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

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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 = 0 - i i 0 , and Z = 1 0 0 - 1 . (1) Exercise 1: Tsirel’son’s 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 | ψ =0 || M | ψ || ||| ψ || (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 Tsirel’son’s (or Cirel’son’s) 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: | C | ≤ 2. So Tsirel’son’s inequality bounds the “amount” of violation that quantum states can have over the CHSH inequality. In fact quantum theory can saturate this bound. Exercise 2: A Quantum Error Detecting Code In this problem we will examine a quantum error detecting code on four qubits.
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