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Holographic entropy inequalities and gapped phases of matter

We extend our studies of holographic entropy inequalities to gapped phases of matter. For any number of regions, we determine the linear entropy inequalities satisfied by systems in which the entanglement entropy satisfies an exact area law. In particular, we find that all holographic entropy inequalities are valid in such systems. In gapped systems with topological order, the"cyclic inequalities"derived recently for the holographic entanglement entropy generalize the Kitaev-Preskill formula for the topological entanglement entropy. Finally, we propose a candidate linear inequality for general 4-party quantum states.


Introduction
In recent years, the study of quantum entanglement and quantum information in general has produced a myriad of applications in high energy physics and condensed matter physics.
A key tool for the quantification of entanglement is, in particular, the entanglement entropy. For general quantum systems, the von Neumann entanglement entropies of subsystems are known to obey subadditivity, the Araki-Lieb inequalities, weak monotonicity, and strong subadditivity, respectively: Such inequalities are important, as they constrain the phase space of entanglement in quantum systems and can in turn be translated into other physical quantities. In particular, in condensed matter physics there exists a conjectured relationship [1,2] between the existence of a gap in a system and whether or not the entanglement entropy in that system obeys an area law [3]. It therefore seems a fruitful direction to explore and better characterize the properties of the entanglement entropy in quantum-mechanical systems.
It should be noted, however, that entropy inequalites for general quantum states are relatively rare; indeed, (1.1)- (1.4) are the only unconditional entropy inequalities known to date. Luckily, there exist classes of quantum systems for which the entanglement entropy is easier to characterize.

Entropy inequalities from holography
In holography, it has been shown that entanglement entropies of regions on the boundary are equal to the areas of the minimal surfaces subtending the boundary region, or in terms of the celebrated Ryu-Takayanagi formula [4,5]: (1.5) The Ryu-Takayanagi formula gives us a powerful new tool for computing entanglement entropies in regimes where such calculations are usually intractible. In higher dimensions, for example, it turns what would be a difficult (if not impossible) conformal field theory calculation into a straightforward minimization of area in a classical metric.
Interestingly, these holographic entenglement entropies obey a larger set of inequalities than those obeyed by the generic quantum mechanical systems. In [6], it was discovered that, indeed, there is a new entanglement entropy inequality which is true for all systems with semi-classical holographic duals, i.e., the conditional mutual information is monogamous, or S(AB) + S(AC) + S(BC) ≥ S(A) + S(B) + S(C) + S(ABC). (1.6) This was done using a method known as inclusion/exclusion, in which minimal surfaces corresponding to the (positively signed) entropic terms on the left hand side are repartitioned into non-minimal surfaces corresponding to the terms on the right hand side. As non-minimal surfaces have more area than minimal surfaces, if such a partitioning can be done, then the inequality is true. A more detailed description of the methodology is available in [6]. Recently, this has also been generalized to higher numbers of regions in [7] by converting the geometric procedure described above to a combinatoric set of contraction mappings from points on a hypercube, which corresponds to the left hand side entanglement entropies, to points on another hypercube, which corresponds to the right hand side entropies. This new method has yielded a new, infinite family of inequalities that has been proven for holographic systems. These so-called "cyclic" inequalities for n = 2k + 1 subsystems take the form This new generalization has also led to the discovery of several further holographic entropy inequalities [7]. Holographic systems are not the only class of quantum systems that obey a more restrictive set of entanglement entropy inequalities. The set of stabilizer states in quantum error correction does so, as well [8][9][10]. It is interesting, however, that the known stablizer inequalities are implied by (weaker than) the holographic inequalities, thus suggesting an nontrivial relationship between these two types of states.
It is important to note that the utilization of holography in the inclusion-exclusion proof technique is actually quite minimal. Instead, holographic entropy inequalities are reduced to linear inequalities between the areas of boundaries of certain bulk regions. However, these inequalities are then proved for arbitrary bulk regions, not only for those that minimize the Ryu-Takayanagi entropy. As suggested in [7], it is therefore natural to expect that they hold likewise in condensed matter systems that satisfy an area law. The exploration of this idea, and the applicability of the resulting inequalities for other condensed matter systems, will be the focus of the remainder of this work. As any system with an exact area law entropy scaling necessarily satisfies these entropy inequalities, they may also provide further indication, particularly in the direction of falsification, as to whether a gapped system indeed implies area law scaling for entanglement entropy.
Another potentially interesting relationship here is to the field of AdS/CMT [11,12]; as we will see, the entanglement entropies for gapped phases of matter with an exact area scaling obey the constraints of general holographic systems, which is suggestive of possibly nontrivial holographic duals of condensed matter systems.

