Topological entanglement negativity in Chern-Simons theories

We study the topological entanglement negativity between two spatial regions in (2+1)-dimensional Chern-Simons gauge theories by using the replica trick and the surgery method. For a bipartitioned or tripartitioned spatial manifold, we show how the topological entanglement negativity depends on the presence of quasiparticles and the choice of ground states. In particular, for two adjacent non-contractible regions on a tripartitioned torus, the entanglement negativity provides a simple way to distinguish Abelian and non-Abelian theories. Our method applies to a Chern-Simons gauge theory defined on an arbitrary oriented (2+1)-dimensional spacetime manifold. Our results agree with the edge theory approach in a recent work [35].

To characterize the quantum entanglement, there are various kinds of entanglement measures. In the case when a system is prepared in a pure state |Ψ and bipartitioned into JHEP09(2016)012 two subsystems A and B, two quantum entanglement measures which turn out to be very useful are the so-called Renyi entropy and von Neumann entropy defined as follows S (n) A = 1 1 − n ln Trρ n A , and S vN A = −Trρ A ln ρ A , (1.1) where n is an integer, and ρ A = Tr B ρ is the reduced density matrix of subsystem A, with ρ = |Ψ Ψ|. The Renyi entropy and von Neumann entropy are related by S vN A = lim n→1 S (n) A . It is noted that when ρ corresponds to a pure state, one has the nice property that S For a mixed state, it is found that the quantum and classical correlations cannot be explicitly separated in these entanglement measures. Now we consider two subsystems A 1 and A 2 which are embedded in a larger system, and therefore ρ A 1 ∪A 2 may correspond to a mixed state. In this case, a useful quantity to study the correlation between A 1 and A 2 is the Renyi mutual information which is symmetric in A 1 and A 2 by definition. Similar to the von Neumann entropy, by taking the n → 1 limit, one can obtain the (von Neumann) mutual information It is found that the mutual information will mix the quantum and classical information together [8], and hence is not a good entanglement measure for mixed states.
In this work, we focus on the topological entanglement negativity in a particular topological quantum field theory (TQFT)-the Chern-Simons theory [52,53]. TQFTs are extensively used in condensed matter physics because of the emergence of topological phases from many-body systems such as the fractional quantum Hall states [45][46][47], gapped quantum spin liquids [48,49], p x + ip y superconductors [50,51] and so on. It is noted that the entanglement negativity for a toric code model, an Abelian topological ordered system, has been studied in previous works [33,34]. Later, an edge theory approach was developed to study the entanglement negativity (and other entanglement measures) in a Chern-Simons theory. Due to the bulk-boundary correspondence in a Chern-Simons theory, we present an alternative approach to study the entanglement negativity from bulk point of view. By applying the replica trick and the surgery method [5,52], we compute the entanglement negativity in a generic Chern-Simons theory.
The rest of the paper is organized as follows. In section 1.1, we introduce the path integral representation of a partially transposed reduced density matrix, which is used to define the entanglement negativity. In section 1.2, we introduce the basic ingredients of Chern-Simons theory and the surgery method. Then by using the surgery method and the replica trick, we calculate the entanglement negativity on various bi-/tri-partitioned manifolds for a Chern-Simons theory in section 2, and study how the entanglement negativity depends on the presence of quasiparticles and the choice of ground states. Then we conclude in section 3. We also include several appendices containing the calculation of the entanglement negativity for a bipartitioned torus, which is helpful to understand the case of a tripartite torus in the main text. To compare with the entanglement negativity, we also calculate the mutual information for various cases in the appendices.

