Local invariants of braiding quantum gates -- associated link polynomials and entangling power

For a generic $n$-qubit system, local invariants under the action of $SL(2,\mathbb{C})^{\otimes n}$ characterize non-local properties of entanglement. In general, such properties are not immediately apparent and hard to construct. Here we consider certain two-qubit Yang-Baxter operators, which we dub of the `X-type', and show that their eigenvalues completely determine the non-local properties of the system. Moreover, we apply the Turaev procedure to these operators and obtain their associated link/knot polynomials. We also compute their entangling power and compare it with that of a generic two-qubit operator.


Introduction
Entanglement, perhaps the most bizarre feature of the quantum world [1,2], plays a crucial role in quantum information processing and quantum computation [3,4]. Its non-local nature goes against our classical intuition, but it can be used to analyze a quantum system in a systematic manner, via group theory and classical invariant theory [5,6]. The parameters appearing in quantum states and quantum operators can in fact be organized by their response under the local action of SL (2, C) ⊗n , for an n-qubit system defined on (C 2 ) ⊗n . This action defines an orbit space of equivalence classes such that the states or operators in a given orbit have the same non-local properties. This analysis has been performed in [7][8][9][10][11][12][13] for local unitaries of pure and mixed states on finite-dimensional Hilbert spaces and in [14] for two-qubit gates.
As is well known from these early works, the systematic computation of these local invariants is a tedious task that gets harder as one increases the number of qubits. Nevertheless, this is important to understand all possible entanglement in a finite quantum system and hence must be tackled.
In this work we explore the possibility of simplifying this task by "creating" quantum systems with braid operators built from Yang-Baxter operators (YBOs), i.e. operators that solve the (spectral parameter-independent) Yang-Baxter equation. In recent years it has been understood that such operators can also act as quantum gates [15][16][17][18][19][20][21][22][23], leading to the speculation of a broad connection between topological and quantum entanglement. We work on the two-qubit space (n = 2), though we expect the properties we find to generalize to higher n. These matrices generate entangled two-qubit states and we denote them X-type operators, for obvious reasons. We find twelve such classes of YBOs that can be both unitary and non-unitary. 1 We organize our results as follows. In Sec. 2 we find one linear and five independent quadratic invariants for an arbitrary two-qubit operator under the action of SL(2, C) ⊗2 . The same procedure can also be carried out for more than two qubits. For the special case of an arbitrary X-type two-qubit operator, we show in Sec. 3 that independent invariants are exhausted by one linear and five quadratic invariants. In Sec. 4 we restrict the X-type operators to braid operators and observe that all the local invariants are expressed solely as functions of the eigenvalues and that the number of independent local invariants coincides with the number of independent eigenvalues in each of twelve possible classes. In Sec. 5 we enhance the X-type braid operators using the procedure outlined in [24] and compute their associated link/knot polynomials. It turns out that the polynomials are not always local invariant, although they can be expressed in terms of the eigenvalues of the braid operators. In Sec. 6 we also consider the entangling powers [25] of the X-type braid operators and compare them with the entangling power of an arbitrary X-type operator. We end with an outlook and discussion in Sec. 7. In App. A we investigate the relation between our classification of X-type YBOs and Hietarinta's classification [26]. For completeness, an analogous computation is presented in App. B for a family of braid operators that is not of the form (1.1).

SL(2, C) ⊗invariants of general two-qubit operators
We consider an operator R acting on two qubits |i 1 i 2 as a 4 × 4 matrix (its row and column are labeled by (i 1 i 2 ) and (ĩ 1ĩ2 ), respectively): When an invertible local operator (ILO) acts on two-qubit states (QR|i 1 i 2 , Q ĩ 1ĩ2 ), we can interpret that R is transformed as QRQ −1 .
