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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.

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  • $\begingroup$ what is this 'both' you're referring to here ? The original and the tranformed matrix? $\endgroup$ – Lelouch Nov 13 '17 at 16:04
  • $\begingroup$ Can you suggest the exact page no. from Nielsen and Chuang $\endgroup$ – Lelouch Nov 13 '17 at 16:19
  • $\begingroup$ 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. $\endgroup$ – user1936752 Nov 14 '17 at 3:09
  • $\begingroup$ 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. $\endgroup$ – user97261 Nov 15 '17 at 17:08
  • $\begingroup$ Sorry, I've literally no idea what you're saying. $\endgroup$ – user1936752 Nov 16 '17 at 3:30

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