Organization
In this work we extend and characterize the realm of applicability of the holographic entanglement entropy inequalities to condensed matter systems. The organization of the paper will be as follows: In section 2, we formally prove the validity of the cyclic inequalities for systems that obey an exact area law. In section 3, we show that these inequalities have a valid interpretation as the topological entanglement entropy in two spatial dimensions. In section 4, we fully characterize the entanglement entropy in systems satisfying an exact area law, and we give a minimal and complete set of entropy inequalities and equalities for any fixed number of regions. We comment about the analogous problem for general quantum system and propose a candidate inequality for four-partite quantum systems. Finally, we conclude in section 5.

Gapped systems with trivial topological order
Here we consider gapped systems with trivial topological order in d+1 dimensions. The entanglement entropy S(A) of a subsystem A, which measures the entanglement between A and its complement A c , is defined to be the von Neumann entropy where ρ A is the reduced density operator for A obtained by tracing out all degrees of freedom outside of A in the many-body ground state of the gapped system. Note that for a bipartite system in a pure state, the reduced density operators obtained by tracing out either part have the same set of eigenvalues, hence the same von Neumann entropy, which can be seen via a Schmidt decomposition [13] of the pure state that we started with. This implies that for any division of the system into subsystems A and its complement A c , is satisfied. For gapped systems with trivial topological order, we assume that the entanglement entropy S(A) of a subsystem A scales as the area of its boundary and neglect any sub-area scaling for the moment. Namely, Note that for any two regions A and B in d spatial dimensions that are non-overlapping except possibly at their boundaries, the entanglement entropy of the combined region AB satisfies Here A ∩ B denotes the area of the codimension 1 hypersurface where regions A and B intersect. The above follows from (2.3) because for any d-dimensional regions A, B, and C, the triple intersection A ∩ B ∩ C is of measure zero. We claim that the cyclic inequality (1.7) for n = 2k + 1 regions is satisfied as a strict equality in this system, namely where the sum is cyclic and all indices are taken modulo n. To prove (2.5), we compute: where Q l = {i + l, . . . , i + k}, N = {1, . . . , n} and {j 1 , j 2 } are unordered pairs that take on the indicated values. Hence to prove that LHS = RHS, we need to show that (2.6) The LHS sums over n(n + 1)/2 = (k + 1)(2k + 1) distinct unordered pairs of indices {j 1 , j 2 }, so it contains (k + 1)(2k + 1) distinct terms. The summation on the RHS also contains (k + 1)(2k + 1) terms. So in order to prove that LHS = RHS, it suffices to prove that any term appearing in the summation on the LHS also appears in the summation on the RHS. This is equivalent to proving that for any unordered pair {j 1 , j 2 }, j 1 , j 2 ∈ {1, . . . , n}, there exists i ∈ {1, . . . , n}, j ∈ {i, . . . , i + k}, such that {j 1 , j 2 } = {i, j}.
In section 4 below, we will generalize this result and identify all entropy inequalities and equalities that are obeyed in systems with an exact area-law scaling. We will find that any holographic entropy inequality is valid for systems with an exact area law.

Construction and validity
Here we consider general gapped systems in 2 + 1 dimensions. It is shown in [14][15][16] that the entanglement entropy of a region A with smooth boundary has the form in the limit L/a → ∞ where a is the correlation length. Here b 0 denotes the number of connected components of ∂A, the boundary of region A. The topological entanglement entropy −γ is a universal constant characterizing the topological state and α is a nonuniversal and ultraviolet divergent coefficient dependent on the short wavelength modes near the boundary of region A. In particular, γ = log D captures the far-IR behaviour of entanglement and the total quantum dimension, D, which can be obtained from topological quantum field theory computations, is related to the number of superselection sectors of the system [14]. We divide the plane into 2k + 2 regions, labeled by In order for the topological entropy S topo defined in (3.3) to be a topological invariant, we require the division of the plane to satisfy i∈I A i = ∅, for all I ⊂ {0, 1, 2, · · · , 2k + 1}, such that 0 ∈ I, and |I| > 3. (

