3. Detail about measurements associated with Kraus operators [8 points].

Prove that if matrices A₁, A2,..., Am are Kraus operators (i.e., they satisfy the equation

Σk=1 A₁ Ak = I) and p is any valid density matrix then, for all k € {1,2,...,m},

AkpA

Tr(AkpA)

(3)

is also a valid density matrix. (A square matrix M is a valid density matrix if, for all

states ), it holds that (M) ≥ 0 and Tr(M)= 1.)