More precisely where untilded (tilded) indices with a = 1, 2, say i a (ĩ a ), are transformed by Q a (Q −1 a ). From this transformation property, one can see that invariants under the action of the ILO can be constructed from a set of Rs by contracting their indices with the four invariant tensors ia ja , ĩ aja , δ iaja , δ˜i a ja , with 01 = − 10 = 1, 00 = 11 = 0. Note that the resulting expressions would also be invariant under the action of a general ILO belonging to C * · SL(2, C) ⊗2 , namely Stochastic Local Operations and Classical Communication (SLOCC) [27]. The factor C * represents the multiplication by a nonzero complex number and does not affect R, since these are similarity transformations.
In the following, we present the invariants at linear and quadratic orders in R. Einstein's convention -repeated indices are understood to be summed over -is used for notational simplicity.
Linear invariant The invariant at linear order is only one and given by (2.4) where Tr denotes the trace taken on the whole Hilbert space of the two qubits.
Quadratic invariants We first list eight invariants which are independent of I 2 1 :
It can be seen that the invariants presented in the previous section are not simply a function of these eigenvalues, but they contain other terms, that have to be invariant combinations on their own. To check this, let us specialize the linear and quadratic invariants to (3.1) where I 2,1 , I 2,2 , I 2,3 , I 2,6 , and I 2,7 are obtained from the above through the identities (2.17).
On the RHS in each formula of (3.3), we can identify the part not expressed by the eigenvalues as an additional invariant combination. 2  are independent SL(2, C) ⊗2 invariants. These can also be expressed as which are clearly functions of the eigenvalues and quadratic local invariants I 2,4 , I 2,5 , I 2,8 , I 2,9 and I 2,10 .
As far as the number of independent invariants is concerned, we have two additional invariants other than the eigenvalues. Note that, in principle, six is the lower bound of the number of the invariants, because there might appear more independent invariants when we consider invariants containing higher powers of R. However, by studying the dimension of orbits of the operator R in (3.1) under the action of SL (2, C) ⊗2 as follows, one can show that the number of independent invariants is precisely six.
The operator R acting on the two qubits i and i + 1 can be expanded in terms of the Pauli-matrix basis as where I is the 2 × 2 unit matrix, and X, Y and Z are the Pauli matrices. l, a 3 , a 6 , b 9 are To study the orbits we consider the Lie algebra generators of SL (2, C) ⊗2 and .

SL(2, C) ⊗2 invariants for X-type YBOs
In this section we consider the invariants for X-type matrices (3.1) that are YBOs, i.e. invertible solutions to the Yang-Baxter equation: In contrast to the additional SL(2, C) ⊗2 invariants (different from the eigenvalues) found for general X-type matrices in the previous section, we find by direct inspection that for these operators here all the quadratic invariants depend only on the eigenvalues.
We list the YBOs (except the trivial one R ∝ 1 4 ) into the following twelve classes and include the corresponding results for the quadratic invariants I 2,4 , I 2,5 , I 2,8 , I 2,9 , I 2,10 with the help of Mathematica: Note that I 2,8 = I 2,10 − I 2,4 and hence there are only three independent local invariants.
This coincides with the number of independent eigenvalues: h 1 , h 8 and √ h 4 h 5 .
• Class 2: In this case also we have the number of independent eigenvalues, namely equal to the number of independent local invariants: I 2,4 , I 2,10 (note in fact that I 2,8 = I 2,10 − I 2,4 ).
• Class 3: We have two independent eigenvalues in this case, h 1 and h 8 , and two independent local invariants as which helps solve for I 2,10 − I 2,9 in terms of I 2,4 and I 2,5 as There are seven other solutions belonging to this class: Here we have two independent eigenvalues, h 1 and −h 1 + h 6 . We can verify that only two of the four local invariants are independent by the expressions implying that I 2,4 and I 2,10 depend on I 2,8 and I 2,9 . Thus we again see that the number of independent eigenvalues is the same as the number of independent local invariants.
