[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 _AB = _BA.
2. Suppose that A V and B W are two subspaces.
   • Prove that _AB = _{A B}.
   • Let and be density matrices. Prove that P( ,A B) = P(,A)P(,B). You may use the fact that Tr(X Y ) = Tr(X)Tr(Y ).