Path integral representation of partially transposed reduced density matrix
In this section, we introduce the path integral representation of a partially transposed reduced density matrix, i.e., ρ T 2 A 1 ∪A 2 [see eq. (1.4)], as well as tr ρ T 2 . The definition in this part applies to a generic quantum field theory.
The density matrix in a thermal state can be expressed as a path integral in the imaginary time interval (0, β) [7,12] with β → ∞ corresponding to the zero temperature limit. Here S E is the Euclidean action and Z = tre −βH is the partition function. x represents the coordinate in the d-dimensional space, and τ is the imaginary time. The rows and columns of the density matrix are represented by the values of fields φ( x, τ ) at τ = 0 and β, respectively. Now we take partial JHEP09(2016)012 transposition corresponding to a subsystem B, then the partial transposed density matrix may be expressed as (1.8) Suppose the system is tripartitioned into A 1 , A 2 and B, then the reduced density matrix for ρ A=A 1 ∪A 2 can be obtained by tracing B, i.e., Then the partial transposed reduced density matrix ρ T 2 A 1 ∪A 2 over the subregion A 2 may be expressed as Then, tr ρ can be obtained by taking n copies of ρ T A 2 A 1 ∪A 2 , and glue them appropriately as follows tr ρ with i ≡ i mod n. As a comparison, it should be noted that tr (ρ A 1 ∪A 2 ) n has the expression (1.12) Once we obtain tr ρ , we can calculate the entanglement negativity based on eq. (1.6).

Chern-Simons theory and surgery
Here we mainly review the properties of Chern-Simons theory that will be used in our study of the entanglement negativity. One may refer to the seminal paper [52] for details of the Chern-Simons theory. The Chern-Simons theory action with a gauge group G on a three-manifold M is given by where 'tr' is the trace over the fundamental representation of the gauge group G, A is the G-connection on a genetic three-manifold M , and k is the coupling constant, which is quantized. The Chern-Simons theory is a topological field theory in the sense that the correlation functions do not depend on the metric of the manifold M . Since the action of the Chern-Simons theory does not contain the metric, the partition function can define a topological invariant of the manifold M . Besides the partition function as an invariant of three-manifolds, invariants of links and knots in three-manifolds can be also defined in the Chern-Simons theory. Such a link or a knot in three-manifolds are the "Wilson line", that traces the holonomy of the gauge connection on an oriented closed curve C in a given irreducible representationR of G, We can compute the correlation functions of non-intersecting links/knots C i , i = 1, · · · , N , with a representationR i to each C i on a three-manifold M , (1.16) (When necessary, we denote the partition function as Z(M, [R 1 , · · · ,R N ] C 1 ,··· ,C N ) where [R 1 , · · · ,R N ] C 1 ,··· ,C N indicates the configuration of the links/knots of Wilson loops.) These links/knots correlation functions can be seen as the partition functions of a Chern-Simons theory on a three-manifold M in the presence of Wilson loops. As shown by Witten [52], the the partition functions are exactly calculable by canonical quantization and the surgery.
The key ingredient of computing the partition function is canonical quantization of a Chern-Simons theory on a three-manifold M with boundary given by a Riemann surface Σ. This canonical quantization will produce a Hilbert space H Σ with an associated state |Ψ M . The dual Hilbert space H * Σ with an associated state Ψ M | state can be obtained by reversing the orientation of the Σ. The partition function of a Chern-Simons theory on a (closed) three-manifold can be computed by performing the Heegaard splitting, which decomposes the three-manifold as the connected sum of two three-manifolds (1.17) When the boundary is a sphere, i.e., Σ = S 2 , the Hilbert space H S 2 is one dimensional. When the boundary Σ = T 2 , which can be seen as the boundary of a solid torus T = D×S 1 , one can obtain a state in H T 2 by inserting a Wilson loop in the representationR i around the non-contractible cycle in the solid torus, The state without the Wilson loop is the vacuum state, denoted as |0 .
The above results allow us to compute the partition function on three-manifolds in the presence of Wilson loops. Let us start with S 2 × S 1 , which can be seen as gluing two solid tori T = D × S 1 with boundaries identified. I.e., S 2 comes from gluing two discs together along their boundary S 1 . The partition function of a Chern-Simons theory in this three-manifold is Performing the modular transformation S: τ → − 1 τ on the second solid torus, where τ is the modular parameter of the torus, and gluing it back, i.e., the non-contractible cycle of the first solid torus is homologous to the contractible cycle of the second solid torus, we get S 3 . We obtain the Chern-Simons partition function 20) where S ij is the element of the modular S matrix. If there is a Wilson loop in the repre-sentationR i in one solid torus, the Chern-Simons partition functions become One can also consider a Wilson loop in the representationR i in a solid torus, which is glued to another solid torus with a Wilson loop in the representationR j . The Chern-Simons partition functions are (1.22) Here we list two main properties of the above results: 1. The normalized vacuum expectation values of disjointed Wilson loops can be factorized, i.e., where the three-manifold M is the connected sum of N three-manifolds M i joined along N − 1 two spheres S 2 . This result comes from the fact that the Hilbert space for S 2 is one-dimensional.
2. If Wilson loops are linked or they are passing through the common boundary S 2 between M i and M j , the factorizability of the partition function is hold when the Hilbert space for S 2 with a pair of charges in the dual representationsR i andR i is one-dimensional. We have where 1(2) contains Wilson loops with a general links/knots configuration C 1 (2) in the shaded region in the M 1(2) manifold shown in figure 1.
A surgery procedure we will frequently use in this work is eq. (1.24). We relate the partition function on a manifold M 1 ∪ M 2 with the partition functions on M 1 and M 2 , by a factor Z(S 3 ,R i ) = S 0i . Notice that we do not consider any links/knots configuration of Wilson loops in our following discussion. This indicates in our discussion, 1(2) in eq. (1.24) only contains unlinked/unknoted Wilson loops.
In addition, the modular S-matrix, which is unitary, is related with the quantum dimension as follows The unitarity condition for the S-matrix implies that (1.26)

