homework3 - CSE 599d Quantum Computing Problem Set 3...

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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.

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homework3 - CSE 599d Quantum Computing Problem Set 3...

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