Entanglement on multiple $S^2$ boundaries in Chern-Simons theory
Siddharth Dwivedi Vivek Kumar Singh P. Ramadevi Yang Zhou Saswati Dhara
 

Fashion Retail: Forecasting Demand for New Items
    Aruna Rajan
    

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§ INTRODUCTION


Quantum entanglement <cit.> is one of the important fundamental aspect of quantum physics and is an active area of research. It is a physical phenomenon which occurs when two particles or systems interact in a way such that the state of one cannot be described independently of the state of the other, even when they are separated by a large distance. Such systems are called as entangled systems. The concept of entanglement has many important applications in various areas such as in quantum information theory, quantum computing <cit.>, quantum teleportation <cit.> etc. It can be understood in a formal framework as following: Consider two systems or parties Alice (A) and Bob (B) with corresponding Hilbert spaces ℋ_A and ℋ_B spanned by the basis vectors |e_i^A⟩ and |e_j^B⟩ respectively. The Hilbert space of the composite system is the direct product space ℋ_A ⊗ℋ_B. A normalized state |Ψ⟩∈ℋ_A ⊗ℋ_B can be written as: 

    |Ψ⟩ = ∑_i, jc_ij |e_i^A⟩⊗ |e_j^B⟩≡∑_i, jc_ij |e_i^A, e_j^B⟩ .
 
If we can write the coefficient c_ij = c_i^A c_j^B, then |Ψ⟩ will be a tensor product of the states describing A and B. These are called separable or non-entangled state. The quantum states that can not be separated in this fashion are precisely the entangled states. One of the famous measure of the entanglement between the systems is the von Neumann entropy, also known as the entanglement entropy. In order to compute this, we associate a projection operator called as density matrix to the state |Ψ⟩ given as ρ = |Ψ⟩⟨Ψ|. Tracing out the Hilbert space of one system (say B) gives a reduced density matrix acting on ℋ_A:

    ρ_A = Tr_B (ρ) = ∑_i⟨ e_i^B| ρ |e_i^B ⟩ .

The entanglement entropy can be calculated from the reduced density matrix as,

    EE = -Tr(ρ_Alnρ_A) = -∑_iλ_i lnλ_i  ,
 
where λ_i are the eigenvalues of ρ_A.

An important question in information theory is to understand the possible patterns of entanglement and to classify the states into different entanglement classes. For example, in two qubit system with Hilbert space ℂ^2 ⊗ℂ^2, there is a class of states called `Bell states' defined up to SLOCC (stochastic local operations and classical communication)  which are maximally entangled:

    |Ψ_Bell⟩ = 1/√(2)( |0^A, 0^B⟩ + |1^A, 1^B⟩) 
     .

The entanglement entropy for Bell state can be calculated as EE_Bell = ln 2 which is the maximum possible entanglement entropy for the reduced density matrix defined on two dimensional Hilbert space. Similarly, in three qubit system ℂ^2 ⊗ℂ^2 ⊗ℂ^2, there are two non-trivial classes of tri-partite entanglement <cit.>: the `GHZ' state (where partial trace over one of the system results in a separable reduced density matrix on the other two systems) and the `W' state (where a partial trace still leaves a non-separable reduced density matrix on the other two systems). A mixed state defined by a density matrix ρ acting on ℋ_A ⊗ℋ_B is said to be separable if it can be decomposed as a convex combination of pure states:

    ρ = ∑_i α_i |Ψ_i^A⟩⟨Ψ_i^A | ⊗ |Φ_i^B⟩⟨Φ_i^B| = ∑_i α_i ρ_i^A ⊗ρ_i^B  ,

where α_i > 0 and ∑_i α_i = 1. But deciding whether such a decomposition exists is a NP hard problem <cit.>. However there are other separability criteria available which provide necessary but not sufficient condition for separability. One widely used concept is the positive partial transpose (PPT) criterion <cit.> which says that if ρ is separable then its partial transpose ρ^Γ_B (with respect to, say B) must be positive semidefinite. The partial transpose can be obtained from the density matrix as following:

    ⟨ e_i^A, e_j^B | ρ^Γ_B |e_k^A, e_l^B ⟩ = ⟨ e_i^A, e_l^B | ρ |e_k^A, e_j^B ⟩ .

If ρ^Γ_B is not positive semidefinite, it necessarily implies that the density matrix ρ is non-separable (i.e. entangled). To capture this information, we define entanglement negativity 𝒩 as,

    𝒩 = ||ρ^Γ||-1/2 = ∑_iμ_i - μ_i/2 ,
 
where μ_i are the eigenvalues of ρ^Γ. A non-zero value of 𝒩 means that ρ^Γ_B has negative eigenvalue and is not positive semidefinite. Thus 𝒩≠ 0 implies that ρ is entangled.

So far, we reviewed the entanglement entropy for any quantum mechanical system  with finite degrees of freedom. Analyzing the entanglement structures in a quantum field theory is a difficult task due to large number of degrees of freedom. However, there is a class of exactly solvable quantum field theories called `topological quantum field theories' where entanglement structures can be examined. In particular, there has been a lot of interest to study entanglement in three dimensional Chern-Simons theory. 

In <cit.>, the SU(N) Chern-Simons functional integral on a three-dimensional manifold with a S^2 boundary is given by a state in the Hilbert space ℋ spanned by n-point correlator conformal blocks of SU(N)_k Wess-Zumino conformal field theory on S^2 with n punctures. The Chern-Simons coupling constant k denotes the level of the SU(N)_k Wess-Zumino conformal field theory. The Wilson lines can be viewed as the world line of quasiparticles creating punctures on the S^2 boundary. Using the correspondence of Chern-Simons theory with the conformal field theory, we associate a primary field ϕ_R_i≡ R_i with every puncture where ϕ_R_i transform under the representation R_i of SU(N). The fusion rules of these primary fields determine the dimensionality of the space of correlator conformal blocks.

In a typical set-up to study entanglement, one considers a boundary Σ with a Hilbert space ℋ_Σ associated with it. The Σ is then partitioned into region A and its complement A̅ as shown in figure <ref>(a). If the Hilbert space can be factorized as ℋ_Σ = ℋ_A⊗ℋ_A̅, then by tracing over one of the factors and computing the reduced density matrix, one can obtain the entanglement structure. The entanglement between connected spatial regions in Chern-Simons theory was studied in <cit.>.

Another set-up is to consider the quantum field theory on a manifold whose boundary consists of disconnected components, i.e. Σ = Σ_1 ∪Σ_2 …∪Σ_n like the one shown in figure <ref>(b). Thus the Hilbert space is the tensor product of Hilbert spaces associated with each component:

    ℋ_Σ = ℋ_Σ_1⊗ℋ_Σ_2⊗…⊗ℋ_Σ_n .

One can then study the entanglement structure by tracing out one or more boundary components which is usually referred as multi-boundary entanglement. 

The multi-boundary entanglement in the context of 1+1 dimensional conformal field theory where Σ was disjoint union of n circles (Σ = S^1 ∪ S^1 ∪…∪ S^1) was investigated in <cit.>. Recently, the study of entanglement for the states on multiple copies of T^2 (i.e. Σ = T^2 ∪ T^2 ∪…∪ T^2) was initiated in <cit.>. The quantum states in these works are obtained by performing the path integral of Chern-Simons theory on link complements with n torus boundaries. This set-up shows some interesting features like periodic behavior of the Rényi entropy for torus link states <cit.>. Further the entanglement structure of torus/hyperbolic link states is similar as that of a GHZ/W state as reflected from the partial tracing giving separable/entangled reduced density matrix <cit.>. We refer the readers for other related works in this set-up <cit.>. 

The entanglement between two S^2 boundaries related by cobordism <cit.> gives a nice pictorial understanding on the entanglement entropy. The qualitative remarks given in <cit.> on the entanglement in the presence of the Wilson lines in Chern-Simons theory creating punctured S^2 boundaries needs to be quantified.

In the present work, we will extend the idea of multi-boundary entanglement to study the structure of the states prepared on n copies of S^2, i.e. we consider a manifold M, obtained by slicing S^3, such that the boundary consists of n number of disconnected S^2 components (shown in figure <ref>(a)):

    Σ = S^2 ∪ S^2 ∪…∪ S^2 = ∪_i=1^n S^2   .


The Hilbert space associated with Σ can thus be given as the tensor product of Hilbert spaces associated with each S^2:

    ℋ_Σ = ℋ_S^2⊗ℋ_S^2⊗…⊗ℋ_S^2 .

It is also possible to have Wilson lines in M which intersect the S^2 boundaries creating marked points or punctures on S^2. The primary fields of SU(N)_k Wess-Zumino conformal field theory transforming in the representation R_i are assigned to each of these marked points. One typical example is shown in figure <ref>(b). In such cases, the physical Hilbert space is the space of conformal blocks with the primary fields inserted at the marked points <cit.>. In the case of four punctured S^2 with representations R_1, R_2, R_3, R_4, the Hilbert space is the SU(N)-invariant subspace:

    ℋ_S^2 = Inv(R_1 ⊗ R_2 ⊗ R_3 ⊗ R_4)  .

In order to have a non-trivial space, the R_i should correspond to the integrable representations of the affine extension of SU(N) i.e. SU(N)_k where k denotes the level. Thus these Hilbert spaces are finite dimensional spaces and are given by the number of times the trivial representation of SU(N) appears in the tensor decomposition of R_1 ⊗ R_2 ⊗ R_3 ⊗ R_4. The quantum states |Ψ⟩∈ℋ_Σ can be constructed from the so called `basic building blocks' by appropriately applying braiding and gluing techniques <cit.>. We shall analyze some of these states in this work which are given in terms of the intermediate representations t taking values in (R_1 ⊗ R_2) ∩ (R̅_3 ⊗R̅_4) (more details are given in section <ref>). Some of the representations t can appear more than once and the number of times they appear denotes their multiplicity. We will show that the multiplicity of such intermediate representations plays a very crucial role in determining the entanglement structure of the states. When each t appears once (multiplicity-free case), the states are GHZ-like and the reduced density matrix is separable. Further if any of the t appears with multiplicity, the reduced density matrix is entangled and the states will have a W-like structure.

As remarked earlier, the authors of <cit.> have studied the entanglement structure on two disjoint S^2 ∪ S^2 boundaries. Basically, they invoke the replica trick to obtain the entropy for the states in ℋ_S^2⊗ℋ_S^2. They show that the states in which the Wilson lines stretch from punctures on one S^2 to the  punctures on other S^2 without braiding (as shown in figure <ref>(a)) are maximally entangled states and are analogous to the Bell states. Any braiding between the Wilson lines in these states, like the one in figure <ref>(b), will not affect the entropy. If the Wilson lines only stretch locally as in figure <ref>(c), the state is a product (unentangled) state. However if the Wilson lines connecting the punctures on one boundary undergo non-
trivial braiding with the Wilson lines connecting the punctures on other boundary as shown in figure <ref>(d), the state can be an entangled state. It would be interesting to study the effect of such braiding (which are referred to as necklaces in <cit.>) on the entropy. In this work, we will systematically perform the calculations of the entanglement entropy by explicitly writing the pure quantum state |Ψ⟩ on n copies of S^2. The total Hilbert space can be bi-partitioned into two parts as:


    ℋ_Σ = ℋ_A ⊗ℋ_B = (⊗_i=1^mℋ_i(S^2))   ⊗  (⊗_i=m+1^nℋ_i(S^2))  .

We will denote this partition as (ℋ_A|ℋ_B) or (m|n-m). By tracing out ℋ_B, we compute the reduced density matrix and the entanglement entropy.

The paper is organized as follows. In section <ref>, we will explain the set-up including a brief review of the Hilbert space and the basis of conformal blocks. In section <ref>, we consider the states which are a n-boundary generalization of the state shown in figure <ref>(a). We compute the entropy and study their large k and large N behavior and give the separability criterion of the reduced density matrices. In section <ref>, we will consider the states in which the Wilson lines undergo horizontal braiding, which is a generalization of the state in figure <ref>(d) to n boundaries. We will show that its entanglement structure shows a periodic behavior as we increase the braiding between the Wilson lines. We give explicit calculation of the entropy and negativity using various examples. We conclude and discuss some future prospects in section <ref>.




§ SET-UP


The canonical quantization of the Chern-Simons theory on a three-manifold M with boundary Σ associates a Hilbert space ℋ_Σ to the boundary. If there are Wilson lines embedded in M which intersect Σ creating marked points or punctures on it, the Hilbert space is the space of conformal blocks with primary fields (transforming in the representations of the gauge group) inserted at these punctures. The usual convention followed is the following. Consider the Wilson line carrying representation R which intersects Σ at point P. We associate the representation R or R̅ to the puncture P, depending upon whether the Wilson line is going out or going into the manifold as shown in figure <ref>(a). Since in the current set-up we are only working when Σ is a Riemann sphere with four punctures, we will briefly explain the Hilbert space associated with it.





 §.§ Hilbert space associated with Riemann sphere and the conformal basis

Consider S^2 with four punctures with four primary fields inserted at these points. Suppose they transform under the representations R_1, R_2, R_3, R_4 respectively of the gauge group G=SU(N) as shown in the figure <ref>(b). Thus the Hilbert space is simply given as the invariant subspace of SU(N):

    ℋ_S^2 = Inv( R_1 ⊗ R_2 ⊗ R_3 ⊗ R_4 )  .
 
This Hilbert space is isomorphic to the space of four-point conformal blocks. The latter has two types of basis as shown in figure <ref>(c) and <ref>(d), usually called as t-channel and s-channel respectively. The basis states are labeled by the intermediate representations which can be obtained as given in the following equation:

    t-channel = R_1 ⊗ R_2_t ⊗ R_3 ⊗ R_4_t'  ;  s-channel = R_2 ⊗ R_3_s ⊗ R_1 ⊗ R_4_s' .
 
In t-channel, we first fuse R_1 and R_2 which gives intermediate representations t ∈ (R_1 ⊗ R_2). Further R_3 and R_4 are fused which gives t' ∈ (R_3 ⊗ R_4). Now since we want R_1 ⊗ R_2 ⊗ R_3 ⊗ R_4 to give the trivial representation of SU(N), we must have t' = t̅ otherwise the Hilbert space will be zero dimensional. Also since t' ∈ (R_3 ⊗ R_4) is equivalent to t̅'̅∈ (R̅_3 ⊗R̅_4), the Hilbert space can be given as

    ℋ_S^2 = {t | t ∈ (R_1 ⊗ R_2) ∩ (R̅_3 ⊗R̅_4) } .
 
This is true when t appears in (R_1 ⊗ R_2) and (R̅_3 ⊗R̅_4) only once. But we can also have t appearing multiple times in these fusion and this will give many more possibilities of getting the trivial representation. For a generic case, the decomposition is given as, 

    R_1⊗ R_2 = ⊕_t (N_R_1 R_2^t)  t   ;  R̅_3 ⊗R̅_4 = ⊕_t (N_R̅_3 R̅_4^t)  t  ,

where N_ab^c is the Verlinde coefficient which tells how many times c appears in the tensor decomposition of a and b. Thus following <cit.>, we will use extra labels i and j to keep track of these multiple occurrences of t where i and j take values 1,2,…,N_R_1 R_2^t and 1,2,…,N_R̅_3 R̅_4^t respectively. Thus the conformal block basis of the Hilbert space can be given as:

    basis(ℋ_S^2) = {  |ϕ_t,ij⟩ |  t ∈ (R_1 ⊗ R_2) ∩ (R̅_3 ⊗R̅_4)  ,  1 ≤ i ≤ N_R_1 R_2^t ,  1 ≤ j ≤ N_R̅_3 R̅_4^t } .

For example, if the representation t_0 appears twice in R_1 ⊗ R_2 and thrice in R̅_3 ⊗R̅_4, there will be 6 basis states corresponding to t_0 given as |ϕ_t_0,11⟩ ,|ϕ_t_0,12⟩ , |ϕ_t_0,13⟩ , |ϕ_t_0,21⟩ , |ϕ_t_0,22⟩ , |ϕ_t_0,23⟩. In general, the four punctured S^2 boundary like the one shown in figure <ref>(b) has the Hilbert space with the following dimension:

    (R_1⊗ R_2 = ⊕_ℓ=1^ N_R_1 R_2^t_ℓ t_ℓ)   &  (R̅_3 ⊗R̅_4 = ⊕_ℓ=1^ N_R̅_3 R̅_4^t_ℓ t_ℓ) ⟹dim(ℋ_S^2) = ∑_ℓ=1^ N_R_1 R_2^t_ℓ N_R̅_3 R̅_4^t_ℓ .

The other conformal block basis is the s-channel of figure <ref>(d) and is given as, 

    basis(ℋ_S^2) = {  |ϕ̂_s,uv⟩ |  s ∈ (R_2 ⊗ R_3) ∩ (R̅_1 ⊗R̅_4)  ,  1 ≤ u ≤ N_R_2 R_3^s ,  1 ≤ v ≤ N_R̅_1 R̅_4^s } ,
 
where u and v are the labels to keep track of multiplicities of s in (R_2 ⊗ R_3) and (R̅_1 ⊗R̅_4) respectively. If the representation occurs once in a tensor product, we will omit the corresponding multiplicity label. Note that we need to make sure that R_1, R_2, R_3, R_4 must be the integrable representations of SU(N)_k for a non-trivial Hilbert space.[See Appendix of <cit.> for a brief review of integrable representations. A representation of SU(N) is integrable if it satisfies l_1 ≤ k, where l_1 is the number of boxes in the first row of its Young tableau.] Moreover the representations t and s in the above discussion should also be integrable to get the full Hilbert space.

The two types of conformal basis given above are orthonormal, i.e.

    ⟨ϕ_t_1, i_1 j_1 | ϕ_t_2, i_2 j_2⟩ = δ_t_1 t_2δ_i_1 i_2δ_j_1 j_2  ;  ⟨ϕ̂_s_1, u_1 v_1 | ϕ̂_s_2, u_2 v_2⟩ = δ_s_1 s_2δ_u_1 u_2δ_v_1 v_2

and are related to each other by unitary transformation which can be achieved by using the fusion matrices (also known as Racah matrices). The transformation rule is given as following: 

   |ϕ_t,ij⟩      = ∑_s ∑_u,v  a_s,uv^t,ij[
[ R_1 R_2; R_3 R_4 ]
] |ϕ̂_s,uv⟩

   |ϕ̂_s,uv⟩      = ∑_t ∑_i,j  a_s,uv^t,ij[
[ R_1 R_2; R_3 R_4 ]
]^* |ϕ_t,ij⟩ ,

   
where s,uv and t,ij label the row and the column of these matrices. The unitary property is given as,

    ∑_s,  u,  v a_s,  uv^t_1,  i_1 j_1[
    [ R_1 R_2; R_3 R_4;     ]] a_s,  uv^t_2,  i_2 j_2[
    [ R_1 R_2; R_3 R_4;     ]]^* = δ_t_1 t_2δ_i_1 i_2δ_j_1 j_2 .