Topological entanglement negativity
Based on the above discussion, we study the topological entanglement negativity between two spatial subregions on various manifolds in this section. The entanglement negativity is calculated in the following steps. (1) We consider tr(ρ T 2 A 1 ∪A 2 ) ne as the partition function on a three-manifold M . (2) We use the surgery method to compute tr(ρ T 2 A 1 ∪A 2 ) ne , and then take n e → 1.
To avoid confusions, a spatial manifold is a two-manifold, which can be viewed as the boundary of the three-dimensional spacetime manifold where the wave function is defined.

Bipartition of a sphere
In this part, for the pedagogical purpose, we consider the simplest case, in which the spatial manifold is a two-sphere S 2 . We consider the general case where there is a quasiparticlē a (a) in the subsystem A (B), whereā is the anti-quasiparticle of a, i.e., a ×ā = I + · · · , with I being the identity operator. A Wilson line in the representationR a connects the quasiparticlesā and a at the two ends, as shown in figure 2(a). For the case without quasiparticles, we can simply setā = a = I at the end. Figure 2(a) represents the wave functional |Ψ , which is defined on a three-ball. It should be noted that the Wilson line in the representationR a is inside the solid ball. For the density matrix ρ = |Ψ Ψ|, we simply need to consider one more 3-ball with two conjugate punctures, which represents Ψ|, as shown in figure 2(b). To study the topological entanglement negativity between A and B, we need to consider the partially transposed density matrix ρ T B (or ρ T A ). Pictorially, this can be operated by switching the submanifold B and B as shown in figure 2(c). Similar graphic representations of ρ T B were also used in the tensor network study of the entanglement negativity. [19,20] Next, to calculate the entanglement negativity between A and B, we will use the replica trick [see eq. (1.6)]. tr ρ T B n can be calculated as follows. First, we make n copies of ρ T B , with each copy represented in figure 2(c). Next, we glue the subregion A (B) in the i-th copy with the subregion A (B ) in the (i + 1)-th (mod n) copy, then we obtain tr ρ T B n . It is emphasized that tr ρ T B n depends on whether n is odd or even. For odd n, i.e., n = n o , the manifold after gluing is a S 3 . On the other hand, for even n, i.e., n = n e , the manifold after gluing are two independent S 3 . Therefore, tr ρ T B n after the normalization has the following expressions where we have considered the fact that tr ρ T B = Z(S 3 ,R a ) = S 0a . Then, according to the definition in eq. (1.6), one can obtain the entanglement negativity as follows For the case without any quasiparticles on the sphere, one simply sets d a = d I = 1, and therefore As a comparison, for odd n, one will obtain the trivial result, i.e., lim no→1 ln tr(ρ T B ) no (trρ T B ) no = 0. It is noted that E AB in eqs. (2.2) and (2.3) are the same as the topological entanglement entropy. This is because for a general pure state, the entanglement negativity for a bipartite system is equal to the 1/2 Renyi entropy, . It is known that for the case in figure 2(a), one has S (n) Here we demonstrate the simplest case of computing the entanglement negativity by the surgery method. As will be shown later, this basic operation provides a building block for the study of more complicated cases.

