see Fig.4 , +∞] and their sandwiched interval admits no island. In this case Is(AB) covers the whole x < 0 region, and there appears an ownerless island The partition point x = q 2 divides the sandwiched interval into As discussed earlier, the appearance of the ownerless island makes the contributions coming from the individual entanglement entropies complicated, and the naive application of the ALC proposal does not suffice. If we naively apply the ALC proposal to calculate the PEE, then we have Solving the balance condition s AA 2 (A) = s BB 2 (B) we find Plugging the above q 2 into the PEE, we find that the BPE is given by Clearly, the BPE calculated in this way does not match with the area of the EWCS. Let us take a deeper look at the s AA 2 (A) constructed from the ALC proposal and write it in terms of the two-body-correlation representation. We find where the ownerless island regions are assigned to s AA 2 (A) and s BB 2 (B) respectively and are chosen to be The above ownerless island regions Io(A) and Io(B) are determined by the balance requirement that gives the minimal BPE. This will be explained in the later sections. Then using the generalized ALC formula, we have which exactly coincide with the EWCS. One can check that the balance requirement s AA 2 (A) = s BB 2 (B) is satisfied by choosing (72).

Balanced partial entanglement in island phases
In this section we consider the BPE in island phase and their correspondence with the EWCS. Compared with the non-island phase, the phase space for the EWCS in island phase is more complicated Figure 5: The geometric picture for the BPE in Phase A1 and D1. In this case, the partition point x = q 1 is settled either in the region x < b 1 (including the island region as there are more saddles. Also there are new phase transitions for the EWCS. These indicate that the phase space of the BPE should also be more complicated, and checking its correspondence with the EWCS becomes more challenging in island phase. For each configuration, there are different ways to assign the contribution from the ownerless island region. We consider all the possible assignments of the ownerless island region and subsequently solve the balance requirements for each assignment. For all the solutions we compute the corresponding BPEs, and choose the minimal one if there are multiple solutions to the balance requirements. Remarkably we find that, the BPEs calculated under different assignments of the ownerless island regions exactly correspond to the different saddles of the EWCS. Also the minimal BPE identically matches the minimal EWCS and hence the correspondence between the BPE and the EWCS holds in island phase. In the following we conduct the analysis for the BPE in all the configurations we have classified in the previous section.

AB with no island
Let us first consider phases-A1 where A and B are adjacent and A ∪ B admits no island. In this case none of A and B or A ∪ B admit islands or their generalized counterparts and therefore no degrees of freedom outside AB contribute to the mixed state correlation between A and B. Let us divide the complement of AB into A 1 ∪ B 1 with the partition point x = q 1 (see the left panel in Fig.5). Note that, in this case the island region x < 0 is included in A 1 B 1 . Based on this partition in phase-A1, the balance requirement is the same as (10) in no-island phase, where Solving the balance condition I(A, BB 1 ) = I(B, AA 1 ) we find a unique solution The corresponding BPE is given by which is exactly the area of EWCS given in (55). For phase-D1, there are two partition points q 1 and q 2 which divide the complement of AB into four regions A 1 B 1 ∪ A 2 B 2 (see the right panel in Fig.5). Also in this case AB admits no island. Using the two-body-correlation representation we have and Solving the two balance conditions I(A, BB 1 B 2 ) = I(A, BB 1 B 2 ) and I(A 2 , BB 1 B 2 ) = I(B 2 , AA 1 A 2 ), we determine the two partition points as follows Then the BPE may be computed by substituting the above balanced partition points into the corresponding PEE as follows which is also identical to the area of EWCS (56). Before we go ahead, we would like to comment on the BPE calculated via the contribution representation in this case. For example, in Phase A1 we can calculate s AA 1 (A) and s BB 1 (B) and then apply the balance requirement s AA 1 (A) = s BB 1 (B) to determine q 1 . One can start from some b 2 close to b 1 such that the AA 1 also admits no island. In this case the island Is(BB 1 ) covers the whole x < 0 region and there is a ownerless island region Io(BB 1 ) = [−b 4 , q] for BB 1 . We can find a solution for the balance requirement when we assign the ownerless island region to B 1 such that Ir(B 1 ) = (−∞, 0) and Ir(B) = . Remarkably, the balance point and BPE calculated in this way coincide with our previous results.
However, when b 2 goes farther from b 1 , the balance solution without any island for AA 1 no longer exists, hence we should consider the AA 1 admitting island. As a result we have Ir(A 1 ) = .
In this case no matter how we assign the contributions from the ownerless island regions Io(AA 1 ) and Io(BB 1 ), the solutions to the balance requirements do not exist. So our previous results are the unique solution to the balance requirements. This is also consistent with the observation that, the EWCS is not a portion of the RT surface for any AA 1 which admits island. Therefore, we may conclude that the two-body-correlation representation of the balance requirements includes partition configurations that cannot be described by the contribution representation.

