UNITARY OPERATORS, ENTANGLEMENT, AND GRAM–SCHMIDT ORTHOGONALIZATION

Abstract

We consider finite-dimensional Hilbert spaces [Formula: see text] with [Formula: see text] with n ≥ 2 and unitary operators. In particular, we consider the case n = 2m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces [Formula: see text] are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for 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++.