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Unformatted text preview: θ 1 + θ 2 along the same axis: R A ( θ 1 ) R A ( θ 1 ) = R A ( θ 1 + θ 2 ) Problem 3 (20 points) Show that the classical Fredkin and Toﬀoli gates perform a fanout operation while their quantum counterparts do not violate the no cloning theorem. Hint: consider a control qubit in a superposition state 1 √ 2 (  i +  1 i ). Problem 4 (30 points) Given a system with density operator ρ and a set of orthonormal vectors  ϕ i i ∈ H n the spectral decomposition of the operator ρ is: ρ = X i λ i  ϕ i ih ϕ i  . The von Neumann entropy is S ( ρ ) =tr( ρ log ρ ) =X i λ i log λ i . 1 Show that when ρ = X i p i ρ i then S ( ρ ) =X i p i log p iX i p i S ( ρ i ) or S ( ρ ) = H ( p i )X i p i S ( ρ i ) with H ( p i ) the Shannon entropy of a random variable X with the probability density function p X = p X ( x = i i ) = p i . 2...
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 Fall '08
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 Computer Science, Linear Algebra, σz, Pauli matrices

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