Adjacent AB with island
In this subsection, we consider the phase-A2 where A = [b 1 , b 2 ] and B = [b 2 , b 4 ] and A ∪ B has an entanglement island. Here the regions A or B individually may or may not admit their own islands. We consider the following three configurations with given assignments for the ownerless island region Io(AB): Note that, there are configurations with I r(A) ⊂ Is(A) (or I r(B) ⊂ Is(B)) when Is(A) = (or Is(B) = ) for certain choice of q. If we take these configurations seriously and solve the balance requirements to get a BPE, then we get the result for the no-island phase. Also this BPE will not be the minimal one. For simplicity, we only consider the cases with Is(A) ⊂ I r(A) and Is(B) ⊂ I r(B) by properly choosing the partition point x = −q. Nevertheless, as we will see, these extra configurations can not give the minimal BPE. In the following, we will systematically solve the balance requirements for each assignment and obtain the corresponding BPE.

Phase-A2a
For Ir(A) = and Ir(B) = [−b 4 , −b 1 ], there is no island contribution to A and all of the island Is(AB) contributes to B. Let us assume that the partition point q 1 lies at 0 < q 1 < b 1 such that . Furthermore we assume AA 1 does not admit an island and hence A 1 also does not receive any island contribution, i.e.
The schematics of the setup is depicted in Fig.6. In this configuration we compute the following two PEEs via the generalized ALC proposal and In (86) we have used the basic proposal 2 to obtain, Solving the balance condition I(A, BIr(B) ∪ B 1 Ir(B 1 )) = I(BIr(B), AA 1 ), we find the condition Plugging (85) or (86), we immediately find the BPE as follows This result coincide with the area of the EWCS saddle in (58). One can then consider the possibility that AA 1 admits an island. In this case Ir(A 1 ) = and we should consider other configurations of Ir(A 1 ) and Ir(B 1 ). Nevertheless, the solution to the balance requirement does not exist in such a configuration. So the result (89) is the only solution for the configurations A2a.

Phase-A2b
In this case, as depicted in Fig.7 In this configuration, we have Is It is interesting that the balance requirement is satisfied for all the choice of q if b 1 < q < b 4 . Since different choices of q give us different BPE, we should choose the minimal one according to the minimal requirement. More explicitly we choose the q that satisfy which has a simple solution, The choice q = b 2 gives the expected minimal BPE which coincide with the EWCS saddle (60) that is anchored on the EoW brane.