 



 §.§ Braiding operators

The braiding operators or the half-monodromy operators, usually denoted as b_i^(±) act on the neighboring i^th and (i+1)^th strands of the Wilson lines by winding the two strands around each other producing a right-handed crossing.[A crossing is called right-handed, if the index finger and thumb of the right hand point in the directions of strands going above and below respectively at the crossing. Similarly, if the index finger and thumb of the left hand point in the directions of strands going above and below respectively, then the crossing is left-handed.] The inverse operator (b_i^(±))^-1 winds the i^th and (i+1)^th strands resulting in a left-handed crossing. The superscripts (+) and (-) are used depending on whether the strands are parallel or anti-parallel as given in the following:

    [ < g r a p h i c s > ]

The basis state |ϕ_t, ij⟩ is an eigenstate of operators b_1^(±) and b_3^(±). Similarly, |ϕ̂_s, uv⟩ is an eigenstate of b_2^(±). The action of the braiding operators on these basis states are given as following:

   b_1^(±)|ϕ_t,ij⟩      = {R_1, R_2, t, i} exp(±iπ(C_R_1+C_R_2-C_t)/k+N) |ϕ_t,ij⟩

   b_2^(±) |ϕ̂_s,uv⟩      = {R_2, R_3, s, u} exp(±iπ(C_R_2+C_R_3-C_s)/k+N) |ϕ̂_s,uv⟩

   b_3^(±) |ϕ_t,ij⟩      = {R̅_3, R̅_4, t, j} exp(±iπ(C_R_3+C_R_4-C_t)/k+N) |ϕ_t,ij⟩ .


Here 3j-phases {R_a, R_b, t, r} only take values ± 1 and are the symmetry phases of the Clebsch-Gordon coefficients when the two coupling representations R_a and R_b are exchanged <cit.>. The term C_R denotes the quadratic Casimir of the representation R given as

    C_R = 1/2( N l + l + ∑_i=1^N-1(l_i^2 - 2i l_i) - l^2/N) ,

where l_i are the number of boxes in the i-th row of Young tableau for R and l is the total number of boxes.





 §.§ Constructing the quantum states using braiding operators and gluing method

The braiding operators can be used to construct new quantum states. A typical example is shown in figure <ref>. Consider the state |α_1⟩ in figure <ref> where Wilson lines carrying representations R_1, R_2, R_3, R_4 stretch from boundary 1 to boundary 2 without any braiding. The Hilbert spaces associated with the two boundaries can be obtained as discussed previously and can be given as

    basis(ℋ_1)    = { |ϕ_t_1, i_1j_1^(1)⟩ |  t_1 ∈ (R̅_3 ⊗R̅_4) ∩ (R̅_1 ⊗R̅_2)  ,  1 ≤ i_1 ≤ N_R̅_3 R̅_4^t_1 ,  1 ≤ j_1 ≤ N_R̅_1 R̅_2^t_1}
    basis(ℋ_2)    = {⟨ϕ_t_2, i_2j_2^(2)|   |  t_2 ∈ (R_3 ⊗ R_4) ∩ (R_1 ⊗ R_2)  ,  1 ≤ i_2 ≤ N_R_3 R_4^t_2 ,  1 ≤ j_2 ≤ N_R_1 R_2^t_2} .
 
Here we used the `ket' and `bra' to denote the basis states since the Hilbert spaces ℋ_1 and ℋ_2 are dual to each other (the two S^2's in this case are oppositely oriented). The state |α_1⟩ can be written as,

    |α_1⟩ = ∑_t_1,  i_1,  j_1  ∑_t_2,  i_2,  j_2 |ϕ_t_1, i_1j_1^(1)⟩⊗⟨ϕ_t_2, i_2j_2|   δ_t_1 t̅_2 δ_i_1 i_2 δ_j_1 j_2 ,

where the delta functions have been used because this state can be seen as S^2 × I with four strands which is equivalent to the identity operation. We can also convert a `bra' to a `ket' which amounts to conjugation and thus will involve two 3j-phases (see ref. <cit.>):

    ⟨ϕ_t_2, i_2j_2|   ⟶  {R_1, R_2, t̅_2, i_2}{R_3, R_4, t̅_2, j_2}  |ϕ_t̅_2, j_2i_2⟩

Thus we can write the state as,

    |α_1⟩   = ∑_t_1,  i_1,  j_1  ∑_t_2,  i_2,  j_2{R_1, R_2, t̅_2, i_2}{R_3, R_4, t̅_2, j_2}  |ϕ_t_1, i_1j_1^(1)⟩⊗ |ϕ_t̅_2, j_2i_2^(2)⟩  δ_t_1 t̅_2 δ_i_1 i_2 δ_j_1 j_2
       = ∑_t∑_i,  j  {R_1, R_2, t, i}{R_3, R_4, t, j}   |ϕ_t,ij^(1)⟩⊗ |ϕ_t,ji^(2)⟩ .

 Now let us apply a series of braiding operators which results in the state |α_2⟩ as shown in figure <ref>. This state can be given as,

    |α_2⟩ = ∑_t∑_i,  j {R_1, R_2, t, i}{R_3, R_4, t, j}   |ϕ_t,ij^(1)⟩⊗( b_3^(-) (b_2^(-))^-1 b_1^(+)) |ϕ_t,ij^(2)⟩ ,

which can be evaluated using eq.(<ref>) and eq.(<ref>). However not all two boundary states can be obtained from |α_1⟩ by applying the braiding operators. For example, consider the state shown in figure <ref>(d). In order to construct this, we need to take a four boundary state and then glue the extra two boundaries by taking its inner product with appropriate states as shown in figure <ref>. The quantum states associated with the three-balls with one, two and four boundaries respectively as shown in figure <ref>(b) serve as the basic building blocks <cit.>. In fact, the four boundary state can be generalized to n boundaries which is the state |Ψ_2⟩ given in section <ref>. Further the state shown in figure <ref>(a) is the n=2 version of the state |Ψ_4⟩ elaborated in section <ref> with one full twist each (i.e. p_1 = p_2 = 1 in figure <ref>). 
 

So far, we have confined to the three-balls with two S^2 boundaries which could have local braiding or necklaces. Partial tracing of the Hilbert space associated with one of the S^2 boundaries is a straightforward exercise in determining the entanglement entropy. Note that the equivalence between the states in figure <ref>(a) and  figure <ref>(b) suggests that we could also consider the states on three-balls with more than two S^2 boundaries and study its entanglement structure from partial tracing some of the S^2 boundaries.




 §.§ Entanglement structure of the states on multiple S^2 boundaries

Given a state |Ψ⟩ on n copies of S^2, we can compute the reduced density matrix by bi-partitioning the total Hilbert space as (ℋ_A|ℋ_B) given in eq.(<ref>) and tracing out the ℋ_B part. Let us take the example of |α_1⟩∈ℋ_1 ⊗ℋ_2 in eq.(<ref>). The total density matrix for this state can be evaluated as,

    ρ_total = |α_1⟩⟨α_1 |/⟨α_1|⟩ .

Tracing out ℋ_2 gives the following reduced density matrix:

    ρ_A = Tr_ℋ_B(ρ_total) = ∑_t'∑_i', j'  ⟨ϕ_t',i'j'^(2)  | ρ_total | ϕ_t',i'j'^(2)⟩ .

Using the orthonormal property of the conformal basis, this can be simplified to

    ρ_A = 1/dim(ℋ_A)(∑_t∑_i, j |ϕ_t,ij^(1)⟩⟨ϕ_t,ij^(1)|)   .

Thus we get the maximum entanglement entropy: EE = lndim(ℋ_A), which is consistent with the result of <cit.> that the quantum state |α_1⟩ is a maximally entangled or Bell state. 

This procedure of tracing out Hilbert space can be generalized for multiple S^2 boundaries. If the basis of the Hilbert space for x^th S^2 boundary is denoted as | ϕ_t_x,i_x j_x^(x)⟩, then tracing out ℋ_B can be achieved as,

    ρ_A = ∑_t_m+1, t_m+2, …, t_n 
     i_m+1, i_m+2, …, i_n  
     j_m+1, j_m+2, …, j_n   ⟨ϕ_t_m+1, i_m+1 j_m+1^(m+1), …, ϕ_t_n, i_n j_n^(n)   | ρ_total | ϕ_t_m+1, i_m+1 j_m+1^(m+1), …, ϕ_t_n, i_n j_n^(n)⟩ ,

where ρ_total is the total density matrix acting on ℋ_A ⊗ℋ_B.
In the next section, we will study the entanglement features of the states which are generalization of the state |α_1⟩ on n copies of S^2.





§ QUANTUM STATES |Ψ_1⟩ AND |Ψ_2⟩


In this section, we consider the states where a pair of Wilson lines stretch from j^th boundary to (j+1)^th boundary as shown in the figure <ref>. Note that |Ψ_1⟩ and |Ψ_2⟩ in figure <ref> differ in the relative orientation of the Wilson lines carrying representations R_2j-1 and R_2j (j=1,2,…,n). When n=2, |Ψ_1⟩ and |Ψ_2⟩ reduce to the state |α_1⟩ of figure <ref> with appropriate orientation of Wilson lines.

These states have been obtained in <cit.> which can be written in conformal basis as: 

    |Ψ_1⟩   = ∑_ℓ=1^_1∑_x_1, …, x_n∏_j=1^n { R_2j-1, R_2j,t_ℓ, x_j }/(dim_q t_ℓ)^n-2/2 |ϕ_t_ℓ,  x_n x_1^(1),ϕ_t_ℓ,  x_1 x_2^(2),…, ϕ_t_ℓ,  x_n-1 x_n^(n)⟩
    
    |Ψ_2⟩   = ∑_ℓ=1^_2∑_y_1, …, y_n∏_j=1^n { R_2j-1, R̅_2j,w_ℓ, y_j }/(dim_q w_ℓ)^n-2/2 |ϕ_w_ℓ,  y_n y_1^(1),ϕ_w_ℓ,  y_1 y_2^(2), …, ϕ_w_ℓ,  y_n-1 y_n^(n)⟩ ,

where the intermediate states in the two cases are labeled by representations t_ℓ and w_ℓ such that t_ℓ∈⋂_j=1^n (R_2j-1⊗ R_2j) and w_ℓ∈⋂_j=1^n (R_2j-1⊗R̅_2j) respectively. The terms dim_q t_ℓ and dim_q w_ℓ are the quantum dimensions of the representations t_ℓ and w_ℓ respectively. Note that both t_ℓ and w_ℓ can occur multiple times in the corresponding tensor decompositions:

    R_2j-1⊗ R_2j = ⊕_ℓ=1^_1 (N_R_2j-1  R_2j^t_ℓ)  t_ℓ  ;   R_2j-1⊗R̅_2j = ⊕_ℓ=1^_2 (N_R_2j-1 R̅_2j^w_ℓ)  w_ℓ

and we need the extra labels x_j and y_j to keep track of multiple occurrences of t_ℓ and w_ℓ respectively. If we define the following:

    N_R_2j-1  R_2j^t_ℓ≡ a_j(t_ℓ)   ;   N_R_2j-1 R̅_2j^w_ℓ≡ b_j(w_ℓ)  ,

then x_j = 1,2,…,a_j(t_ℓ) and y_j = 1,2,…,b_j(w_ℓ) for a given t_ℓ and w_ℓ respectively.

The states |Ψ_1⟩ and |Ψ_2⟩ live in the total Hilbert spaces ℋ(Ψ_1) and ℋ(Ψ_2) respectively which are the product of n Hilbert spaces:

ℋ(Ψ_1)    = ⊗_j=1^n ℋ_j(Ψ_1)       ;  dim(ℋ_j(Ψ_1))    = ∑_ℓ=1^_1 a_j-1(t_ℓ)   a_j(t_ℓ) 

ℋ(Ψ_2)    = ⊗_j=1^n ℋ_j(Ψ_2)      ;  dim(ℋ_j(Ψ_2))    = ∑_ℓ=1^_2 b_j-1(w_ℓ)   b_j(w_ℓ)  ,

where ℋ_j(Ψ_1) and ℋ_j(Ψ_2) are Hilbert spaces associated with S^2 having four point conformal block given by representations {R̅_2j-3, R̅_2j-2, R_2j, R_2j-1} and {R̅_2j-3, R_2j-2, R̅_2j, R_2j-1} respectively which can be seen from the figure <ref>.  Since the 3j-phases { R_2j-1, R_2j,t_ℓ, x_j } and { R_2j-1, R̅_2j,w_ℓ, y_j } only take values ± 1, they will not affect the entanglement structure of the quantum states |Ψ_1⟩ and |Ψ_2⟩ and can be removed by applying U(1) transformation (i.e. exp(i π) if phase is -1). Bi-partitioning the total Hilbert space as (ℋ_A|ℋ_B) = (m|n-m) and tracing out ℋ_B gives the reduced density matrix for the two states:

    ρ_A(Ψ_1)    = 1/⟨Ψ_1|⟩∑_ℓ=1^_1∑_x_0, x_1, …,x_m 
     x_0', x_1', …, x_m' f(t_ℓ)   δ_x_0 x_0'δ_x_m x_m'⊗_j=1^m |ϕ_t_ℓ,  x_j-1 x_j^(j)⟩⟨ϕ_t_ℓ,  x'_j-1 x'_j^(j)| 
    ρ_A(Ψ_2)    = 1/⟨Ψ_2|⟩∑_ℓ=1^_2∑_y_0, y_1, …,y_m 
     y_0', y_1', …, y_m' g(w_ℓ)  δ_y_0 y_0'δ_y_m y_m'⊗_j=1^m |ϕ_w_ℓ,  y_j-1 y_j^(j)⟩⟨ϕ_w_ℓ,  y'_j-1 y'_j^(j) |  ,

where we have defined x_0 ≡ x_n and y_0 ≡ y_n. The functions f(t_ℓ) and g(w_ℓ) are defined as,

    f(t_ℓ) = (dim_q t_ℓ)^2-n(∏_j=m+1^n-1 a_j(t_ℓ) )   ;   g(w_ℓ) = (dim_q w_ℓ)^2-n(∏_j=m+1^n-1 b_j(w_ℓ) )

and the trace part is,

    ⟨Ψ_1|=⟩∑_ℓ=1^_1( (dim_q t_ℓ)^2-n∏_j=1^n a_j(t_ℓ) ) ;  ⟨Ψ_2|=⟩∑_ℓ=1^_2( (dim_q w_ℓ)^2-n∏_j=1^n b_j(w_ℓ) )  .

These can be written as block diagonal matrices where each block correspond to t_ℓ and w_ℓ respectively as following: 

    ρ_A(Ψ_1)    = 1/⟨Ψ_1|⟩(
    [   f(t_1)  P(t_1)                                                   ;                    f(t_2)  P(t_2)                                  ;                                                  ⋱                 ;                                                    f(t__1)  P(t__1) ]) 
    ρ_A(Ψ_2)    = 1/⟨Ψ_2|⟩(
    [   g(w_1)  Q(w_1)                                                   ;                    g(w_2)  Q(w_2)                                  ;                                                  ⋱                 ;                                                    g(w__2)  Q(w__2) ])   .

Let us pause for a moment and try to understand the structure of these density matrices. Take for example ρ_A(Ψ_1). From eq.(<ref>), we can see that its ℓ^th block will correspond to the row |t_ℓ, t_ℓ, …, t_ℓ_m⟩ and the column ⟨t_ℓ, t_ℓ, …, t_ℓ_m | of the matrix ρ_A(Ψ_1) and there are precisely _1 number of non-zero blocks. There are other (_1^m - _1) number of blocks for the states like |t_1, t_2, …, t_m⟩ (and its permutations) which are all 0 and have not been included in the eq.(<ref>). Now let us see the structure of each block. The matrix P(t_ℓ) corresponding to the block t_ℓ is made of the following part of eq.(<ref>):

    P(t_ℓ) = ∑_x_0, x_1, …,x_m 
     x_0', x_1' …, x_m'  δ_x_0 x_0'  δ_x_m x_m' |x_0 x_1, x_1 x_2, …, x_m-1 x_m⟩⟨x_0' x_1', x_1' x_2', …, x_m-1' x_m' |  ,
 
where x_j takes values 1, 2, …, a_j(t_ℓ). Thus we can think of P(t_ℓ) to be acting on the following m qudit Hilbert space:

    ℋ_P(t_ℓ) = ℂ^a_0 a_1⊗ℂ^a_1 a_2⊗…⊗ℂ^a_m-1 a_m .

A possible choice of the matrix form for P(t_ℓ) in this space can be given as the following Kronecker product: 

    P(t_ℓ) =  Θ_1(t_ℓ) ⊗Θ_2(t_ℓ) ⊗…⊗Θ_m-1(t_ℓ) ⊗1_a_0⊗1_a_m ,

where the two identity matrices appear because of the delta functions in eq.(<ref>) and Θ_j(t_ℓ) is a a_j^2(t_ℓ) × a_j^2(t_ℓ) matrix which can be written as the following block matrix:

    Θ_j(t_ℓ) = (
    [      J_11      J_12         …   J_1 a_j;      J_21      J_22         …   J_2 a_j;         ⋮         ⋮         ⋱         ⋮;   J_a_j 1    J_a_j2         … J_a_j a_j ])  ,

Each block J_pq is a a_j(t_ℓ) × a_j(t_ℓ) single-entry matrix whose (p,q) element is 1 and all the other elements are 0. Further Θ_j(t_ℓ) has only one non-zero eigenvalue which is a_j(t_ℓ) and this means that the matrix P(t_ℓ) has only one non-zero eigenvalue: a_1(t_ℓ) a_2(t_ℓ) … a_m-1(t_ℓ) which is repeated with a multiplicity a_0(t_ℓ)   a_m(t_ℓ). Thus the block f(t_ℓ) P(t_ℓ) in eq.(<ref>) will have the following eigenvalues (after substituting the value of f(t_ℓ)),

    λ_t_ℓ = (dim_q t_ℓ)^2-n/⟨Ψ_1|⟩∏_j=1^n a_j(t_ℓ)/a_m(t_ℓ)  a_n(t_ℓ)  ,  mul(λ_t_ℓ) = a_m(t_ℓ)  a_n(t_ℓ)  ,

where λ_t_ℓ is repeated with multiplicity mul(λ_t_ℓ) given above. Having found all the eigenvalues of the reduced density matrix, the entanglement entropy can be given as,

    EE(Ψ_1) = - ∑_ℓ=1^_1mul(λ_t_ℓ) λ_t_ℓlnλ_t_ℓ .

A similar analysis can be done for the quantum state |Ψ_2⟩ and the eigenvalues can be obtained as,

    λ_w_ℓ = (dim_q w_ℓ)^2-n/⟨Ψ_2|⟩∏_j=1^n b_j(w_ℓ)/b_m(w_ℓ)  b_n(w_ℓ)  ,  mul(λ_w_ℓ) = b_m(w_ℓ)  b_n(w_ℓ)  ,

where λ_w_ℓ is repeated with multiplicity mul(λ_w_ℓ) and thus the entanglement entropy is:

    EE(Ψ_2) = - ∑_ℓ=1^_2mul(λ_w_ℓ) λ_w_ℓlnλ_w_ℓ .

For n=2, all the eigenvalues of the reduced density matrix become equal. Thus the entanglement entropy is maximum and hence these states become Bell states.

   EE(Ψ_1^(n=2))    = ln( ∑_ℓ=1^_1 a_1(t_ℓ)  a_2(t_ℓ) ) = lndim(ℋ_A)         ⟹  |Ψ_1^(n=2)⟩= |Ψ_Bell ⟩

   EE(Ψ_2^(n=2))    = ln( ∑_ℓ=1^_2 b_1(w_ℓ)  b_2(w_ℓ) ) = lndim(ℋ_A)         ⟹  |Ψ_2^(n=2)⟩= |Ψ_Bell ⟩ .
	

Note that this entanglement entropy for two S^2 boundaries is the logarithm of the Chern-Simons partition function on S^2 × S^1 in the presence of Wilson lines carrying representations R_1, R_2, R_3 and R_4 <cit.>. More precisely, we have:

    EE(Ψ_1^(n=2))    = ln Z(S^2 × S^1 ;  R_1,R_2,R̅_3,R̅_4)  
    EE(Ψ_2^(n=2))    = ln Z(S^2 × S^1 ;  R_1, R̅_2, R̅_3, R_4)  .


