Entanglement contour from subset entanglement entropies

Entanglement contour characterizes the spatial structure of entanglement and quantifies the contribution from the degrees of freedom in any subset of the region $\mathcal{A}$ to the total entanglement entropy $S_{\mathcal{A}}$. Recently in \cite{Wen:2018whg} the author gave a simple proposal for the entanglement contour (or partial entanglement entropy) in two-dimensional theories, which involves the entanglement entropies of all the subsets inside $\mathcal{A}$. In this paper we explicitly study this proposal and show it satisfies many rational requirements for entanglement contour. Together with the holographic picture constructed with the modular planes \cite{Wen:2018whg}, we propose the correspondence between bulk geodesic chords and boundary partial entanglement entropies, which can be considered as a finer version of the Ryu-Takayanagi (RT) formula. As an example we calculate the fine correspondence between the points on $\mathcal{A}$ and the points on the RT surface $\mathcal{E}_{\mathcal{A}}$ for the BTZ black hole. We also give a strategy to extract the local modular flow from our proposal.


I. INTRODUCTION
The entanglement entropy for quantum many body systems, which is defined as the von Neumann entropy of the reduced density matrix, has become a quite hot topic in the study of modern theoretical physics. On the one hand, it can be used to distinguish new topological phases and characterize critical points, e.g., [2][3][4][5][6]. On the other hand, in the context of AdS/CFT [7][8][9] the Ryu-Takayanagi (RT) formula [10,11] (see also [12][13][14][15] for its extension to holographies beyond AdS/CFT) relates quantum entanglement to spacetime geometry, thus entanglement entropy becomes an important tool to study quantum gravity and holography itself.
However entanglement entropy only contains part of the information in the reduced density matrix. The authors of [16] considered the possibility of the existence of a function f A (i) which captures how much the degrees of freedom in site i contribute to the entanglement entropy S A of the region A. In other words f A (i), which we call the entanglement contour function following [16], characterizes the spatial structure of the entanglement in A. In the continuum limit, we denote this function as f A (x 1 , · · · x d ), where x i with 1 ≤ i ≤ d − 1, are the coordinates that parameterize A, and d is the spacetime dimension of the field theory. By definition it should satisfies the following basic requirements Also in [16], a set of requirements for the entanglement contour is proposed. However they cannot uniquely determine the contour function and the fundamental relationship between entanglement contour and the reduced * 101012491@seu.edu.cn density matrix is still not clear. Only few attempts to construct the contour functions have been explored in lattice models [16][17][18][19][20]. We will indirectly study the contour function f A (x) through the partial entanglement entropy s A (A 2 ) for any subset A 2 of A, which captures the contribution from A 2 to S A and is defined by Knowing the s A (A 2 ) for an arbitrary A 2 is equivalent to knowing the contour function. We focus on one dimensional region A and its connected subsets in d = 2 theories. In these cases for an arbitrary subset A 2 , in general it can divide A into three subsets A 1 , A 2 and A 3 (see the upper figure in Fig.1). When A 2 shares boundary with A then A 1 (or A 3 ) vanishes, and the partition only involves two subsets. In [1] it is proposed that the partial entanglement entropy s A (A 2 ) of the subset A 2 can be written as a linear combination of the entanglement entropies of certain subsets in A. In this paper we will show that this proposal satisfies rational physical requirements for entanglement contour. Furthermore we combine it with the holographic picture of the modular planes [1] to study a finer correspondence between the geodesic chords and boundary partial entanglement entropies. We also try to recover the modular flow from the partial entanglement entropies.

