Quantum Computing based on Tensor Products ... - Cinvestav
Quantum Computing based on Tensor Products ... - Cinvestav
Quantum Computing based on Tensor Products ... - Cinvestav
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Hilbert Spaces<br />
Basic noti<strong>on</strong>s<br />
Complex field. C<br />
Vector space. H: N<strong>on</strong>-empty set. 0 ∈ H<br />
Additi<strong>on</strong>. + : H × H → H. (H, +) Abelian group<br />
Scalar multiplicati<strong>on</strong>. · : C × H → H. Distributive w.r.t. additi<strong>on</strong><br />
Inner product. 〈·|·〉 : H × H → C. Sesquilinear form. Positive definite.<br />
Norm. � · �2 : H → R + , x ↦→ � · �2 = √ 〈x|x〉.<br />
Completeness. Every Cauchy sequence is c<strong>on</strong>vergent.<br />
Autoduality. For each T ∈ H ∗ exists y ∈ H: T(x) = 〈y|x〉.<br />
Morales-Luna (CINVESTAV) QC <str<strong>on</strong>g>based</str<strong>on</strong>g> <strong>on</strong> <strong>Tensor</strong> <strong>Products</strong> 5-th Int. WS App. Cat. Th. 14 / 38
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