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Unformatted text preview: Homework 9 Solutions Problem 1 (i)First of all, prove that for any arbitrary density matrices i and j , Tr ( i j ) 1 (1) with the upper limit being reached if and only if i = j is a pure density matrix. Each density matrix has its own spectral representation. ( i ) = X n i n | i n ih i n | (2) Thus, Tr ( i j ) = X n,m i n j m Tr ( | i n ih i n | j m ih j m | ) = X n,m i n j m |h i n | j m i| 2 = X n,m i n j m = 1 (3) This inequality becomes equality if and only if |h i n | j m i| = 1 for all i n j m 6 = 0. Since the eigenvectors have unit norm, | i n i and | j m i 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 = X i a i i , < a i 1 , X i a i = 1 (4) where is a pure density matrix and there are at least 2 distinct i , Tr( 2 ) = X i,j a i a j Tr ( i...
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