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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 ρ and σ in their eigenbasis, as ρ = ∑ i p i  i ih i  and σ = ∑ a q a  a ih a  , we have, H ( ρ k σ ) = tr ρ (log ρ log σ ) = 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 ), Jensen’s 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 ( ρ k σ ) = 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 ( ρ k σ ) ≥ 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 ρ = σ . Problem 2 (a) Consider the relative entropy of ρ AB and the product state ρ A ⊗ ρ B . Using the positivity of relative entropy, we have, H ( ρ AB k ρ A ⊗ ρ B ) ≥ ⇒ Tr[ ρ AB log ρ AB ] Tr[ ρ AB log( ρ A ⊗ ρ B )] ≥ (6) Using log( M + N ) = log...
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 Spring '09
 Linear Algebra, Día, Hilbert space, density matrix, Dia qa

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