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HW2 - | Problem 3(30 points Prove Schwarz inequality for...

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Department of Electrical Engineering and Computer Science University of Central Florida COT 6600– Quantum Computing Fall 2010 (dcm) Homework Assignment 2. – Due Wednesday November 5, 2008. Problem 1 (30 points) Prove the following properties of a tensor product: A ( B C ) = ( A B ) C ( A B ) = B A ( A · A 0 ) ( B · B 0 ) = ( A B ) · ( A 0 B 0 ) where A, A 0 , B, B 0 and C are matrices of appropriate sizes and A · B denotes the product of matrices A and B and A B their tensor product. Problem 2 (10 points) Show that unitary matrices preserve inner products and thus they preserve norms. If U is a unitary matrix then: V, V 0 C n = ( UV, UV 0 ) = ( V, V 0 ) and V, C n = |
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Unformatted text preview: | . Problem 3 (30 points) Prove Schwarz inequality for the inner product of two linear operators A and B : tr( A · A † ) · tr( B · B † ) ≥| tr( A · B † ) | 2 Problem 4 (10 points) Show that A is Hermitian (self-adjoint) if and only if: A T = A * with A T the transpose of A and A * the complex conjugate of A . Problem 5 (20 points) Consider the following three matrices: S x = ± 0 1 1 0 ¶ , S y = ± 0 -i i ¶ , S z = ± 1 0 -1 ¶ . Construct the commutators [ S x ,S y ] , [ S y ,S z ] and [ S z ,S x ]....
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