Unformatted text preview: 1 Â¶ . Problem 3 (20 points) Consider an operator A in H 4 and four orthonormal eigenvectors:  Ïˆ i i = Î± i 00 Î± i 01 Î± i 10 Î± i 11 1 â‰¤ i â‰¤ 4 . Construct the spectral decomposition of A . Problem 4 (20 points) Prove that the Hadamard operator on one qubit may be written as H = 1 âˆš 2 [(  i +  1 i ) h  +(  i  1 i ) h 1  ] Problem 5 (20 points) A quantum system is prepared in state  a i with probability 1 2 and in state  b i with probability 1 2 , where  a i = r 3 4  i + r 1 4  1 i and  b i = r 3 4  i r 1 4  1 i Calculate the corresponding density matrix Ï ....
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 Fall '08
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 Computer Science, Hilbert space, Quantum computing, Hadamard, Department of Electrical Engineering and Computer Science, Hadamard operator

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