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Unitary Operators, Entanglement, And Gram–Schmidt Orthogonalization

Author

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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)

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++.

Suggested Citation

  • Yorick Hardy & Willi-Hans Steeb, 2009. "Unitary Operators, Entanglement, And Gram–Schmidt Orthogonalization," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 891-899.
    • Handle: RePEc:wsi:ijmpcx:v:20:y:2009:i:06:n:s0129183109014060
    DOI: 10.1142/S0129183109014060
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