# Why do unitary transformations on ensembles of states result in the same density matrix?

This is from Nielsen and Chuang. If we have an ensemble of pure states that obey the following relationship for all $i, j$

$\vert \psi_i \rangle = \sum_{j} u_{ij}\vert \phi_j \rangle$

where $u_{ij}$ corresponds to the entries of a unitary matrix, we will obtain the same density matrix for both. That is, we will have

$\rho = \sum_{i}\vert \psi_i \rangle\langle \psi_i \vert = \sum_{i}\vert \phi_i \rangle\langle \phi_i \vert$

The proof is straightforward and given in the book but I have no idea what the correct way to interpret this result is. Any help is appreciated.

• what is this 'both' you're referring to here ? The original and the tranformed matrix? – Lelouch Nov 13 '17 at 16:04
• Can you suggest the exact page no. from Nielsen and Chuang – Lelouch Nov 13 '17 at 16:19
• This is theorem 2.6 (unitarity freedom in the ensemble for density matrices). It's page 103 in my edition but might be different in yours. – user1936752 Nov 14 '17 at 3:09
• Equal energy and volume does not imply parallel transport. That is their unities are unique. Unitary ensembles can represent the same system in two different configurations i.e 6D Boltzmann space-time however are still subject to identical normalization. – user97261 Nov 15 '17 at 17:08
• Sorry, I've literally no idea what you're saying. – user1936752 Nov 16 '17 at 3:30