Unitary Braid Matrices: Bridge between Topological and Quantum Entanglements

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (non-local) unitary actions on separable pure product states of three identical subsystems (say, the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body entanglements (in three 2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0,1).Thus braiding operators corresponding to over- and under-crossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements. For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest. A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.


Introduction (two faces of unitary braid matrices)
The third Reidemeister move in the theory of knots and links impose equivalence between two specific sequences of over-and under-crossing of three braids. Such sequences can be repeated and closing the ends of the braids one obtains topologically entangled Borromean rings whose history reaches back far into the past (see fig. 7 of Ref. 1). Braid matrices, "Baxterized" to depend on spectral (rapidity) parameters satisfy an equation -the braid equation -which corresponds precisely to the above mentioned Reidemeister move. They provide matricial representations of a particular type of topological entanglements. When a braid matrix is also unitary it can also be implemented to induce unitary transformations in base spaces of corresponding dimensions representing possible quantum states of an object (say, the spin projections of particles).
Let us make this more precise. Let R (θ) be a unitary N 2 × N 2 matrix, I the N × N unit matrix and R 12 = R ⊗ I, R 23 = I ⊗ R, (1.1) R 12 , R 23 act on triple tensor products V N ⊗ V N ⊗ V N of N-dimensional vector spaces V N .
To be a braid matrix R (θ) must satisfy The indices (12), (23) correspond to successive crossings (braid 2 overcrossing braid 1 and undercrossing braid 3). The equality sign imposes the essential Reidemeister constraint (the 3-rd move). This will be the link of our matrices to topological entanglement. The role of braid matrices satisfying unitarity R (θ) + = R (θ) (1.3) in quantum entanglements have been noted and discussed by Kauffman and Lomonaco [1] in their paper "Braiding operators are universal quantum gates". A large number of relevant sources are cited in Ref 1. We have presented before two quite distinct classes of unitary N 2 × N 2 braid matrices [2,3], one real and for even N and the other complex, for All N. There is no upper limit to N. Refs. 2, 3 cite other sources.
In the following sections we will systematically, explicitly derive the measures of 2-body and 3-body entanglements generated by the action of the braiding operator B = R 12 (θ) R 23 (θ + θ ′ ) R 12 (θ ′ ) = R 23 (θ ′ ) R 12 (θ + θ ′ ) R 23 (θ) ( The measures of 2-tangles and 3-tangles derived in Ref. 4 will be used throughout. Topological and quantum entanglements, two domains of B, will thus be brought together. One essential point must be noted: The unitary matrix R is not locally unitary. R cannot be expressed as R 1 ⊗ R 2 acting on V N ⊗ V N where R 1 , R 2 are each a unitary N × N matrix acting on V N . Such an R would have been trivial in the context of braiding. Nor would they have induced quantum entanglements acting on a product state |a ⊗ |b in V ⊗ V. Non-local unitarity is crucial in the action of B. Local unitary transformations can however be used to classify already entangled states, as has been done systematically by Carteret and Sulbery [5]. We aim at generating quantum entanglements. We close the introduction with some notations we will use throughout. For N = 2, V 2 is usually taken to be spanned by the spin projections of spin- With passage to higher dimensions in mind we will often use the state vectors For all N, For convenience we will continue to use the terminology of spin projections. But the indices above can also correspond to other suitably enumerated quantum states of a system.