II. THE PARTIAL ENTANGLEMENT ENTROPY PROPOSAL AND ITS PROPERTIES
The proposal [1] for the partial entanglement entropy s A (A 2 ) of A is given by The first two terms in the bracket are the entanglement entropies of the unions between A 2 and the other two subsets respectively, the last two terms are the entanglement entropies of the other two subsets. This proposal also applies to A 1 and A 3 . For example, consider the partial entanglement entropy s A (A 1 ), the left subset of A 1 vanishes and the right subset becomes A 2 ∪ A 3 , so according to the proposal (4) we get For consistency, one can easily check that Since the partial entanglement entropy (4) is a linear combination of subset entanglement entropies which obey many inequalities [21,23]. It will be interesting to see which inequalities the partial entanglement entropy will satisfy. In the following we list some of the important properties satisfied by s A (A 2 ): 1. Additivity and normalization: When we split A 2 into arbitrary two parts A 2 = A a 2 ∪ A b 2 , by def-inition we always have 2. Positivity: The strong subadditivity [24,25] 5. Lower bound: According to the monogamy of mutual information [26] we have where the last inequality comes from the Araki-Lieb triangle inequality S A ≥ |S A1∪A3 − S A2 )|.
6. Symmetry: Given region A and a partition of A, we consider a symmetry transformation T under which ρ A = T ρ A T + is invariant. Acting T to any subset we have T A i = A i . Since T is a symmetry, the subsets A i and A i should play the equivalent role in A. So we should have S Ai = S A i , or further more S Ai∪Aj = S A i ∪A j . This means thus In other words the partial entanglement entropies (or entanglement contour) respect the symmetry T .
The derivation of the above properties is quite general because it only depends on our definition and the general inequalities satisfied by entanglement entropies. However the derivation of the additivity only applies to one dimensional regions. According to our proposal, for one-dimensional A we always have . It can only be extended to the cases in higher dimensions with rotation symmetries or translation symmetries in the other d − 2 spatial dimensions, thus the partition is still effectively the same as the cases with only one spatial dimension. In the cases with rotation symmetries the region A and its subsets A i should all be spheres or spherical shells (or annulus), while in the cases with translation symmetries the region and subsets should all be infinitely long strips (see the two lower figures in Fig.1). However, in general the partitions in higher dimensions will be more complicated than the d = 2 case and can involve more than three subsets (see Fig.2). In these cases the relationship between the partial entanglement entropy and the subset entanglement entropies should satisfy a more general formula and the satisfactory of additivity will need further assumptions on the entanglement entropies. We leave this for future investigations. The upper bound (9) indicates that for any site or subset A 2 inside A, s A (A 2 ) should not be larger than the entanglement entropy of the subset. This is reasonable when we interpret s A (A 2 ) as the partial entanglement entropy. Because, unlike S A2 , s A (A 2 ) does not count the entanglement between A 2 and other subsets inside A, thus should be no larger.
The lower bound (11) is much less obvious. We can test its rationality using the Bit thread picture [27], which quantifies the quantum entanglement with a set of "bit threads" with a cross-sectional area of 4 Planck areas and can only end on the boundary or the horizon. In this picture S A is the maximum possible number of threads emanating from A and end on A c . Also the partial entanglement entropy s A (A 2 ) have a quite natural definition as the number of threads emanating from A 2 and end on A c , when the number of threads from A to A c is maximal. So far all the configurations that maximize the flux of bit threads on E A are considered to be degenerate in the bit thread picture as they give the same S A . However different degenerate configurations will give different s A (A 2 ) thus the entanglement contour can be different. In the bit thread picture s A (A 2 ) has upper and lower bounds. For example, s A (A 2 ) reaches its upper bound when the number of bit threads emanating from A 2 is maximized and they all cross E A . So we have On the other hand when s A (A 1 ∪ A 3 ) reaches its upper bound s A (A 2 ) will reach its lower bound, which is given by It is interesting that, since S A2 ≥ S A −S A1∪A3 , the lower bound (11) for s A (A 2 ) defined by (4) is stronger than the bound (15) we get from the bit thread picture. This is not surprising since the bit threads have the largest freedom to move in the bulk thus should give the weakest bound for the partial entanglement entropy. This implies the freedom for the bit threads may be confined in some way. We note that the properties of s A (A i ) depend on whether A i shares a boundary with A or not. This is because when we consider subsets that share boundary with A, the number of relevant subsets will reduce thus some inequalities saturate. Using S A = S A c and S A1∪A3 = S A2∪A c , we can write the lower bound (11) as This means the partial entanglement entropy s A (A 2 ) is always larger than half of the mutual information between A 2 and A c . More interestingly, according to (5) and (6) the inequality saturates for the partial entanglement entropies of the subsets that share boundary with A, i.e., The naive reason for the saturation of (17) is that, when we consider the partial entanglement entropy of A 1 (or A 3 ), A 2 can be absorbed by A 3 (or A 1 ) thus only two subsets are relevant. The inequality (10) saturates when one of the subsets vanishes.
