Abstract
Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the (d,m,l)-generalized Yang-Baxter equation, for m/2≤l≤m, which allows to systematically construct such braiding operators. This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics. We obtain families of unitary and non-unitary braiding operators that generate the full braid group. Explicit examples are given for a 2-, 3-, and 4-qubit system, including the classification of the entangled states generated by these operators based on Stochastic Local Operations and Classical Communication.
Highlights
This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics
We study the Stochastic Local Operations and Classical Communication (SLOCC) classes of entangled states generated by these matrices
We review the notion of set partitions and partition algebras following [29]
Summary
We review the notion of set partitions and partition algebras following [29]. We present just the bare minimum needed in this work, pointing the reader to that reference for more details. As an example consider the following diagram showing the partition of a set with k = 7, this represents the set partition {{1, 3, 5, 4 , 5 } , {2, 3 } , {4, 6, 7, 6 } , {1 , 2 } , {7 }}. D1 ◦ d2, place d1 above d2 and trace the lines to obtain the new partition. Using these diagrams one can verify that the i i+1i+2 pi pi+. Linear combinations of elements of Ak with coefficients being complex numbers form the partition algebra CAk(1)
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