3.2)
In other words, there is no point on the plane that is shared by A 0 and more than two other regions. We define the topological entropy S topo for 2k + 2 regions as where all indices are taken modulo (2k + 1). Note that our definition of the topological entropy reduces to the Kitaev-Preskill one [14] when k = 1 (i.e., a division of the plane into 4 regions). Also note that our calculation in section 2 implies that for gapped systems with trivial topological order, S topo = 0, that is, the dependence of S topo on the length of the boundaries cancels out. To see that S topo is a topological invariant, consider deforming the boundary between regions labeled by the index set J ⊂ {0, 1, 2, . . . , 2k + 1}, i.e., points in the set We note the following properties of the entanglement entropy before proceeding to the main arguments: • Under general deformations of the plane, the change in the entanglement entropy of any region A, ∆S(A), is equal to the change in the entanglement entropy of the complement of A, ∆S(A c ). This follows as a consequence of (2.2).
• For any k / ∈ J, points of deformation (points in S) are far from A k . Therefore, we expect ∆S(A k ) = 0, provided that all regions are large compared to the correlation length. In the same spirit, ∆S(A) = 0 for any region A that is a union of such A k 's. It then follows if A is appended to any region B, the change in entanglement of that region is unaffected, namely, ∆S(B ∪ A) = ∆S(B). Now we argue for the topological invariance of S topo . Possible deformations of the regions are classified into the following two cases: In this case, where in the last step, we cyclically left permute the summands in the first summation by k steps, which leaves the summation invariant.
(a) |J| = 2. Denote the only nonzero element in J as j. In this case, Denote the nonzero elements in J as j 1 and j 2 . Moreover, since j 1 , j 2 ∈ {1, 2, · · · , 2k + 1}, |j 1 − j 2 | ≤ k, i.e., they are separated by a distance of at most k (note j 1 , j 2 are mod (2k + 1) integers). Without loss of generality, we assume j 2 = j 1 + l, for some 0 < l ≤ k. We further write j 1 as j for simplicity. In this case, To see that S topo is a universal quantity, we consider the same argument in [14], where a smooth deformation of the local Hamiltonian during which no quantum critical points are encountered. Since the Hamiltonian is local, any smooth deformations of the Hamiltonian can be written as a sum of smooth deformations of local terms. Moreover, by utilizing the topological invariance of S topo , we may deform the regions in the following ways while keeping S topo invariant: • Stretch the boundaries of the regions so that L → ∞, and the entanglement entropy of a region takes the form of (3.1).
• Deform the boundaries of the regions so that all deformation of the Hamiltonian happens locally in the bulk of the regions.
We further assume that the correlation length remains small compared to the size of the regions throughout the deformation. Hence any local deformations of the Hamiltonian in the bulk only has miniscule effects for the ground state near the boundary. As a result, the entanglement entropies of the deformed regions (and hence S topo ) should not be affected by such local deformations of the Hamiltonian. Thus we conclude that the topological entropy we defined in ( For a general division of the plane into 2k + 2 regions that satisfies condition (3.2), we can compute the topological entropy: where all indices are taken modulo (2k + 1), and b 0 [∂A] denotes the zeroth Betti number (the number of connected components) of the boundary of a region A.
Hence S topo is proportional to the topological entanglement entropy −γ, with the proportionality constant determined by the topology of the regions. This implies that we can extract the topological entanglement entropy of a 2 + 1 dimensional topologically ordered system with a mass gap by suitably divide the system into 2k + 2 regions, and compute the topological entropy S topo .