Tripartition of a sphere
In this section, we study the entanglement negativity between A 1 and A 2 for a tripartite spatial manifold S 2 , where the sphere is divided into A 1 , A 2 and B. In particular, we are mainly interested in two cases: (1) A 1 and A 2 are adjacent, as shown in figure 3(a), and (2) A 1 and A 2 are disjoint, as shown in figure 4(a).

Case of adjacent A 1 and A 2
First we consider the case where A 1 and A 2 are adjacent to each other, as shown in figure 3(a). There is a Wilson line in representationR a which threads through both the JHEP09(2016)012 A 1 A 2 interface and the A 1 B interface. Again, for the case without any quasiparticle on the sphere, one can simply setR a =R I at the end.
For convenience, we deform the three-dimensional spacetime manifold in figure 3(a), without changing the topology, to two three-balls connected by a tube, as shown in figure 3(b). Then the reduced density matrix ρ A 1 ∪A 2 can be obtained by tracing over the B part, as shown in figure 3(c). Based on ρ A 1 ∪A 2 , one can easily obtain ρ T A 2 A 1 ∪A 2 by switching A 2 and A 2 , as shown in figure 3(d). One can find that the operation of partial transposition here is the same as that in figure 2.
Now we are ready to calculate tr(ρ A 1 ∪A 2 ) n as follows. We make n copies of ρ A 1 ∪A 2 , and glue the region A 1 (A 2 ) in the i-th copy with A 1 (A 2 ) in the (i + 1)-th (mod n) copy. Similar with the case of a bipartitioned sphere, the result depends on whether n is odd or even as follows.
For odd n, i.e., n = n o , the resulting manifold is two S 3 connected by n o tubes. Each tube is contributed by the one that connects A 1 and A 1 in figure 3(d). Then, by using the surgery procedure in figure 1, we cut all the tubes that connect the two S 3 , with each tube contributing a factor Z(S 3 ,R a ) −1 . Therefore, one can obtain tr ρ For even n, i.e., n = n e , the resulting manifold is three S 3 connected by n e tubes. The extra S 3 has the same origin as the case of a bipartitiond sphere in figure 2. Similar with the n = n o case, by cutting each tube, one can obtain tr ρ Then the entanglement negativity between A 1 and A 2 can be expressed as which is the same as eq. (2.2). In other words, for a tripartitioned S 2 as shown in figure 3, the existence of region B does not affect the entanglement negativity between A 1 and A 2 , i.e.,

Case of disjoint A 1 and A 2
Here, we consider the case that A 1 and A 2 are disjoint, as shown in figure 4(a). We also include a quasiparticle a (anti-quasiparticleā) in region A 2 (A 1 ). Therefore, a Wilson line in representationR a threads through both the A 1 B interface and the A 2 B interface. The three-manifold in figure 4(a) is equivalent to two three-balls connected by a tube, as shown in figure 4(b). Based on this, one can obtain the reduced density matrix ρ A 1 ∪A 2 , JHEP09(2016)012 as shown in figure 4(c). To get the partially transposed reduced density matrix ρ one simply needs to switch A 2 with A 2 , as shown in figure 4(d).
We now calculate the entanglement negativity between A 1 and A 2 . As before, we make n copies of ρ figure 4(d). Then we glue region A 1 (A 2 ) in the i-th copy with the region A 1 (A 2 ) in the (i + 1)-th (mod n) copy. In this way, we obtain tr ρ It can be found that the resulting manifold is two S 3 connected by n tubes, which is independent of whether n is even or odd. By considering the surgery procedure in figure 1, one can cut all the tubes that connect the two S 3 , with each tube contributing a factor Z(S 3 ,R a ). Then one can obtain tr ρ for both n = n o and n = n e . Therefore, one can obtain the entanglement negativity between A 1 and A 2 as follows I.e., there is no entanglement negativity between A 1 and A 2 in this case. It is noted that the topological mutual information between A 1 and A 2 for this case is also zero [see eq. (B.6)].