Once again we have two independent eigenvalues, h 1 and −h 1 + h 6 . We see that only two of the four local invariants are independent from the expressions, where we can solve for I 2,9 − I 2,10 in terms of I 2,4 and I 2,5 , which in turn implies that the two eigenvalues, λ + and λ − are functions of I 2,4 and I 2,5 . Thus we have the same number of independent local invariants and independent eigenvalues.
with the quadratic invariants We can show that only two of these four local invariants are independent, I 2,4 and I 2,8 as can be seen from the expressions which helps solve for the independent eigenvalues, λ + and λ − in terms of I 2,4 and I 2,8 .
belongs to this class.
The eigenvalues are λ (4.14) Clearly in this case we have two independent eigenvalues, h 1 + h 3 and h 1 − h 3 and two independent local invariants, I 2,4 and I 2,9 as I 2, Here we only have one local invariant which is consistent with the number of independent eigenvalues, depending on h 1 .
Once again we have a single local invariant and a single independent eigenvalue.
There are three other solutions belonging to this class: The eigenvalues are λ 1± = λ 2± = ±h 1 (≡ ±λ) with the quadratic invariants There is a single local invariant and a single independent eigenvalue depending on h 1 .
There are three other solutions in this class: The eigenvalues are λ 1± = λ 2± = h 8 (≡ λ) with the quadratic invariants The number of local invariants coincides with the number of independent eigenvalues.
There are seven other solutions belonging to this class: with the quadratic invariants The number of local invariants coincides with the number of independent eigenvalues.
This classification is based on the pattern of eigenvalues and quadratic invariants, which is different from the criterium used by Hietarinta [26]. The relation between the two classifications is detailed in App. A.

Link invariants for X-type YBOs
A theorem due to Alexander [28] states that every knot/link embedded in S 2 can be obtained as a closure of a braid group element. In order for this to be valid, the braid group generators must satisfy two additional moves, apart from the three usual Reidemeister moves, 3 called the Markov moves. This leads to the enhancement procedure of Turaev and the subsequent computation of knot/link polynomials [24], which we perform in the following.
Definition : where, as above, tr 2 denotes the partial trace on the second qubit space. Let B n be the n-strand braid group generated by σ 1 , · · · , σ n−1 . Link polynomials for a braid group element ξ ∈ B n are then obtained as where w(ξ) = (the number of positive crossings) − (the number of negative crossings) is the writhe of the link, and ρ is a representation of B n constructed from the YBO R as We take V = C ⊗n for qubit systems and n = 2 for two-qubit systems. I = 1 2 and R i,i+1 denotes R acting on the i-th and (i + 1)-th qubits.
Note that the polynomials obtained from (5.4) are not always invariant under the local action of SL(2, C) ⊗n due to the presence of µ ⊗n . In the case when µ = I, the link polynomials are local invariants. As we explicitly see in (5.11) and (5.12), the link polynomials (5.4) with µ = I are not expected to be local invariants even if they are expressible only in terms of the eigenvalues. We can say that any local invariant constructed from an X-type YBO is expressed 3 Recall that the second and third Reidemeister moves represent the relations, as a function depending only on eigenvalues of the YBO. However, the converse is not true in general.
We now enhance the twelve classes of X-type braid operators 4 obtained in Sec. 4 and obtain the associated link invariants.
• Class 1 : In this case we can enhance the braid operator when µ = I, µ = Z, µ = I ± Z.
1. µ = I, x = ±h 1 , y = ±1 and h 8 = h 1 . For example the link invariants corresponding to a two-strand braid group element ξ = σ k (k ∈ Z) are given by that distinguish links with even linking numbers. At h 8 = h 1 the braid operator has [29,30]: with l = ±i λ 1 λ 2 and m = ∓i λ 1 −λ 2 √ λ 1 λ 2 . From these we can obtain the relations Notice that this algebra is similar to the Brauer algebra with the braid operator g i replacing the permutation operator. The Skein relations for the braid operator in this case can be read off from (5.7) as g i and g −1 i can be thought of as positive and negative crossings respectively. This helps us to obtain link invariants in a combinatorial 2. µ = Z, x = ±h 1 , y = ±1 and h 8 = −h 1 . The link invariants corresponding to a two-strand braid group element ξ = σ k 1 (k ∈ Z) become that distinguish links with even linking numbers. In this case, the braid operator has four eigenvalues ±λ 1 = ±h 1 , ± λ 2 = ± √ h 4 h 5 and can be used to obtain G 2link invariants [32]. Note that (5.10) is not a local invariant although it can be expressed in terms of the eigenvalues. Actually, under the transformation (2.3) with for k even, and for k odd.