The vanishing PEE in island phase
One can also consider other configurations for the phase A2b, for example where a portion of A 1 is transferred to B 1 compared with the previous configuration (see Fig.8), If A 1 admit island, i.e. q 1 ≤ r * b 1 , then we have In these configurations, we may also set q = b 2 such that Ir(A) and Ir(B) are not changed. Subsequently, we find that the PEEs are given by It is obvious that, such kind of configurations with q = b 2 and A 1 admitting an island satisfy the balance requirements and give the same BPE as in (96). This means that the balance point that gives the minimal BPE is highly non-unique. In other words, it does not change the BPE whether we assign the region E = [−q 1 , q 1 ] to A 1 Ir(A 1 ) or B 1 Ir(B 1 ), which indicates that the PEE between the regions E and AIr(A)BIr(B) is zero, Note that, when q = b 2 the configurations has reflection symmetry and there is no ownerless islands. Also the calculation only involves entanglement entropies for regions with islands, hence  the results do not rely on the two basic proposals (40) and (45). This is important as it indicates that the vanishing PEE (100) can be derived independently from the basic proposals for PEE. Furthermore, we can take the limit b 4 → ∞ and denote F = ABIr . Under this limit, the configuration is always in Phase-A2b with a proper choice for b 2 , and the equation (100) still holds. So we can make the following generic statement: • For any region E and F which is separated by a region A and its (generalized) island Ir(A), we have which is crucial for us to derive (45) from (40). According to the additivity and positivity properties, the PEE between any subsets of E and F also vanishes.

Phase-A2c
For the assignment Ir(A) = [−b 4 , −b 1 ] and Ir(B) = , the analysis is symmetric to the case of Phase-A2a in the exchange of A and B. In this case, assuming that BB 1 does not admit an island, we should solve the balance requirement I(AIr(A), BB 1 ) = I(B, AIr(A)A 1 Ir(A 1 ). We can find a solution satisfying both the balance requirement and the minimal requirement, which gives the partition point x = q 1 > b 4 as follows, The BPE in this case is given by which exactly matches with the EWCS saddle (62) that is anchored on the piece of the RT surface E AB emanating from x = b 4 .

Minimizing the BPE in Phase-A2
Now we compare the three BPEs in the phase-A2, which correspond to the three saddle EWCSs, and choose the minimal one. When we computed the BPE in Phase-A2a, we assumed that AA 1 does not admit an island and hence have not considered the possibility that there may be balance points when AA 1 admits island. Here we can exclude this possibility by showing that when the Phase-A2a gives smaller BPE than the Phase-A2b, AA 1 should not admit island. A similar statement also applies to BB 1 in Phase-A2c.
In the case of phase-A2a, we know A does not admit an island and hence b 1 /b 2 > r * with r * ≡ 1 − 2 e 2κ + e 4κ + 2e 2κ . Since the solution q 1 = b 2 1 /b 2 can be written as we conclude that A 1 also does not admit an island. Nevertheless the assumption that AA 1 does not admit island may not be satisfied. Now we compare the BPE in Phase A2a and A2b, and find that the critical point between these two phases is When b 2 < b 1 / r * the Phase-A2a gives smaller BPE, and furthermore we have which confirms our assumption that AA 1 does not admit island. Similarly one can compare the BPE in Phase-A2b and Phase-A2c and find the critical point to be When b 4 < b 2 / r * the Phase-A2c gives smaller BPE, and furthermore we have which confirms our assumption that BB 1 does not admit island.
When AB admits an island, we require b 1 ≤ r * b 4 and we can always find a b 2 inside (b 1 , b 4 ) satisfying the following inequality In other words, we can confirm that when AB admits island, there always exists a b 2 such that the BPE or EWCS is in Phase-A2b. In the following, we will systematically investigate these sub-phases, solve the balance requirements for each case and subsequently obtain the corresponding BPEs.

Phase-D2a
For Ir(A) = and Ir(B) = [−b 4 , −b 1 ], we assume that AA 1 A 2 has no island, such that we have Ir(A 1 ) = Ir(A 2 ) = , and the two balance points q 1 and q 2 lie at where the four PEEs in the above equations are calculated by: and The solution to the two balance conditions are given by The corresponding BPE(A : B) may be obtained as follows which exactly matches with the area of EWCS in Phase D2a given in (59). Interestingly, in the holographic geometric picture, these two balance partition points are exactly located where the RT surface extending from the EWCS ends on the asymptotic boundary, as shown in Fig.10.

