For a mixed state ρ, the expected value of A in the state ρ is tr ⁡ ⁡ ( A ρ ρ ) , and the probability of obtaining an eigenvalue a n in a discrete, nondegenerate spectrum of the corresponding observable A is given by P ( a n ) = tr ⁡ ⁡ ( | a n ⟩ ⟩ ⟨ ⟨ a n | ρ ρ ) = ⟨ ⟨ a n | ρ ρ | a n ⟩ ⟩ .