In the later sections, we will analyze the reduced density matrices for n≥3 to get more insight into the structure of the states |Ψ_1⟩ and |Ψ_2⟩.




 §.§ Multiplicity free case

When the representations t_ℓ and w_ℓ appear without any multiplicity in any of the tensor product (SU(2) gauge group for example), then the eigenvalues of the reduced density matrices labeled by t_ℓ and w_ℓ are given as:

    λ_t_ℓ(Ψ_1) = (dim_q t_ℓ)^2-n/∑_ℓ'=1^_1 (dim_q t_ℓ')^2-n  ;  λ_w_ℓ(Ψ_2) = (dim_q w_ℓ)^2-n/∑_ℓ'=1^_2 (dim_q w_ℓ')^2-n

and the corresponding entanglement entropies will be,

    EE(Ψ_1) = -∑_ℓ=1^_1λ_t_ℓlnλ_t_ℓ  ;  EE(Ψ_2) = -∑_ℓ=1^_2λ_w_ℓlnλ_w_ℓ .





 §.§ Separability criteria of the reduced density matrix

In this section, we will study the reduced density matrices in eq.(<ref>) acting on the Hilbert space ℋ_A and find out whether they are separable or entangled. For this, let us first compute the entanglement negativity. Consider the bi-partition of ℋ_A into two parts:

    ℋ_A = ℋ_A_1⊗ℋ_A_2 = (⊗_i=1^rℋ_i )  ⊗(⊗_i=r+1^mℋ_i )  .

The partial transpose of the reduced density matrix ρ_A(Ψ_1) with respect to ℋ_A_2 can be given as,

    ρ^Γ(Ψ_1) = 1/⟨Ψ_1|⟩(
    [   f(t_1)   P^Γ(t_1)                   0                   …                   0;                   0   f(t_2)   P^Γ(t_2)                   …                   0;                   ⋮                   ⋮                   ⋱                   ⋮;                   0                   0                   … f(t__1)   P^Γ(t__1) ])  ,

where P^Γ(t_ℓ) denotes the partial transpose of P(t_ℓ). As given in eq.(<ref>), P(t_ℓ) denotes the following part of the density matrix, where we have explicitly shown the breaking of the basis for ℋ_A_1 and ℋ_A_2:

    P(t_ℓ) = ∑_x_0, x_1, …,x_m 
     x_0', x_1' …, x_m'  δ_x_0 x_0'  δ_x_m x_m'    |  x_0  x_1,  x_1  x_2, …,  x_r-1 x_r_A_1,  x_r  x_r+1, …,  x_m-1  x_m_A_2  ⟩
       ⟨  x_0'  x_1',  x_1'  x_2', …, x_r-1' x_r'_A_1,  x_r'  x_r+1', …,  x_m-1'  x_m'_A_2   |  .
 
As given previously, we can write P(t_ℓ) as the Kronecker product,

    P(t_ℓ) =  Θ_1⊗…⊗Θ_r-1⊗Θ_r⊗Θ_r+1⊗…⊗Θ_m-1⊗1_a_0⊗1_a_m .

An important role is played by the matrix Θ_r here.  Notice that in eq.(<ref>), the last label of the basis set of ℋ_A_1 and the first label of the basis set of ℋ_A_2 are the same, namely x_r. Since Θ_r is the matrix with x_r and x_r' labeling its rows and columns, it will transform non-trivially under the operation of partial transpose. When we apply the partial transpose with respect to A_2, we get,

    P^Γ(t_ℓ) = ∑_x_0, x_1, …,x_m 
     x_0', x_1' …, x_m'  δ_x_0 x_0'  δ_x_m x_m'    |  x_0  x_1,  x_1  x_2, …,  x_r-1 x_r_A_1,  x_r'  x_r+1', …,  x_m-1'  x_m'_A_2  ⟩
       ⟨  x_0'  x_1',  x_1'  x_2', …, x_r-1' x_r'_A_1,  x_r  x_r+1, …,  x_m-1  x_m_A_2   |
 
and hence P^Γ(t_ℓ) can be written as following,

    P^Γ(t_ℓ) =  Θ_1⊗…⊗Θ_r-1⊗Θ_r^Γ⊗Θ_r+1^T ⊗…⊗Θ_m-1^T ⊗1_a_0⊗1_a_m^T  ,

where superscript T denotes the usual transpose which does not affect the eigenvalues. The matrix Θ_r^Γ can be obtained from Θ_r as following:

    Θ_r(t_ℓ) = (
    [      J_11      J_12         …   J_1 a_r;      J_21      J_22         …   J_2 a_r;         ⋮         ⋮         ⋱         ⋮;   J_a_r 1   J_a_r 2         … J_a_r a_r ])   ;  Θ_r^Γ(t_ℓ) = (
    [      J_11      J_21         …   J_a_r 1;      J_12      J_22         …   J_a_r 2;         ⋮         ⋮         ⋱         ⋮;   J_1 a_r   J_2 a_r         … J_a_r a_r ])  ,

where J_pq is the single-entry matrix defined earlier. The eigenvalues of Θ_r^Γ are ± 1 with multiplicities given as following,

    λ(Θ_r^Γ) = {-1, 1}  ;  mul(-1) = a_r(a_r-1)/2  ,  mul(1) = a_r(a_r+1)/2 .

We only need to consider the negative eigenvalues to calculate entanglement negativity. The block of ρ^Γ(Ψ_1) corresponding to representation t_ℓ which is f(t_ℓ)   P^Γ(t_ℓ) will have the following negative eigenvalues,

    λ_t_ℓ^Γ = - f_t_ℓa_1(t_ℓ) a_2(t_ℓ) … a_m-1(t_ℓ)/a_r(t_ℓ)  ,  mul(λ_t_ℓ^Γ) = a_0(t_ℓ) a_m(t_ℓ) a_r(t_ℓ) (a_r(t_ℓ) - 1/2)  .

With all these negative eigenvalues, the entanglement negativity can be computed,

    𝒩(Ψ_1) = 1/⟨Ψ_1|⟩∑_ℓ=1^_1 (dim_q t_ℓ)^2-n(a_r(t_ℓ) - 1/2)∏_j=1^n a_j(t_ℓ)  .

Since the quantum dimension and a_j(t_ℓ) are always positive for any t_ℓ, the negativity will vanish if and only if a_r(t_ℓ) = 1 for all t_ℓ, i.e. if and only if the fusion of R_2r-1 and R_2r gives t_ℓ without multiplicity. A similar analysis can be done for the reduced density matrix  ρ(Ψ_2) and the negativity can be obtained as,

    𝒩(Ψ_2) = 1/⟨Ψ_2|⟩∑_ℓ=1^_2 (dim_q w_ℓ)^2-n(b_r(w_ℓ) - 1/2) ∏_j=1^n b_j(w_ℓ)  .
 
Thus a_r(t_ℓ) = 1 and b_r(w_ℓ) = 1 for all values of t_ℓ and w_ℓ, are both necessary and sufficient for the respective entanglement negativities 𝒩(Ψ_1) and 𝒩(Ψ_2) to vanish. Further, if the reduced density matrix is separable then negativity vanishes. This implies that a_r(t_ℓ) and b_r(w_ℓ) should be 1 for all the representations t_ℓ and w_ℓ.

Now consider the case when all the multiplicities in the tensor decompositions of representations associated with the remaining S^2 boundaries 1,2,…,m are unit, i.e. a_0(t_ℓ) = a_1(t_ℓ) = … = a_m(t_ℓ) = 1 and b_0(t_ℓ) = b_1(t_ℓ) = … = b_m(t_ℓ) = 1 for all the representations t_ℓ and w_ℓ. In such a case the reduced density matrices can be written as,

    ρ(Ψ_1) = ∑_ℓ=1^_1f(t_ℓ)/⟨Ψ_1|⟩  ρ_t_ℓ^A_1⊗ρ_t_ℓ^A_2  ;  ρ(Ψ_2) = ∑_ℓ=1^_2g(w_ℓ)/⟨Ψ_2|⟩  ρ_w_ℓ^A_1⊗ρ_w_ℓ^A_2 ,

where ρ_X^A_1 and ρ_X^A_2 are the density matrices acting on Hilbert spaces ℋ_A_1 and ℋ_A_2 respectively and they correspond to the pure states:

    ρ_X^A_1   = |X^(1), X^(2), …, X^(r)⟩⟨X^(1), X^(2), …, X^(r) | 
    ρ_X^A_2   = |X^(r+1), X^(r+2), …, X^(m)⟩⟨X^(r+1), X^(r+2), …, X^(m) |  .

The coefficients are non-negative and they add up to 1:

    ∑_ℓ=1^_1f(t_ℓ)/⟨Ψ_1|⟩ = 1 = ∑_ℓ=1^_2g(w_ℓ)/⟨Ψ_2|⟩ .

Thus we see that the reduced density matrices are separable. From the above discussion, we can propose the following. 

Proposition. The necessary condition for the reduced density matrix ρ, corresponding to the states |Ψ_1⟩ or |Ψ_2⟩,  acting on (⊗_i=1^rℋ_i )  ⊗(⊗_i=r+1^mℋ_i )  to be separable is R_2r-1⊗ R_2r = ⊕_ℓ t_ℓ. On the other hand, the tensor product R_2i-1⊗  R_2i=⊕_ℓ t_ℓ for each of the m boundaries gives a sufficient condition for separability of ρ, i.e,
[box=]align
   ρ= separable      ⟹  a_r(t_ℓ) = 1,    ∀t_ℓ

a_0(t_ℓ) = a_1(t_ℓ) = …= a_m(t_ℓ) = 1,    ∀t_ℓ     ⟹  ρ= separable  .
	

For the group SU(2), the decomposition of the tensor product of any two representations is always multiplicity free. Thus one immediate result follows from the above discussion:

Corollary. For SU(2) Chern-Simons theory, the states |Ψ_1⟩ and |Ψ_2⟩ are GHZ-like and the reduced density matrices are separable.

Further if we consider all the representations (of SU(N)) to be the same, then all the multiplicities will be equal (i.e. a_i(t_ℓ) = a_j(t_ℓ) and b_i(t_ℓ) = b_j(t_ℓ) for all values of i and j). Thus t_ℓ and w_ℓ appearing without multiplicity in R ⊗ R and R ⊗R̅ becomes a necessary and sufficient condition for reduced density matrices to be separable.  

    R ⊗ R = ⊕_ℓ=1^_1 t_ℓ⟺ρ(Ψ_1) = separable  ;   R ⊗R̅ = ⊕_ℓ=1^_2 w_ℓ⟺ρ(Ψ_2) = separable .

From the analysis presented in this section, we can say that the multiplicity plays a crucial role in determining the entanglement structure of the quantum states |Ψ_1⟩ and |Ψ_2⟩. When there are no multiplicities, these states have a GHZ-like entanglement structure because the reduced density matrices are separable. However when the multiplicities are involved (a_r > 1 and b_r>1 to be precise), these states have a W-like entanglement structure and the reduced density matrices are non-separable.

So far in this section, we studied the entanglement properties of the states |Ψ_1⟩ and |Ψ_2⟩ and obtained the entropy values for a generic case which depends on the choice of representations carried by the Wilson lines. It also depends on the rank of the gauge group SU(N) as well as the Chern-Simons level k. In the following subsection, we will consider explicit examples giving the analytic expressions of entropy as a function of N and k. Further we will investigate how the entropy behaves in large N or large k limit and we provide various numerical plots supporting the results.  




 §.§ SU(N)_k entanglement entropy and its large k and large N limit

The spectrum of the reduced density matrix and hence the entanglement entropy depends on the quantum dimensions of various representations. For the SU(N) group, these can be evaluated from the corresponding Young tableau using the q-numbers denoted as [x]. For example: 

    dim_q( (2,1) ) = [N][N+1][N-1]/[1][1][3] .

The q-numbers are written in terms of variable q which is the primitive (k+N)^th root of unity: q=exp(2π ik+N). Thus the q-numbers for our computation will be given as,

    [x] = q^x/2 - q^-x/2/q^1/2 - q^-1/2 = (π/k+N) sin(π  x/k+N)  .

These numbers have the property that [N+α] = [k-α]. Thus the large k limit or the large N limit (with the other parameter fixed) of these numbers are given as following:

    lim_k →∞[α] = lim_N →∞[α] = α  ;  lim_k →∞[N±α] = N±α  ;  lim_N →∞[N ±α] = k ∓α ,

where α is a finite positive integer. Hence in the k →∞ limit, dim_q R →dim(R) which is the usual dimension of the representation R. The corresponding N →∞ limit can be obtained by simply replacing each (N±α) factor in dim(R) by (k∓α). As an example, we will have:

    lim_k →∞dim_q( (3) ) = N(N+1)(N+2)/6  ;  lim_N →∞dim_q( (3) ) = k(k-1)(k-2)/6 .

Using this we can see that in the large k or large N limit, the eigenvalues of the reduced density matrix are rational numbers. The k →∞ limit of the entanglement entropy can be obtained by simply replacing dim_q t →dim(t) and dim_q w →dim(w) in the eq.(<ref>) and eq.(<ref>) respectively. The N →∞ limit can be computed as mentioned above where we first replace dim_q t →dim(t) and dim_q w →dim(w) and then replace all (N±α) → (k∓α) in the expressions of dim(t) and dim(w) respectively.

In the following, we give some examples explicitly showing the computation of entanglement entropies and their large k and large N behavior.




  §.§.§ Wilson lines carrying symmetric representations of SU(N)

Let us compute the entropy when Wilson lines in figure <ref> carry symmetric representations of SU(N) as following:

    R_2j-1 = a  ,   R_2j = b ,
    
where we assume that a ≤ b without loss of generality. We will use the Young tableau notation in which a representation R is specified by a set of non-negative integers: R = (l_1, l_2,…,l_N-1) where l_i denotes the number of boxes in the i-th row of Young tableau. From the group theory, we know the following tensor decompositions for a ≤ b:

t    ∈(R_2j-1 ⊗R_2j)       = ⊕_ℓ=0^a   (a+b-ℓ, ℓ)       ≡⊕_ℓ=0^a  t_ℓ 

w    ∈(R_2j-1 ⊗R̅_2j)       = ⊕_ℓ=0^a   (b-a+2ℓ, b-a+ℓ,…,b-a+ℓ)       ≡⊕_ℓ=0^a  w_ℓ  .


We should mention at this point that we want all the representations to be integrable (as already discussed in section 2) for which we must have k ≥ (a+b).[The maximum number of boxes in the first row of Young tableau happens for the representations t_0 and w_a in the eq.(<ref>) which is (a+b). Thus k ≥ (a+b) will ensure that all the representations are integrable.]. The quantum dimensions needed for our computation are given as, 

    dim_q t_ℓ   = [N+ℓ-2]! [N+a+b-ℓ-1]! [a+b-2ℓ+1]/[ℓ]! [a+b-ℓ+1]! [N-1]! [N-2]!
    dim_q w_ℓ   = [N+ℓ-2]! [N+b-a+ℓ-2]! [N+b-a+2ℓ-1]/[ℓ]! [b-a+ℓ]! [N-1]! [N-2]! ,

where the factorial is defined as [x]! = ∏_i=1^x [i] with [0]!=1. The 3j-phases for symmetric representations as given in <cit.> are:

    {R_2j-1, R_2j, t_ℓ} = (-1)^a+b/2-ℓ  ,  {R_2j-1, R̅_2j, w_ℓ} = (-1)^b-a/2-ℓ .
 
Thus the quantum states for this example, up to overall phase, will be:

    |Ψ_1⟩ = ∑_ℓ=0^a  (-1)^nℓ/(dim_q t_ℓ)^n-2/2 |ϕ_ℓ^(1),…, ϕ_ℓ^(n)⟩ ,    |Ψ_2⟩ = ∑_ℓ=0^a  (-1)^nℓ/(dim_q w_ℓ)^n-2/2 |ϕ_ℓ^(1),…, ϕ_ℓ^(n)⟩ .

The eigenvalues of the reduced density matrix for the bi-partition (m|n-m) will be,

    λ_ρ_A(Ψ_1)    = 1/∑_ℓ=0^a (dim_q t_ℓ)^2-n{ (dim_q t_0)^2-n, (dim_q t_1)^2-n, …, (dim_q t_a)^2-n}
    λ_ρ_A(Ψ_2)    = 1/∑_ℓ=0^a (dim_q w_ℓ)^2-n{ (dim_q w_0)^2-n, (dim_q w_1)^2-n, …, (dim_q w_a)^2-n} .

When a=1, i.e. R_2j-1 is a fundamental representation, each Hilbert space will be two dimensional and the eigenvalues are given as,

    λ_ρ_A(Ψ_1)    = {α_1^n-2/α_1^n-2 + β_1^n-2  ,  β_1^n-2/α_1^n-2 + β_1^n-2}
    λ_ρ_A(Ψ_2)    = {α_2^n-2/α_2^n-2 + β_2^n-2  ,  β_2^n-2/α_2^n-2 + β_2^n-2} ,
 
where we have defined,
 
α_1    = sin(π(N-1)/k+N) sin(πb/k+N)   ;        β_1 = sin(π/k+N) sin(π(b+N)/k+N)  
 
α_2    = sin(π/k+N) sin(πb/k+N)    ;        β_2 = sin(π(N-1)/k+N) sin(π(b+N)/k+N)  .
 
Note that the two eigenvalues may become equal at certain values of N and k, i.e. α_1 = β_1 and α_2 = β_2 respectively. In such a case, the entanglement entropy will be maximum (ln 2) and the two states |Ψ_1⟩ and |Ψ_2⟩ will be the maximally entangled GHZ states. The values at which this may happen are:

    |Ψ_1⟩ = |GHZ⟩: SU(2)_k=2b
    SU(3)_k=9 for b=2
    SU(N)_k=N for b=1|Ψ_2⟩ = |GHZ⟩: SU(2)_k=2b
    SU(3)_k=9 for b=7
    SU(b+1)_k=b+1 .
 
Thus if we consider the case where all the Wilson lines carry the fundamental representation of SU(N), then the state |Ψ_1⟩ will be a GHZ state for the group SU(N)_k=N and the state |Ψ_2⟩ will be a GHZ state for the group SU(2)_k=2.

In figure <ref>, we have plotted the behavior of entanglement entropy for the state |Ψ_1⟩ when b=1, i.e. when all the representations R_i are fundamental.   

We can see that entropy decreases and eventually goes to 0 as we increase the number of boundaries n (for a fixed k and N). However, when either k or N increases (with other parameters fixed), the entropy first increases, becomes maximum at k=N (which is consistent from eq.(<ref>)) and then gradually decreases and converges to a finite value as k →∞ or N →∞. The third and the fifth plots in figure <ref> also show that when both k as well as N become large, the entanglement entropy converges to its maximum value ln 2. This behavior can be seen from eqn.(<ref>). When either k →∞ or N →∞ (and the other parameter is fixed), the entropy converges as following:

    lim_k →∞EE(Ψ_1)    = x_1^n log(x_1^2-n+1)+x_1^2 log(x_1^n-2+1)x_1^n+x_1^2, lim_N →∞EE(Ψ_1) = y_1^n log(y_1^2-n+1)+y_1^2 log(y_1^n-2+1)y_1^n+y_1^2
    lim_k →∞EE(Ψ_2)    = x_2^n log(x_2^2-n+1)+x_2^2 log(x_2^n-2+1)x_2^n+x_2^2, lim_N →∞EE(Ψ_2) = y_2^n log(y_2^2-n+1)+y_2^2 log(y_2^n-2+1)y_2^n+y_2^2,

where we have,

    x_1 = N+b/b(N-1),   y_1 = k-b/b(k+1),   x_2 = (N+b)(N-1)/b,   y_2 = (k-b)(k+1)/b .