Phase-D2b
In where the four PEEs in the above equations are calculated as follows and Similar to the adjacent cases, here the above two balance conditions coincide and are given by This means that there are infinite number of solutions to the balance requirements. Again, combining the balance requirements with the minimal requirement, the balance point should further satisfy the following extremal condition, Solving these constraints, we arrive at Then the BPE for this phase is given by which coincides with the area of EWCS in Phase D2b (61). Similar to the Phase-A2b, we can transfer a portion, for example [−q 1 , q 1 ] of A 1 Ir(A 1 ) to B 1 Ir(B 1 ) as long as A 1 admit island. In such configurations the BPE is the same as the above result.

Phase-D2c
For Ir(A) = [−b 4 , −b 1 ], Ir(B) = , the configuration is symmetric to Phase-D2a in the exchange of A and B (see Fig.12). Following the same arguments as in phase-D2a, we arrive at two balance points and the corresponding BPE(A : B) is given by which matches with the area of the EWCS given in (63), for the Phase D2c.

Minimizing the BPE
Now we compare the above three BPEs, which correspond to the three saddle EWCS, and choose the minimal one. The critical point between phase-D2a and phase-D2b is given by When A admits island we set otherwise we set Also we apply a similar prescription to set Ir(B) and Ir(B 1 ). According to our discussion for the Phase-A2b, the balance requirement is always satisfied, and Then we test the other balance requirement Let us first consider the cases where both of A and B do not admit island and hence Ir(A) = Ir(B) = .
We find that =0 , where we have used the basic proposal 2. It is obvious that the second balance requirement is also satisfied. Hence the BPE between A and B vanishes, Since the BPE should be non-negative, the above BPE is the minimal one. One can further check the cases where A or B admits island and get the same vanishing BPE. Then the vanishing BPE exactly matches to the vanishing EWCS.

BPE from minimizing the crossing PEE
The crossing PEE at the balance point has been shown to be minimal in vacuum CFTs [71]. Now we show that this also holds in the island phase. We pick phase-A2a and phase-D2a as examples.
For phase-A2a, the crossing PEEs are given by and .
(142) Then the total crossing PEE is One easily finds that the extremal points q 1 for the crossing PEE are Since 0 < q 1 < b 1 , we have q 1 = b 2 1 /b 2 as the point that minimizes the total crossing PEE. This is exactly the balanced point and the minimized total crossing PEE is given by which may be identified as the lower bound of Markov gap h(A : B) [103]. In non-island phase for adjacent intervals, the non-crossing PEE part in BPE exactly coincides with half the mutual information so that the crossing PEE part in BPE gives the Markov gap [71]. However, in island phase, the non-crossing PEE part in BPE is never equal to half the mutual information. For phase-A2a, we have the non-crossing PEE while half of the mutual information is given by Thus the crossing PEE for phase-A2a should not exactly give the Markov gap. Now we consider an example of disjoint phases, namely the phase-D2a. The crossing PEEs in this phase are given by , , Then the total crossing PEE is Again, one may find that the total crossing PEE is minimized at the points which are exactly the balance points.

Partial entanglement entropy and its geometric picture in island phase
In this section, we calculate the contributions to various PEEs via the generalized ALC formula under the assignment of the ownerless island that gives the minimal BPE. We will see that the contributions s AB (A) and s AB (B) correspond to the two portions of the RT surface of AB, which are divided by the point at which the EWCS Σ AB anchors on the RT surface E AB .

