Topological entanglement entropy meets holographic entropy inequalities

In the literature, there is no consensus on the precise definition of TEE. We will review the three commonly used definitions as follows:

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  The first definition comes from the area law, Eq. (1). The additive constant term $-\gamma<0$ is defined as the TEE, and is believed to capture some global features of entanglement in the ground state. We will use this quantity $\gamma$ as the definition of TEE, i.e, $S_{\text{topo}}=\gamma$. Our sense of ‘topological’ means that $S_{\text{topo}}$ depends only on global features and not the system’s dimensions, when talking about area law. In the context of anyons, we mean that $S_{\text{topo}}$ only depends on the total quantum dimension $\mathcal{D}$ and $\gamma=\log{\mathcal{D}}$. It is in some sense a strong assumption that the ground state is described by a TQFT. Nevertheless, we will proceed with this.

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  The second definition comes from the work of Levin and Wen Levin and Wen (2006) where TEE is defined as a tripartite information quantity called the conditional mutual information, Eq. (2),

  $\displaystyle S_{\text{topo}}^{\text{LW}}=$

  $\displaystyle S(AB)+S(BC)-S(B)-S(ABC)\,,$

  (2)

  computed on a geometry as shown in Fig. 1b. In the Eq. (2), $S(X)$ is the VN entropy associated with region $X$. However, on changing this particular geometry (for example, permuting labels of regions or choosing a disk) renders this quantity useless. We would like to point out that the information quantity used in this case i.e., conditional mutual information is related to the entropy inequality called the strong sub-additivity (SSA) Lieb and Ruskai (1973).

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  The work of Kitaev and Preskill Kitaev and Preskill (2006) motivated a third definition of the TEE involving a different tripartite information quantity related to an entropy inequality called monogamy of mutual information (MMI) Hayden et al. (2013) as follows,

  	$\displaystyle S_{\text{topo}}^{\text{KP}}=$	$\displaystyle S(AB)+S(AC)+S(BC)$		(3)
  		$\displaystyle-S(A)-S(B)-S(C)-S(ABC),$

  where we choose the sign convention to set $S_{\text{topo}}\geq 0$. This was first evaluated on a disk (see Fig. 1a and a non-simply connected geometry, Fig. 1b. It was found to be invariant under permutation of labels of regions. The topological property of $S_{\text{topo}}^{\text{KP}}$ holds for different geometries, however, the value $S_{\text{topo}}^{\text{KP}}$ depends on the topology (more specifically, on the number of connected components of the geometry). We call such quantities fixed-topology TEE probes. The Ref. Bao et al. (2015a) gives an explicit formula of $S_{\text{topo}}^{\text{KP}}$ in terms of the zeroth Betti number.

  

A holography-inspired approach Bao et al. (2015a) showed that the cyclic family of holographic entropy inequalities (HEI) Bao et al. (2015b) gives the TEE, $\gamma$. This was demonstrated by explicitly by applying the area law, Eq. (1). In this current work, we extend the previously done analysis Bao et al. (2015a, b) in the following sense,

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  We don’t explicitly use the area law, Eq. (1) and instead perform a TQFT calculation on a 2d disk-like geometry. We demonstrate that the TEE depends only on the total quantum dimension $\mathcal{D}$, in agreement with the results of Ref. Kitaev and Preskill (2006); Levin and Wen (2006); Bao et al. (2015a).

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  We show that all HEI Bao et al. (2024a) are satisfied by gapped Hamiltonians with unique ground state, Eq. (1) and, especially the facet inequalities, can be used to evaluate the TEE, $\gamma$.

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  We further show that a certain class of entanglement entropy inequalities that need not hold true for holographic quantum states, could hold true for the 2d-disk geometry Kitaev and Preskill (2006). Hence, these entropy relations can also be used as probes for TEE.

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  We contrast the definitions of TEE in Eq. (2) and Eq. (3) and classify all possible definitions of $S_{\text{topo}}$ into two classes, suggesting necessary conditions on the subtraction scheme Kim et al. (2023a); Levin (2024) of entanglement entropies.

The organization of this paper is as follows: In section, Sec. II, we begin by reviewing the necessary TQFT tools for our discussion. Our main results are in section, Sec. III, where we consider the HEIs in the context of TEE and use these results in subsection IV.1 to discuss entanglement-based probes of topological phases. Finally, we discuss our results and outline some future directions in section, Sec. IV. Throughout this paper, we will neglect the spurious TEE which often arise in the symmetry protected topological phases Zou and Haah (2016); Santos et al. (2018); Williamson et al. (2019); Stephen et al. (2019); Kato and Brandão (2020); Kim et al. (2023b). However, we will bring them into the discussion towards the end in the section, Sec. IV.