Unitary braid matrices and their actions
(I) Real, unitary, even-dimensional braid matrices: A class of real, unitary, (2n) 2 × (2n) 2 dimensional braid matrices [2] is given by and ((K, J) are (2n) × (2n) matrices given by with i = 2n+1−i and I the (2n)×(2n) unit matrix. We always denote by (ij) the matrix with a single non-zero element, unity on row i and column j. A detailed study of this class can be found in Ref.
2. An equivalent construction, without explicit introduction of the tensor product structure (K ⊗ J), can be found in Ref. 6. From (2.3), and using (2.4) one obtains unitarity and the explicit evaluation (A misprint in the overall factor in (2.16) of Ref. 2 is corrected above. We have also set z ′′ (1 + zz ′ ) = (z + z ′ ) in the second term there.) We now consider the action of B on basis states |abc = |a ⊗ |b ⊗ |c , Similarly (b, c) can be j, j , k, k respectively over the same domain. The notations (2.8), (2.9) make the formalism much more compact. Using (2.3-9) one obtains Noting that one immediately verifies that, consistently with unitarity of B, On the left of (2.10), |abc is by definition a separable product of pure states, hence unentangled. One the right the superposition can be shown to imply entanglement. On assuming it can be expressed as a product ( x i |x i ) ⊗ ( y i |y i ) ⊗ ( z i |z i ) one runs into contradictions. In Sec. 3 we will go much further. We will obtain explicitly the intrinsic 3-body entanglement (3-tangle)and the 2-body entanglements (2-tangles) of the three subsystems. They will be expressed in terms of (f 0 , f 1 , f 2 , f 3 ) of (2.11).
(II) Complex, even dimensional, multi-parameter unitary braid matrices: In a series of papers (some of which are cited in Ref. 3) we have constructed a class multiparameter braid matrices, the number of such parameters increasing as N 2 with the dimension N 2 × N 2 of the matrix for both even and odd dimensions (N = 2, 3, 4, 5, 6, · · · ). It was then noted [2,3] that for all these parameters pure imaginary this class corresponds to unitary braid matrices. (One may also consider real parameters with θ imaginary.) In this subsection we restrict our considerations to even-dimensional matrices. For N odd special features arise which are best treated separately (Sec. 4). The even dimensional, unitary (2n) 2 × (2n) 2 matrix is given by the definitions of the projectors being

Consider again the action of
(2.16) One obtains (compare (2.10), (2.11)) B |abc = f 0 |abc + f 1 abc + f 2 |abc + f 3 abc (2.17) but now with coefficients defined below. Set with only the sum (θ + θ ′ ) as factor above (in contrast to tanh θ, tanh θ ′ , tanh (θ + θ ′ ) all playing roles in the previous case). In terms of (λ, µ) one has satisfying the unitarity constraints Again one can easily verify that the right side of (2.17) represents an entangled states (as for (2.10)). We will obtain explicitly the 3-tangle and the 2-tangle in Sec. 3. Repeated actions of B with different parameters will modify the coefficients as are coefficients due to the action of B ′ alone. This set may belong to the same class as (f 0 , f 1 , f 2 , f 3 ) or to the other one of our two classes (see (2.11) and (2.19)). The process can be repeated remaining always in the closed subspace |abc , abc , |abc , abc . (Starting with the same |a but with different |b ′ , |c ′ , for example, and continuing thus one can entangle each individual state with all the others successively in the total base space. See the relevant remarks in sec. 6.) The essential features of the aspects that interest us principally (analyzed in sec. 3) can be shown to be conserved under iterations indicated above. For that reason, and also for simplicity, we will restrict our study (sec. 3) to the two sets of coefficients (2.11) and (2.19).

