UNITARY OPERATORS, ENTANGLEMENT, AND GRAM–SCHMIDT ORTHOGONALIZATION

Yorick Hardy and Willi-Hans Steeb ()
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Yorick Hardy: International School for Scientific Computing, University of Johannesburg, South Africa
Willi-Hans Steeb: International School for Scientific Computing, University of Johannesburg, South Africa

International Journal of Modern Physics C (IJMPC), 2009, vol. 20, issue 06, 891-899

Abstract: We consider finite-dimensional Hilbert spaces${\mathcal H}$with$\dim ({\mathcal H}) =n$withn ≥ 2and unitary operators. In particular, we consider the casen = 2m, wherem ≥ 2in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces${\mathcal H}$are connected by a unitary transformation (n×nunitary matrices), i.e.ψ = Uϕ, whereUis a unitary operatorUU*= I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operatorUfor finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.

Keywords: Unitary transformation; Hilbert space; modified Gram–Schmidt orthonormalization technique; Bell states; Symbolic C++ (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1142/S0129183109014060

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