QUANTUM ALGORITHMS
Definition. Let V be a vector space and let A,B ≤ V be subspaces.
• We say that A is orthogonal to B if for every |ai ∈ A and |bi ∈ B we have ha | bi = 0.
• Define the sum of A and B to be A + B = n|ai + |bi | |ai ∈ A,|bi ∈ Bo.
1. Suppose that A and B are orthogonal to each other.
(i) What is dim(A + B)?
(ii) Show that P(|vi,A + B) = P(|vi,A) + P(|vi,B).
(iii) Show that ΠAΠB = ΠBΠA.
2. Suppose that A ≤ V and B ≤ W are two subspaces.
(i) Prove that ΠA⊗B = ΠA ⊗ ΠB.
(ii) Let ρ and τ be density matrices. Prove that P(ρ ⊗ τ,A ⊗ B) = P(ρ,A)P(τ,B). You may use the fact that Tr(X ⊗ Y ) = Tr(X)Tr(Y ).
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