soln1

soln1 - EE 520 Quantum Information Processing Solution to...

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EE 520: Quantum Information Processing Solution to Homework # 1 Exercise 2.2 In terms of the basis {| 0 i , | 1 i} ˆ A = ± 0 1 1 0 ! = | 1 ih 0 | + | 0 ih 1 | . We can transform ˆ A by choosing a different basis. For instance, in the basis |±i = ( | 0 i ± | 1 i ) / 2 ˆ A = ± 1 0 0 - 1 ! = | + ih + | - |-ih-| . Exercise 2.9 ˆ X = | 1 ih 0 | + | 0 ih 1 | , ˆ Y = i | 1 ih 0 | - i | 0 ih 1 | , ˆ Z = | 0 ih 0 | - | 1 ih 1 | , Exercise 2.10 In the | v i i basis, the dyad | v j ih v k | is the matrix with the jk element equal to 1 and all other elements 0. Exercise 2.11 All three of the Pauli matrices have eigenvalues ± 1, which means they all have the same diagonal representation ± 1 0 0 - 1 ! The ± 1 eigenvectors for ˆ X, ˆ Y , ˆ Z (respectively) are { ( | 0 i ± | 1 i ) / 2 } , { ( | 0 i ± i | 1 i ) / 2 } , and {| 0 i , | 1 i} (up to a phase). 1
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Exercise 2.17 If a normal matrix ˆ H is Hermitian ˆ H = ˆ H , then it is Hermitian in any basis: ( ˆ U ˆ H ˆ U ) = ˆ U ˆ H ˆ U . In particular, it must be Hermitian in the diagonal representation, which implies that all of its eigenvalues λ i (along the diagonal) must satisfy λ i = λ * i . The converse is even simpler. If ˆ H is normal and has real eigenvalues, then in the diagonal representation it is clearly Hermitian (since λ i = λ * i , which means that it is Hermitian in any basis (as shown above). Exercise 2.19
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This note was uploaded on 10/15/2008 for the course EE 520 taught by Professor Brun during the Fall '08 term at USC.

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soln1 - EE 520 Quantum Information Processing Solution to...

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