HW9%20solutions - Homework 9 Solutions Problem 1(i)First of...

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Homework 9 Solutions Problem 1 (i)First of all, prove that for any arbitrary density matrices ρ i and ρ j , Tr ( ρ i ρ j ) 1 (1) with the upper limit being reached if and only if ρ i = ρ j is a pure density matrix. Each density matrix has its own spectral representation. ρ ( i ) = n ρ i n | φ i n φ i n | (2) Thus, Tr ( ρ i ρ j ) = n,m ρ i n ρ j m Tr ( | φ i n φ i n | φ j m φ j m | ) = n,m ρ i n ρ j m | φ i n | φ j m | 2 = n,m ρ i n ρ j m = 1 (3) This inequality becomes equality if and only if | φ i n | φ j m | = 1 for all ρ i n ρ j m = 0. Since the eigenvectors have unit norm, | φ i n and | φ j m differ by at most a phase factor. Each set of eigenvectors is orthogonal, and there is only one n and one m that contributes to the double sum above. (ii)Substituting ρ = i a i ρ i , 0 < a i 1 , i a i = 1 (4) where ρ is a pure density matrix and there are at least 2 distinct ρ i , Tr( ρ 2 ) = i,j a i a j Tr ( ρ i ρ j ) i,j a i a j = 1 (5) with the equality reached when ρ i = ρ j is a pure state. Since there are at least 2 distinct ρ i , the trace is strictly less than one.
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