On the Unitary Representations of the Braid Group B 6

We consider a non-abelian leakage-free qudit system that consists of two qubits each composed of three anyons. For this system, we need to have a non-abelian four dimensional unitary representation of the braid group B6 to obtain a totally leakage-free braiding. The obtained representation is denoted by ρ. We first prove that ρ is irreducible. Next, we find the points y ∈ C∗ at which the representation ρ is equivalent to the tensor product of a one dimensional representation χ(y) and μ̂6(±i), an irreducible four dimensional representation of the braid group B6. The representation μ̂6(±i) was constructed by E. Formanek to classify the irreducible representations of the braid group Bn of low degree. Finally, we prove that the representation χ(y) ⊗ μ̂6(±i) is a unitary relative to a hermitian positive definite matrix.


Introduction
Due to Artin, the braid group B n is represented in the group Aut(F n ) of automorphisms of the free group F n generated by x 1 , . . . , x n . The matrix representation of B n was published by W. Burau in 1936. This representation was known as a Burau representation. Since then, other matrix representations of B n have been constructed. For more details, see [1].
Braid group unitary representations have been essential in topological quantum computations. To understand the d-dimensional systems in which anyons are exchanged, a lot of work has been made. The exchange of n anyons inside the qudit system, the d-dimensional analogues of qubits, has been governed by the braid group B n which has n − 1 generators τ 1 , . . . , τ n−1 . Here, τ i exchanges the particle i with its neighbor, particle i + 1.
When the topological charge of the qudits changes due to the braiding of the anyons from different qudits, a leakage of some of the information will occur in the computational Hilbert space, the fusion space of the anyons.
The leakage-free braiding of anyons has been under investigation for a while. To perform universal quantum computation without any leakage, the requirement would be to consider two-qubit gates. This would be very restrictive and this property can only be realized for two-qubit systems related to the Ising-like anyons model [2].
R. Ainsworth and J.K. Slingerland showed that a non-abelian, leakage-free qudit of dimension d involving n anyons is equivalent to a non-abelian d-dimensional representation of the braid group B n . Here, n is the sum of the number of anyons n 1 inside the first qudit and the number of anyons n 2 inside the second qudit. As for the dimension d of the representation of B n , it is the product of the dimensions d 1 and d 2 of the Hilbert spaces of the individual qudits.
Moreover, it was proved in [2] that the number of anyons per qubit is either 3 or 4. Thus, there are mainly 3 different types of two-qubit systems and a 4-dimensional representation of the corresponding braid group is constructed for each. Taking into account E. Formanek's result that there is no d-dimensional representation of B n with d ≺ n − 2, it was verified in [2] that the only possible type of two-qubit system is having 2 qubits of which each is composed of 3 anyons.
This system is a non-abelian leakage-free qudit system of dimension 4 involving 6 anyons. It is equivalent to a non-abelian 4-dimensional representation of the braid group B 6 . This representation is denoted by ρ. Since the number of anyons is 6, there are 5 elementary exchanges τ 1 , . . . , τ 5 . The exchanges τ 1 , τ 2 , τ 4 , and τ 5 satisfy the following relations: where ρ 1 and ρ 2 are the d 1 and d 2 -dimensional representations of B n 1 and B n 2 on the Hilbert spaces of the first and second qudit respectively. I d 1 and I d 2 are the d 1 and d 2 -dimensional identity matrices respectively. Here, n 1 = n 2 = 3 and d 1 = d 2 = 2.
The matrix ρ(τ 3 ) is constructed by imposing braid group relations. For more details, see [2]. In our work, we consider the unitary representation ρ and the irreducible representationμ 6 (±i) which is defined by E. Formanek in [3]. Both representations are 4-dimensional representations of the braid group B 6 .
First, we prove that the unitary representation ρ : B 6 → GL 4 (C) is irreducible.
As the representation ρ is proved to be irreducible, it follows that it is equivalent to the tensor product of a one-dimensional representation χ(y) and the irreducible 4-dimensional representation µ 6 (±i), where y ∈ C * . For more details, see [3].
We then determine the points y ∈ C * at which the two representations ρ and χ(y) ⊗μ 6 (±i) are equivalent.
Finally, we show that the representation χ(y) ⊗μ 6 (±i) is a unitary relative to a hermitian positive definite matrix.

