dorsal/arxiv
View SchemaProjecting onto Qubit Irreps of Young Diagrams
| Authors | Stephen S Bullock |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0608150 |
| URL | https://arxiv.org/abs/quant-ph/0608150 |
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.
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"abstract": "Let K be the diagonal subgroup of U(2)^{(x)n}. We may view the one-qubit\nstate-space H_1 as a standard representation of U(2) and the n-qubit state\nspace H_n=(H_1)^{(x) n} as the n-fold tensor product of standard\nrepresentations. Representation theory then decomposes H_n into irreducible\nsubrepresentations of K parametrized by combinatorial objects known as Young\ndiagrams. We argue that n-1 classically controlled measurement circuits, each a\nFredkin-gate interferometer, may be used to form a projection operator onto a\nrandom Young diagram irrep within H_n. For H_2, the two irreps happen to be\northogonal and correspond to the symmetric and wedge product. The latter is\nspanned by ket{Psi^-}, and the standard two-qubit swap interferometer requiring\na single Fredkin gate suffices in this case. In the n-qubit case, it is\npossible to extract many copies of ket{Psi^-}. Thus applying this process using\nnondestructive Fredkin interferometers allows for the creation of entangled\nbits (e-bits) using fully mixed states and von Neumann measurements.",
"arxiv_id": "quant-ph/0608150",
"authors": [
"Stephen S Bullock"
],
"categories": [
"quant-ph"
],
"title": "Projecting onto Qubit Irreps of Young Diagrams",
"url": "https://arxiv.org/abs/quant-ph/0608150"
},
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