It is easy to see that Since (16) is just a result of the monogamy of mutual information. In summary, s A (A 1 ) (or s A (A 3 )) can be considered as a mutual information while s A (A 2 ) should be considered as the increment of the mutual information.

III. EVIDENCES FOR THE ENTANGLEMENT CONTOUR PROPOSAL
Here we give an argument for the justification of our proposal (4). We consider a subset A 2 and the accompanying partition of A. Assuming that the entanglement entropies for all the subsets of A can determine the entanglement contour, a natural universal ansatz for the partial entanglement entropy s A (A 2 ) could be a linear combination of the entanglement entropies of all the relevant subsets inside A [28], The coefficients are constants and do not depend on the choice of A 2 . Imposing the additivity (7) and normalization, one can uniquely determine the coefficients in the ansatz and get (4). Also we can test (4) in some cases where the contour function can be constructed holographically. In the context of AdS 3 /CFT 2 , a holographic picture for the entanglement contour of a single interval is given in [1]. The boundary modular flow lines in the causal development D A are the integral curves along the boundary modular flow. It is shown that the entanglement wedge can be sliced by the modular planes, which are defined as the orbits of the boundary modular flow lines under the bulk modular flow. Each modular plane will intersect with the interval A on a point A(x 0 ) and its RT surface E A on E A (x 0 ). Furthermore, the cyclic gluing of the point A(x 0 ) on A turns on the nonzero contribution to the S A on E A (x 0 ), thus gives a fine correspondence between the points on A and E A . In the same sense, the partial entanglement entropy s A (A i ) is just given by where E i is just the part of E A that correspond the subset A i under this fine correspondence. See Fig.3 for a graphical description of this construction. Also the similar construction is conducted for WCFT in [15] in the context of AdS 3 /WCFT correspondence [30,31]. When the fine correspondence is constructed, the partial entanglement entropy can be calculated by the length of the corresponding bulk geodesic cords.
On the other hand, since the entanglement entropies for arbitrary single intervals are known in both of CFT 2 and WCFT, the partial entanglement entropies can be calculated using our proposal (4). As a non-trivial test, the results match with each other.

IV. CORRESPONDENCE BETWEEN BULK GEODESIC CHORDS AND BOUNDARY PARTIAL ENTANGLEMENT ENTROPIES
The fine correspondence relate the length of any geodesic chords E i to the partial entanglement entropy of a subset A i , and furthermore to a linear combination of subset entanglement entropies according to our proposal (4). This is indeed a holographic reconstruction of bulk geometric quantities using boundary entanglement entropies, which is similar to the reconstruction using the kinematic space formalism [32][33][34][35] (see [22,36] for related discussion).
However the exploration of the fine correspondence follow the strategy of [1] relies on the explicit information of the bulk and boundary modular flows and is quite complicated to carry out. When all the subset entanglement entropies are known, we show that using our proposal (4) to calculate s A (A i ) and imposing the matching condition (20) for the bulk geodesic chords, we can determine the fine correspondence in a much simpler way. The strategy is to consider a pair of points A(x 0 ) and E A (x 0 ) on A and E A that lead to the partitions A = A 1 ∪A 3 , E A = E 1 ∪E 3 . Then these two points are correspond to each other when they satisfy the matching condition (20). The right hand side of (20) can be easily calculated by integrating the length of the geodesic chords.