Two adjacent non-contractible regions on a torus with non-contractible B
Here, we focus on the spatial manifold of a torus, T 2 . For the simplest case of a bipartite torus, one can refer to the appendix A, where the operation is straightforward and helpful for understanding the more complicated cases. We first consider two adjacent non-contractible regions A 1 and A 2 on a torus with a non-contractible region B, but with different number of components for the interface, as shown in figure 5(a) and figure 6(a). In figure 5(a), the two adjacent regions A 1 and A 2 share a one-component A 1 A 2 interface, and in figure 6(a), the two adjacent regions A 1 and A 2 share a two-component A 1 A 2 interface. In the following, we will study the entanglement negativity between A 1 and A 2 for these two cases separately.

One-component interface
For the configuration in figure 5(a), it is equivalent to three 3-balls connected by three tubes, as shown in figure 5(b). Then one can obtain the reduced density matrix ρ A 1 ∪A 2 by tracing over the B part, as shown in figure 5(c). The partial transposition of the reduced density matrix ρ A 1 ∪A 2 is fulfilled by switching A 2 with A 2 , as shown in figure 5(d).
Generally, the Wilson loop can fluctuate among different representations. For simplicity, we first consider the case in which the Wilson loop is in a definite representationR a . To study tr(ρ figure 5(d). Then we glue region A 1 (A 2 ) in the i-th copy with A 1 (A 2 ) in the (i + 1)-th (mod n) copy, based on which we obtain tr(ρ The result after gluing depends on whether n is odd or even as follows: for odd n, i.e., n = n o , the resulting manifold is three S 3 connected by 3n o tubes. The 3n o tubes are contributed by the ones connecting and B − B, respectively. The tube connecting B − B corresponds to the vertical tube in figure 5(d). Then, by using the surgery procedure in figure 1, one can cut all the 3n o tubes, with each tube contributing a factor Z(S 3 ,R a ). Then one can obtain tr ρ

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where we have used the fact Z(S 2 × S 1 ,R a ,R a ) = 1. On the other hand, for even n, i.e., n = n e , the resulting manifold is four S 3 connected by 3n e tubes, where the extra S 3 is caused by the partial transposition. Similar with the case of n = n o , the 3n e tubes are contributed by the ones connecting respectively. By cutting all the 3n e tubes with surgery, one can immediately obtain tr ρ (2.11) It is then straightforward to show that for a general state |ψ = j ψ j |R j , i.e., the Wilson loop is in a superposition of different representationsR j , one has tr ρ Then, one can obtain the entanglement negativity between A 1 and A 2 as follows By imposing the normalization condition j |ψ j | 2 = 1, E A 1 A 2 can be simplified as (2.14) Several comments on the above result are in orders: 1. By comparing with eq. (A.5) for the case of a bipartitioned torus where B = ∅, it is found that , which is different from the result for a tripartitioned sphere in eq. (2.7). The reason is that for a torus geometry, as the region B shrinks to ∅, the component of the A 1 A 2 interface changes from one to two.
2. It is found that E A 1 A 2 in eq. (2.14) can be used to distinguish an Abelian theory from a non-Abelian theory. For an Abelian theory, we have d j = 1 for arbitrary representationsR j , and therefore E A 1 A 2 = − ln D, which is independent of the choice of ground states. On the other hand, for a non-Abelian theory, there exists at least one representationR j so that d j > 1. Therefore, for a non-Abelian theory, E A 1 A 2 depends on the choice of ground states. A1∪A2 , in which we do partial transposition over A 2 , i.e., we switch A 2 with A 2 in (c).

