hw5sol_fa07 - C/CS/Phy191 Problem Set 5 Out: October 21,...

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Unformatted text preview: C/CS/Phy191 Problem Set 5 Out: October 21, 2007 1. The uncertainty principle bounds how well a quantum state can be localized simultaneously in the standard basis and the Fourier basis. In this question, we will derive an uncertainty principle for a discrete system of n-qubits. Let | i = x { , 1 } n x | x i be the state of an n-qubit system. A measure of the spread of | i is S ( | i ) x | x | . For example, for a completely localized state | i = | y i ( y { , 1 } n ), the spread is S ( | i ) = 1. For a maximally spread state | i = 1 2 n x | x i , S ( | i ) = 2 n 1 2 n = 2 n . a) Prove that for any quantum state | i on n qubits, S ( | i ) 2 n / 2 . (Hint: use the Cauchy-Schwarz inequality, v w k v kk w k .) Answer : We need to show that if x { , 1 } n | x | 2 = 1, then x | x | 2 n / 2 . Using the Cauchy-Schwarz inequality v w k v kk w k , we get x | x | = x ( | x | 1 ) x | x | 2 1 / 2 x 1 2 1 / 2 = 1 2 n / 2 = 2 n / 2 , with equality iff | x | = 1 / 2 n / 2 for all x . b) Suppose that | x | a for all x . Prove that S ( | i ) 1 a . (Hint: think about normalization....) Answer : Using the normalization condition, 1 = x | x | 2 x a | x | = aS ( ) . (Notice that this inequality is an equality iff all x are zero or exactly a that is, to minimize the spread, concentrate the probability mass as much as possible while still satisfying the constraint | x | a .) Now let H n | i = x x | x i , where (by homework 2) x = 1 2 n / 2 y ( 1 ) x y y . ( x y n i = 1 x i y i .) c) In Problem Set 3 you showed that H n x = y ( 1 ) x y y ( x y n i = 1 x i y i ). Hence we can obtain the action of H n on as H n | i = x x | x i , where x = 1 2 n / 2 y ( 1 ) x y y . Use this to prove that for all y , | y | 1 2 n / 2 S ( | i ) . (Hint: use the triangle inequality.) Answer : Using the triangle inequality | a + b | | a | + | b | , | y | = 1 2 n / 2 | y ( 1 ) x y y | 1 2 n / 2 y | ( 1 ) x y y | = 1 2 n / 2 y | y | = 1 2 n / 2 S ( ) ....
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hw5sol_fa07 - C/CS/Phy191 Problem Set 5 Out: October 21,...

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