I was studying quantum information from Nielsen and Chuang's book and I got a little bit confused because sometimes they use the terms "spectral decomposition" and "orthonormal decomposition" for what seems to me to be the same thing.
Correct me if I'm wrong but the spectral decomposition of, let's say, the density matrix is
$$\rho = \sum_{i} \lambda_{i} |i\rangle\langle i|$$ where the $\lambda_i$ are the eigenvalues of $\rho$ and $|i\rangle$ are the eigenvectors.
Isn't here the orthonormal decomposition of $\rho$ the same as the spectral decompositon? Or when we do the orthonormal decomposition the $\lambda_i$ aren't necessarily the eigenvalues and the $|i\rangle$ aren't necessarily the eigenvectors?
If there's any kind of difference between those two concept, can you please enlighten me? And can you tell me in what context should I use one instead of the other?
Best Answer
Quote from the 2011 Edition: *"Diagonal representations are sometimes also known as orthonormal decompositions."
So: Yes, they are the same as an eigenvalue decomposition (in Nielsen & Chuang), at least for hermitian matrices (which is the context in which they use it).