HW3 - -1 Problem 3(20 points Consider an operator A in H...

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Department of Electrical Engineering and Computer Science University of Central Florida COT 6600– Quantum Computing Fall 2010 (dcm) Homework Assignment 3. – Due Monday, October 26, 2010. Problem 1 (20 points) A linear operator A is said to be normal if it commutes with its adjoint [ A, A ] = 0. Let | φ a i be an eigenvector of A , A | φ a i = a | φ a i . 1) Show that Hermitian and unitary operators are “normal”. 2) Given that A | φ a i = a * | φ a i show that any two eigenvectors | φ a 1 i and | φ a 2 i of A where a 1 6 = a 2 , must be orthogonal, i.e., h φ a 1 | φ a 2 i = 0. Problem 2 (20 points) Given the state vector of a system of two qubits: | ψ i = 1 3 ( | 00 i + | 01 i + | 10 i ) calculate the new state | φ i of the system obtained after two Hadamard transformations: | φ i = ( H H ) | ψ i with the Hadamard operator: H = 1 2 1 1
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Unformatted text preview: -1 ¶ . Problem 3 (20 points) Consider an operator A in H 4 and four orthonormal eigenvectors: | ψ i i = α i 00 α i 01 α i 10 α i 11 1 ≤ i ≤ 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 √ 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 ρ ....
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