Two-component interface
Next, we consider two adjacent non-contractible regions A 1 and A 2 on a spatial manifold T 2 , with a two-component A 1 A 2 interface, as shown in figure 6(a). The configuration in figure 6(a) is equivalent to four 3-balls connected by four tubes in topology, as shown in figure 6(b). Then it is straightforward to obtain the reduced density matrix ρ A 1 ∪A 2 by tracing out the B part, as shown in figure 6(c). Next, for the partially transposed reduced density matrix ρ T A 2 A 1 ∪A 2 , we simply need to switch A 2 with A 2 , as shown in figure 6(d).
As in the previous part, we first consider the simple case that the Wilson loop is in a definite representationR a . To calculate tr(ρ T A 2 A 1 ∪A 2 ) n , we make n copies of ρ T 2 A 1 ∪A 2 in figure 6(d). Then by gluing the region A 1 (A 2 ) in the i-th copy with the region A 1 (A 2 ) in the (i + 1)-th (mod n) copy, we can obtain tr(ρ T A 2 A 1 ∪A 2 ) n . As before, the resulting manifold depends on whether n is odd or even, as follows. For odd n, i.e., n = n o , the resulting manifold is four S 3 connected by 4n o tubes. The 4n o tubes are contributed by the ones connecting A 2 − A 2 , A 1 − A 1 , A 1 − A 1 , A 2 − A 2 and B − B, respectively. By using the surgery procedure in figure 1, we can cut all the 4n o tubes, with each tube contributing JHEP09(2016)012 Z(S 3 ,R a ). Therefore, one can obtain tr ρ On the other hand, for even n, i.e., n = n e , the resulting manifold is six S 3 connected by 3n e tubes, where the extra two S 3 is caused by the partial transposition. Similar with the case of n = n o , the 4n e tubes are contributed by the ones connecting respectively. With the surgery method, one can immediately obtain tr ρ (2.16) It is then straightforward to show that for a general state |ψ = j ψ j |R j , one has tr ρ Then, one can obtain the entanglement negativity between A 1 and A 2 as follows By imposing the normalization condition j |ψ j | 2 = 1, E A 1 A 2 can be simplified as Compared with the case of one-component A 1 A 2 interface in the previous part [see eq. (2.14)], here the power of S 0j is changed from 1 to 2, which is caused by changing the number of components in A 1 A 2 interface. In addition, similar with the result in eq. (2.14), E A 1 A 2 in eq. (2.19) can also be used to distinguish an Abelian theory from a non-Abelian theory by studying its dependence on the choice of ground states.

Two adjacent non-contractible regions on a torus with contractible B
Here, we study the entanglement negativity between two adjacent non-contractible regions A 1 and A 2 on a spatial manifold T 2 , with a contractible region B, as shown in figure 7(a). For convenience, we deform the three-manifold in figure 7(a) into the three-manifold in figure 7(b), where there are four S 3 connected by four tubes, which can be further deformed into the three-manifold in figure 7(c). Then it is straightforward to obtain the reduced density matrix ρ A 1 A 2 by tracing out the B part, as shown in figure 8(a). To obtain the partially transposed reduced density matrix ρ A 1 ∪A 2 , we simply need to switch A 2 with A 2 in ρ A 1 ∪A 2 , as shown in figure 8(b).
As before, for simplicity, we first consider the case in which the Wilson loop is in a definite representationR a . To obtain tr(ρ T A 2 A 1 ∪A 2 ) n , we make n copies of ρ figure 8(b). Then we glue the region A 1 (A 2 ) in the i-th copy with the region A 1 (A 2 ) in the (i + 1)-th (mod n) copy, and obtain tr(ρ T A 2 A 1 ∪A 2 ) n . Since the configuration in figure 8(b) is already very complicated, it is helpful for the readers to understand the gluing based on the case of a bipartite torus [see figure 10(c)], considering that the limit B → ∅ in figure 7(a) corresponds to a bipartitioned torus.
The gluing result depends on whether n is odd or even as follows. For odd n, i.e., n = n o , the resulting manifold is four S 3 connected by 4n o tubes. One should be very careful here. For convenience, we label the four rows of 3-balls in figure 8(d) as the first, second, third and fourth rows of 3-balls from top to bottom. In the resulting manifold after gluing, two S 3 are contributed by the 3-balls in the first and fourth rows in figure 8(d). It is noted that there is no Wilson line threading through these two S 3 , and therefore each of them contributes Z(S 3 ) after the surgery. The other two S 3 are contributed by the 3-balls in the second and third rows. Since there are Wilson lines threading through these two S 3 , each of them contributes Z(S 3 ,R a ) after the surgery. tubes. Therefore, after the surgery procedure in figure 1, each of these tubes contributes Z(S 3 ). The other 2n o tubes are contributed by the tubes that connect A 1 − A 1 (A 1 − A 1 ) in the second row, and the ones that connect A 2 − A 2 (A 2 − A 2 ) in the third row. For these 2n o tubes, since there are Wilson lines threading through them, each tube contributes a factor Z(S 3 ,R a ) after the surgery.
Based on the above analysis, one can obtain tr ρ On the other hand, for even n, i.e., n = n e , the resulting manifold is six S 3 connected by 4n e tubes. Compared with the case of n = n o , the extra two S 3 are introduced by the partial transposition, which is similar to the case of a bipartite torus in figure 10.
In particular, for the extra two S 3 , there are Wilson lines threading through them, and therefore each of them contributes Z(S 3 ,R a ) after the surgery. Therefore, one can obtain tr ρ (2.21)