Here we would like to give a non-trivial example: the CFT 2 defined on a compact space with finite temperature which holographically duals to a BTZ black hole. The modular flow in this case is not clear thus the construction [1] using modular planes does not work here. Though the computation of the entanglement entropy on the field theory side is formidable, the holographic computation can be performed via the RT formula. Consider a static BTZ black hole Here 0 ≤ x ≤ 2π is the angular coordinate and the radius of the boundary circle is taken to be unity. Given an interval A : 0 ≤ x ≤ x 0 that is small enough thus the RT surface is connected, which is settled at t = 0 and characterized by [38,39] . (22) Note that when A becomes larger than some critical point, the geodesic that computed the entanglement entropy S A transfer from the connected geodesic to the two component disconnected geodesics [37]. In this paper we leave the cases with disconnected RT surface for future discussion. The holographic entanglement entropy for A is then given by where β = 2π r+ . Consider a point at x = x 1 on A that divide A into We apply (23) to the subsets A 1 and A 3 to calculate their entanglement entropies and use (5) to calculate the partial entanglement entropy, then we get This furthermore determines the entanglement contour function on A Then we calculate the right hand side of (20) by integrating ds along the RT surface from x =¯ to x =x 1 Here we use (¯ , r I ) to denote the coordinate of the point where we cut off the RT surface. Since the cutoffs satisfy r ∞ = 1 , we have¯ = π β coth πx0 β 2 , such that Matching (28) with (25) we get the fine correspondence (see Fig.4) between the points on A and the points on Consider the following partition of A The length of the geodesics chord E 2 :x 1x2 , wherex i and x i are related by the fine correspondence (29), is given by It is easy to see that the above result is just the partial entanglement entropy s A (A 2 ) calculated by the subset entanglement entropies via (4). So the subset A 2 is cor- The gray disk at the center represents the black hole.
Here we show the fine correspondence between the points on A and the points on EA. The x coordinates of the two of points that are corresponded to each other are related by (29) respond to the geodesic E 2 via the fine correspondence (29), in the same sense as the RT formula.

V. EXTRACTING LOCAL MODULAR FLOWS FROM THE PARTIAL ENTANGLEMENT ENTROPY
The entanglement contour contains much more information than entanglement entropy alone. Then it is interesting to ask whether the entanglement contour proposed by (4) contains all the information needed to reconstruct the reduced density matrix (or modular Hamiltonian). In this section we show that, for the cases where the modular Hamiltonian is local thus generate a local modular (geometric) flow, the modular flow can be extracted from the partial entanglement entropy via our proposal. This involves a remarkable property of the partial entanglement entropy found in [1]. Which is, for an arbitrary subset A i , we have where A any spacelike region that has the same causal development as A (D A = D A ), and A i is the subset of A which intersect with the same class of modular flow lines (or modular planes) as A i . More explicitly, a pair of boundary modular flow lines can be determined by the two end points of A i as their orbits under the boundary modular flow. The A i whose end points are settle on the same pair of modular flow lines will satisfy (32). This property is obvious in the construction of [1] since the cyclic gluing of both A i and A i will turn on the replica story on the same class of modular planes thus correspond to the same bulk geodesic chord E 2 . Here we take (32) as a property of the modular flow and try to reconstruct the modular flow lines by applying (4)-(6) to both sides of (32). In 2-dimensional field theory, given an interval A, it will be more convenient to consider the partition A = A 1 ∪ A 3 via an arbitrary point O on A. In most cases the condition that D A = D A requires that the interval A should share end points with A, i.e., ∂ ± A = ∂ ± A (however this is not true for warped CFT, see [13,15]). Then for an arbitrary A , there exist a point O which divide And O is on the modular flow line that passes O. Furthermore the modular flow line that passes O consist of all the points O that satisfy (32). In other words the trajectory of the modular flow that pass through O consist of all the possible O that satisfy (32) . It is easy to check that the modular flow lines of CFT 2 and warped CFT can be reconstructed in this way following the calculations in [1,15]. Here we give another non-trivial example, which is reconstructing the modular flow lines of the field theories invariant under the BMS 3 group, which is conjectured to be the field theory dual of the 3-dimensional flat space [40][41][42]. Consider the vacuum state of the BMSFT that lives on a null plane with a spacelike coordinate φ and null coordinate u. For an arbitrary interval A : (− , the entanglement entropy is given by [14,[43][44][45] where c M is the central charge. Note that, for simplicity we have set the other central charge c L = 0 thus the theory duals to Einstein gravity. The corresponding modular flow is found in [14] using the Rindler method, Integrating along the modular flow k t we get the modular flow lines where c 1 is the integration constant that characterizes all the modular flow lines. Note that it is quite complicated to get the modular flow in this way because the construction of the Rindler transformation is usually a formidable task. Then we try to reproduce the modular flow lines (35) using our partial entanglement entropy proposal and the property (32). Assuming the modular flow line is parametrized by the function u = u(φ). The point O that satisfies (32) should be on the modular flow line thus given by (u(φ), φ). According to (33) and (5), we have the following subset entanglement entropies, and furthermore the partial entanglement entropy The condition (32) indicates that, s A (A 1 ) should be invariant under a translation of O along on the modular flow line, or Plugging (37) in to the above equation we get This reproduces the modular flow lines (35) by a redefinition of the constant c 1 = c 2 l 2 φ .

