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Unformatted text preview: CSE 599d Quantum Computing Problem Set 3 Author: Dave Bacon ( Department of Computer Science & Engineering, University of Washington ) Due: February 17, 2006 For this problem set recall that the Pauli X , Y , and Z are X = 0 1 1 0 , Y = i i , and Z = 1 1 . (1) Exercise 1: Tsirelsons Inequality Suppose that A , A , B , B are operators on some Hilbert space H which satisfy A 2 = A 2 = B 2 = B 2 = I and [ A,B ] = [ A,B ] = [ A ,B ] = [ A ,B ] = 0 (where the commutator is [ M,N ] = MN NM .) (a) Define C = AB + AB + A B A B . Show that C 2 = 4 I [ A,A ][ B,B ]. (b) The sup norm of an operator M is defined as  M  sup = sup  i6 =0  M  i  i (2) where   is the standard norm on our Hilbert space. Prove that  M + N  sup  M  sup +  N  sup (3) and  MN  sup  M  sup  N  sup (4) (c) Use these properties of the sup norm to show that  C  sup 2 2 (5) This is Tsirelsons (or Cirelsons) inequality. Suppose we are working on a Hilbert space of two qubits. If we take A = A 1 I , A = A 2 I , B = I B 1 , and B = I B 2 , then this expression is  A 1 B 1 + A 1 B 1 + A 2 B 1 A 2 B 2  sup 2 2 (6) Recall that from class we saw that for local hidden variable theories satisfy the CHSH inequality: h C i 2. So Tsirelsons inequality bounds the amount of violation that quantum states can have over the CHSH inequality. In fact quantum theory can saturate this bound.fact quantum theory can saturate this bound....
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This note was uploaded on 11/06/2011 for the course CSE 599 taught by Professor Staff during the Fall '08 term at University of Washington.
 Fall '08
 Staff
 Computer Science

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