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It is emphasized that the square term [· · · ] 2 in the first row arises from the fact that the two sets of Wilson loops (red and green) in figure 8(b), after gluing 2n e copies, are independent to each other. This square term is absent in eq. (2.20), because the two sets of Wilson loops are glued to each other for n = n o .
It is straightforward to check that for a general state |ψ = i ψ i |R i , one has tr ρ Then, one can obtain the entanglement negativity between A 1 and A 2 as follows By imposing the normalization condition j |ψ j | 2 = 1, E A 1 A 2 can be simplified as The result is the same as eq. (A.5) for a bipartite torus, i.e., for the configuration in figure 7(a). For this case, the entanglement negativity between A 1 and A 2 depends on the choice of ground states for both Abelian and non-Abelian Chern-Simons theories.

Two disjoint non-contractible regions on a torus
Finally, we demonstrate the vanishing entanglement negativity for two disjoint noncontractible regions A 1 and A 2 on a spatial manifold T 2 , as shown in figure 9(a), in which the regions A 1 and A 2 are separated by non-contractible regions B. The configuration in figure 9(a) is topologically equivalent to four 3-balls connected by four tubes appropriately, as shown in figure 9(b). Then it is straightforward to obtain the reduced density matrix ρ A 1 ∪A 2 by tracing out the B part, as shown in figure 9(c). The partially transposed reduced density matrix ρ figure 9(d) is obtained by switching A 2 and A 2 in figure 9(c). As before, we first consider the simple case that the Wilson loop is in a definite represen-tationR a . By repeating the gluing and surgery procedures as before, it is straightforward where the partial transposition is over degrees of freedom in A 2 , i.e., we switch A 2 with A 2 in ρ A1∪A2 in (c).
to check that tr ρ The result is independent of whether n is odd or even. For a general state |ψ = i ψ i |R i , one can find that tr ρ .

(2.26)
Then, one can obtain the entanglement negativity between A 1 and A 2 as follows i.e., there is no entanglement negativity between two disjoint non-contractible regions on a torus.