When both k →∞ and N →∞ (in no particular order), the entropy converges to:

    lim_k →∞
     N →∞EE(Ψ_1) = b^n log(b^2-n+1)+b^2 log(b^n-2+1)/b^n+b^2  ;  lim_k →∞
     N →∞EE(Ψ_2) = ln(2) δ_n,2 .

The first term for b=1 becomes ln 2. Thus the state |Ψ_1⟩ becomes a GHZ state and  |Ψ_2⟩ becomes a non-entangled state in the large level and large rank limit:

    Wilson lines in fundamental representation of SU(N): lim_N →∞lim_k →∞ |Ψ_1⟩ = |GHZ⟩ .

For completeness, we have also given various plots in figure <ref>, showing the behavior of entanglement entropy for the state |Ψ_2⟩ for fundamental representations of SU(N).   


Note that using the level-rank duality <cit.>, our entropy results for representations R can be extended to the representations R^T by simply exchanging N ↔ k.[We thank Howard Schnitzer for pointing this out.] Here R^T represents a transposed representation where the Young tableau for R^T is obtained by transposing the Young tableau of R. As an example, 

    R = (3,2)  ⟹    R^T = (2,2,1) .

Now consider the decomposition of tensor product of symmetric representations in eq.(<ref>). For the transposed representations, this decomposition can be given as,

    (R_2j-1⊗ R_2j) = ⊕_ℓ=0^a  t_ℓ     ⟹   (R_2j-1^T ⊗ R_2j^T) = ⊕_ℓ=0^a  t_ℓ^T 
    
    (R_2j-1⊗R̅_2j) = ⊕_ℓ=0^a   w_ℓ     ⟹   (R_2j-1^T ⊗R̅_2j^T) = ⊕_ℓ=0^a   w_ℓ^T  .
 
Moreover, using the property [N+α] = [k-α] of the q-number for any integer α, it is not difficult to see that the quantum dimensions of t_ℓ and w_ℓ in SU(N)_k will be the same as the quantum dimensions of t_ℓ^T and w_ℓ^T respectively in SU(k)_N. Thus the SU(N)_k entropy obtained for the representations R_2j-1 and R_2j will be equal to the SU(k)_N entropy for the representations R_2j-1^T and R_2j^T.

The tensor decomposition of two symmetric representations of SU(N) is multiplicity free. Thus the reduced density matrices for this example will be separable with vanishing entanglement negativity. However while taking the tensor product of mixed representations of SU(N), we will encounter multiplicity. In the following, we will consider two examples of mixed representations and give the results for entropy and negativity. 




  §.§.§ Wilson lines carrying mixed representations of SU(N)

Let us first consider the case when all the Wilson lines carry the adjoint representation of SU(N). Since adjoint representation is self conjugate, the two states |Ψ_1⟩ and |Ψ_2⟩ will be the same. We will have the following decomposition for N ≥ 4:[For SU(2), adj⊗adj = ∙⊕(2)⊕(4) which has already been covered in previous section (see eq.(<ref>)). For SU(3), we have to discard representation t_2 in eq.(<ref>).]

    adj⊗adj = t_1 ⊕ t_2 ⊕ t_3 ⊕ t_4 ⊕ t_5 ⊕ 2 t_6  ,

where various representations are given as following:

    {t_1, t_2, t_3, t_4, t_5, t_6 } = {∙,  (2^2, 1^N-4),  (3, 1^N-3),  (3^2, 2^N-3),  (4, 2^N-2),  (2, 1^N-2) } .

In writing these representations, we have used a compact notation to denote the number of boxes in various rows of the Young tableau, where (l_i^x) means that l_i is repeated x times. For example, (2^2, 1^2) denotes the representation (2,2,1,1) in which the first two rows of the Young tableau have two boxes each and the third and fourth rows have one box each. 
The quantum dimensions of these representations will be,

    dim_q t_2 = [N+1][N-3][N]^2/[2]^2[N-2],  dim_q t_3 = dim_q t_4 = [N+2][N-2][N+1][N-1]/[2]^2
       dim_q t_5 = [N+3][N-1][N]^2/[2]^2,  dim_q t_6 = [N+1][N-1],  dim_q t_1 = 1  .

Thus the quantum state |Ψ_1⟩ = |Ψ_2⟩≡ |Ψ⟩ for this case can be written as,

    |Ψ⟩ = ∑_ℓ=1^5 {adj, adj,t_ℓ}^n /(dim_q t_ℓ)^n-2/2⊗_i=1^n |ϕ_ℓ^(i)⟩ + ∑_x_1, …, x_n∏_j=1^n {adj, adj,t_6, x_j }/(dim_q t_6)^n-2/2⊗_i=1^n |ϕ_6,  x_i-1 x_i^(i)⟩ ,

where each variable x_i takes value 1 and 2. The non-vanishing eigenvalues of the reduced density matrix will be,

    λ_ρ_A = 1/⟨Ψ|⟩{λ_t_1, λ_t_2, λ_t_3, λ_t_4, λ_t_5, λ_t_6, λ_t_6, λ_t_6, λ_t_6} ,

where ⟨Ψ|=⟩λ_t_1+λ_t_2+λ_t_3+λ_t_4+λ_t_5+4λ_t_6 and,

    λ_t_ℓ = 1/(dim_q t_ℓ)^n-2   (for ℓ=1,2,3,4,5)   ;  λ_t_6 = 2^n-2/(dim_q t_6)^n-2 .

The entanglement entropy will be maximum for n=2 which is ln 9 (for SU(3), it is ln 8) and decreases and goes to 0 as n increases as shown in the first two plots of figure <ref>. Similarly for a given n, the entropy decreases as k or N increases (with other parameter fixed) but converges to a fixed value as k →∞ or N →∞ respectively which is clear from the plots in figure <ref>.

These limiting values can be obtained from the eigenvalues of the reduced density matrix which converge to the following when k →∞ for a fixed N:

    lim_k →∞λ_ρ_A   = 1/trace_1{ 1, 4^n-2 (N-2)^n-2(N^4-2 N^3-3 N^2)^n-2, 4^n-2(N^4-5 N^2+4)^n-2, 4^n-2(N^4-5 N^2+4)^n-2, . 
       . 4^n-2(N^4+2 N^3-3 N^2)^n-2, 2^n-2(N^2-1)^n-2, 2^n-2(N^2-1)^n-2, 2^n-2(N^2-1)^n-2, 2^n-2(N^2-1)^n-2} .

Similarly when N →∞ for a fixed k, the eigenvalues converge to:

    lim_N →∞λ_ρ_A   = 1/trace_2{1,4^n-2 (k+2)^n-2(k^4+2 k^3-3 k^2)^n-2,4^n-2(k^4-5 k^2+4)^n-2,4^n-2(k^4-5 k^2+4)^n-2, . 
       . 4^n-2(k^4-2 k^3-3 k^2)^n-2, 2^n-2(k^2-1)^n-2,2^n-2(k^2-1)^n-2,2^n-2(k^2-1)^n-2,2^n-2(k^2-1)^n-2} ,

where trace_1 and trace_2 are appropriate factors which make sure that the reduced density matrix has unit trace. When both k →∞ and N →∞ (in no particular order), then EE→ 0 and the state |Ψ⟩ tends to an unentangled state:

    lim_k →∞
     N →∞EE(Ψ) = 0  .

Moreover, using the results of the earlier sections, we can also compute the entanglement negativity which comes out to be:

    𝒩 = 2^n-1  (dim_q t_6)^2-n/⟨Ψ|⟩ = 2^n-1  [N+1]^2-n  [N-1]^2-n/⟨Ψ|⟩ .

We can see that 𝒩≠ 0 and hence the state |Ψ⟩ has a W-like entanglement structure with non-separable reduced density matrix.

Next consider an example when all the Wilson lines carry the mixed representation (2,1) = (2,1) of SU(N). In this case, we will have:

    R_2j-1⊗ R_2j   = t_1 ⊕ t_2 ⊕ t_3 ⊕ t_4 ⊕ t_5 ⊕ t_6 ⊕ 2 t_7 
    
    R_2j-1⊗R̅_2j   = w_1 ⊕ w_2 ⊕ w_3 ⊕ w_4 ⊕ w_5 ⊕ w_6 ⊕ 2 w_7  .

where various representations t_ℓ∈ (R_2j-1⊗ R_2j) and w_ℓ∈ (R_2j-1⊗R̅_2j) are given as following:[For SU(3), representations t_2, t_3 and w_2, w_6 do not appear in the tensor decomposition.]

    {t_1 … t_7 }   = { (2^3),  (2^2, 1^2),  (3, 1^3),  (3^2),  (4, 1^2),  (4, 2),  (3, 2, 1) }
    {w_1 … w_7 }   = {∙, (2^2, 1^N-4), (3, 1^N-3), (3^2, 2^N-3), (4, 2^N-2), (4, 3, 2^N-4, 1), (2, 1^N-2) } .

The non-vanishing eigenvalues of the reduced density matrix are:

    λ(Ψ_1)    = 1/⟨Ψ_1|⟩{λ_t_1, λ_t_2, λ_t_3, λ_t_4, λ_t_5, λ_t_6, λ_t_7, λ_t_7, λ_t_7, λ_t_7}
    λ(Ψ_2)    = 1/⟨Ψ_2|⟩{λ_w_1, λ_w_2, λ_w_3, λ_w_4, λ_w_5, λ_w_6, λ_w_7, λ_w_7, λ_w_7, λ_w_7} ,

where ⟨Ψ_1|=⟩λ_t_1 + λ_t_2 + λ_t_3 + λ_t_4 + λ_t_5 + λ_t_6 + 4λ_t_7 and ⟨Ψ_2|=⟩λ_w_1 + λ_w_2 + λ_w_3 + λ_w_4 + λ_w_5 + λ_w_6 + 4λ_w_7 and we have:

λ_t_ℓ    = 1/(dim_q t_ℓ)^n-2   (for  ℓ=1,2,3,4,5,6)      ;  λ_t_7    = 2^n-2/(dim_q t_7)^n-2 

λ_w_ℓ    = 1/(dim_q w_ℓ)^n-2   (for  ℓ=1,2,3,4,5,6)      ;  λ_w_7    = 2^n-2/(dim_q w_7)^n-2  .

When k →∞, the quantum dimensions become the usual dimension which for the SU(N) group are given as:

    dim(t_1) = (N^2-N) (N+1)!144 (N-3)!  ,  dim(t_2) = N (N+1)!80 (N-4)!  ,  dim(t_3) = (N+2)!72 (N-4)!  ,  dim(t_4) = (N^2+N) (N+2)!144 (N-2)!
       dim(t_5) = (N+3)!72 (N-3)!  ,  dim(t_6) = N (N+3)!80 (N-2)!  ,  dim(t_7) = N (N+2)!45 (N-3)!  ,  dim(w_1) = 1   ,  dim(w_2) = N^2(N-3)(N+1)4
       dim(w_3) = dim(w_4) = (N-2)(N-1)(N+1)(N+2)4  ,  dim(w_5) = N^2(N-1)(N+3)4
       dim(w_6) = (N-3)(N-1)^2(N+1)^2(N+3)9  ,  dim(w_7) = (N-1)(N+1)  .

The entanglement entropy can be computed for k →∞ by simply replacing dim_q t_ℓ and dim_q w_ℓ by dim(t_ℓ)  and dim(w_ℓ) respectively as given in eq.(<ref>). Similarly the N →∞ limit of the entropy can be obtained by replacing the quantum dimensions by usual dimensions and then replacing each (N ± x) factor in eq.(<ref>) by (k ∓ x). When both k and N goes to infinity (in no particular order), the eigenvalues of the reduced density matrix converge to the following:

    lim_k →∞
     N →∞λ(Ψ_1^(n ≥ 2))    = {5^2 9^nα, 9^2 5^nα, 5^2 2^2-n 9^nα, 5^2 9^nα, 5^2 2^2-n 9^nα, 9^2 5^nα, 8^2-n 45^nα, 8^2-n 45^nα, 8^2-n 45^nα, 8^2-n 45^nα}
    lim_k →∞
     N →∞λ(Ψ_2^(n ≥ 3))    = {1,0,0,0,0,0,0,0,0,0 };  λ(Ψ_2^(n = 2)) = {110, 110, 110, 110, 110, 110, 110, 110, 110, 110} ,

where α≡ 162 × 5^n+50 × 9^n+25 × 2^3-n 9^n+2^8-3 n 45^n. Thus in this limit, the state |Ψ_2^(n ≥ 3)⟩ becomes an unentangled state. The state |Ψ_1⟩ however remains entangled and the entropy decreases monotonically as n increases and converges to ln 2 as n becomes very large as shown in figure <ref>:


    (lim_k →∞
     N →∞EE(Ψ_1))_n →∞ = ln(2)  .

The entanglement negativity in this example can be evaluated as:

    𝒩(Ψ_1) = 2^n-1  (dim_q t_7)^2-n/trace_1  ;  𝒩(Ψ_2) = 2^n-1  (dim_q w_7)^2-n/trace_2 .

Clearly 𝒩≠ 0 and the states have a W-like entanglement structure with non-separable reduced density matrices.




 §.§ Braiding between the Wilson lines for states |Ψ_1⟩ and |Ψ_2⟩

The braiding between the Wilson lines for the quantum states |Ψ_1⟩ and |Ψ_2⟩ can be achieved by acting the braiding operators (b_1, b_2 or b_3) on the basis states.[One such example was shown in the figure <ref> for the state |α_1⟩ which is the n=2 case of |Ψ_1⟩.] Let us consider |Ψ_1⟩ first. Any braiding between the Wilson lines carrying representations R_2j-1 and R_2j can be seen as the action of b_1^(+) operator on the j^th boundary. Similarly the braiding between the Wilson lines R_2j-2 and R_2j-3 can be seen as the action of b_3^(+) operator on the j^th boundary. The action of these operators on the j^th basis is given as:

    b_1^(+) |ϕ_t,  x_j-1 x_j^(j)⟩   = {R_2j-1, R_2j, t, x_j-1}exp(±iπ (C_R_2j-1+C_R_2j-C_t)/k+N) |ϕ_t,  x_j-1 x_j^(j)⟩
    
    b_3^(+) |ϕ_t,  x_j-1 x_j^(j)⟩   = {R_2j-2, R_2j-3, t, x_j}exp(±iπ (C_R_2j-2+C_R_2j-3-C_t)/k+N) |ϕ_t,  x_j-1 x_j^(j)⟩ .

Thus the new basis state is related to the old basis by U(1) transformation, which does not affect the entanglement structure. It is also possible to have braiding between the Wilson lines R_2j and R_2j-2 which can be obtained by applying b_2^(-) on the j^th basis state. Since the operator b_2 does not act directly on the `t' basis, we need to change it to `s' basis using the transformation rules given in eq.(<ref>) and then apply b_2. Then we can transform back to the `t' using eq.(<ref>). The end result will be the following basis state:

    b_2^(-) |ϕ_t,  x_j-1 x_j^(j)⟩   = ∑_s,u,v
    t', x_j-1',  x_j' a_s,uv^t, x_j-1 x_j[
    [  R_2j-1    R_2j; R̅_2j-2 R̅_2j-3 ]]   a_s,uv^t', x_j-1' x_j'[
    [  R_2j-1    R_2j; R̅_2j-2 R̅_2j-3 ]]^*
       ×{R_2j, R̅_2j-2, t, x_j}exp(±iπ (C_R_2j+C_R̅_2j-2-C_s)/k+N) |ϕ_t',  x_j-1' x_j'^(j)⟩ .

If we denote the Racah matrix a_s,uv^t,ij as ℛ and the matrix consisting of the eigenvalues of the braiding operator b_2 as Λ, we can write the above equation as,

    b_2^(-) |ϕ_t,  x_j-1 x_j^(j)⟩ = ∑_t',  x_j-1',  x_j'(ℛ^†Λℛ)_t', x_j-1' x_j'^t, x_j-1 x_j |ϕ_t',  x_j-1' x_j'^(j)⟩ .

where the subscript and superscript in (ℛ^†Λℛ) label the row and column respectively of the matrix ℛ^†Λℛ. Since ℛ and Λ are the unitary matrices, so is ℛ^†Λℛ and hence the new basis is related to old one by unitary transformation. A similar analysis can also be done for the state |Ψ_2⟩. This means that any braiding between the Wilson lines for the states |Ψ_1⟩ and |Ψ_2⟩ will not affect the entanglement structure of these states. This is in accordance with <cit.> where, using the replica trick, the authors show that the entanglement structure of the state |α_1⟩ of figure <ref> does not change under the braiding between the Wilson lines.  

Thus we see that if the Wilson lines only braid locally, the entanglement structure of the quantum states remains same. In order to see the affect of the braiding on the entanglement entropy, we must consider the states where Wilson lines connecting punctures on one of the S^2 braid with the Wilson lines connecting the punctures on a different S^2, which we call as horizontal braiding (necklaces). In the next section, we study the states having horizontal braiding between the Wilson lines and study its affect on the entanglement structure.




§ HORIZONTAL BRAIDING AND ITS AFFECT ON ENTANGLEMENT ENTROPY


In this section, we will consider the states with horizontal braiding and study their entanglement structure by explicitly evaluating the entropy and negativity. Particularly, we will work out the states corresponding to the necklaces in the following subsections.




 §.§ Quantum state |Ψ_3⟩ on S^2 ∪ S^2

Consider the state |Ψ_3⟩ shown in figure <ref>. The Wilson lines carrying representations R_1 and R_2 undergo horizontal braiding 2p number of times (i.e. twisted p number of times) while the other two Wilson lines with representations R_3 and R_4 go straight from one boundary to the other. This state can be constructed as given in Appendix A and is given as:


    |Ψ_3⟩    = ∑_ℓ=1^∑_r_1, r_2, r_3{ R_1, R̅_1, t_ℓ, r_1 }{ R_2, R̅_2,t_ℓ, r_2 }{ R_4, R̅_3, t_ℓ, r_3 }{ R_2}{ R_3}/√(dim_q t_ℓ) √((dim_q R_1)(dim_q R_2)(dim_q R_3)(dim_q R_4))
       × F(t_ℓ, r_1, r_2, p)    | ϕ_t_ℓ,  r_3  r_1^(1), ϕ_t_ℓ,  r_2 r_3^(2)⟩ .

Here the representation t_ℓ∈ (R_1⊗R̅_1) ∩ (R_2⊗R̅_2) ∩ (R_4⊗R̅_3) which may appear in each of this decomposition multiple times. We denote the corresponding multiplicities as a_1(t_ℓ) ≡ N_R_1 R̅_1^t_ℓ, a_2(t_ℓ) ≡ N_R_2 R̅_2^t_ℓ and a_3(t_ℓ) ≡ N_R_4 R̅_3^t_ℓ respectively. The labels r_1, r_2, r_3 are used to keep track of these multiple occurrences where r_i takes values from 1 to a_i(t_ℓ). The notation { R }≡{ R, R̅, 0} is used for the 2j-phase and the function F is defined as, 

    F(t_ℓ, r_1, r_2, p) = ∑_s,  u{R̅_1, R_2, s, u }√(dim_q s)   a_s,  uu^t_ℓ,  r_1 r_2[
    [  R_1 R̅_1;  R_2 R̅_2;      ]] exp[2 π i   p ( C_s - C_R̅_1 - C_R_2)/k+N]  ,

where the summation is over representation s ∈ (R̅_1⊗ R_2) and u is the corresponding multiplicity label. Note that the factors like {R_2}, {R_3}, dim_q R_i are constant and will be canceled out while normalizing |Ψ_3⟩. Also since the 3j-phases in eq.(<ref>) are overall phases, they do not affect the entanglement structure. The state |Ψ_3⟩∈ℋ_1 ⊗ℋ_2, where ℋ_1 and ℋ_2 are the Hilbert spaces associated with the two S^2 boundaries. After tracing out ℋ_2, we get the following reduced density matrix:

    ρ_A = ∑_ℓ=1^∑_r_1,r_1', r_2, r_3 F(t_ℓ, r_1, r_2, p)   F^*(t_ℓ, r_1', r_2,p)/(dim_q t_ℓ)  ⟨Ψ_3|⟩|ϕ_t_ℓ,  r_3 r_1^(1)⟩⟨ϕ_t_ℓ,  r_3 r_1'^(1)| ,

where the trace is given as,

    ⟨Ψ_3|=⟩∑_ℓ=1^∑_r_1, r_2, r_3|F(t_ℓ, r_1, r_2, p)|^2/dim_q t_ℓ = ∑_ℓ=1^∑_r_1, r_2a_3(t_ℓ) |F(t_ℓ, r_1, r_2, p)|^2/dim_q t_ℓ .