Preliminaries
Definition 1 (See [4]). The braid group on n strings, B n , is the abstract group with presentation The Hecke algebra representation of B 6 was constructed by V.F.R. Jones in [5]. E. Formanek obtained a low-degree representation of B 6 by conjugating the representation constructed by V.F.R. Jones by a certain permutation matrix. For more details, see [3].
is the reduced Burau representation, andβ n (z) is the composition factor of the reduced Burau representation.

Irreducibility of
The construction of a two-qubit system with a minimum amount of leakage has been of great interest. The only two-qubit system that can be realized without leakage is the system of two 3-anyon qubits. This system is equivalent to a 4-dimensional representation of the braid group B 6 . This representation which was constructed in [2] is denoted by ρ.
In this section, we prove that ρ : B 6 → GL 4 (C) is irreducible. We denote τ i , the exchange of the i th and (i + 1) th anyon, by σ i where 1 ≤ i ≤ 5.
That is, a must be a primitive eighth root of unity. Furthermore, Note that since a is a primitive eighth root of unity, a 8 = 1 and a 2 1. Then, a 3 a. Consequently, a 3 ā, f f 3 , andf f 3 . This emphasizes that the defined matrices ρ(σ i ), 1 ≤ i ≤ 5, are well-defined.
Now we study the irreducibility of ρ. For simplicity, we denote ρ(σ i ) by σ i for 1 ≤ i ≤ 5. For simplicity, we take i = 1. Since S is invariant, it follows that σ 2 (e 1 ) = This implies that c = 0, a contradiction.
For simplicity, we take i = 1. Since S is invariant, it follows that σ 2 (e 1 + ue 2 ) = This implies that c = 0, a contradiction.
Thus, there are no non trivial proper invariant subspaces of dimension 1. For simplicity, we take i = 1. Since S is invariant, it follows that σ 2 (e 1 ) = This implies that c = 0, a contradiction.
For simplicity, we take i = 1. Since S is invariant, it follows that σ 4 (e 1 ) = This implies that e = 0. But, e = c. Thus, c = 0, a contradiction.
For simplicity, we take i = 3. Since S is invariant, it follows that σ 4 (e 3 ) = This implies that e = 0. But, e = c. Thus, c = 0, a contradiction.
For simplicity, we take i = 1. Since S is invariant, it follows that σ 4 (e 1 ) = This implies that e = 0. But, e = c. Thus, c = 0, a contradiction.
Thus, there are no non trivial proper invariant subspaces of dimension 2. Now, we state the theorem of irreducibility. Clearly, the representation ρ is unitary, that is σ i σ * i = I 4 for 1 ≤ i ≤ 5. We note that if the representation is unitary, then the orthogonal complement of a proper invariant subspace is again a proper invariant subspace. As there is no proper invariant subspace of dimension 1, there is no proper invariant subspace of dimension 3.
As a result, all the possible proper subspaces are not invariant. Consequently, ρ is irreducible.

The
Representations ρ and χ(y) ⊗μ 6 (±i) Are Equivalent By Theorem 3, the representation ρ is irreducible. The eigen values of ρ(σ i ) for 1 ≤ i ≤ 5 are different from those ofβ 4 (z), the composition factor of the reduced Burau representation. Therefore, the representation ρ is not equivalent to the tensor product of a one dimensional representation χ(y) andβ 4 (z). That is, ρ is not of a Burau type.
Moreover, ρ is a 4-dimensional representation. Consequently, Theorem 2 implies that the representation ρ is equivalent to the representation χ(y) ⊗μ 6 (±i) for some y ∈ C * . Note that, by Theorem 1, the representationμ 6 (z) is irreducible for z = ±i since the roots of the polynomial t 2 + 1 are clearly ±i.
In this section, we determine the points y ∈ C * at which the representations ρ and χ(y) ⊗μ 6 (±i) are equivalent.
As a result, the two considered representations are equivalent at the following points: where i is the complex number such that i 2 = −1.
In this section, we find the matrix M and we prove that M is a hermitian and positive definite.
Let M * be the complex conjugate transpose of M. Clearly, M * = M. This implies that the invertible matrix M is hermitian.
By computations, the eigen values of the matrix M are 2 + √ 2 and 2 − √ 2. Clearly, both values are positive. Consequently, M is a positive definite matrix.
As a result, the representationμ 6 (±i) is a unitary relative to an invertible hermitian positive definite matrix M.
Note that the unitarity of the representationμ 6 (±i) relative to the matrix M clearly implies that the representation χ(y) ⊗μ 6 (±i) is also a unitary relative to the same matrix M.