Quantum Institute: Visitor Schedule

The Quantum Lunch is regularly held on Thursdays in the Theoretical Division Conference Room, TA-3, Building 123, Room 121. For more information, contact Diego Dalvit.

November 29, 2007
Thursday, 12:30 PM

John Yard,
T-CNLS

Applications of Schur Duality to Quantum Information and Computation

Abstract

Decomposing tensor representations is at the heart of much of quantum mechanics. For instance, the tensor product of two spin-1/2 representations of SU(2) decomposes as a direct sum of a singlet and a triplet subspace. More generally, the n'th tensor power of the standard representation of SU(d) decomposes as a direct sum of tensor products of irreps of SU(d) and the symmetric group of permutations on n letters, yielding a dual relationship (known as Schur duality) between representations of SU(d) and of the symmetric group. In this talk, I will give applications of Schur duality and its generalizations to quantum information theory and quantum computation. For this, I will first show how Schur duality leads to efficient protocols for processing quantum data. I will then discuss a deformed version of this duality where the unitary group is replaced by its corresponding quantum group and the symmetric group by the braid group on n strands, showing how it leads to definitions of the Jones and HOMFLY polynomial invariants of knots and links, together with algorithms for their approximation on a quantum computer. I will then discussion an analogous decomposition occurring when the unitary group is replaced by an orthogonal or symplectic group and the symmetric group by a Brauer algebra. I will then outline how the quantum group versions of these decompositions lead to definitions of the Kauffman polynomial link invariant, concluding with a discussion of how the Kauffman polynomial can be approximated on a quantum computer.