We can write the result in a matrix form as following. First let us define A(t_ℓ) whose rows and columns are labeled by r_1 and r_1' respectively. So this will be a square matrix of order a_1(t_ℓ) whose elements are given as,

    A(t_ℓ)_r_1, r_1' = ∑_r_2 F(t_ℓ, r_1, r_2, p)   F^*(t_ℓ, r_1', r_2, p)  .

The summation over r_3 in eq.(<ref>) gives the identity matrix 1 of order a_3(t_ℓ). Thus the summation over various multiplicity labels in eq.(<ref>) can be written as the Kronecker product A(t_ℓ) ⊗1≡ M(t_ℓ). Thus the final matrix can be given in a block diagonal form as,

    ρ_A = 1/⟨Ψ_3|⟩(
    [ M(t_1)/dim_q t_1                                                   ;                  M(t_2)/dim_q t_2                                  ;                                                  ⋱                 ;                                                      M(t_)/dim_q t_;                  ])  .
 
If λ_i(A(t_ℓ)) denotes the eigenvalue of A(t_ℓ), then matrix M(t_ℓ) will also have same eigenvalues where each eigenvalue will have multiplicity a_3(t_ℓ). Thus the eigenvalue corresponding to the block t_ℓ in the reduced density matrix can be given as,

    λ_i(t_ℓ) = λ_i(A(t_ℓ))/⟨Ψ_3|(⟩dim_q t_ℓ)  ;  mul(λ_i(t_ℓ)) = a_3(t_ℓ)  .

Thus the entanglement entropy is given as,

    EE = -∑_ℓ=1^∑_i a_3(t_ℓ) λ_i(t_ℓ) lnλ_i(t_ℓ) .

When there is no multiplicity, i.e. when a_1(t_ℓ) = a_2(t_ℓ) = a_3(t_ℓ) = 1, the entropy will become:

    EE = -∑_ℓ=1^|F(t_ℓ, p)|^2/⟨Ψ_3|(⟩dim_q t_ℓ)ln( |F(t_ℓ, p)|^2/⟨Ψ_3|(⟩dim_q t_ℓ));  ⟨Ψ_3|=⟩∑_ℓ'=1^|F(t_ℓ', p)|^2/dim_q t_ℓ' .





  §.§.§ Periodic entanglement structure

The entanglement structure for the state |Ψ_3⟩ is periodic in the twist number p. Consider the function F given in eq.(<ref>). When p is 0, the exponential factor becomes unit. In such a case, using the identity of the Racah matrix given in eq.(<ref>) and its unitary property in eq.(<ref>),  we get:

    F(t_ℓ, r_1, r_2, 0) = √(dim_q R_1)√(dim_q R_2)/{ R_2} δ_t_ℓ, 0 δ_r_1, 1 δ_r_2, 1 .

This sets t_ℓ=0. Since the trivial representation always occurs once in the tensor decomposition of R ⊗R̅ for any R, all the multiplicities become one and the quantum state of eq.(<ref>), after normalization, becomes a product state (which is consistent with the arguments presented in <cit.>):

    [ < g r a p h i c s > ] = |ϕ_0^(1)⟩⊗ |ϕ_0^(2)⟩ .

Further, we show that the state becomes product state not only at p = 0 but also for non-zero values of p. For example, the exponential factor in eq.(<ref>) becomes unit when p is a multiple of 2N(k+N) and |Ψ_3⟩ will be non-entangled.[The quadratic Casimirs in general have a denominator 2N, see for example eq.(<ref>).] In general, this happens whenever p is a multiple of , where  is an integer which depends on the values of N and k (and also the choice of representations R_1, R_2, R_3, R_4) such that  is a divisor of 2N(k+N). In fact, the entanglement structure repeats itself in a periodic way when we increase p as following: 

    EE(p) = EE(p+ℤ)   ;  EE(p=0) = 0  .

Moreover, there is a symmetry around p= in the sense that:

    EE(p) = EE( - p )  .

We will denote  as `periodicity' which is the fundamental period of the periodic behavior of the entanglement structure. Its value can be obtained on a case by case basis. In the following we compute the entanglement entropy for the fundamental representations of SU(N) showing this behavior and will give the explicit values of .




  §.§.§ Fundamental representation of SU(N)

When all the Wilson lines carry the fundamental representation, each Hilbert space is two dimensional. The representation t_ℓ∈(1)⊗(̅1̅)̅ = ∙⊕adj. We assume k ≥ 2 so that both the representations are integrable. The quantum dimensions and quadratic Casimirs are given as,

    dim_q (∙) = 1,  dim_q (adj) = [N+1][N-1]   ;   C_∙ = 0,   C_adj = N   .

We also need the Racah matrix which can be taken from <cit.>:

    a_ts[
    [    (1) (̅1̅)̅;    (1) (̅1̅)̅ ]] = (
    [                                     ∙                adj;                  ∙             -1/[N] √([N+1] [N-1])/[N];                adj √([N+1] [N-1])/[N]              1/[N] ])  .

The phase factors are given as {(1), (̅1̅)̅, ∙} = 1 and {(1), (̅1̅)̅, adj} = -1. Using various information, we can compute the function F as (ignoring the constant factors C_R̅_1 and C_R_2 in the exponential):

    F(∙,  p)    = -(1 + [N+1][N-1] exp[2π i  pN/k+N]) 1/[N]
    
    F(adj,  p)    = (1 - exp[2π i  pN/k+N]) √([N+1] [N-1])/[N] .
 
The two eigenvalues of the reduced density matrix will be given as,

    λ(ρ_A) = {1-x,  x},    x = 4 sin^2(π p N/k+N)/2 ([N-1] [N+1]-1) cos(2 π p N/k+N)+[N-1]^2 [N+1]^2+3

and the entropy will be:

    EE = -(1-x)ln(1-x) - xln x  .

Interestingly, the entropy is invariant under the exchange N ⟷ k. Moreover, we can see that the spectrum of ρ_A and hence the entropy has a periodic behavior in twist number and the fundamental period is given as[We find that for SU(2), the entanglement entropy vanishes at k=2.]

    = k+N/gcd(k,N) .

Since we are working in a two dimensional Hilbert space, EE≤ln 2. Further, whenever EE = ln 2 (which happens when x=1/2), the state |Ψ_3⟩ will be a maximally entangled Bell state. Doing a numerical check, we find the following values at which this can happen:

    SU(2)_k=4K   with twists p=K, K+1 (mod 2K+1)
    SU(3)_k=3   with twists p=1 (mod 2)
    SU(4M)_k=2   with twists p=M, M+1 (mod 2M+1)  ⟹ |Ψ_3⟩ = |Ψ_Bell⟩ ,

where K≥ 1 and M ≥ 1 are integers. In figure <ref>, we have plotted the variation of entropy for various values of k, N and p. The first three plots show the periodic behavior in entropy as p increases (for fixed values of k and N). We find that whenever the periodicity  is even, the entropy is maximum for p = /2 which is evident from the plots.
 

We also find the asymptotic behavior of the entanglement entropy for large k or large N (with other parameters fixed) values which is given as:

    EE(k ≫ 1) ∼4 p^2 π^2/k^2 N^2(1 - ln[ 4 p^2 π^2/k^2 N^2]  )   ;  EE(N ≫ 1) ∼4 p^2 π^2/k^2 N^2(1 - ln[ 4 p^2 π^2/k^2 N^2]  )  .

From this, we can see that the state is no longer entangled (EE→ 0) as k →∞ or N →∞. The last four plots in figure <ref> show that entropy decreases as k or N increases which is in accordance with the eq.(<ref>).

So far we have considered the braiding between the Wilson lines R_1 and R_2. It is also possible to have horizontal braiding between the Wilson lines R_3 and R_4 (similar to the figure <ref>(a)). We will study the entanglement structure of this kind of state in the next subsection.




 §.§ Quantum state |Ψ_4⟩ on n copies of S^2

In this section, we will analyze the entanglement structure of the state |Ψ_4⟩ on n copies of S^2 as shown in figure <ref>, where the Wilson lines connecting the punctures on one boundary undergo non-trivial braiding with the Wilson lines connecting the punctures on other boundary. We have also shown the state corresponding to n=2 in figure <ref>. The Wilson lines carrying representations R_2j-1 and R_2j are braided 2p_j number of times.

This state can be constructed as given in the appendix A and can be written as:

    |Ψ_4⟩    = ∑_ℓ=1^∑_x_1, x_2, …, x_2n∏_j=1^2n{R_j, R̅_j, t_ℓ, x_j }/(dim_q t_ℓ)^n-1(∏_j=1^n { R_2j}) 
       ×(∏_j=1^n F_j(t_ℓ, x_2j-1, x_2j, p_j) ) |ϕ_t_ℓ;  x_2n, x_1^(1), ϕ_t_ℓ;  x_2, x_3^(2), …, ϕ_t_ℓ;  x_2n-2, x_2n-1^(n)⟩ .

Here the representation t_ℓ∈⋂_j=1^2n (R_j⊗R̅_j) which may appear in the decomposition of (R_j⊗R̅_j) multiple times. We denote this multiplicity as a_j(t_ℓ) = N_R_j R̅_j^t_ℓ. To keep track of this, we have used the label x_j which runs from 1 to a_j(t_ℓ) for a particular representation t_ℓ. The notation { R }≡{ R, R̅, 0} is used for the 2j-phase and the function F is defined as,

    F_j(t_ℓ, x_2j-1, x_2j, p_j)    = ∑_s_j,  y_j{R̅_2j-1, R_2j, s_j, y_j }√(dim_q s_j)   a_s_j,  y_j,y_j^t_ℓ,  x_2j-1, x_2j[
    [  R_2j-1 R̅_2j-1;    R_2j   R̅_2j;         ]] 
       ×exp[2i π  p_j ( C_s_j - C_R̅_2j-1 - C_R_2j)/k+N]  ,

where the summation is over representation s_j ∈ (R̅_2j-1⊗ R_2j) and y_j is the corresponding multiplicity label which keeps track of multiple occurrences of s_j in (R̅_2j-1⊗ R_2j). 







The state |Ψ_4⟩ lives in the total Hilbert space ℋ which can be given as the tensor product ℋ = ⊗_j=1^n ℋ_j,  with ℋ_j denoting the Hilbert space associated with the j^th S^2 with four point conformal block given by representations {R̅_2j-2, R_2j-2, R̅_2j-1, R_2j-1}. The dimension of each Hilbert space will be: dim(ℋ_j) = ∑_ℓ=1^ a_2j-2(t_ℓ)   a_2j-1(t_ℓ). Similar to the previous examples, we bi-partition the total Hilbert space as (ℋ_A|ℋ_B) or (m|n-m) and trace out ℋ_B to get the following reduced density matrix:

    ρ_A     = ∑_ℓ=1^∑_x_1,  x_2, …,  x_2n
     x_1',  x_2', …,  x_2m-1', x_2n'(dim_q t_ℓ)^2-2n/⟨Ψ_4|⟩∏_j=0^m F_j(t_ℓ, x_2j-1, x_2j, p_j)   F_j^*(t_ℓ, x_2j-1', x_2j', p_j) 
       ×∏_j=m+1^n-1 F_j(t_ℓ, x_2j-1, x_2j, p_j)   F_j^*(t_ℓ, x_2j-1, x_2j, p_j)  δ_x_2m,x_2m' δ_x_2n-1,x_2n-1'
       × |ϕ_t_ℓ;  x_0, x_1^(1), ϕ_t_ℓ;  x_2, x_3^(2), …, ϕ_t_ℓ;  x_2m-2, x_2m-1^(m)⟩⟨ϕ_t_ℓ;  x_0', x_1'^(1), ϕ_t_ℓ;  x_2', x_3'^(2), …, ϕ_t_ℓ;  x_2m-2', x_2m-1'^(m) |  ,

where we have used the notations F_0,  x_-1,  x_0 and p_0 to denote F_n,  x_2n-1,  x_2n and p_n respectively and we have defined trace as:

    ⟨Ψ_4|=⟩∑_ℓ=1^  ∑_x_1,  x_2, …,  x_2n∏_j=1^n |F_j(t_ℓ, x_2j-1, x_2j, p_j) |^2/(dim_q t_ℓ)^2n-2 .

We can write ρ_A into a block diagonal form, where each block corresponds to a particular representation t_ℓ:[We should remind again that in the eq.(<ref>), we have only included the blocks which are non-zero. The actual matrix is much bigger having order = dim(ℋ_A).]

    ρ_A = 1/⟨Ψ_4|⟩(
    [ ∏_j=m+1^n-1α_j(t_1)(dim_q t_1)^2n-2 M_t_1                                                                                                                              ;                                           ∏_j=m+1^n-1α_j(t_2)(dim_q t_2)^2n-2 M_t_2                                                                                    ;                                                                                                                             ⋱                                          ;                                                                                                                                  ∏_j=m+1^n-1α_j(t_)(dim_q t_)^2n-2 M_t_ ])  .

Here M_t_1, M_t_2, …, M_t_ are in general matrices and the factor α_j(t_ℓ) is defined as,

    α_j(t_ℓ) = ∑_x_2j-1,  x_2j| F_j(t_ℓ, x_2j-1, x_2j, p_j) |^2  .

The matrix M_t_ℓ for the representation t_ℓ can be written as Kronecker product:

    M_t_ℓ = X(t_ℓ) ⊗ Y_1(t_ℓ) ⊗ Y_2(t_ℓ) ⊗…⊗ Y_m-1(t_ℓ) ⊗ Z(t_ℓ)  .

The elements of X(t_ℓ) and Z(t_ℓ) are given as,

    X_x_2n,  x_2n'   = ∑_x_2n-1 F_n(t_ℓ, x_2n-1, x_2n, p_n)   F_n^*(t_ℓ, x_2n-1, x_2n', p_n) 
    
    Z_x_2m-1,  x_2m-1'   = ∑_x_2m F_m(t_ℓ, x_2m-1, x_2m, p_m)   F_m^*(t_ℓ, x_2m-1', x_2m, p_m)  .
 
The structure of Y_i(t_ℓ) is more complicated which can be written as a block matrix:

    (Y_i)_x_2i-1,  x_2i-1' = (B_i)^x_2i-1,  x_2i-1' ,

where x_2i-1 and x_2i-1' label the row and column of Y_i(t_ℓ) respectively. Each element of Y_i(t_ℓ) is a matrix (B_i)^x_2i-1,  x_2i-1' and thus Y_i(t_ℓ) will take the following form:

    Y_i(t_ℓ) = (
    [            (B_i)^1,1            (B_i)^1,2                    …      (B_i)^1, a_2i-1;            (B_i)^2,1            (B_i)^2,2                    …      (B_i)^2, a_2i-1;                    ⋮                    ⋮                    ⋱                    ⋮;       (B_i)^a_2i-1,1       (B_i)^a_2i-1,2                    … (B_i)^a_2i-1, a_2i-1 ])  .

The rows and columns of (B_i)^α, β are specified by the values of x_2i and x_2i' respectively and its elements are,

    (B_i)^α, β_x_2i,  x_2i' = F_i(t_ℓ, α, x_2i, p_i)   F_i^*(t_ℓ, β, x_2i', p_i)  .

Thus M_t_ℓ will be a square matrix of order:

    order(M_t_ℓ) = ∏_j=0^2m-1 a_j(t_ℓ)  ,

where a_0 = a_2n. For example, if the representation t=t_0 appears in each of (R_j ⊗R̅_j) with multiplicity two, then each multiplicity label will take two values 1 and 2. Thus for m=2 case, the matrix M_t_0 will be a square matrix of order 16 which will be given as,

    M_t_0 = (
    [ X_11 X_12; X_21 X_22;      ]) ⊗(
    [ (
    [ B_11^11 B_12^11; B_21^11 B_22^11;         ]) (
    [ B_11^12 B_12^12; B_21^12 B_22^12;         ]); (
    [ B_11^21 B_12^21; B_21^21 B_22^21;         ]) (
    [ B_11^22 B_12^22; B_21^22 B_22^22;         ]);                                                  ]) ⊗(
    [ Z_11 Z_12; Z_21 Z_22;      ])  .

Having written M_t_ℓ as Kronecker product, its eigenvalues can be computed as:

    λ_i(M_t_ℓ) = λ^X_i_0 λ^Y_1_i_1 λ^Y_2_i_2 … λ^Y_m-1_i_m-1 λ^Z_i_m ,
 
which means that the eigenvalues of M_t_ℓ are all the possible products of various eigenvalues of matrices X, Y_1, …, Y_m-1, Z. Thus the eigenvalues of the reduced density matrix corresponding to the block of representation t_ℓ can be computed as,

    λ_i(t_ℓ) = 1/⟨Ψ_4|⟩(∏_j=m+1^n-1α_j(t_ℓ)(dim_q t_ℓ)^2n-2 λ_i(M_t_ℓ) )  .
 
The entanglement entropy can now be computed:

    EE = -∑_ℓ=1^(∑_i λ_i(t_ℓ) lnλ_i(t_ℓ) ) .
 




  §.§.§ Multiplicity free case

When there is no multiplicity in the representation t_ℓ in any of the fusion (for example, if we are working with SU(2) gauge group), we do not need additional multiplicity labels. In such a case, each of the matrix M_t_ℓ in eq.(<ref>) is simply a number given as,

    M_t_ℓ = ∏_j=0^m F_j(t_ℓ, p_j)   F_j^*(t_ℓ, p_j)  .

Thus the reduced density matrix becomes,

    ρ_A = 1/⟨Ψ_4|⟩(
    [ ∏_j=1^n| F_j(t_1, p_j) |^2(dim_q t_1)^2n-2                                                                                                                                 ;                                            ∏_j=1^n| F_j(t_2, p_j) |^2(dim_q t_2)^2n-2                                                                                      ;                                                                                                                                ⋱                                           ;                                                                                                                                    ∏_j=1^n| F_j(t_, p_j) |^2(dim_q t_)^2n-2 ])  .

This is a diagonal matrix and the eigenvalues, labeled by representation t_ℓ, can be given as:

    λ(t_ℓ) = (dim_q t_ℓ)^2-2n∏_j=1^n| F_j(t_ℓ, p_j) |^2/∑_ℓ'=1^( (dim_q t_ℓ')^2-2n∏_j=1^n| F_j(t_ℓ', p_j) |^2 )

and the entanglement entropy will be,

    EE = -∑_ℓ=1^λ(t_ℓ) lnλ(t_ℓ)  .





