soln9_09

soln9_09 - Ph219a/CS219a Solutions to Hw 9 June 5, 2009...

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Unformatted text preview: Ph219a/CS219a Solutions to Hw 9 June 5, 2009 Problem 1 (a) Clearly for x = 1, ln x = x- 1 = 0. Since the function f ( x ) = ln x is strictly concave, ln x x- 1 for x 6 = 1, if x- 1 is a tangent at x = 1. It is easy to see that this is indeed the case, since their slopes match at x = 1: d (ln x ) dx | x =1 = 1 x | x =1 = 1 and d ( x- 1) dx = 1. (b) Using the above inequality, we have, ln p ( x ) q ( x ) 1- q ( x ) p ( x ) H ( p k q ) = X x p ( x )log p ( x ) q ( x ) X x p ( x ) 1- q ( x ) p ( x ) = 0 (1) Equality holds iff q ( x ) p ( x ) = 1 for all x , ie. iff the distributions { p ( x ) } and { q ( x ) } are identical. (Note that in the second step, we have omitted a factor of ln2 in going from the natural loga- rithm to the log function.) (c) Expanding and in their eigenbasis, as = i p i | i ih i | and = a q a | a ih a | , we have, H ( k ) = tr (log - log ) = X i p i log p i- X i,a p i log q a h i | a ih a | i i = X i p i log p i- X a D ia log q a ! (2) where we have defined the matrix with elements D ia = |h i | a i| 2 . It is easy to see that D is a doubly stochastic matrix: a D ia = a h i | a ih a | i i = h i | i i = 1 = h a | a i = i D ia . 1 (d) Since the log function is strictly concave, and the set a D ia = 1 for each i (implying D ia 1 i,a ), Jensens inequality directly gives, log X a D ia q a ! X a D ia log q a (3) with equality only if D ia = 1 for some a . (e) Putting Eqns.(2) and (3) together, we have, H ( k ) = X i p i log p i- X a D ia log q a ! X i p i log p i- log X a D ia q a ! = X i p i (log p i- log r i ) = H ( p k r ) (4) with r i = a D ia q a , and equality iff D ia = 1 for some a , for each i . (f) Now that we have expressed the quantum relative entropy as a classical relative entropy between two classical distributions p = { p i } and r = { r i } , we can use the result of part(b), so that H ( k ) H ( p k r ) (5) The second inequality saturates iff p i = r i = a D ia q a , and the first inequality saturates iff D ia = 1 for some a , for each i . Thus D has to be a permutation matrix, which implies that the set { p i } and { q a } are the same upto a permutation. Thus, equality holds iff = . Problem 2 (a) Consider the relative entropy of AB and the product state A B . Using the positivity of relative entropy, we have, H ( AB k A B ) Tr[ AB log AB ]- Tr[ AB log( A B )] (6) Using log( M + N ) = log...
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soln9_09 - Ph219a/CS219a Solutions to Hw 9 June 5, 2009...

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