The link invariants obtained in this case are just constants: ±1 (k odd). The link invariants obtained in this case are the same constants as (5.13).
For 3 and 4, the braid operators have four different eigenvalues, and G 2 -link invariants are expected to be obtained.
• Class 2 : In this case the braid operator can be enhanced using only µ = I. We then have x = ±h 3 , y = ±1. The link invariants obtained are similar to the Class 1 counterpart as seen for an element of the two-strand braid group, ξ = σ k 1 (k ∈ Z): (k odd), (5.14) that distinguish links with even linking numbers. The braid operator has three eigenval- implying that a scaled version of this braid operator, • Class 3 : Here enhancement occurs when µ = Z and µ The link invariants are constants in this case: The link invariants are the same constants as above.
As there are two distinct eigenvalues in these cases, {h 1 , h 8 }, each with multiplicity two, we expect to realize the Hecke algebra, H n (q), generated by invertible σ i , using this braid operator [33]. This happens either for In this case enhancement is possible when µ = I, µ = Z, µ = I ± Z and when µ = I + h 6 2h 1 −h 6 Z. 1. µ = I, x = ±h 1 , y = ±1 and h 6 = 0. We obtain constant link invariants: L R σ k 1 = 4 for k even and ±2 for k odd.
. In this case we obtain nontrivial link polynomials as seen in a two-strand braid group element ξ = σ k 1 (k ∈ Z): , (5.16) and in a three-strand braid group element ξ = σ k 1 1 σ k 2 2 (k 1 , k 2 ∈ Z): Since there are two distinct eigenvalues in these cases, {h 1 , −h 1 + h 6 } with multiplicities three and one respectively, we expect to realize the Hecke algebra (5.15), H n (q), with this braid operator and this indeed happens for either • Class 5 : Here enhancement occurs for µ = Z and µ = I ± Z.
In this case the two-strand braid group elements give vanishing link invariants. We can also see that elements of the threestrand braid group vanish: 5 L R σ k 1 σ l 2 = L R σ k 1 σ l 2 σ m 1 σ n 2 = 0 for k, l, m, n ∈ Z. 2. µ = I + Z, and x = ±h 1 , y = ±2. In this case, we obtain constant link invariants: In this case we also obtain constant link invariants: L R σ k 1 = (∓1) k . 5 Note that L R σ k 1 σ l 2 σ n 1 and L R σ k 2 σ l 1 σ n 2 reduce to L R σ k+n 1 σ l 2 and L R σ l 1 σ k+n 2 respectively, because R commutes with µ ⊗ µ.
For these cases, R 5 has two different eigenvalues {h 1 , −h 1 + h 6 } with multiplicity two for each. We see that the Hecke algebra (5.15), H n (q), is realized by • Class 6 : Enhancement is possible for the following five cases (λ ± = 1 2 h 1 + h 8 ± 2(h 2 1 + h 2 8 ) denote the eigenvalues of R): For the cases 2-5, we obtain the same result for the link invariants: L R σ k 1 = (±1) k . Each of the eigenvalues λ ± has multiplicity two. The braid operator can be used to realize the Hecke algebra by The case µ = I alone enhances this operator when x = ±(h 1 + h 3 ), y = ±1. We obtain non-trivial link invariants in this case as seen for two-strand and three-strand braid group elements: .