Computation of quantum entanglements
We now extract from (2.10), (2.11) and (2.17), (2.19) respectively the quantum entanglements generated by B acting on the pure product state |abc , where |abc can be any triplet selected from the N 3 dimensional base space. For spin 1 2 particles For higher spins (with |a now written as |−a ) and similarly for (|b , |c ). Under the action of the braiding operator B (defined in (1. For higher spin, as noted before, one can start with the same |a but (|b , |c ) chosen from all the other possibilities. But for each initial choice one remains, under the action B, in the subspace given by the right hand of (3.3). This is the very special, fundamental, property of our unitary matrices. This allows us, even for higher spins, to implement systematically the formalism and concepts of Coffman, Kundu and Wootters (CKW) concerning 3particle entanglements (and corresponding 2-particle ones for the three subsystems) in Ref. 4. This we now proceed to do. The density matrix corresponding (3.3) is Tracing out c one obtains Tracing out b in ρ 12 one obtains a diagonal One can write down (ρ 13 , ρ 13 ), (ρ 2 , ρ 3 ) from symmetry. Thus, for example, and so on. The spin-flipped matrix can now be obtained for (ρ 12 , ρ 23 , ρ 13 ) and then the products (ρ AB ρ AB ). Thus The products (ρ 13 ρ 13 ), (ρ 23 ρ 23 ) are related to the result above through evident permutations of the indices (1, 2, 3). The eigenstates can be read off as . The eigenvalues of (ρ 12 ρ 12 ) are (The ordering of the first two roots depends on values of the parameters in B.) Taking square roots the "concurrence" is Note that the product (λ 1 λ 2 ) is the same for the three subsystems. Thus from (17) and (24) of CKW, the 3-tangle is The invariance of τ 123 under permutations of the particles (1, 2, 3) (i. e. (a, b, c)) is evident above. Having obtained the results in terms of (f 0 , f 1 , f 2 , f 3 ) we proceed below to study them for our two classes implementing (2.11) and (2.19). We start with τ 123 since the crucial role of B is to entangle 3 particles.
Away from such points, for the generic case, keeping in mind (2.12) we define From (2.11), (3.12), (3.13), (3.14) the 2-particle concurrences are and (3.27) and For (z = 1, z ′ = 0, z ′′ = 1) and also for (z = 0, z ′ = 1, z ′′ = 1) C 12 = C 23 = C 13 = 0, while τ 123 = 1 attaining the maximum. It is the situation one finds in GHZ state though that is quite different otherwise. (We refer to the comments below (24) of CKW). One can also compare a Borromean ring (Ref. 1, sec. 8.3, for example). If any one of the three entangled braids is cut, the remaining two fall apart-they are no longer entangled.

(II):
We now obtain the quantum entanglements induced by the action of B in a product state |abc for our complex, multiparameter, unitary braid operator. From (2.19), satisfying Here, from (2.18) The 3-tangle is now from (3.15), along with (3.31), (3.12), (3.13), (3.14) the 2-tangles are Let us take a closer look at these results.
1. The vanishing of C 13 is related to the fact that in the action of B on |abc , the terms involving m ab act on |ab and those involving m bc on |bc . Thus |b is acted on by both parts while |a and |c are decoupled in the above sense. One can alter the actions on them independently by varying the two sets of parameters. One the other hand the presence of |b generates a coupling with |a on one hand and with |c on the other. A parallel feature was absent for our class (I). There apart from (θ, θ ′ ) there are no free parameters (like the m's for class (II)). And z ′′ = (z + z ′ ) (1 + zz ′ ) −1 combines (z, z ′ ) nonlinearly.

Odd dimensions
The real matrices (2.1-3), our class (I), have no odd dimensional counterparts. The complex, multiparameter matrices (2.14), (2.15) are not thus restricted. In fact the odd dimensional sequences based on "nested sequences of projectors" were the first to be constructed. The lowest odd dimensional (9 × 9) case, with imaginary parameters for unitarity, is explicitly presented in sec 11 of Ref. 2. The crucial difference with N = 2n is that for N = (2n − 1), (n = 2, 3, · · · ) i = (2n − 1) − i + 1 = 2n − i and hence n = 2n − n = n. The action of B on the states (4.6) can now be studied. From our point of view (links with quantum entanglements) not only |nnn but also (|nnc ,|nbn ,|ann ) are trivial since one obtains under action of B superposition of states |nn ⊗ (|c , |c ) and so on. Only one spin is affected. Entanglement is not produced. The states (|nbc , |abn ) can also be set aside, under the action of B the state |n remains a bystander to 2-particle entanglements of |bc and |ab . The state |anc deserves a closer look. One obtains The coefficients (f 0 , f 1 , f 2 , f 3 ) are obtained by setting, in (2.18), (2.19), But now tracing out indices (with n = n) leads to differences. As compared to (3.6), now (with b = b = n) One has 2 × 2 matrix now. One the other hand tracing out n in ρ 12 This is no longer diagonal like (3.7). We will not analyze such cases further in this paper. For N = 2n, at the centre of the matrix R is the square lattice with corners, which can be denoted as (nn, nn, nn, nn). For N = 2n − 1, (since n = n) this reduces to the point (nn) common to the diagonal and the anti-diagonal. This is the source of difference. When this common point is not involved in |abc the results correspond for even and odd dimensions.

