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Unformatted text preview: CSE 599d Quantum Computing Problem Set 1 Author: Dave Bacon ( Department of Computer Science & Engineering, University of Washington ) Due: January 20, 2006 Exercise 1: Majorization and Random Permutations Let x = ( x 1 ,x 2 ,...,x n ) denote a vector of n real numbers, x ∈ R n . Define x ↓ as the vector x sorted such that the components of the vector are in decreasing order, x ↓ = ( x ↓ 1 ,x ↓ 2 ,...,x ↓ n ) where x ↓ 1 ≥ x ↓ 2 ≥ ··· ≥ x ↓ n . Thus, for example, x ↓ 1 is the largest component of x . We say that the vector x is majorized by the vector y if ∑ k i =1 x ↓ i ≤ ∑ k i =1 y ↓ i for all k < n (i.e. k = 1 , 2 ,...,n- 1) and ∑ n i =1 x ↓ i = ∑ n i =1 y ↓ i . When x is majorized by y we write x ≺ y . (a) Suppose that p ∈ R n is such that p i ≥ 0 and ∑ n i =1 p i = 1 (we say that p is a vector of probabilities). There is a single vector of probabilities which is majorized by all other vectors of probabilities. What is this vector and prove that it is the only vector which has this property. (b) An n × n matrix A = ( a ij ) is called doubly stochastic if a ij ≥ 0 for all i and j , ∑ n i =1 a ij = 1 for all j and ∑ n j =1 a ij = 1 for all i . Show that every convex combination of a doubly stochastic matrices is a doubly stochastic matrix (recall that a convex combination of matrices A 1 ,A 2 ,...,A m is a sum of these matrices, ∑ m j =1 q j A j with q j ≥ 0 and ∑ m j =1 q j = 1.) (c) Prove that if Ax ≺ x for all x then A must be doubly stochastic (hint consider the vector from part (a) as well as vectors like (0 , , 1 , ,..., 0).) (d) Prove that if A is doubly stochastic then Ax ≺ x for all vectors x ....
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- Fall '08
- Computer Science, Black Box, Pauli, Wolfgang Pauli, Pauli matrices