Consider a five-party information quantity, $\mathcal{Q}_{5}$, which is given by Eq. (9). The quantity $\mathcal{Q}_{5}$ along with other information quantities formed using the cyclic family of holographic inequalities were shown to give the TEE Bao et al. (2015a) by assuming the area law, Eq. (1). Here, we verify the claim by calculating $\mathcal{Q}_{5}$ explicitly using TQFT, Eqs. (10)–(11).

In fact, it is straightforward to do a similar proof for the family of cyclic inequalities, see Sec. S2 of SM ¹¹1See Supplementary Material for details.. Having illustrated an example, we will now generalize our claims about HEI. We will begin by giving some definitions.

Given $n$ parties, one can construct an entropy basis of all possible ($2^{n}-1$) subregion entanglement entropies and order it lexicographically. For example, if $n=3$ and the subregions are $\{A,B,C\}$, then the entropy basis is $\{S_{A},S_{B},S_{C},S_{AB},S_{AC},S_{BC},S_{ABC}\}$. An entropy vector is a vector in this basis. We will call it a holographic entropy vector when the system is a holographic system, i.e, the $n$ subregions are chosen on a boundary $\partial X$ of a bulk manifold $X$. The holographic entropy cone is the space of all allowed holographic entropy vectors. It is a convex, rational polyhedral cone, which can be characterized by the facets of the cone. Thus, knowing all the facet HEIs is sufficient for most purposes and any other non-facet HEI will be redundant.

We will now define the $n$-partite information quantity $\mathcal{I}_{n}$ introduced already, which we will call multi-information. Given $n$ regions $A_{i}$, where $i=1,\dots,n$, we define $\mathcal{I}_{n}$ to be,

which for up to $n=3$ are given by (using shorthand $S(A_{i})\equiv S_{A_{i}}$),

Note that $\mathcal{I}_{n}$ is generally sign indefinite for $n>3$ in holographic systems and the results from Kitaev and Preskill (2006) indicates that $\mathcal{I}_{3}$ is sign-definite. Here, we find that $\mathcal{I}_{n}$ is sign definite and equals $-\log{\mathcal{D}}$, for all $n\geq 3$. In the Sec. S1 of SM Note (1) we explicitly calculate $\mathcal{I}_{3}$, $\mathcal{I}_{4}$ and $\mathcal{I}_{5}$. We introduce another quantity $\mathcal{I}_{m,n}$, which is defined same as $\mathcal{I}_{m}$ but for $m\leq n$, where $n$ is the total number of subregions, such that $\mathcal{I}_{n,n}\equiv\mathcal{I}_{n}$. For example, consider a system with $5$ subregions ($A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ and $A_{5}$) and arbitrarily choose $3$ subregions, say, $A_{1}$, $A_{2}$ and $A_{3}$ then $\mathcal{I}_{3,5}=\mathcal{I}_{3}$, Eq. (14). The set of multi-information $\{\mathcal{I}_{i,n}\}_{i=3}^{n}$ forms a basis for the facet HEI He et al. (2019) involving $n-$parties (excluding subaddivity). With the above definitions, we can now formulate our following conjectures and propositions.

See the supplementary material for more discussions around the conjectures III.1 and III.2.

This is proved in Ref. Bao et al. (2024a). For more details, see sec. S3 of SM.

We will take a step further and discuss the case when these inequalities are saturated. Now consider the 2d planar geometry where each region is disconnected. In this case, each $k$-party entanglement term contributes $k$ number of $-\gamma$ terms as there are $k$ boundaries. We know from He et al. (2020) that all facet HEIs of the HEC are balanced, i.e, each party appears equal number of times with positive and negative coefficients. As a result, the topological terms cancel and the HEIs are strictly zero.

We have an alternative proof for the same based on conjecture III.3 stated after lemma III.1. We give a third proof of this argument in sec. S3 of SM exploiting the structural forms of HEIs. We have verified numerically that this holds for all known superbalanced facet HEIs.

We will now look at the true HEIs that are not facets of the HEC. For example, consider the following information quantity,

Calculating the information quantity $\mathcal{Q}_{6,1}$ on the disk-like geometry, Fig. 3 using Eq. (7), we find that $\mathcal{Q}_{6,1}$ is not topological, Eq. (18). However, it is possible to find some fixed configuration(s) of geometry where $Q_{6,1}$ is topological and gives the TEE. We classify all such information quantities that are topological only under specific choice of geometries as fixed-geometry TEE probes (in contrast to fixed-topology TEE probes). The definition of $S_{\text{topo}}$ using strong sub-additivity in Levin and Wen (2006) is another example of this class. More explicitly, we can check that on a $2d-$disk, we have

whereas the same quantity evaluated on a torus-like geometry with the subregion $B$ disconnected gives the TEE (see Fig. 1b).

