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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>B</mi> <mn>6</mn> </msub> </semantics> </math> </inline-formula> to obtain a totally leakage-free braiding. The obtained representation is denoted by <inline-formula> <math display="inline"> <semantics> <mi>&#961;</mi> </semantics> </math> </inline-formula>. We first prove that <inline-formula> <math display="inline"> <semantics> <mi>&#961;</mi> </semantics> </math> </inline-formula> is irreducible. Next, we find the points <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>&#8712;</mo> <msup> <mi mathvariant="double-struck">C</mi> <mo>*</mo> </msup> </mrow> </semantics> </math> </inline-formula> at which the representation <inline-formula> <math display="inline"> <semantics> <mi>&#961;</mi> </semantics> </math> </inline-formula> is equivalent to the tensor product of a one dimensional representation <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#967;</mi> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi>&#956;</mi> <mo>^</mo> </mover> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mo>&#177;</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, an irreducible four dimensional representation of the braid group <inline-formula> <math display="inline"> <semantics> <msub> <mi>B</mi> <mn>6</mn> </msub> </semantics> </math> </inline-formula>. The representation <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi>&#956;</mi> <mo>^</mo> </mover> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mo>&#177;</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> was constructed by E. Formanek to classify the irreducible representations of the braid group <inline-formula> <math display="inline"> <semantics> <msub> <mi>B</mi> <mi>n</mi> </msub> </semantics> </math> </inline-formula> of low degree. Finally, we prove that the representation <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#967;</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&otimes;</mo> <msub> <mover accent="true"> <mi>&#956;</mi> <mo>^</mo> </mover> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mo>&#177;</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is a unitary relative to a hermitian positive definite matrix.

Abstract licence: CC BY

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