JHEP09(2016)012 3 Concluding remarks
In this work, by using the surgery method and the replica trick, we compute the topological entanglement negativity between two spatial regions for Chern-Simons field theories.
We study examples on various manifolds with different bipartitions or tripartitions. In particular, we study how the entanglement negativity depends on the distributions of quasiparticles and the choice of ground states. For two adjacent non-contractible regions on a tripartitioned torus, the entanglement negativity is dependent (independent) on the choice of ground states for non-Ablelian (Abelian) theories. Therefore, it provides a simple way to distinguish Abelian and non-Abelian theories. Our method applies to arbitrary oriented (2+1) dimensional manifolds with arbitrary ways of bipartions/tripartitions.
All the cases studied in this work agree with the results obtained by using a complimentary approach, the edge theory approach, presented in ref. [35]. Here we would like to give some remarks on comparing the method in this work and the edge theory approach: (1) For the edge theory approach, it is unnecessary to know the (2+1) dimensional spacetime manifold and gluing pictures, which are usually complicated. By expressing the (Ishibashi) edge states at the entanglement cut, a straightforward calculation can be performed, which is usually tedious. (2) On the other hand, for the surgery approach shown in this work, the only complication is understanding the 3-manifold. However, this method is elegant, in the sense that once the corresponding 3-manifold for the partially transposed reduced density matrix is known, the results can be directly read off. Thus, these two methods are complimentary and have their own merits.
Finally, we close by pointing out a future problem: It is interesting to generalize our method(s) to higher dimensions, such as (3+1) dimensions. Most recently, the surgery of (3+1)-dimensional manifolds (with particle and loop excitations) was discussed in ref. [54,55]. One can study the entanglement entropy and negativity of (3+1)-dimensional TQFTs, once the corresponding partition function can be evaluated.

A Topological entanglement negativity: bipartitioned torus
Here, we consider a bipartitioned torus, as shown in figure 10(a), which is topologically equivalent to two 3-balls connected by two tubes as shown in figure 10(b). It is straightforward to obtain the partially transposed reduced density matrix ρ T B AB as shown in figure 10(c). For the first step, we consider the simplest case, i.e., the Wilson loop is in a definite representationR a . We follow the replica trick introduced in the main text: first, we make n copies of ρ T B AB as in figure 9(c). Then, we glue the region A (B) in the i-th copy with the region A(B ) in the (i + 1)-th (mod n) copy, based on which we obtain tr(ρ T B AB ) n . One can find that the resulting manifold depends on whether n is odd or even. For odd n, i.e., n = n o , we obtain two S 3 connected by 2n o tubes. It should be noted that these tubes are contributed by those connecting contributing Z(S 3 ,R a ). Then we can obtain where we have used the fact that trρ T B A∪B = Z(S 2 × S 1 ,R a ,R a ) = 1. On the other hand, for even n, i.e., n = n e , the resulting manifold is composed of two independent manifolds, with each manifold being two S 3 connected by n e tubes. By using the same surgery procedure as above, we can obtain It is then straightforward to show that for a general pure state |ψ = j ψ j |R j , i.e., the Wilson loop is in a superposition of different representations, one has Then one can obtain the entanglement negativity between A and B as follows By imposing the normalization condition j |ψ j | 2 = 1, E AB can be simplified as which is the same as the 1 2 -Renyi entropy S ), as expected.

B Topological mutual information between two regions for various cases
Here, we make a comparison between the mutual information and the entanglement negativity between two regions A 1 and A 2 for various cases. Observing the topology in figure 11(a), it is straightforward to check that S (n) A . Next, to calculate the entanglement entropy for A 1 , we deform the configuration of |Ψ JHEP09(2016)012 into figure 11(c), based on which one can obtain ρ A 1 in figure 11(d). Then, one has and S (n) Therefore, the mutual information between A 1 and A 2 is given by which is independent of n.

B.1.2 Case of disjoint A 1 and A 2
This case corresponds to the configuration in figure 4, and can be easily studied based on the previous results. It can be found that S A in eq. (B.2). In addition, since the total system stays in a pure state, then we have S (n) B , which has the same expression as eq. (B.4). Then one has B.2 Two adjacent non-contractible regions on a torus with non-contractible B

B.2.1 One-component interface
This case corresponds to the configuration in figure 5(a), and can be studied based on the result in ref. [5]. For a general state |Ψ = a ψ a |R a , it is found that S (n) This case corresponds to the configuration in figure 7(a). The entanglement entropy S A 1 ∪A 2 has already been calculated in ref. [5], and has the following simple expression Now we need to calculate S (n) A 1 (or S (n) A 2 ). As shown in figure 12, for convenience, we denote the compliment part of A 1 as A 1 = B. Shown in figure 12(b) is a deformation of |Ψ , based on which we can obtain ρ A 1 in figure 12(c). Then it can be checked that It is noted that although I A 1 ,A 2 is independent of the quantum dimension, I A 1 ,A 2 with n > 1 depends on the quantum dimension explicitly.