• Class 8 : We obtain just constant link invariants: L R σ k 1 = (±1) k 2 cos π 4 k . There are two distinct eigenvalues, (1 ± i)h 1 leading to a realization of the Hecke algebra either when σ i = − 1−i 2h 1 R 8 at q = i or when In this case enhancement is only possible with µ = I and x = ±h 1 , y = ±1. The link invariants obtained are just constant: L R σ k 1 = 4 for k even and ±2 for k odd. There are two distinct eigenvalues in this case, {h 1 , −h 1 } with multiplicities of three and one respectively. However, there is no realization of the Hecke algebra in this case.
• Class 10 : We can enhance this braid operator when µ = Z and µ = ∓i 2h 1 The link invariants are constant: L R σ k 1 = 1 for k even and ∓1 for k odd.
The link invariants give the same constants as above.
In each of these three cases the braid operator has two distinct eigenvalues, ±h 1 , each with multiplicity two. The Hecke algebra (5.15) is realized by σ i = ± 1 h 1 R 10 with q = 1.
Then, the relation reduces to σ 2 i = 1. 6 • Class 11 : In this case enhancement is only possible for µ = Z at x = ±ih 8 , y = ±i. The link invariants L R σ k 1 vanish. The braid operator has a single eigenvalue, h 8 with multiplicity four and the braid operator can be used to realize the Hecke algebra after scaling it with a factor − 1 h 8 at q = −1. Then, the relation reduces to (σ i + 1) 2 = 0.
• Class 12 : Enhancement is possible for the following five cases: For the cases 2-5, we obtain the same result for the link invariants: L R σ k 1 = (±1) k . The braid operator has a single eigenvalue, 1−i 2 h 1 , of multiplicity four, and can be used to realize the Hecke algebra at q = −1 after scaling it with a factor, − 1+i h 1 . Again, the relation reduces to (σ i + 1) 2 = 0.

Entangling power
We have seen in Sec. 4 that the independent local invariants for the X-type YBOs are functions of just their independent eigenvalues, implying that in these systems the quantum entangle-ment and its non-local properties are obtained in terms of the eigenvalues of the "entanglers".
A subtle feature, as we observed, is that this is not true for an entangler that is not a YBO.
As a further check of this, we compute here the entangling powers [25] of the X-type YBOs and compare it with the entangling power of an arbitrary X-type entangler.
The entangling power for an operator U is defined as where the overline denotes an average over some distribution of the product states, |ψ 1 ⊗ |ψ 2 and E denotes an entanglement measure for two-qubit states. To determine the entanglement measure in a two-qubit space, we look for independent local invariants under the action of SL(2, C) ⊗2 . The entanglement measure we choose to compute the entangling power is expected to be a function of only these local invariants.

Invariant of two-qubit states under SL(2, C) ⊗2
A two-qubit state with coefficients t i 1 i 2 is changed by an ILO Q = Q 1 ⊗ Q 2 ∈ SL(2, C) ⊗2 as which amounts to the change of the coefficients: Invariant quantities under the change (6.4) can be constructed by contracting indices of the coefficients by invariant tensors ia ja (a = 1, 2) for SL(2, C) at the a-th qubit. The invariant of the lowest order is quadratic in t: One can show that there is no independent invariant at higher orders in t as follows. It is easy to see that we cannot construct invariants of odd orders in t. Any invariant of the 2N -th order in t can be expressed as where (K 2N −2 ) i 2 j 2 denotes a polynomial of the (2N − 2)-th order in t with indices other than i 2 and j 2 contracted. We assume that invariants up to the order less than 2N are functions of J 2 . Due to the identity we obtain Note that i 2 j 2 (K 2N −2 ) i 2 j 2 is an invariant of the (2N − 2)-th order and thus a function of J 2 by the assumption. Hence, J 2N is also a function of J 2 , which completes a proof by the induction.
As another proof, we show that there is just a single local invariant for a two-qubit state, by considering the infinitesimal action of SL(2, C) ⊗2 on an arbitrary two-qubit state, as we mentioned below (3.6). This is obtained from the expressions and with i and i + 1 denoting the first and second qubits respectively. It can be checked that only three of these six vectors are linearly independent. The three vectors generate a threedimensional hypersurface in the four dimensions spanned by α 1 , · · · , α 4 . A single direction perpendicular to the hypersurface corresponds to a single local invariant.