Examples
Here we consider a few examples which elucidate some of the general constructions in section 3.1. Figures 1 and 2 illustrate two possible divisions of the plane into 2k + 2 regions that satisfy condition (3.2).
There are three types of deformations to the regions for both divisions. We consider the change in S topo under such deformations.
First, consider deforming the boundary between two regions in figures 1 and 2. This can either be the boundary between two slices of the pie, say A 1 and A 2 , or the boundary between a slice of the pie and the complement of the pie, say A 1 and A 0 . In the former case, region A 0 is not involved in the deformation, so case 1 of our general analysis for the topological invariance of S topo implies that ∆S topo = 0. In the latter case, there is one more region (A 1 ) besides A 0 that are involved in the deformation, so case 2a of our general analysis implies ∆S topo = 0.
Next, consider deforming a triple point where three regions meet. Without loss of generality, consider the three regions A 1 , A 2 and A 0 . There are two more regions (A 1 and A 2 ) besides A 0 that are involved in the deformation, so case 2b of our general analysis implies that ∆S topo = 0.
Note that for the division in figure 1, one could also deform the center of the pie chart, which is seemingly different from other points in the plane. All 2k + 1 regions but A 0 are involved in the deformation. However, since A 0 is not involved, case 1 of our general analysis still applies in this case, and ∆S topo = 0.
To compute the topological entropy S topo for these two divisions, we apply the general formula (3.4). For pie-chart divisions in figures 1 and 2

Beyond area-law scaling
In the above sections, we have considered systems where the entanglement entropy is in the form of (3.1). In general, (3.1) will be supplemented with various corrections. We here consider a few examples and examine the behavior of the topological entropy (3.3) as a function of k in systems that deviates from exact area-law scaling.
Since the correction leads to imperfect cancellation of the local contributions to entanglement entropy, (3.3) thus yields the topological entropy up to some local correction  factors. For the following examples, we restrict ourselves to the pie-chart division in 2 + 1 dimensions in figure 1 with radius R in units of the lattice size a. For the sake of simplicity, the 2k + 1 slices are divided evenly.
For a generic 2 + 1 dimensional gapped system with non-trivial topological order, the entanglement entropy of some region A with perimeter L large compared to the correlation length is given by where the correction from (3.1) assumes the form of β p /L p , for all odd integers p.
The O(L −p ) correction to the topological entropy (3.3) is given by where x = (1 + 2/π)k + 1/π. For k ≥ 1, (∂|∆S p topo |/∂k) < 0 for each p. The higher k expressions are therefore slightly less sensitive to deviations of the local piece from perfect area-law scaling.
More generally, we may consider entropy scaling S A = αL − γ + f (L/a) where we recast other small deviations in the local piece of entanglement entropy into the form of f (L/a). Let = L/a for a lattice with spacing a, we here briefly sketch the behaviors for corrections f ( ) = log( ) and f ( ) = log( ) [17,18]. Note that near criticality [18], log( ) scaling becomes dominant in the local piece of entanglement entropy. For systems sufficiently far from a phase transition, we can treat them as an order correction in the area-law systems.
For f ( ) = log( ), where x = (1 + 2/π)k + 1/π. And for f ( ) = log( ), In such cases, (∂|∆S topo |/∂k) > 0 for regions with boundary much larger than the correlation length. which renders higher k definitions of the topological entropy S topo more sensitive to local corrections to entanglement entropy. In principle, for some realistic condensed matter system with non-trivial topological order, the generalized definition (3.3) offers a wider range of selection where one can choose the optimal k for purposes of studying both the topological entanglement entropy and local deviations from area-law scaling.

