[SOLVED] Quantum Homework 10

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 ∈ B^o.

1. Suppose that A and B are orthogonal to each other.
   • What is dim(A + B)?
   • Show that P(|vi,A + B) = P(|vi,A) + P(|vi,B).
   • Show that Π_AΠ_B = Π_BΠ_A.
2. Suppose that A ≤ V and B ≤ W are two subspaces.
   • Prove that Π_A⊗_B = Π_A ⊗ Π_B.
   • 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 ).