Note that a general ILO belongs to C * · SL(2, C) ⊗2 rather than SL(2, C) ⊗2 . Due to the overall factor C * (multiplication by a nonzero complex number), only the value of J 2 being zero or non-zero has an SLOCC-invariant meaning and labels SLOCC classes. For instance, J 2 = 0(= 0) indicates the Bell-state class (the product-state class).

Entangling power of X-type YBOs
We discuss the entangling power of the twelve classes of X-type YBOs separately. We see that although the entangling power is not always a function only of eigenvalues for general YBOs, it is always so for unitary YBOs.
• Class 1 : The YBO R 1 has four free parameters, h 1 , h 4 , h 5 , h 8 , and its eigenvalues are λ 1+ = h 1 , The entangling power (6.15) reads e P (R 1 ) = 1 64 • Class 2 : Free parameters of the YBO R 2 are h 2 , h 3 , h 7 , and its eigenvalues are ±λ 1 = ± √ h 2 h 7 , is a function of only the local invariants, expressed only in terms of the eigenvalues.
These properties are not changed by enhancement or by imposing the unitary condition |h 2 | = |h 3 | = |h 7 | = 1.
• Class 3 : The YBO R 3 is a function of h 1 , h 7 , h 8 , and its eigenvalues are λ + = h 1 , λ − = h 8 , which are not changed by the enhancement. The entangling power is computed to be e P (R 3 ) = 9 64 which is now dependent on h 7 , a parameter that changes under the local action of SL(2, C) ⊗2 . R 3 is unitary for h 1 = −h 8 = e iϕ 1 and h 7 = 0, turning it into a special case of Class 1. Then (6.20) becomes a constant 1 16 , which is a trivial function of the eigenvalues.
• Class 4 : The YBO R 4 has three parameters h 1 , h 4 and h 6 , with its eigenvalues λ 1 = h 1 and λ 2 = −h 1 + h 6 , which is kept intact by enhancement. The entangling power is computed to be e P (R 4 ) = 9 64 |h 4 h 6 | 2 + 1 64 which implies that the unitary R 4 is not an entangler.
• Class 5 : The YBO R 5 is again a function of h 1 , h 4 and h 6 , with its eigenvalues λ + = h 1 and λ − = −h 1 + h 6 , before and after enhancement. The entangling power becomes e P (R 5 ) = 9 64 |h 4 h 6 | 2 + 1 16 The YBO R 6 has three parameters h 1 , h 2 , h 8 , and its eigenvalues are given by λ ± = 1 2 h 1 + h 8 ± 2(h 2 1 + h 2 8 ) , before and after the enhancement. The entangling power becomes e P (R 6 ) = 9 64 |h 2 h 8 | 2 + 1 16 which is now dependent on h 2 , a parameter that changes under the SL(2, C) ⊗2 . h 1 and h 8 are expressed by the eigenvalues as In this case R 6 cannot be unitary for any choice of the parameters.
• Class 7 : The YBO R 7 is a function of h 1 , h 2 and h 3 , with its eigenvalues λ + = h 1 + h 3 and , which is not changed by enhancement. The entangling power is computed to be e P (R 7 ) = 9 64  64 , which is a trivial function of the eigenvalues. Note that this coincides with the entangling power of the Bell matrix and is the largest possible entangling power in a two-qubit system.
• Class 9 : R 9 is a function of h 1 and h 7 with its eigenvalues ±h 1 , which is not changed by enhancement. The entangling power is computed to be e P (R 9 ) = 9 64 which is now dependent on h 7 , a parameter that changes under the local action of SL(2, C) ⊗2 . Again the second line in (6.15) vanishes.
R 9 becomes unitary when h 1 = e iϕ 1 and h 7 = 0, making it a special case of Class 1. Then the entangling power vanishes, implying that the unitary R 9 is not an entangler.