  §.§.§ Separability criteria of the reduced density matrix

Let us study the reduced density matrix in eq.(<ref>) which acts on the Hilbert space ℋ_A to get more insight into the entanglement structure of the state |Ψ_4⟩. We want to see if this reduced density matrix is separable or not for which we would like to calculate entanglement negativity. Consider the further bi-partitioning of the Hilbert space ℋ_A into two parts:

    ℋ_A = ℋ_A_1⊗ℋ_A_2 = (⊗_i=1^rℋ_i )  ⊗(⊗_i=r+1^mℋ_i )  .

The partial transpose of the reduced density matrix with respect to ℋ_A_2 can be given as,

    ρ^Γ   = 1/⟨Ψ_4|⟩(
    [ ∏_j=m+1^n-1α_j(t_1)(dim_q t_1)^2n-2 M_t_1^Γ                                                                                                                                    ;                                             ∏_j=m+1^n-1α_j(t_2)(dim_q t_2)^2n-2 M_t_2^Γ                                                                                        ;                                                                                                                                   ⋱                                            ;                                                                                                                                        ∏_j=m+1^n-1α_j(t_)(dim_q t_)^2n-2 M_t_^Γ ])  ,

where M_t_ℓ^Γ denotes the partial transpose of M_t_ℓ. To see how M_t_ℓ^Γ differs from M_t_ℓ, we need to find which multiplicity labels (corresponding to t_ℓ) are exchanged while partial transposing. In the following, we have explicitly shown the breaking of the multiplicity labels of eq.(<ref>) on the Hilbert spaces ℋ_A_1 and ℋ_A_2: 

    |  x_0  x_1,  x_2  x_3, …,  x_2r-2 x_2r-1_A_1,  x_2r  x_2r+1, …,  x_2m-2  x_2m-1_A_2  ⟩
       ⟨  x_0'  x_1',  x_2'  x_3', …, x_2r-2' x_2r-1'_A_1,  x_2r'  x_2r+1', …,  x_2m-2'  x_2m-1'_A_2   |  .
 
For the partial transpose with respect to ℋ_A_2, we need to swap the basis labels corresponding to ℋ_A_2 in the `ket' and `bra', i.e. x_2r↔ x_2r',  x_2r+1↔ x_2r+1', …,  x_2m-1↔ x_2m-1'. In the process of partial transpose, the matrices which are constructed from the basis labels of only ℋ_A_1 will be unaffected, while those matrices which are constructed from the basis labels of only ℋ_A_2 will be transposed. Only those matrices will transform non-trivially which are made up from the labels of both ℋ_A_1 and ℋ_A_2. In eq.(<ref>), we have given M_t_ℓ as Kronecker product of various matrices. Thus it is easy to see that its partial transpose will be given as,

    M_t_ℓ^Γ = X ⊗ Y_1 ⊗…⊗ Y_r-1⊗ Y_r^Γ⊗ Y_r+1^T ⊗…⊗ Y_m-1^T ⊗ Z^T  ,

where the superscript T denotes the usual transpose of a matrix which does not change its eigenvalues. The matrix Y_r^Γ can be obtained from Y_r as,

    Y_r^Γ(t_ℓ) = (
    [            [(B_r)^1,1]^T            [(B_r)^1,2]^T                        …      [(B_r)^1, a_2r-1]^T;            [(B_r)^2,1]^T            [(B_r)^2,2]^T                        …      [(B_r)^2, a_2r-1]^T;                        ⋮                        ⋮                        ⋱                        ⋮;       [(B_r)^a_2r-1,1]^T       [(B_r)^a_2r-1,2]^T                        … [(B_r)^a_2r-1, a_2r-1]^T ])  ,

where each block of Y_r has been transposed. If the matrix Y_r^Γ(t_ℓ) has negative eigenvalue for any t_ℓ, we get a non-zero negativity and the reduced density matrix is non-separable. In fact, we will give an example in later section which shows that the negativity is non-zero but currently we do not have a proof whether Y_r^Γ(t_ℓ) will always have negative eigenvalues. On the other hand, when a_2r(t_ℓ) = 1, each of the (B_r)^i,j is just a number and thus Y_r^Γ(t_ℓ) = Y_r(t_ℓ). In such a case, there are no negative eigenvalues of Y_r^Γ(t_ℓ) and the entanglement negativity vanishes. Moreover, when the multiplicities in the decomposition of the tensor product of representations for the conformal blocks associated with the remaining S^2 boundaries 1,2,…,m are unit, i.e. when a_0(t_ℓ) = a_1(t_ℓ) = …= a_2m-1(t_ℓ) = 1 for all t_ℓ, the reduced density matrix can be written as,

    ρ = ∑_ℓ=1^g(t_ℓ) ∏_j=m+1^n-1α_j(t_ℓ)⟨Ψ_4|(⟩dim_q t_ℓ)^2n-2  ρ_t_ℓ^A_1⊗ρ_t_ℓ^A_2 ,

where

    g(t_ℓ) = (∑_x_2m| F_m(t_ℓ,x_2m, p_m) |^2 )   (∑_x_2n-1| F_n(t_ℓ,x_2n-1, p_m) |^2 )

and ρ_t_ℓ^A_1 and ρ_t_ℓ^A_2 are the density matrices acting on Hilbert spaces ℋ_A_1 and ℋ_A_2 respectively which correspond to the pure states:

    ρ_t_ℓ^A_1   = |t_ℓ^(1), t_ℓ^(2), …, t_ℓ^(r)⟩⟨t_ℓ^(1), t_ℓ^(2), …, t_ℓ^(r) | 
    ρ_t_ℓ^A_2   = |t_ℓ^(r+1), t_ℓ^(r+2), …, t_ℓ^(m)⟩⟨t_ℓ^(r+1), t_ℓ^(r+2), …, t_ℓ^(m) |  .

One can check that the coefficients are positive and add up to 1:

    ⟨Ψ_4|=⟩∑_ℓ'=1^( g(t_ℓ') ∏_j=m+1^n-1α_j(t_ℓ')/(dim_q t_ℓ')^2n-2)   ⟹  ∑_ℓ=1^( g(t_ℓ) ∏_j=m+1^n-1α_j(t_ℓ)⟨Ψ_4|(⟩dim_q t_ℓ)^2n-2) = 1  .

Thus the reduced density matrix is separable. In summary, we can say the following which is similar to our proposition in eq.(<ref>):
[box=]align
  a_0(t_ℓ) = a_1(t_ℓ) = …= a_2m-1(t_ℓ) = 1,    ∀t_ℓ     ⟹  ρ= separable 

	a_2r(t_ℓ) > 1,    for some  t_ℓ  ?⟹   𝒩 ≠0      ⟹  ρ= non-separable
	

Note that we have put a `?' on one of the arrows because we do not have a formal proof for a generic case. In the later section, we will study an example which gives non-zero negativity and supports this statement. Thus, once again we see that indeed the multiplicity plays a crucial role in determining whether the state |Ψ_4⟩ has a GHZ-like or W-like entanglement structure.

Just like the state |Ψ_3⟩, the entanglement structure of the state |Ψ_4⟩ also shows a periodic behavior as we increase the number of twists in the braiding, which we will discuss in the following.




  §.§.§ Periodic entanglement structure

Here we will show the periodic behavior of entanglement structure for the state |Ψ_4⟩. Consider the function F_j given in eq.(<ref>). Following the previous arguments, when p_j is 0 for some value of j, the exponential factor becomes unit and the function F_j becomes:

    F_j = √(dim_q R_2j-1)√(dim_q R_2j)/{ R_2j} δ_t_ℓ, 0 δ_x_2j-1, 1 δ_x_2j, 1 .

This sets t_ℓ=0 and hence all the multiplicities become one. In such a case, the quantum state of eq.(<ref>), after normalization, will be given as

    |Ψ_4⟩ = |ϕ_0^(1), ϕ_0^(2), …, ϕ_0^(n)⟩

which is a non-entangled state and the entanglement entropy (as well as negativity) vanishes. Specifically for n=2, this analysis tells that the following states are non-entangled states:

    [ < g r a p h i c s > ] = [ < g r a p h i c s > ] = [ < g r a p h i c s > ] = |ϕ_0^(1)⟩⊗ |ϕ_0^(2)⟩ .

Further, the state becomes product state for non-zero values of p_j. For example, the exponential factor in eq.(<ref>) becomes unit when p_j is a multiple of 2N(k+N) and |Ψ_4⟩ will be non-entangled. In general, this happens whenever p_j is a multiple of  such that  is a divisor of 2N(k+N). The entanglement structure repeats itself in a periodic way when we increase any of the twist number as following: 

EE(p_j)    = EE(p_j+ℤ)      ;  EE(p_j=0)    = 0 
 
𝒩(p_j)    = 𝒩(p_j+ℤ)      ;  𝒩(p_j=0)    = 0  .


Further, there is a symmetry around p_j=:

    EE(p_j) = EE( - p_j )   ;  𝒩(p_j) = 𝒩( - p_j )  .

We denote  as `periodicity' which is the fundamental period of this periodic behavior of the entanglement structure. In the following sections, we will compute the entanglement entropy and negativity for various examples showing this behavior and give the explicit values of .




  §.§.§ Wilson lines carrying symmetric representations of SU(N)

In this section we will study the entanglement structure when all the representations are symmetric representations of SU(N). Since we will be dealing with symmetric representations and their conjugates, we will use the following notation to write them:  

    a = a Λ_1 ≡ [a;0]   ;  b = b Λ_N-1≡ [0;b] 
       ba = a Λ_1 + b Λ_N-1≡ [a;b]  ,
 
where Λ_1, Λ_2, …, Λ_N-1 are the fundamental weights of SU(N). The notation [a;0] will denote a symmetric representation whose Young diagram has one row with a number of boxes. Its conjugate is denoted as [0;a] whose Young tableau is given in the above equation where centertableaux, boxsize=1.1em∙ represents a vertical row with (N-1) boxes. 
From the group theory we know the following decomposition, which will be helpful in the computation:

    a ≤ b :      [a;0] ⊗ [0;b] = ⊕_ℓ=0^a   [ℓ;b-a+ℓ] 
    
    a ≥ b :      [a;0] ⊗ [0;b] = ⊕_ℓ=0^b   [a-b+ℓ;ℓ]  .

The quantum dimension and the quadratic Casimir of the representation [x;y] is given as,

    dim_q [x;y]    = [N+x-2]! [N+y-2]! [N+x+y-1]/[x]! [y]! [N-1]! [N-2]!
    
    C_[x;y]   = (N-1) (Nx+Ny+x^2+y^2)+2 x y/2 N .

Since the decomposition of the tensor product of a symmetric representation and its conjugate does not give representations with multiple occurrences, we are dealing with a multiplicity free case and we can omit all the multiplicity labels. The 3j-phase for a representation t_ℓ∈ [a;0] ⊗ [0;b] can be taken as <cit.>:

    {[a;0]  ,  [0;b]  ,  t_ℓ} = (-1)^a-b+ ℓ ,

where t_ℓ is either [a-b+ℓ;ℓ] or [ℓ;b-a+ℓ] depending on whether a ≥ b or a ≤ b. 

Let us first consider the simplest case where all the representations R_j are fundamental representations of SU(N), i.e.,

    R_1 = R_2 = … = R_2n= (1)≡ [1;0]  .

Both representations t_ℓ and s_ℓ take two values:

    t_ℓ, s_ℓ∈ (R ⊗R̅) = [0;0] ⊕ [1;1]  .

We assume k ≥ 2 so that both the representations are integrable. The Racah matrix required for this computation is given in eq.(<ref>). Using various information, we can compute the function F_j as:

    F_j(t_0,p_j)    = -exp(2 i π  p_j/k N+N^2) ( exp(-2 i π  N p_j/k+N) + [N+1][N-1]/[N]) 
    
    F_j(t_1,p_j)    = -exp(2 i π  p_j/k N+N^2) (1-exp(-2 i π  N p_j/k+N) ) √([N+1] [N-1])/[N] .
 
The total Hilbert space is 2^n dimensional and we break it into two parts as ℋ = ℋ_A ⊗ℋ_B corresponding to the bi-partition (m|n-m). Tracing out ℋ_B assigns an entanglement structure to the state |Ψ_4⟩ and we get the following entanglement entropy:

    EE   = (∏_j=1^nf_p_j)/(∏_j=1^n f_p_j) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j)ln((∏_j=1^n f_p_j) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j)/(∏_j=1^nf_p_j)) 
       + [N-1]^2[N+1]^2(∏_j=1^n g_p_j)/(∏_j=1^n f_p_j) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j)ln((∏_j=1^n f_p_j) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j)/[N-1]^2[N+1]^2(∏_j=1^n g_p_j)),

where we have defined the following:

    f_p    = [N+1]^3[N-1]^3 + 2cos(2 π  N p/k+N)[N+1]^2[N-1]^2 + [N+1][N-1] 
    
    g_p    = 4sin ^2(π  N p/k+N) .

The entanglement structure is periodic in any of the twist number p_j with a fundamental period  i.e. 

    EE(p_j) = EE(p_j+ℤ)   ;  EE(0)=0   ,

where the period is given as,[For SU(2), the entanglement entropy vanishes at k=2.]

    = k+N/gcd(k,N) .


This behavior can be seen in the first three plots of figure <ref>. The last four plots show that the entropy remains same under the exchange N⟷ k just like the earlier case of the state |Ψ_3⟩. This can be explicitly seen from the eq.(<ref>) noting that f_p and g_p are invariant under N⟷ k. Since the reduced density matrix has only two non-zero eigenvalues, EE≤ln 2. We also did the numerical analysis to find the values of N, k and p_j at which the entropy is maximum (ln 2). We find that it is possible only for n = 2 in which case the state will be maximally entangled Bell state. The values at which this can happen are given below:

    SU(2)_k=4K   with twists p_j=K, K+1 (mod 2K+1)
    SU(3)_k=3   with twists p_j=1 (mod 2)
    SU(4M)_k=2   with twists p_j=M, M+1 (mod 2M+1)  ⟹ |Ψ_4^(n=2)⟩ = |Ψ_Bell⟩ ,

where K≥ 1 and M ≥ 1 are integers and the twist values are given modulo the periodicity. Note that these are the same values at which the state |Ψ_3⟩ was found to be a Bell state (see eq.(<ref>)).

For large values of k or N (with other parameters fixed), the asymptotic behavior of the entanglement entropy is given as,

    EE(k ≫ 1)    ∼4^n π^2n(∏_j=1^n p_j^2)/k^2n N^2n(N^2-1)^n-2(1 - ln[ 4^n π^2n(∏_j=1^n p_j^2)/k^2n N^2n(N^2-1)^n-2]  ) 
    EE(N ≫ 1)    ∼4^n π^2n(∏_j=1^n p_j^2)/k^2n N^2n(k^2-1)^n-2(1 - ln[ 4^n π^2n(∏_j=1^n p_j^2)/k^2n N^2n(k^2-1)^n-2]  )

and thus the state is no longer entangled (EE→ 0) as k →∞ or N →∞. 



Next consider the case where the Wilson lines carrying fundamental representation braid with the Wilson lines carrying symmetric representation, i.e. our choice is:

    R_1 = R_3 = … = R_2n-1= [1;0]   ;   R_2 = R_4 = … = R_2n= [a;0]  ,

where a≥ 1 with a=1 corresponding to the fundamental representation which has been already discussed above. The Hilbert space in this case is still two dimensional with the representations t_ℓ and s_ℓ taking the following values:

    t_ℓ   ∈ (R_2j-1⊗R̅_2j-1) ∩ (R_2j⊗R̅_2j) = [0;0] ⊕ [1;1] 
    
    s_ℓ   ∈ (R̅_2j-1⊗ R_2j) = [a-1;0] ⊕ [a;1]  .

Moreover since we want all the representations appearing in the tensor product to be integrable, we must have k ≥ (a+1). The required fusion matrix for our computation can be taken from <cit.>:

    a_ts[
    [  R_2j-1 R̅_2j-1;    R_2j   R̅_2j ]] = (
    [                                           s_0=[a-1;0]                  s_1=[a;1];                  t_0=[0;0]        -√([a]/[N] [N+a-1]) √([N+a] [N-1]/[N] [N+a-1]);                  t_1=[1;1] √([N+a] [N-1]/[N] [N+a-1])         √([a]/[N] [N+a-1]) ])  .

The entropy for fundamental representation given in eq.(<ref>) can be generalized for this case and the result is given as,

    EE   = (∏_j=1^nf_p_j^(a))/(∏_j=1^n f_p_j^(a)) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j^(a))ln((∏_j=1^n f_p_j^(a)) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j^(a))/(∏_j=1^nf_p_j^(a))) 
       + [N-1]^2[N+1]^2(∏_j=1^n g_p_j^(a))/(∏_j=1^n f_p_j^(a)) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j^(a))ln((∏_j=1^n f_p_j^(a)) + [N-1]^2[N+1]^2(∏_j=1^n g_p_j^(a))/[N-1]^2[N+1]^2(∏_j=1^n g_p_j^(a))),

where we have defined the following:

    f_p^(a)   = [N+a]^2[N+1][N-1]^3 + 2cos(2 π p(N+a-1)/k+N)[N+a][N+1][N-1]^2[a] + [N+1][N-1][a]^2 
    
    g_p^(a)   = 4[N+a][a]/[N+1]sin ^2(π  p (N+a-1)/k+N) .

Again the entanglement structure is periodic in any of the twist number p_j with a fundamental period  with the following properties  

    EE(p_j) = EE(p_j+ℤ)   ;  EE(0)=0   ;  EE(p_j) = EE(-p_j)

where the period is given as,

    = k+N/gcd(k-a+1, N+a-1) .

For fixed values of p_j, the asymptotic behavior of the entropy for large k (finite N) and large N (finite k) are given below:

    EE(k ≫ 1)    ∼4^n π^2n a^n (N+a)^n  (∏_j=1^n p_j^2)/k^2n N^2n(N-1)^n-2(N+1)^2n-2(1 - ln[ 4^n π^2n a^n (N+a)^n  (∏_j=1^n p_j^2)/k^2n N^2n(N-1)^n-2(N+1)^2n-2]  ) 
    EE(N ≫ 1)    ∼4^n π^2n a^n (k-a)^n  (∏_j=1^n p_j^2)/k^2n N^2n(k+1)^n-2(k-1)^2n-2(1 - ln[ 4^n π^2n a^n (k-a)^n  (∏_j=1^n p_j^2)/k^2n N^2n(k+1)^n-2(k-1)^2n-2]  )  .

Just like the earlier case, we find that the entropy may obtain its maximum value ln 2 only when n=2 and this can happen for the following values of a, N, k and p_j:

   SU(2)_k=4K      : a=1    &    a=(4K-1)        ;  p_j=K    &    K+1   (mod  ) 

   SU(3)_k=2K+1      : a=K      ;  p_j=1   (mod  )   

   SU(4M)_k=4K-2      : a=(4K-3)      ;  p_j=(K+M-1)    &    (K+M)   (mod  )  ,


where K≥ 1 and M ≥ 1 are integers and the twist numbers are given modulo appropriate periodicity values . Thus these values of the parameters will describe the state |Ψ_4⟩ as a Bell state.

For the examples in this section, we have restricted to the symmetric representations of SU(N), the tensor decompositions of which are always multiplicity free. Unfortunately the Racah matrices are not known in general and we are limited in our choice of representations. In the next example, we will consider the (2,1) representation of SU(N), the Racah matrices of which were obtained in <cit.>. In this case, we can get t_ℓ with multiplicity which enables us to show the explicit results of entanglement negativity.




