# Are there constraints on the density matrix during decoherence?

A privileged basis emerges during measurement. starting from a pure vector state $|V \rangle$ in a n dimensional hilbert space it gives after décoherence a density matrix $p_1 |e_1 \rangle \langle e_1| + p_2 |e_2 \rangle \langle e_2|+ p_3 |e_3 \rangle \langle e_3|+....$ where the $e_i$ are orthogonal.

i am interested in the shape of this basis before the end when it is not yet orthogonal. we have n(n-1)/2 possible couples of different $(e_i (t),e_j (t))$ with an angle $\alpha_{ij}$ evolving from 0 to $\pi / 2$ at the end. this number is related to the number of the indepedent off diagonal terms of the matrix. i wonder if there are constraints during it. Are all possible paths possible from pure density matrix to the diagonalized matrix? $\\$ Edit I think that for any measurement the density matrix obeys Lindblad equation and as t tends to infinity $\dot \rho$ tends to zero.