Selected topics in Mathematical Physics: Quantum Information Theory Talk 5: Averages over the unitary group
نویسنده
چکیده
In this talk, it is shown how to calculate averages over the Unitary group by applying mathematical tools presented in the previous talk. Particulary, Schur’s Lemma is utilized to two useful examples. Eventually, the results are used to show that a random pure state on a finite-dimensional bipartite Hilbert space is close to maximally entangled. 1 Representations of the Unitary group and Schur’s Lemma There are two common representations of the Unitary group: The standard representation. In the standard representation of U(n) on Cn every unitary matrix U ∈ U(n) is mapped onto itself, i.e. U 7→ U . This is a irreducible representation. The representation on Cn⊗Cn. A very useful representation when considering bipartite systems is the map U 7→ U ⊗ U . This is a reducible representation of U(n). The groupmodule Cn ⊗ Cn can be decomposed into the following two subspaces Csym := {|ψ〉 ∈ C ⊗ C : F(|ψ〉) = |ψ〉} (1) Cantisym := {|ψ〉 ∈ C ⊗ C : F(|ψ〉) = − |ψ〉} (2) They are defined through the ”flip operator” F on Cn ⊗ Cn, with: F(|ψ〉 ⊗ |φ〉) = |φ〉 ⊗ |ψ〉 (3) These two spaces are invariant and cannot be decomposed into more subspaces. One important result of group theory is Schur’s Lemma. There are different formulations depicting similar ideas that are called by the same name. The version presented here will be the most useful for this talks purpose. Schur’s Lemma. Let g 7→ Ug ∈ U(X) be a unitary representation of the group G on the finite-dimensional complex space X. Then there is a unique decomposition of X into subspaces Xi, so that these subspaces are mutually orthogonal. I.e.: X = m ⊗ i=1 Xi, Xi ⊥ Xj(i 6= j) (4)
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