  §.§.§ Wilson lines carrying mixed representation of SU(N)

Let us consider the case when all the Wilson lines carry (2,1) = (2,1) representation of SU(N), i.e. we will set

    R_1 = R_2 = … = R_2n = R = (2,1) .

In this case, the representations t_ℓ, s_ℓ∈ (R ⊗R̅) which is given as:

    R ⊗R̅ = t_1 ⊕ t_2 ⊕ t_3 ⊕ t_4 ⊕ t_5 ⊕ t_6 ⊕ 2 t_7  ,

where various representations are given as following:

t_1 = ∙,   t_2    = (2^2, 1^N-4),      t_3    = (3, 1^N-3),      t_4    = (3^2, 2^N-3) 

t_5    = (4, 2^N-2),     t_6    = (4, 3, 2^N-4, 1),     t_7    = (2, 1^N-2)  .

The representation t_7 occurs with multiplicity 2 while others appear once. The Racah matrix elements are given in the appendix B. The multiplicity labels will be omitted for t_1, t_2, t_3, t_4, t_5, t_6 since they occur only once. The 3j-phases required for the computation can be taken as following:

{R̅, R, t_1}    = 1,      {R̅, R, t_2}    = 1,      {R̅, R, t_3}    = -1,      {R̅, R, t_4}    = -1 

{R̅, R, t_5}    = 1,     {R̅, R, t_6}    = 1,      {R̅, R, t_7, 1}    = 1,      {R̅, R, t_7, 2}    = -1  .

The quantum dimensions of various representations are:

dim_q t_1 = 1,   dim_q t_2    = [N-3][N]^2[N+1]/[2]^2,      dim_q t_3    = dim_q t_4 = [N-2][N-1][N+1][N+2]/[2]^2, 

dim_q t_5    = [N-1][N]^2[N+3]/[2]^2,      dim_q t_6    = [N-3][N-1]^2[N+1]^2[N+3]/[3]^2, 

dim_q t_7    = [N-1][N+1]  .

We also require the quadratic Casimirs:

    C_t_1 = 0,  C_t_2 = 2(N-1),  C_t_3 = 2N,  C_t_4 = 2N,  C_t_5 = 2(N+1),  C_t_6 = 3N,  C_t_7 = N  .

The function F in the eq.(<ref>) for various representations will be given as,

    F_j(t_y, p_j)    = ∑_ℓ=1^6 {R̅, R, t_ℓ}√(dim_q t_ℓ) ℛ_ℓ y  Ω_j(t_ℓ) + √(dim_q t_7)(ℛ_7y-ℛ_8y) Ω_j(t_7) 
    
    F_j(t_7,1,1,p_j)    = ∑_ℓ=1^6 {R̅, R, t_ℓ}√(dim_q t_ℓ) ℛ_ℓ 7  Ω_j(t_ℓ) + √(dim_q t_7) (ℛ_77-ℛ_87) Ω_j(t_7) 
    
    F_j(t_7,2,2,p_j)    = ∑_ℓ=1^6 {R̅, R, t_ℓ}√(dim_q t_ℓ) ℛ_ℓ 8  Ω_j(t_ℓ) + √(dim_q t_7) (ℛ_78-ℛ_88) Ω_j(t_7) 
    
    F_j(t_7,1,2,p_j)    = ∑_ℓ=1^6 {R̅, R, t_ℓ}√(dim_q t_ℓ) ℛ_ℓ 9  Ω_j(t_ℓ) + √(dim_q t_7) (ℛ_79-ℛ_89) Ω_j(t_7) 
    
    F_j(t_7,2,1,p_j)    = ∑_ℓ=1^6 {R̅, R, t_ℓ}√(dim_q t_ℓ) ℛ_ℓ,10 Ω_j(t_ℓ) + √(dim_q t_7) (ℛ_7,10-ℛ_8,10) Ω_j(t_7) ,

where t_y in the first equation can take values t_y = t_1, t_2, t_3, t_4, t_5, t_6. The ℛ_ij denotes the element of the Racah matrix ℛ given in the appendix B in eq.(<ref>) and we have defined[We have discarded the terms Ω_j(R)^-1 and Ω_j(R̅)^-1 in F_j because these are overall factors and will be canceled out while computing F_jF_j^*.]:

    Ω_j(t) = exp[2 iπ  p_j  C_t/k+N]  .

Each of the Hilbert space in this case is ten dimensional and hence the total Hilbert space has dimension: dim(ℋ) = 10^n. We divide this Hilbert space into ℋ_A ⊗ℋ_B corresponding to the bi-partition (m|n-m). Tracing out ℋ_B gives the reduced density matrix acting on ℋ_A (dim(ℋ_A) = 10^m) whose block diagonal form is given as:

    ρ_A = 1/⟨Ψ_4|⟩(
    [   ∏_j=1^n| F_j(t_1, p_j) |^2(dim_q t_1)^2n-2                                                                                                                                                                                    ;                                                                                         ⋱                                                                                                                                       ;                                                                                                                                        ∏_j=1^n| F_j(t_6, p_j) |^2(dim_q   t_6)^2n-2                                             ;                                                                                                                                                                                           ∏_j=m+1^n-1α_j(t_7)(dim_q  t_7)^2n-2 M ])  ,

where 

    α_j(t_7) = | F_j(t_7, 1, 1, p_j) |^2 + | F_j(t_7, 1, 2, p_j) |^2 + | F_j(t_7, 2, 1, p_j) |^2 + | F_j(t_7, 2, 2, p_j) |^2

and M is a square matrix of order 2^2m given as Kronecker product:[Note that since dim(ℋ_A) = 10^m, the density matrix ρ_A is also of the order 10^m. However many of its blocks are 0 and we have only written the non-zero blocks in eq.(<ref>).]

    M = X ⊗ Y_1 ⊗ Y_2 ⊗…⊗ Y_m-1⊗ Z  .

Here X and Z are 2 × 2 matrices given as,

    X    = (
    [                  ∑_i=1^2 | F_n(t_7, i, 1, p_n) |^2 ∑_i=1^2 F_n(t_7, i, 1, p_n)  F_n^*(t_7, i, 2, p_n); ∑_i=1^2 F_n(t_7, i, 2, p_n)  F_n^*(t_7, i, 1, p_n)                  ∑_i=1^2 | F_n(t_7, i, 2, p_n) |^2 ]) 
     
    Z    = (
    [                  ∑_i=1^2 | F_m(t_7, 1, i, p_m) |^2 ∑_i=1^2 F_m(t_7, 1, i, p_m)  F_m^*(t_7, 2, i, p_m); ∑_i=1^2 F_m(t_7, 2, i, p_n)  F_m^*(t_7, 1, i, p_m)                  ∑_i=1^2 | F_m(t_7, 2, i, p_m) |^2 ])  .
 
Also the matrix Y_i is a 4× 4 matrix:

    Y_i = (
    [ F_i^*(t_7, 1, 1, p_i) F_i^*(t_7, 1, 2, p_i) F_i^*(t_7, 2, 1, p_i) F_i^*(t_7, 2, 2, p_i);                       ]) ⊗(
    [ F_i(t_7, 1, 1, p_i); F_i(t_7, 1, 2, p_i); F_i(t_7, 2, 1, p_i); F_i(t_7, 2, 2, p_i);                     ])  
     .




Let us first do the computation for n=2 with the bi-partition (1|1), i.e. m=1. The reduced density matrix in this case will be,

    ρ_A = 1/⟨Ψ_4|⟩(
    [ | F_1(t_1, p_1) |^2 | F_2(t_1, p_2) |^2(dim_q t_1)^2                                                                                                                                                                                                                    ;                                                                                                         ⋱                                                                                                                                                               ;                                                                                                                                                                | F_1(t_6, p_1) |^2 | F_2(t_6, p_2) |^2(dim_q t_6)^2                                                     ;                                                                                                                                                                                                                                                      X ⊗ Z(dim_q  t_7)^2 ])  ,

where X and Z can be obtained from eq.(<ref>) by setting m=1 and n=2. The entanglement structure will be invariant under the exchange p_1 ↔ p_2 and is periodic in p_1 or p_2 with properties:

    EE_A(p_j) = EE_A(p_j+ℤ)   ;  EE_A(0)=0   ;  EE_A(p_j) = EE_A(-p_j)  ,

where the period is given as,

    = 
    (k+N),   for N= odd
    
    (k+N),   for N= even & gcd(k,N)=1
    (k+N)2,   for N= even & gcd(k,N)>1 .

This periodic behavior can be seen from the first two plots of figure <ref>. We have also given various other plots in figure <ref> showing the variation of SU(N)_k entanglement entropy as a function of k for fixed N and vice versa. We can see that EE→ 0 as k →∞ or N →∞ and the state |Ψ_4^(n=2)⟩ becomes a non-entangled state in this limit.


Next, let us study the entanglement structure of the state |Ψ_4^(n=3)⟩. The total Hilbert space is ℋ = ℋ_1 ⊗ℋ_2 ⊗ℋ_3 with dim(ℋ) = 1000 where each of ℋ_i is the space associated with S^2 with four point conformal block carrying representations {R̅, R, R̅, R } with R = (2,1). There are two inequivalent ways of tracing out Hilbert spaces. In the first case, we trace out ℋ_2 ⊗ℋ_3 which correspond to m=1. The reduced density matrices can be obtained from the general form given in eq.(<ref>) by substituting n=3 and m=1. The entanglement entropy has the same periodicity and follows the same pattern as in the case of (1|1) bi-partition for the n=2 case. The only difference is that the entanglement entropies in this case are smaller than the corresponding eigenvalues of the (1|1) bi-partition. This is in fact a general trend, where the entanglement entropy for (1|n-1) bi-partition decreases as n increases (keeping other parameters fixed).

The second way of tracing out the Hilbert space is to trace out only ℋ_3 which correspond to m=2 and gives a reduced density matrix defined on ℋ_1 ⊗ℋ_2 which is:[For SU(3), the representations t_2 and t_6 are no longer present in the tensor decomposition of R ⊗R̅ and we must set the corresponding blocks to 0 in eq.(<ref>).]

    ρ = 1/⟨Ψ_4|⟩(
    [ ∏_j=1^3| F_j(t_1, p_j) |^2(dim_q t_1)^4                                                                                                                                                                ;                                                                               ⋱                                                                                                                        ;                                                                                                                         ∏_j=1^3| F_j(t_6, p_j) |^2(dim_q t_6)^4                                        ;                                                                                                                                                                               X ⊗ Y_1 ⊗ Z(dim_q  t_7)^4 ])  .

To keep the analysis simple, let us set p_1 = p_2 = p_3 = p. The explicit form of matrices X, Y_1 and Z can be obtained from eq.(<ref>) and eq.(<ref>) by setting m=2 and n=3 where we note that F_1 = F_2 = F_3 = F since we have set p_1 = p_2 = p_3. We are also interested in computing the entanglement negativity. The partial transpose of ρ with respect to ℋ_2 can be obtained from the general form given in eq.(<ref>). The matrices X and Z for this case are given as,

    X = (
    [ ∑_i=1^2 F_i1 F_i1^* ∑_i=1^2 F_i1 F_i2^*; ∑_i=1^2 F_i2 F_i1^* ∑_i=1^2 F_i2 F_i2^* ])   ;   Z = (
    [ ∑_i=1^2 F_1i F_1i^* ∑_i=1^2 F_1i F_2i^*; ∑_i=1^2 F_2i F_1i^* ∑_i=1^2 F_2i F_2i^* ])  .

Also the matrix Y_1 and its partial transpose are:

    Y_1 = (
    [ F_11F_11^* F_11F_12^* F_11F_21^* F_11F_22^*; F_12F_11^* F_12F_12^* F_12F_21^* F_12F_22^*; F_21F_11^* F_21F_12^* F_21F_21^* F_21F_22^*; F_22F_11^* F_22F_12^* F_22F_21^* F_22F_22^*;            ]), Y_1^Γ = (
    [ F_11F_11^* F_12F_11^* F_11F_21^* F_12F_21^*; F_11F_12^* F_12F_12^* F_11F_22^* F_12F_22^*; F_21F_11^* F_22F_11^* F_21F_21^* F_22F_21^*; F_21F_12^* F_22F_12^* F_21F_22^* F_22F_22^*;            ])  ,
 
where we have defined F_αβ≡ F(t_7, α, β, p) which are given in eq.(<ref>). Both the entanglement entropy and negativity are periodic in twist number p:

EE_A(p)    = EE_A(p_j+ℤ)      ;  EE_A(0)    = 0      ;  EE_A(p)    = EE_A(-p) 

𝒩(p)    = 𝒩(p+ℤ)      ;  𝒩(0)    = 0      ;  𝒩(p)    = 𝒩(-p)  ,


where the period  is the same as given in eq.(<ref>). 

We have given some plots in figure <ref> showing the variation of SU(N)_k entanglement entropy and entanglement negativity as a function of k for fixed N and vice versa. We can see that the entanglement negativity is non-zero. Thus the reduced density matrix is non-separable on ℋ_1 ⊗ℋ_2 and the state |Ψ_4^(n=3)⟩ has a W-like entanglement structure.




§ CONCLUSION AND DISCUSSION


The theme of this paper was to understand the entanglement structure between n disjoint S^2 boundaries of a three-manifold described by SU(N) Chern-Simons theory. We have focused on S^2 with four punctures corresponding to the four point conformal blocks carrying the integrable representations of SU(N) gauge group with Chern-Simons level k. The punctures are connected through the Wilson lines embedded in the bulk of the three-manifold. The quantum states considered in this work live in the total Hilbert space which is a tensor product of the Hilbert spaces associated with each S^2. The states are labeled by those representations `t' of SU(N) which appear in the fusion of the representations associated with various conformal blocks. Further, since t may appear multiple times in the fusion, the states also carry extra labels to keep track of the multiple occurrences of t. In fact, the multiplicity of t plays a very crucial role in determining the entanglement structure of these quantum states. When there is no multiplicity, the states have a GHZ-like entanglement structure, i.e., the reduced density matrices (obtained by tracing one or more ℋ_S^2) are fully separable. We proved this by decomposing the reduced density matrix into a convex combination of pure product states (see eqn(<ref>) and eqn(<ref>)). On the other hand, when there is multiplicity, we show that the entanglement negativity is non-zero for the states |Ψ_1⟩ and |Ψ_2⟩ (eq.(<ref>) and eq.(<ref>) respectively). The entanglement negativity for the state |Ψ_4⟩ for n=3 is worked out for the (2,1) representations placed on the Wilson lines and is shown to be non-zero in the plots of figure <ref>. This exercise shows that the multiplicity is responsible for the  reduced density matrix to be non-separable, i.e. entangled and the states will have a W-like entanglement structure. 

The entanglement structure of the states |Ψ_1⟩ and |Ψ_2⟩ remains invariant under the braiding between the Wilson lines. We explicitly show that any braiding can be seen as a unitary transformation on the basis states and does not change the entropy. Our computation for general n is consistent with the diagrammatic replica method described in <cit.> for n=2. We also consider the two boundary states |Ψ_3⟩ and |Ψ_4⟩ which have necklace braiding in which the Wilson lines connecting the punctures of j^th boundary braid 2p_j times (i.e. p_j number of full twists) with the Wilson lines connecting the punctures of (j+1)^th boundary. This type of braiding affects the entanglement structure and we give various examples showing its effect on the entropy. In fact we have generalized |Ψ_4⟩ to n boundaries and gave various plots showing the variation of entropy with n in figure <ref>. We find that these states have a periodic structure in any of the twist number p_j, i.e. the spectrum of the reduced density matrix remains same if we increase p_j to (p_j + ℤ).  Moreover when any of the p_j is 0 or a multiple of , the state becomes a non-entangled state and the entropy vanishes. We call the integer  as `periodicity' which is the fundamental period of this periodic behavior and its value depends on N and k. We give various examples showing this periodic behavior in the entropy and explicitly obtained the values of .

For two boundaries (n=2), we find that the states |Ψ_1⟩ and |Ψ_2⟩ are Bell states (which is in agreement with <cit.>). Further we show that it may be possible for the states |Ψ_3⟩ and |Ψ_4⟩ to be the Bell states for particular values of N and k as listed in eq.(<ref>) and eq.(<ref>) for symmetric representations. For n ≥ 3 boundaries, we find that it is possible for the states |Ψ_1⟩ and |Ψ_2⟩ to be maximally entangled GHZ states for a specific choice of rank and level of the gauge group as given in eq.(<ref>). For example, when all the Wilson lines carry fundamental representation, |Ψ_1⟩ will be a GHZ state for SU(N)_k=N and |Ψ_2⟩ will be a GHZ state for SU(2)_k=2.

For each of the examples considered in this paper, we also analyzed the large k and large N behavior of the entanglement entropies. For the states |Ψ_1⟩ and |Ψ_2⟩, the large k limit of the entropy can be obtained by simply replacing dim_q(t) →dim(t). We also gave a prescription on how to compute the N →∞ (for finite value of k) limit of the entropy using the identities of q-numbers. In these limits, the entropy converges to a finite value. We also obtained the entropy values when both N →∞ and k →∞ (in no particular order). We find that for symmetric representations and (2,1) representations, the entropy for the state |Ψ_1⟩ converges to a non-zero value but the entropy of |Ψ_2⟩ (for n>2)  goes to 0 (eq.(<ref>) and eq.(<ref>)). In the case of adjoint representation (where |Ψ_1⟩ = |Ψ_2⟩), the entropy vanishes in this limit (eq.(<ref>)). Moreover the large k and large N values of entropy for the states |Ψ_3⟩ and |Ψ_4⟩ tends to 0 as k →∞ or N →∞ which is clear from the plots shown in figures <ref>, <ref>, <ref> and <ref>.

Entanglement entropy for three-manifolds with two S^2 boundaries having more than four punctures, related by cobordism,  needs to be investigated. There could be local braiding and necklaces of Wilson lines inside such a three-manifold. The computation of quantum states will involve six or higher point conformal blocks. Further, we need to define the quantum states for three-manifolds with n copies of S^2 boundaries with six or more punctures to determine the  necklace states. It appears that in such cases, the local braiding will give entanglement entropy involving the partition function on S^2 × S^1 in the presence of Wilson lines (similar to the eq.(<ref>)) <cit.>. However, such a relation between the entanglement entropy and the partition function on S^2 × S^1 in the presence of necklaces needs to be explored. We will pursue these aspects in future.

In <cit.>, the entanglement entropy by partial tracing within an S^2 boundary divided into two spatial regions was discussed. It would be interesting to study the entanglement structure obtained by the additional division of each S^2 into sub-regions for the three-manifolds with n copies of S^2. We hope to report progress in future work.

Recently, the entanglement structure in Chern-Simons theory is discussed within topological string context <cit.> exploiting Gopakumar-Vafa duality. The implication of our results on the dual closed topological strings is another plausible direction to pursue.
 




Acknowledgements
We would like to thank Abhijit Gadde, Shiraz Minwalla, R.K. Kaul, Piotr Sulkowski, T.R. Govindarajan and Lata Kh Joshi for useful discussion and correspondence. The work of VKS is supported by the ERC Starting Grant no. 335739 “Quantum fields and knot homologies” funded by the European Research Council under the European Union's Seventh Framework Programme. PR and Saswati would like to acknowledge DST-RFBR grant (INT/RUS/RFBR/P-309) for support. Saswati would like to thank CSIR for the research fellowship. PR would like to thank ICTP where part of this work was done during her visit as senior associate.





§ COMPUTING THE STATES |Ψ_3⟩ AND |Ψ_4⟩


The state |Ψ_3⟩ given in eq.(<ref>) can be obtained by considering the state |Ψ_2⟩ of eq.(<ref>) on four S^2 boundaries and gluing the extra two copies of S^2 with appropriate quantum states as shown in figure <ref>. The boundaries labeled 3', 4' and 5' have opposite orientations to that of boundaries labeled 3, 4 and 5 respectively. Gluing these oppositely oriented boundaries as shown in the figure will give the final state which is a two boundary state.

