Projecting onto Qubit Irreps of Young Diagrams

Abstract

Let K be the diagonal subgroup of U(2)^{(x)n}. We may view the one-qubit state-space H_1 as a standard representation of U(2) and the n-qubit state space H_n=(H_1)^{(x) n} as the n-fold tensor product of standard representations. Representation theory then decomposes H_n into irreducible subrepresentations of K parametrized by combinatorial objects known as Young diagrams. We argue that n-1 classically controlled measurement circuits, each a Fredkin-gate interferometer, may be used to form a projection operator onto a random Young diagram irrep within H_n. For H_2, the two irreps happen to be orthogonal and correspond to the symmetric and wedge product. The latter is spanned by ket{Psi^-}, and the standard two-qubit swap interferometer requiring a single Fredkin gate suffices in this case. In the n-qubit case, it is possible to extract many copies of ket{Psi^-}. Thus applying this process using nondestructive Fredkin interferometers allows for the creation of entangled bits (e-bits) using fully mixed states and von Neumann measurements.