UNITARY OPERATORS, ENTANGLEMENT, AND GRAM-SCHMIDT ORTHOGONALIZATION

We consider finite-dimensional Hilbert spaces H with dim(H) = n with n >= 2 and unitary operators. In particular, we consider the case n = 2(m), where m >= 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states psi and phi in these Hilbert spaces H are connected by a unitary transformation (n x n unitary matrices), i.e. psi = U phi, where U is a unitary operator UU* = I. Given the normalized states psi and phi, 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++.