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[itex]\rho[/itex] is diagonal in that basis. What are the conditions on [itex]\rho[/itex] such that the basis that diagonalizes it is unique?

It's easy enough to work out the answer in the simplest case, of a two-dimensional basis: Then [itex]\rho[/itex] can be represented as a 2x2 matrix. It will have two eigenvalues, [itex]p[/itex] and [itex]q[/itex]. If [itex]p = q[/itex], then [itex]\rho[/itex] is diagonal in every basis. If [itex]p \neq q[/itex], then there is a unique basis (up to permutations of elements and overall phase factor) that diagonalizes [itex]\rho[/itex].

Is there some result that is similar for bases of arbitrary dimension? What's the condition on [itex]\rho[/itex] such that there is a unique way to diagonalize it?