VI. DISCUSSION
Based on our proposal (4) and the inequalities satisfied by entanglement entropies, we showed that the partial entanglement entropies satisfy many rational requirements for entanglement contour (including all the requirements proposed in [16]). We gave some arguments for the justification of our proposal. We also discussed the relation between the partial entanglement entropy and mutual information. Note that the holographic picture using the modular planes indicates that the partial entanglement entropy is invariant under the modular flow of A in the sense of (32), which is not obvious from the field theory side. However we have shown that, this is also true for the s A (A 2 ) we proposed in the cases where the modular flow in known (see also [1,15]). Furthermore we can even use this property to reconstruct the local modular flows in various 2-dimensional field theories. This implies our simple proposal (4) has deep physical meaning that remains to be discovered. Combine our proposal with the holographic picture, we get the correspondence between bulk geodesic chords and boundary partial entanglement entropies, which can be considered as a finer version of the RT formula or its analogues [12][13][14][15] in holographies beyond AdS/CFT. One can consult [36] for an interesting application of this correspondence to interpret the bulk volume with boundary entanglement entropies.
The property (32) should hold at least for holographic theories with local modular Hamiltonians, it will be interesting to test it in more general theories. In this paper we only considered A that is connected and with one spatial dimension, it will be interesting to study the cases where A is disconnected or in higher dimensions. In higher dimensions the partition will be more complicated. Furthermore since we need to keep the subleading contributions in the entanglement entropies in our proposal, the choice of the regulation scheme for all the subset entanglement entropies will be very subtle [46]. So far we only use the entanglement entropies of the subsets to calculate the partial entanglement entropies. On the other way around, if we can calculate the entanglement contour using the fine correspondence in holography [1,22] or other methods for lattice models like [16][17][18][19][20], we can calculate the entanglement entropies of the subsets with less symmetries via (4)-(6) [46]. For example we can calculate the entanglement entropy of a annulus with the entanglement contour of a disk. The other vital question is that, does the entanglement contour exist in cases with non-local modular Hamiltonian? If it exists then how can we extend the above discussions to the non-local cases? A possible entry for this problem is to follow the extension [47] of the reconstruction of bulk operators from the local to non-local cases.
The dynamical aspects of the entanglement contour are explored in [16,22] which shows the entanglement contour capture much more information than entanglement entropy in the dynamical situations. Also by definition the holographic picture of entanglement contour [1,15] should be closely related to the other holographic formalisms that can give a finer description of holographic entanglement, like the tensor network [48] and the bit threads picture [27] (for related discussions see [22]). The partial entanglement entropy can also be closely related to the holographic entanglement of purification [49] [50], or to the logarithmic negativity following the discussions in [51].
Note that during the accomplishment of this paper, [22] appears which has some overlap with our paper.

VII. ACKNOWLEDGEMENT
We would like to thank Jonah Kudler-Flam, Muxin Han and Gang Yang for helpful discussions. We would also like to thank Chang-pu Sun and Chuan-jie Zhu for support. This work is supported by NSFC Grant No.11805109.