All entropy inequalities for systems with an exact area law
We will now derive the full set of constraints satisfied by the entanglement entropy in systems with an exact area law, S(A) ∼ ∂A. We begin with a useful construction from [7] that allows us to reduce from continuous geometries to a combinatorial problem. For simplicity, we shall assume that the system lives on a manifold without boundary. Let A 1 , . . . , A n be disjoint (apart from their boundaries) regions in a system with an exact area law. We introduce the purifying region A n+1 as the closure of (A 1 ∪ · · · ∪ A n ) c . The entropy of an arbitrary composite region A I = i∈I A i for I ⊆ [n + 1] := {1, . . . , n + 1} can then be evaluated in the following way, where I c denotes the complement of I in [n + 1].
We now consider the undirected complete graph on n + 1 vertices, equipped with the edge weights w(i, j) = ∂A i ∩ ∂A j . Let w(I, J) = i∈I,j∈J w(i, j) denote the total weight of all edges between two disjoint subsets I and J, and δ(I) = w(I, I c ) the cut function. Then it follows from the above that S(A I ) ∼ ∂A I = δ(I).
Conversely, for any given undirected graph with non-negative edge weights we can always construct a geometry and associated regions A 1 , . . . , A n+1 such that ∂A I = δ(I) (cf. [7]). Therefore, proving entropy inequalities for systems with an exact area law is completely equivalent to proving linear inequalities for the cut function.
To determine if a linear inequality holds for the cut function in an arbitrary undirected weighted graph, we expand: But note that β ij are precisely the entropies of a Bell pair shared between subsystems i and j in an (n + 1)-partite pure state. In view of our reduction from systems with an exact area law to graphs, we thus obtain the following result: In section 2, we had proved that the cyclic inequalities (1.7) hold with equality for system that satisfy an exact area law. It follows immediately from lemma 4.1 that we can test the validity of an arbitrary entropy equality by verifying that they hold with equality when evaluated for Bell pairs. In [7], it was observed that this is the case for the cyclic inequalities (1.7) as well as four other holographic entropy inequalities established therein. It follows that all these inequalities hold with equality for systems satisfying an exact area law. In particular, this confirms our explicit derivation for the cyclic inequalities in section 2.
In principle, lemma 4.1 solves completely the problem of characterizing the entanglement entropy in systems with an exact area law. We will now describe the set of all possible entanglement entropies more concretely. For this, it is useful to observe that, for any fixed number of regions n, the collection of valid entropy inequalities I c I S(A I ) ≥ 0 cuts out a convex cone. This cone consists of all vectors s = (S(A I )) ∅ =I⊆[n] ∈ R 2 n −1 formed from the entanglement entropies obtained by varying over arbitrary regions A 1 , . . . , A n and all systems satisfying an exact area law. Following [7,9,19,20], we shall call it the area-law entropy cone. Like any convex cone, it can be dually described in terms of its extreme rays, which we obtain immediately from lemma 4.1: The extreme rays of the area-law entropy cone for n regions are given by the entropy vectors β ij of Bell pairs shared between any two subsystems in the purified (n + 1)partite system.
Since the entropies of Bell pairs can be realized holographically, we may think of the area-law entropy cone as a degeneration of the holographic entropy cone defined in [7]. It is arguably the smallest entropy cone that can capture bipartite entanglement.
The following theorem then gives a complete characterization of the entanglement entropy in systems with an exact area law: by the standard identity for an alternating sum of binomial coefficients, which is applicable since |V \ {i, j}| ≥ 3 − 2 = 1. The fact that the equalities in (3) are all linearly independent can easily be seen by induction on |V |. It follows from the above that the area-law cone is contained in a linear subspace of dimension 2 n − 1 − n k=3 n k = n + n 2 = n+1 2 . We will now show that this is indeed the dimension of the area-law cone. For this, it suffices to observe that (β ij ) k + (β ij ) l − (β ij ) {k,l} = 2 if {i, j} = {k, l}, and otherwise zero. This not only implies that the n+1 2 many extreme rays β ij are all linearly independent, but also that the area-law entropy cone is cut out by the inequalities on the subspace defined by the multivariate information equalities (3). For l ≤ n, these are just the inequalities in (1), while for l = n + 1 we obtain the inequalities in (2) by using the relation S(A I ) = S(A I c ). It is clear from the above that the entropy (in)equalities (1)-(3) form a minimal set.
Geometrically, the area-law entropy cone is an "orthant" of dimension n+1 2 , as follows from the proof of the theorem, where we have shown that the extreme rays are linearly independent. We note that the set of defining (in)equalities is in general not unique as the entropy cone has positive codimension for n > 2.

Generating entropy equalities from graphs
Our method can be easily adapted to include information about the spatial connectivity of the regions A 1 , . . . , A n that enter an (in)equality: If we can guarantee that A i ∩ A j = ∅ then we do not need to consider the corresponding Bell pair β ij when verifying an entropy  Proof. It suffices to argue that the difference to the multivariate information vanishes given our assumption on the spatial connectivity of the regions A 1 , . . . , A n . For this, note that, since j and k are in different connected components so that (j, k) ∈ E and therefore ∂A j ∩ ∂A k = ∅ by our assumption.
For the graph displayed in figure 5 below we recover the statement derived above that I(A : C|B) = 0 for systems with A ∩ C = ∅. We note that the connectivity assumption in lemma 4.4 is equivalent to requiring that I(A i : A j ) = 0 for all {i, j} ∈ E. We may therefore think of 4.4 as a constrained entropy equality in the sense of [21]. Below we list all non-trivial constrained entropy equalities obtained by lemma 4.4 from graphs with four vertices: The last equality is in fact unconditionally true for system with an exact area law; it is the fourpartite information equality from theorem 4.3.