Now we will look at another example where the given inequality does not hold for all holographic states, but the information quantity defined using it, $\mathcal{Q}_{6,2}$ turns out to probe the TEE, Eq. (21). The factor of $2$ in front of $\log{\mathcal{D}}$ comes from the sum of coefficients being $-2$ [we will illuminate this point in III.4]

This is surprising as it naively seems to suggest that the space of allowed entropy vector space of the unique ground state of a gapped TQFT is strictly contained within the HEC. However, the space of allowed entropy vector space in the case of degenerate ground states is expected to be larger than the HEC (and thus those states can violate the HEIs). Alternatively, it might be the case, that inclusion of the spurious entanglement entropy may violate these inequalities that seem to naively hold. We postpone these interesting questions for our future work and instead generalize the above observations below. We will need a few more definitions before we do that.

For example, consider the monogamy of mutual information (MMI), Eq. (3) involving three subregions. Each subregion $A,B,C$ are balanced, implying $1$-balanced, every $2$-clusters $AB,AC,BC$ are also balanced, implying $2$-balanced. However, the $3$-cluster $ABC$ is not balanced and so on. Note that $1$-balance and $2$-balance can be alternatively referred as balance and superbalance, respectively. We will make the following conjecture III.3 based on empirical observations.

Now we will state our general proposition applicable to all facet and non-facet inequalities.

This explains why facet HEIs $(n\geq 3)$ are both valid and topological. We will make this more concrete below.

The proof is given in Ref. He et al. (2020).

Since, all facet HEIs (except SA) satisfy lemma III.1 and propositions III.2,III.4, therefore, all facet HEIs are both valid and topological.

One may question the validity of the conjecture III.3. Empirically, we have checked all known (over 2000) holographic entropy inequalities $(n\geq 3)$ in the 2d disk geometry and found them to be both valid and topological. We have also tested them over many superbalanced information quantities that are holographically false to gather empirical evidence in favour of proposition III.4 and have not found any counterexample.

We will try to reason the validity of conjecture III.3 from the perspective of entanglement entropy in quantum field theories Casini and Huerta (2004, 2023); Hubeny et al. (2019). In a local QFT, the entanglement entropy of any finite region diverges. Balanced information quantities cancel these divergences, however, it treats the purifying region $O$ as a separate entity. Superbalanced quantities on the other hand treat the purifier $O$ at the same footing as other regions. This give rise to a scheme-independent cancellation of divergences. It has been argued that superbalanced quantities characterize certain topological properties of the configurations Hubeny et al. (2019). We believe that this picture is captured clearly in the TQFT setting where the total quantum dimension is related to the TEE. In the following section, we will suggest how our results are useful in the context of entanglement-based probes of topological phases.

Topological phase is associated with the existence of anyons. The TEE captures this anyonic content exactly because TEE is a function of the total quantum dimension of a given gapped Hamiltonian. Therefore calculating TEE of a system in a unique ground state is one of the possible pathways to detect topological order. Coming up with information quantities that can capture this long-range correlations, equivalently, the topological order is a non-trivial task. Our analysis sheds light on how one can write down some information quantities that can probe the TEE on $2d-$disk-like geometry. We showed that an information quantity which is superbalanced (see definition III.3), can be used as a probe for the TEE. Hence, using the proposed prescription one can write down information quantities that can successfully probe the TEE and hence detect any topological order. Thus, a general formula for calculating the TEE is given by a superbalanced information quantity $\mathcal{Q}$ can be dubbed as,

where $c$ is the sum of coefficients in $\mathcal{Q}$. If one is interested in obtaining $S_{\text{topo}}$, then

where the formula Eq. (26) (and Eq. (27)) is applicable for the $2d$ disk geometry. By changing the topology, especially, connected components of the system, the formula needs to be appropriately modified taking the topology into account.

As shown in the Sec. S1 and Sec. S2 of SM Note (1), the multi-information $\mathcal{I}_{n}$ and cyclic entropy inequalities $\mathcal{Q}_{2n+1}$ are good candidates for probing TEE . While the cyclic inequalities are motivated from holography, the multi-information doesn’t have a supporting holographic counterpart. Both of these quantities are generalization of the MMI used in Kitaev and Preskill (2006), which is given by $\mathcal{Q}_{3}$ and $\mathcal{I}_{3}$, respectively. The advantage of generalizing these quantities is that if the physical system in consideration is naturally divided into more than three regions, these quantities give a direct approach to calculate the TEE. Another advantage of using other facet HEI probes is that, it might be the case that for a non-trivial topology, one has the pre-factor vanishing for some inequalities having the same sum of coefficients, say $c$, in those cases, one has to choose a different probe, having a sum of coefficients, $c^{\prime}$ (and $c^{\prime}\neq c$).

