Linear isometry of Hilbert spaces
= Linear isometry of Hilbert spaces
{title2=$V^\dagger V=I$}
A linear map $V:\mathcal H\to\mathcal K$ is an isometry when it preserves inner products, equivalently $V^\dagger V=I$. It need not be surjective. It maps an <orthonormal basis> to an orthonormal family, and $\rho\mapsto V\rho V^\dagger$ preserves the nonzero <eigenvalues> and the <Von Neumann entropy> of a <density operator>.