The quantum state β in figure <ref> can be obtained by acting the b_2 braiding operator 2p times on one of the basis states:

    β   =  ∑_t, r_1, r_2⟨ϕ_t, r_1 r_2^(3') | ⊗(b_2^(-))^2p |ϕ_t, r_1 r_2^(4)⟩
       = ∑_t, r_1, r_2∑_s, u, v  a_s, u v^t, r_1 r_2[
    [  R_1 R̅_1;  R_2 R̅_2 ]] ⟨ϕ_t, r_1 r_2^(3') | ⊗(b_2^(-))^2p_1 |ϕ̂_s, u v^(4)⟩ ,

where we have used the transformation rules of eq.(<ref>) in the second equation to change the basis, so that the b_2 operator can be applied. Here t ∈ (R_1 ⊗R̅_1) ∩ (R_2 ⊗R̅_2) and r_1, r_2 are the corresponding multiplicity labels. Similarly s ∈ (R̅_1 ⊗R_2) with u, v denoting the multiplicity labels. Using the braiding eigenvalues from eq.(<ref>), we obtain:[Since the braiding operator is acting even number of times, the phase factor {R̅_1, R_2, s, u} in the eigenvalue of b_2 given in eq.(<ref>) is raised to even power and hence becomes 1.]

    β = ∑_t, r_1, r_2
    s, u, v a_s, uv^t, r_1 r_2[
    [  R_1 R̅_1;  R_2 R̅_2 ]] exp(2π i  p  (C_s - C_R̅_1 - C_R_2)/k+N) ⟨ϕ_t, r_1 r_2^(3') | ⊗ |ϕ̂_s, u v^(4)⟩ .

The quantum state α in figure <ref> can be simply given as:

    ⟨α | = ⟨ϕ_0, 11^(4') | = ∑_s', u', v' a_s', u'v'^0, 11[
    [  R_1 R̅_1;  R_2 R̅_2 ]]^*⟨ϕ̂_s', u' v'^(4') |  .

Gluing of the boundaries 4 and 4' can be achieved by taking the inner products of quantum states α and β which gives, 

    ⟨γ | ≡⟨α | β⟩ = ∑_t, r_1, r_2 F(t, r_1, r_2, p)  ⟨ϕ_t, r_1 r_2^(3') |  ,

where we have defined:

    F(t, r_1, r_2, p) = ∑_s,u,v  a_s, uv^0, 11[
    [  R_1 R̅_1;  R_2 R̅_2 ]]^* a_s, uv^t, r_1 r_2[
    [  R_1 R̅_1;  R_2 R̅_2 ]] exp(2π i  p (C_s - C_R̅_1 - C_R_2)/k+N)  .

The state ⟨μ | in figure <ref> is written as,

    ⟨μ | = ⟨ϕ̂_0, 1 1^(5') | = ∑_t', r_3', r_4' a_0, 11^t', r_3' r_4'[
    [  R_4 R̅_3;  R_3 R̅_4 ]] ⟨ϕ_t', r_3' r_4'^(5') |  .

The four boundary state in figure <ref> is the |Ψ_2⟩ of eq.(<ref>) for n=4 with appropriate representations and can be written as:

    |Ψ_2⟩   = ∑_w, y_1,y_2,y_3,y_4{R_1, R̅_1, w, y_1}{R_2, R̅_2, w, y_2}{R_4, R̅_3, w, y_3}{R_4, R̅_3, w, y_4}/dim_q w
       ×    |ϕ_w, y_4 y_1^(1), ϕ_w, y_1 y_2^(3), ϕ_w, y_2 y_3^(2), ϕ_w, y_3 y_4^(5)⟩ ,

where w ∈ (R_1 ⊗R̅_1) ∩ (R_2 ⊗R̅_2) ∩ (R_4 ⊗R̅_3) ∩ (R_4 ⊗R̅_3) and y_1, y_2, y_3, y_4 are corresponding multiplicity labels. The final state can be obtained by gluing the boundary 3' with 3 and 5' with 5 which can be achieved by taking the inner product ⟨γ, μ | Ψ_2 ⟩. Using the following property of the Racah matrix

    a_s, uv^0, 11[
    [  P P̅;  Q Q̅ ]]    = {Q} {P̅,Q, s, u }/√((dim_q   P) (dim_q   Q))  √(dim_q   s)  δ_uv
    
    a_0, 11^t, ij[
    [  P  Q; Q̅ P̅ ]]    = {Q} {P,Q,t,i }/√((dim_q   P) (dim_q   Q))  √(dim_q   t)  δ_ij ,

the inner product gives the required state: 

    |Ψ_3⟩ = ∑_t,  r_1,r_2,r_3{R_1, R̅_1, t, r_1}{R_2, R̅_2, t, r_2}{R_4, R̅_3, t, r_3}{R_2}{R_3}/√(dim_q t)  √(dim_q R_1  dim_q R_2 dim_q R_3  dim_q R_4) F(t, r_1, r_2, p) |ϕ_t, r_3 r_1^(1), ϕ_t, r_2 r_3^(2)⟩

and the function F can be further simplified using eq.(<ref>) as,

    F(t, r_1, r_2, p) = ∑_s,u {R̅_1, R_2, s, u}√(dim_q s)  a_s, uu^t, r_1 r_2[
    [  R_1 R̅_1;  R_2 R̅_2 ]] exp(2π i  p  (C_s - C_R̅_1 - C_R_2)/k+N)  .

This is the state given in eq.(<ref>) where the factors like {R_2}, {R_3} and (dim_q   R_j) are constant terms and will be canceled out while normalizing |Ψ_3⟩.

Next we want to compute the state |Ψ_4⟩ given in eq.(<ref>) on n number of boundaries. For this, we need to consider the state |Ψ_2⟩ of eq.(<ref>) on 2n copies of S^2 and glue the extra n copies with S^2 boundaries of appropriate quantum states. We will show the computation of |Ψ_4⟩ for n=2 and n=3 from where it can be generalized to any n. The figure <ref> gives the construction of |Ψ_3⟩ for n=2. The boundaries labeled 3', 4', 5', and 6' have opposite orientations to that of boundaries labeled 3, 4, 5, and 6 respectively. Gluing these oppositely oriented boundaries as shown in the figure will give the final state which is a two boundary state.

The quantum state β_1 is similar to the state β shown in the figure <ref> and can be directly written:

    β_1 = ∑_t, r_1, r_2
    s_1, u_1, v_1 a_s_1, u_1v_1^t, r_1 r_2[
    [  R_1 R̅_1;  R_2 R̅_2 ]] exp(2π i  p_1 (C_s_1 - C_R̅_1 - C_R_2)/k+N) ⟨ϕ_t, r_1 r_2^(3') | ⊗ |ϕ̂_s_1, u_1 v_1^(4)⟩ ,

where t ∈ (R_1 ⊗R̅_1) ∩ (R_2 ⊗R̅_2) and r_1, r_2 are the corresponding multiplicity labels. Similarly s_1 ∈ (R̅_1 ⊗R_2) with u_1, v_1 denoting the multiplicity labels. The quantum state α_1 is same as the state α in the figure <ref>:

    ⟨α_1 | = ⟨ϕ_0, 11^(4') | = ∑_s_1, u_1, v_1 a_s_1, u_1v_1^0, 11[
    [  R_1 R̅_1;  R_2 R̅_2 ]]^*⟨ϕ̂_s, u v^(4') |  .

Gluing of the boundaries 4 and 4' by taking the inner products of states α_1 and β_1 gives, 

    ⟨γ_1 | ≡⟨α_1 | β_1 ⟩ = ∑_t, r_1, r_2 F_1(t, r_1, r_2, p_1) ⟨ϕ_t, r_1 r_2^(3') |  ,

where we have defined:

    F_1(t, r_1, r_2, p_1) = ∑_s_1,u_1,v_1  a_s_1, u_1v_1^0, 11[
    [  R_1 R̅_1;  R_2 R̅_2 ]]^* a_s_1, u_1v_1^t, r_1 r_2[
    [  R_1 R̅_1;  R_2 R̅_2 ]] exp(2π i  p_1 (C_s_1 - C_R̅_1 - C_R_2)/k+N)  .

Similarly, the state ⟨γ_2| can be obtained by gluing boundaries 6 and 6':

    ⟨γ_2 | ≡⟨α_2 | β_2 ⟩ = ∑_t', r_3, r_4 F_2(t', r_3, r_4, p_2) ⟨ϕ_t', r_3 r_4^(5') |  ,

where t' ∈ (R_3 ⊗R̅_3) ∩ (R_4 ⊗R̅_4) with r_3, r_4 as the corresponding multiplicity labels and the function F_2 is defined as:

    F_2(t', r_3, r_4, p_2) = ∑_s_2,u_2, v_2  a_s_2, u_2v_2^0, 11[
    [  R_3 R̅_3;  R_4 R̅_4 ]]^* a_s_2, u_2v_2^t', r_3 r_4[
    [  R_3 R̅_3;  R_4 R̅_4 ]] exp(2π i  p_2 (C_s_2 - C_R̅_3 - C_R_4)/k+N)  .

Here representation s_2 ∈ (R̅_3 ⊗ R_4) with u_2 and v_2 as the multiplicity labels. The four boundary state in figure <ref> can be written using eq.(<ref>) as:

    |Ψ_2⟩ = ∑_w, y_1,y_2,y_3,y_4∏_j=1^4 {R_j, R̅_j, w, y_j}/dim_q w |ϕ_w, y_4 y_1^(1), ϕ_w, y_1 y_2^(3), ϕ_w, y_2 y_3^(2), ϕ_w, y_3 y_4^(5)⟩ ,

where w ∈ (R_1 ⊗R̅_1) ∩ (R_2 ⊗R̅_2) ∩ (R_3 ⊗R̅_3) ∩ (R_4 ⊗R̅_4) and y_1, y_2, y_3, y_4 are corresponding multiplicity labels. The final state can be obtained by gluing the boundary 3' with 3 and 5' with 5 which can be achieved by taking the inner product ⟨γ_1, γ_2 | Ψ_2 ⟩. Thus we will get the required state:  

    |Ψ_4^2bdy⟩ = ∑_t,  r_1,r_2,r_3,r_4∏_j=1^4 {R_j, R̅_j, t, r_j}/dim_q t F_1(t, r_1, r_2, p_1)  F_2(t, r_3, r_4, p_2) |ϕ_t, r_4 r_1^(1), ϕ_t, r_2 r_3^(2)⟩ .

In a similar way, one can also obtain the state |Ψ_4⟩ for n=3 as shown in the figure <ref> by gluing the boundaries of various states. 

The six boundary state is the |Ψ_2⟩ of eq.(<ref>) for n=6 and is given as:

    |Ψ_2⟩ = ∑_t, r_1,…,r_6∏_j=1^6 {R_j, R̅_j, t, r_j}/(dim_q t)^2 |ϕ_t, r_6 r_1^(1), ϕ_t, r_1 r_2^(4), ϕ_t, r_2 r_3^(2), ϕ_t, r_3 r_4^(5), ϕ_t, r_4 r_5^(3), ϕ_t, r_5 r_6^(6)⟩ .

To get the desired state, we need to compute the inner product ⟨γ_1, γ_2, γ_3 | Ψ_2 ⟩ where,

    ⟨γ_1, γ_2, γ_3 | = ∑_t_1, t_2, t_3 
     x_1,…, x_6  ∏_j=1^3 F_j(t_j, x_2j-1, x_2j, p_j)  ⟨ϕ_t_1, x_1 x_2^(4'), ϕ_t_2, x_3 x_4^(5'), ϕ_t_3, x_5 x_6^(6') |  .

Computing the inner product gives the required state:

    |Ψ_4^3bdy⟩ = ∑_t,  r_1,…,r_6∏_j=1^6 {R_j, R̅_j, t, r_j}/(dim_q t)^2∏_j=1^3 F_j(t, r_2j-1, r_2j, p_j)   ⊗_i=1^3 |ϕ_t, r_2i-2  r_2i-1^(i)⟩ .

This procedure can be generalized to n boundaries and we get:

    |Ψ_4⟩ = ∑_t,  r_1,…,r_2n∏_j=1^2n{R_j, R̅_j, t, r_j}/(dim_q t)^n-1∏_j=1^n F_j(t, r_2j-1, r_2j, p_j)   ⊗_i=1^n |ϕ_t, r_2i-2  r_2i-1^(i)⟩ ,

where the function F is given as (after using eq.(<ref>)),

    F_j(t, r_2j-1, r_2j, p_j)    = ∑_s_j,u_j {R̅_2j-1, R_2j, s_j, u_j }√(dim_q   s_j)   a_s_j, u_j v_j^t, r_2j-1 r_2j[
    [  R_2j-1 R̅_2j-1;    R_2j   R̅_2j ]] 
       ×exp(2π i  p_j (C_s_j - C_R̅_2j-1 - C_R_2j)/k+N)  .

Here s_j ∈ (R̅_2j-1⊗ R_2j) with corresponding multiplicity tracking label as u_j.




§ RACAH MATRIX FOR (2,1) REPRESENTATION OF SU(N)


In this section, we tabulate the type I Racah matrix elements for R = (2,1) which were obtained in <cit.> and are denoted as,

    a_t_i,  r_i, r_i'^t_j,  r_j, r_j'[
    [  R R̅;  R R̅ ]]  ,

where t_i,  r_i,  r_i' labels the row and t_j, r_j,  r_j' labels the column of the Racah matrix (see <cit.> for more details). The representations t_ℓ∈ (R ⊗R̅) = t_1 ⊕ t_2 ⊕ t_3 ⊕ t_4 ⊕ t_5 ⊕ t_6 ⊕ 2 t_7 and are given as,

t_1 = ∙,   t_2    = (2^2, 1^N-4),      t_3    = (3, 1^N-3),      t_4    = (3^2, 2^N-3) 

t_5    = (4, 2^N-2),     t_6    = (4, 3, 2^N-4, 1),     t_7    = (2, 1^N-2)  .

Since only the representation t_7 appears twice, the Racah matrix (which we shall denote by ℛ) will be of order 10, where the ten rows are labeled as: 

    t_1,  t_2,  t_3,  t_4,  t_5,  t_6,  (t_7)_11,  (t_7)_22,  (t_7)_12,  (t_7)_21 .

We will also use the same ordering for the columns. The elements of the Racah matrix ℛ are given as,

    ℛ_αβ = √(dim_q t_α) √(dim_q t_β)  𝒬_αβ ,

where α and β take values from 1 to 10 with the identification t_8 = t_9 = t_10 = t_7. The elements 𝒬_αβ can be organized into a 10 × 10 matrix which is given as:

    𝒬 = (
    [               t_1      t_2      t_3      t_4      t_5      t_6 (t_7)_11 (t_7)_22 (t_7)_12 (t_7)_21;      t_1      Λ_1      Λ_1     -Λ_1     -Λ_1      Λ_1      Λ_1      Λ_1     -Λ_1        0        0;      t_2      Λ_1     Λ_14     Λ_16     Λ_16     Λ_11     Λ_18     Λ_23     Λ_10     -Λ_6     -Λ_6;      t_3     -Λ_1     Λ_16     Λ_15     Λ_15     Λ_12        0     Λ_22      Λ_3     -Λ_7     -Λ_8;      t_4     -Λ_1     Λ_16     Λ_15     Λ_15     Λ_12        0     Λ_22      Λ_3     -Λ_8     -Λ_7;      t_5      Λ_1     Λ_11     Λ_12     Λ_12     Λ_13     Λ_17     Λ_21      Λ_9     -Λ_5     -Λ_5;      t_6      Λ_1     Λ_18        0        0     Λ_17     Λ_19     Λ_11        0        0        0; (t_7)_11      Λ_1     Λ_23     Λ_22     Λ_22     Λ_21     Λ_11     Λ_20      Λ_4      Λ_2      Λ_2; (t_7)_22     -Λ_1     Λ_10      Λ_3      Λ_3      Λ_9        0      Λ_4      Λ_3        0        0; (t_7)_12        0      Λ_6      Λ_8      Λ_7      Λ_5        0     -Λ_2        0      Λ_3      Λ_3; (t_7)_21        0      Λ_6      Λ_7      Λ_8      Λ_5        0     -Λ_2        0      Λ_3      Λ_3;          ])  .

The various terms used here are given in the following:

    Λ_1    = [3]ω, Λ_2 = i[2 N]ω√(μ)[N], Λ_3 = -1ω, Λ_4 =-[4][2]ω, Λ_5 = i√([N-2])[N+1]ω [N]√([N+2]), Λ_6 = -i[N-1]√([N+2])ω√([N-2])[N]
    Λ_7    = i[N+1]ω√(μ), Λ_8 = -i[N-1]ω√(μ), Λ_9 = -[N-2]ω [N] , Λ_10 = -[N+2]ω [N], Λ_11 = -[3]^2ωμ, Λ_12 = [3][N-2]ωμ
    Λ_13   = [2]^2[N-1]+[2]^2[N+1]-[N-4][N][N+3][N-2][N-1][N]^3[N+1][N+2][N+3], Λ_14 = [2]^2[N-1]+[2]^2[N+1]-[N+4][N][N-3][N-3][N-2][N-1][N]^3[N+1][N+2], Λ_15 = [3]ωμ
    Λ_16   = [3][N+2]ωμ [N], Λ_17 = [2][3]^2ωμ [N][N+3], Λ_18 = [2][3]^2ωμ [N][N-3], Λ_19 = -[3]^3[N]ω^2 μ [N+3][N-3]
    Λ_20   = [N][N+1][N+2]   X_20ω^3 μ^2 [2][N][N+1]^2+ω^4 μ [2][N+2]^2, Λ_21 = [N][N+1]   X_21ω^3 μ [2], Λ_22 = X_22ω^2 μ [2], Λ_23 = X_23ω^2 μ [2][N+1]

where we have defined ω = [N-1][N][N+1] and μ = [N-2][N+2], with [x] denoting the q-number defined in eq.(<ref>). The other factors involved in above equation are given below: 

    X_20   = [2]^2 [3]^2 [2N]^2 - ( 2 ω^4 μ [3][N]^3 [N+1]^3 + 2 ω^4 μ [3][N]^3 [N-1]^3 - ω^3 μ^2 [2][N+2][N]^2 [N+1]^2 - ω^3 μ^2 [2][N-2][N]^2 [N-1]^2) ( [N+2][N+1] + [N-2][N-1]) 
       + ω^2 μ [2] [3]^2 ( [N-3][N-2] + [N+3][N+2]) 
    
    X_21   = -[2]^2 [3]^2 - 2 ω^2 μ [3]^2[N]^2 [N+1]^2 - 2 ω^2 μ [3][N]^2 [N-1]^2 + μ^3 [2] [N+1][N+2] + ωμ [3]^2 [N-3][N+1] [N-2] + ωμ [3]^2 [N+3][N+1] [N+2]
    
    X_22   = μ [2][N]([3]^2 - μ) + μ [3][N-3]( 1 - [3][N][N-2]) + μ [3][N+3]( 1 - [3][N][N+2]) 
    
    X_23   = -[2]^2 [3]^2 - 2 ω^2 μ [3]^2[N]^2 [N-1]^2 - 2 ω^2 μ [3][N]^2 [N+1]^2 + μ^3 [2] [N-1][N-2] + ωμ [3]^2 [N-3][N-1] [N-2] + ωμ [3]^2 [N+3][N-1] [N+2] .



JHEP
 