To reiterate our classification of TEE probes, we have defined all superbalanced information quantities as fixed-topology TEE probes while all balanced information quantities are termed as fixed-geometry TEE probes, for which the 2d disk geometry is not suitable.

We will take inspiration from holography to comment on multi-information $\mathcal{I}_{n}$ being the potential signal of multipartite entanglement in gapped Hamiltonians. Recent works in holography Balasubramanian et al. (2024); Gadde et al. (2024) propose various information quantities as measures (and signals) of multipartite entanglement entropy, each with their pros and cons. In particular, holographically, $\mathcal{I}_{n}$ is zero on separable quantum states and sign-indefinite for entangled states. Unfortunately, it is zero, for some states with true multiparite entanglement. Therefore, it is a good but not the best signal for probing multipartite entanglement in holography. In gapped Hamiltonians with unique ground states (i.e. topologies with 2d-disk), $\mathcal{I}_{n}=-\gamma$ (for $n\geq 3$), suggestive of $n$-party global correlations in the ground state. Of course, if one picks a geometry in which all subregions are disconnected, then $\mathcal{I}_{n}$ vanishes identically. So, we already know that $\mathcal{I}_{n}$ is a good signal. It would be interesting to import the alternative definitions of signals of multipartite entanglement from holography and probe them in topological systems. We will report results in this direction in future work.

It was shown previously that, by including spurious contributions to entanglement entropy, the equality $\gamma=\log{\mathcal{D}}$ ceases to hold Zou and Haah (2016); Santos et al. (2018); Williamson et al. (2019); Stephen et al. (2019); Kato and Brandão (2020); Kim et al. (2023b). One limitation of our proof method is that we have first proved that the information quantities formed using HEIs are topological, i.e., they satisfy the equality $\gamma=\log{\mathcal{D}}$, and then use it to show that the HEIs are valid. One may argue that if instead one has an inequality $\gamma\geq\log{\mathcal{D}}$, are the facet HEIs still valid? Since the sum of coefficients for a facet HEI is negative, if there are physical processes that produce a positive contribution to the entanglement entropy (such as spurious contributions) for every or some subregions, depending on the relative magnitude of those contributions, it is a priori not clear if these inequalities continue to hold and one might expect to see violations similar to the case of quantum extremal surfaces in holography Akers et al. (2021). Perhaps, these exotic systems will shed more light on the nature of the HEC, if they cease to hold true for superbalanced information quantities with negative sum of coefficients that are not valid for holographic states, but continue to hold for facet HEIs. This is a very interesting question, and we will take a deep dive on this question in a future work. The answer will also shed light on the space of quantum states of gapped Hamiltonians with unique ground states.

Another limitation is that we have used the geometry of a 2d-disk for our proof. Does the proof hold when one considers other topologies? It is clear that the geometry where each region is disconnected from each other, all superbalanced quantities will identically evaluate to zero. However, we would like to believe that the facet HEIs would always give a non-negative value of the entanglement entropy for the subtraction scheme. It would be interesting if there is a geometry that contradicts this. For the case of connected geometries, where some HEIs may yield zero, others may not. In summary, we would like to conjecture that the full holographic entropy cone, inclusive of all facet HEIs is a probe of global correlations in gapped Hamiltonians. We also think that one may be able to come up with geometries that violate some or all of the superbalanced information quantities with negative sum of coefficients that do not hold holographically.

Lastly, we would like to learn beyond holography and gapped Hamiltonians, and ask which inequalities hold true, in general, for all quantum states. All quantum entanglement entropy inequalities (QEEI) must be obeyed by the GHZ states. This puts a lower bound on the sum of the coefficients of the QEEIs Bao et al. (2024a) to be zero. On the other hand, the 2d-disk geometry considered in our analysis puts the upper bound on the sum of coefficients to zero for the validity of such inequalities. Therefore, one only needs to look at those inequalities whose sum of coefficients are identically zero. We believe that these inequalities form the boundary that separates inequalities that are valid for topological systems from those that do not. Although one can easily find geometries that violate superbalanced inequalities that do not hold in holography whose sum of coefficients is zero ²²2We thank Michael Levin for working out this example with us., we leave it as a future exercise, to ascertain if the same is true for inequalities that hold holographically. This may shed light on whether the space of ground states of gapped Hamiltonians is closed or open.