Entanglement via a special coupling of 3 spins
This section is a brief digression. We restrict our remarks here to 3 spin 1 2 particles. For this case, in the study of entanglements, basic roles are usually attributed to the states (See Ref. 7 and sources cited there.) Their local unitary transformations can also be considered [5]. Our approach via the braiding operator B led to states of the type (f 0 |000 + f 1 |011 + f 2 |101 + f 3 |110 ) and (g 0 |111 + g 1 |100 + g 2 |010 + g 3 |001 ) with normalized coefficients (with parameter dependence) given in sec 2. The states |000 and |111 of |GHZ are separately superposed respectively with those of W , |W respectively and these 4-term in the superpositions have been thoroughly studied in the preceding sections. Though we are particularly interested in such cases it is interesting to point out the direct relations of |GHZ , W and |W to a

Discussions
We want to emphasize how in larger dimensions each object (say, spin states of component particles) is seen to be entangled with all the others through a full exploitation of our formalism. Since our approach is via the braiding operator B (defined in (1.4)) we start by picking out a triplet |a ⊗ |b ⊗ |c ≡ |abc , (6.1) where (a, b, c) is any element among the basis states spanning V N ⊗ V N ⊗ V N and obtain the entangled superpositions studied (sec. 3) B |abc = f 0 |abc + f 1 abc + f 2 |abc + f 3 abc .
where |a = |N − a + 1 and so on. The states (|a , |a ), |b , b , (|c , |c ), of the subsystems are involved above. But now one can start again with any triplet |ab ′ c ′ , |a ′′ bc ′′ , · · · , (b ′ = b or b and so on) covering thus systematically all possible choices in V N ⊗ V N ⊗ V N and then implement the action of B. Thus at the end each |a will be entangled with each |b and each |c . At each step a quadruplet |abc , abc , |abc , abc will be involved, this being the essential property of both classes of unitary braid matrices we propose (with non-zero terms on the diagonal and the anti-diagonal only). Consider the simplest non-trivial case. For three spin half particles the two quadruplets will be (|000 , |011 , |101 , |110 ) (|111 , |100 , |010 , |001 ). But already it is evident that starting by turns with, say (|000 , |001 ) finally |a = |0 will be entangled with |bc = (|00 , |01 , |10 , |11 ) i.e. with all possible states of |b and |c . This is a general feature.
In sec .5 we have contrasted our typical superposition (6.2), to the prominent roles in the study of 3-particle entanglements of the states |GHZ , |W , W given in (5.1)-(5.3). It is implicit in our formalism that the maximum 3-tangle is obtained for namely 1 2 (|111 + |100 + |010 + |001 ) and 1 2 (|000 + |011 + |101 + |110 ). The (d 1 , d 2 , d 3 ) defined in (21) of CKW [4] are for both the cases above The value (6.4) are the same as for |GHZ . For the maximal superposition (normalized sum of the quadruplets with all coefficients equal) 1 √ 8 (|000 + |011 + |101 + |111 + |100 + |010 + |001 ) there is a striking change. Now, (d 1 , d 2 , d 3 ) = 1 2 6 (4, 6, 2) (6.6) τ 123 = 1 2 6 (4 − 2 · 6 + 4 · 2) = 0. (6.7) One reaches now the lower bound. CKW notes below (25) "It would be very interesting to know which of the results of this paper generalize to larger objects or to larger collections of objects". Our formalism furnishes one possible approach to many component objects and their collections. We have not answered the question whether the entanglement in larger dimensions can be formulated in a systematically hierarchical fashion, involving simultaneously more and more objects. Our motivation has been "entangling topological and quantum entanglements" via the braiding operator B corresponding to third Reidemeister move. Having constructed unitary classes B are were able to implement them to generate quantum entanglements.
Three particle entanglements were emphasized before [10]. We have studied, for our cases, the permutation invariant measures of entanglement of Ref. 4. The crucial feature of our treatment is the study of B acting on V × V × V rather than R acting on V × V . The treatment starting in sec. 2 (II) displays one aspect of the multiple possibilities inherent in our multi-parameter models. This can be put side by side with their role (for real parameters) in statistical models [11]. At the end of sec. 3 we briefly evoke possible periodicity in the space of parameters. Introducing a magnetic field (and a simple generalization of the formalism of Ref. 12) one would obtain periodicity in time of our entangled 3-particles states. This will be studied elsewhere.