Adjacent AB with island
When BPE for phase-A2a minimizes, we have Ir

Discussion
In this paper, we have explored the entanglement structure in the island phase in the context of partial entanglement entropy. Despite several primitive attempts [95][96][97], this remains quite an unexplored aspect of entanglement islands. Based on the claim [49] that a system in island phase has self-encoding property, we conclude that calculating the entanglement entropy of a region A involves the degrees of freedom in the island Is(A), which is outside this region. This property essentially changes the way we evaluate the contribution to entanglement entropy S A from subsets in A. Firstly, the island Is(A) should be understood as a window through which A can entangle with degrees of freedom outside A ∪ Is(A). Secondly, when we consider the contribution from a subregion α, we should also include the contribution from Is(α), or the generalized (or reflected) island Ir(α) if there are ownerless island regions. With the island contributions taken into account, we find a generalized version of the ALC proposal to construct the PEE and a generalized version of the balance requirement to define the BPE in the island phase. For configurations without ownerless islands, the assignments of the island regions to the subsets is clear with no ambiguity. Nevertheless, in configurations with ownerless island regions, there is no intrinsic rule to clarify these assignments. For any choice of the assignment, we can solve the generalized balance requirements and calculate the BPE. Remarkably we find that, the BPEs for different assignments of the ownerless island correspond to different saddles of the EWCS. Then it is natural to choose the assignment that gives the minimal BPE. Furthermore, with the assignment of ownerless island settled, we calculate the contributions s AB (A) and s AB (B) and explore their geometric picture, which is consistent with the geometric picture in non-island phase.
Our results are based on several proposals for the island phases which are summarized in the following.
• As in the no-island phases, the PEE structure of the island phase is also described by the two-point PEEs I(x, y) which is unaffected by how we divide the system.
• When we consider any type of correlation between two spacelike separated regions A and B, we should also take into account the contributions from their island regions as well as from the ownerless islands.
• The holographic Weyl transformed CFT 2 is a non-gravitational toy model that admits island phases [49].
• The basic proposal 1 relates an arbitrary two-point function for the twist operators in the Weyl transformed CFT 2 to the PEE between the region enclosed by the two points and its complement.
Similar results may also be obtained in the toy model of JT gravity coupled to a non-gravitational CFT 2 bath [11] and the toy model of AdS/BCFT [12], if one admits the generalization of the twopoint functions to non-reflection symmetric configurations 8 and assumes the basic proposal 1. In this paper we mentioned the self-encoding property [49] of the island phase several times. This property gives us the guidelines on how one should take into account the contribution from the island regions when we study correlations between spacelike separated regions. Also, it helps us better understand the physical meaning of the two-point functions of the twist operators in the Weyl transformed CFT 2 .
The physical meaning of this paper is multifaceted. On one hand, our results give a non-trivial test to the correspondence between the BPE and the EWCS, and to the purification independence of the BPE. On the other hand, they indicate that the above listed proposals are highly consistent. These proposals not only give a finer description for the entanglement structure of the island phases, but also generalize the concept of entanglement islands to the context of non-gravitational systems. Testing and proving the above proposals or conjectures from other perspectives will be important future directions.
The BPE and EWCS are closely related to the reflected entropy. In holographic setups, evidence is given in [55] for the correspondence between the EWCS and the reflected entropy. Also in [70] it was shown that the BPE in the canonical purification reduces to the reflected entropy as the reflection symmetry in the canonical purification automatically satisfies the balance requirements. In other words, the reflection symmetry is a solution to the balance requirements. However, an explicit example has been provided in [104], where the reflected entropy is not monotonically decreasing under partial trace. This indicates that the reflected entropy is not a physical measure of mixed-state correlations. Given the proposal that the BPE is a generalization of the reflected entropy to generic purifications, this criticism also applies to the BPE.
Here we give some arguments for why the study of the BPE or the reflected entropy is still important. Firstly, for a generic quantum system, it is possible that the reflection symmetric configuration in the canonical purification is not the balance point that gives the minimal BPE. In this case, the BPE differs from the reflected entropy. Although this may not happen for quantum field theories where entanglement decreases with distance, it could happen in the system studied in [104], which is a discrete few-body system with no concept of distance. More importantly, as was pointed out in [71], the BPE can also be defined based on an optimization problem, which concerns minimizing the crossing PEE. So far, the BPE under this definition coincides with the one defined through the balance requirements for all the cases studied in [71,72] and this paper (see Sec.5.5). Nevertheless, it is important to address that the minimization of the crossing PEE is a priori independent of the balance requirements, hence it is possible that the BPE defined by these two requirements are not equivalent in general. It is possible that the BPE defined from minimizing the crossing PEE is monotonically decreasing under partial trace. We leave this for future investigations.