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 to obtain a totally leakage-free braiding. The obtained representation is denoted by . We first prove that is irreducible. Next, we find the points at which the representation is equivalent to the tensor product of a one dimensional representation and , an irreducible four dimensional representation of the braid group . The representation was constructed by E. Formanek to classify the irreducible representations of the braid group of low degree. Finally, we prove that the representation is a unitary relative to a hermitian positive definite matrix.
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