The search for general quantum entropy inequalities
The simplistic method of graph combinatorics has thus far managed to reproduce the forms of many familiar entropic inequalities. In light of the observation, one may suspect that it could also be useful in generating other generic n-party quantum inequalities. In light of the above speculation, we attempt a simple search for 4-party linear quantum inequalities beyond the von Neumann entropy inequalities (1.1)-(1.4).

A B C D E AB AC AD AE BC BD BE CD CE DE
Due to the convexity of the cones, we know that all points inside an entropy cone can be written as a linear combination of its extremal rays. The entropy cone Γ 4 produced by all von Neumann entropy inequalities is known and is characterized by the extremal rays listed in table 1 [22].
It is shown in [21], however, that families 7 and 8 are not physically constructible. Therefore, it is suspected that Γ * 4 should be a proper subset of Γ 4 for n ≥ 4 [21,22], 1 so that additional 4-party entropy inequalities may be needed to complete the entropy cone. 2 By searching through the integral linear combinations of the constrained entropy equalities constructed from graphs as described in section 4.1 above, we have generated a set of inequalities that satisfy families 1 through 6 but can violate families 7 and 8 for certain permutations. After a cursory search, we found two such candidate inequalities (up 1 In fact, studies in the classical Shannon entropy reveal that additional entropy inequalities, such as Zhang-Yeung Inequality, are needed in addition to the Shannon-type entropies. 2 There is also the possibility of a characterization in terms of non-linear inequalities. However, that is beyond the scope of this work. AC AD AE BC BD BE CD CE DE  1 1 2 2 2 2  2  2  2  2  2  2  2  2  2   Table 2. A candidate extremal ray for the 4-party quantum entropy cone proposed by [22]. It is known that the Ingleton inequality does not hold for general quantum states (not even for classical probability distributions), but that it is a valid inequality for the subclass of stabilizer states [8,9]. Inequality (4.6) on the other hand seems to be independent of the other 4-party linear candidates and to the best of our knowledge, has not been tested to a greater extent. All tests we've conducted so far on this inequality return the same result as the quantum analogue of Zhang-Yeung inequality. It will be worthwhile to generate random 5-partite quantum pure states and numerically check if the inequality can be violated. Note that such checks do not constitute a proof. However, it can be useful in finding a counterexample.

Conclusion and future directions
We here restate our findings: 1. We have completely characterized the entropy (in)equalities obeyed by systems in which the entanglement entropy satisfies an exact area law. We find that such an entropy inequality is valid if and only if it is valid for the entropies of Bell pairs shared between arbitrary subsystems. In particular, all holographic entropy inequalities, such as the cyclic inequalities established recently in [7], are satisfied by systems with an exact area law. These (in)equalities may provide constraining tests to determine whether certain condensed matter systems satisfy an area law.
2. The cyclic (in)equalities in two-dimensional systems with non-trivial topological order can be seen as a generalization of [14] which extracts the topological entanglement entropy using higher number of partitions. These higher k generalizations of S topo are sensitive (or insensitive) to different types of deviations from area-law scaling.
3. A graph representation for constrained entropy equalities for systems with an exact area-law scaling is found. As this construction recovers a wide class of entropy equalities including strong subadditivity, it may be suspected that further quantum inequalities may also be found in the set of graph-generated equalities. Following this approach, we have found a candidate linear entropy inequality for general 4-party quantum states.
As we have seen, the graph representation of entropies in area-law systems used in section 4 offers surprisingly powerful insights. In the absence of the minimization that appears in holography, several holographic inequalities now hold exactly as equalities for systems satisfying an exact area law, and we may understand the entropy cone spanned by the area law systems as a particular degeneration of the holographic entropy cone [7].The method may also provide useful insight for the long-standing problem of finding linear inequalities for the entropies of general multipartite quantum states. In this regard, we also note that generalizations of the graph-theoretical approach is much desirable. One such generalization will involve constructing different graphs for a quantum state with holographic dual. We suspect that the geometry of AdS or its dual kinematic space can be effectively captured by analyzing generalized graph representations for these states. In particular, machineries developed in spectral graph drawing may be used to recover the emergent geometry for more general states.