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Unformatted text preview: February 16, 2006 Physics 681481; CS 483: Discussion of #2 I. Constructing a spooky 2Qbit state We can write the state  Ψ i as  Ψ i = 1 √ 3 (  00 i +  01 i +  10 i ) = ( 1 ⊗ H )( q 2 3  i i + q 1 3  1 i H  i ) = ( 1 ⊗ H ) C H h ( q 2 3  i + q 1 3  1 i )  i i = ( 1 ⊗ H ) C H ( w ⊗ 1 )  i i , (1) where w is any oneQbit unitary transformation that takes  i into q 2 3  i + q 1 3  1 i . To construct a controlledHadamard C H from a controlledNOT C , note that the NOT operation X is x · σ while the Hadamard transformation is H = 1 √ 2 ( X + Z ) = 1 √ 2 ( x + z ) · σ. It follows from Section A2 of the appendix to chapter 1 that H = uXu † , (2) where u is the oneQbit unitary transformation associated with any rotation that takes x into 1 √ 2 ( x + z ). Since we also have 1 = uu † , it follows that C H = ( 1 ⊗ u ) C ( 1 ⊗ u † ) . (3) So (1) reduces to the compact form  Ψ i = ( 1 ⊗ Hu ) C ( w ⊗ u † )  i i , (4) which produces the state...
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 Spring '08
 Ginsparg
 Linear Algebra, unitary transformation, Qbits

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