• Class 10 : Again, R 10 is a function of just h 1 and h 7 , with its eigenvalues ±h 1 , before and after enhancement. The entangling power is given by R 11 has free parameters h 7 and h 8 , and its eigenvalue is h 8 , which is not affected by enhancement. The entangling power is e P (R 11 ) = 9 64 where h 7 changes under the SL(2, C) ⊗2 . In this case R 11 cannot be unitary.
• Class 12 : R 12 is a function of h 1 and h 2 , with its eigenvalue 1−i 2 h 1 , before and after the enhancement. The entangling power is e P (R 12 ) = 9 64 where h 2 changes under the SL(2, C) ⊗2 . R 12 cannot be unitary.

Outlook
Quantum gates realized using braid operators are expected to create a robust entangled state from a product state. The entangled states thus obtained depend on parameters forming local invariants and are insensitive to local perturbations. Such parameters should characterize non-local properties of quantum entanglement. This criterion can be used to exclude braid operators that do not possess this property. To achieve this, it is essential to identify the complete set of parameters of local invariants for a braiding quantum gate that would determine the quantum entanglement of these systems. For the twelve classes of the X-type two-qubit braid operators considered in this paper, we found that the complete set is fixed by the independent eigenvalues of these operators. This is in marked contrast with the case of a generic two-qubit operator, whose eigenvalues alone are not sufficient to determine the entanglement measures of the system.
One of possible future directions would be to analyze robustness of entanglement [37] for braiding quantum gates and to understand how topological properties coming from the braid contribute to the robustness of the quantum entanglement. In addition, it would be crucial to check these features for multi-qubit braid operators that can be constructed using the generalized Yang-Baxter equation [35,36] for which several solutions have been found [38][39][40][41][42].

A Relation to the classification by Hietarinta
This rather technical appendix is devoted to a comparison between our results and the ones obtained by Hietarinta in [26].

A.1 Classification by Hietarinta
We start by summarizing Hietarinta's classification. In [26], all solutions to the constant algebraic Yang-Baxter equation: that is identical to (4.1).
Relevant results in [26] are summarized for solutions to (A.4) as follows. The continuous with κ a complex factor and Q an invertible 2 × 2 matrix, map a solution to a solution. Each of the following discrete transformations For X-type solutions that we consider in the text, the classification is valid with setting p = q = 0 in R H2,3 : • Class 2: This falls into R H1,4 with k = h 3 , p = h 2 and q = h 7 .
The seven other solutions given in the text are equivalent to the representative as 3.
• Class 7: This falls into R H1,4 with k = h 1 + h 3 and p = q = h 1 − h 3 by the transformation (A.5): The other solution is not equivalent to R 7 . Actually, we can see that R 7-1 falls into R H3,1 with p = q = h 1 − h 3 and k = s = h 1 + h 3 by κ(Q ⊗ Q)R H3,1 (Q ⊗ Q) −1 = R 7-1 with κ = 1 and Q = . However, R 7 and R 7-1 belong to the same class in our classification, since they have the same eigenvalues and quadratic invariants. We explicitly see that they are SL(2, C) ⊗2 -equivalent: (Q 1 ⊗ , R 9-1 becomes R 9 by (A.6) with h 2 → h 7 , whereas R 9-2 and R 9-3 are not equivalent to the representative R 9 . Actually, R 9-2 falls into R H2,3 with k = h 1 and s = h 7 by the transformation (A.6), and R 9-3 becomes R 9-2 by (A.6) with h 2 → h 7 . However, these two groups are SL(2, C) ⊗2 equivalent: (Q 1 ⊗ Q 2 )R 9 (Q 1 ⊗ Q 2 ) −1 = R 9-2 with (A.11). Among all the two-qubit braid operators in (A.9), the X-type braid operators analyzed in this paper do not fully capture solutions of the form R H2,3 . For completeness we analyze that case here.
We see that the result continues to possess the same features as the X-type braid operators.