Liberté, égalité, fidelité

I realized I did not make clear that the e.vals have to be real and positive. Sorry!

But I did some further numerics and you are right in general. I generated X = AB with A, B >= 0 and used partial trace. X can have complex evals in general.

And yes, it's good to see these kinds of posts here!

Thanks for the reply! But the example you posted has complex e.vals which are conjugates of each other with +ve real part, hence the trace and det condition will be satisfied without the e.vals being (real) positive.

Train your biceps with some dagger curls! Although \ddagger curls would be easier for balance.

Haha, took it almost verbatim from the Preliminaries of my thesis!

Here is a 'simple' 3-line proof for this statement!

(Although the simplicity hides behind the form and properties of the matrix geometric mean.)