So let's say there are $2$ experimentalists who have density matrix systems $A$ and $B$. They both agree that for the experiment they need identical density matrices $\rho_A = \rho_B$ which is a mixed state. My question is how do they agree upon $\rho_A = \rho_B$? (Like the classical probabilities might be approximately the same but not exact)
I mean if they do a measurement they change the density matrix and get a pure state. They can't use Noether's theorem of energy for time invariance as the measurement is immune to that since the measurement can have different outcomes to the same initial condition (Born rule).
Now one can argue that the production of the density matrices uses some kind Noether invariant for example it would mean they that they were created at spatially different locations but are the same due construction invariance. However, $A$ can still say his measurement of the invariant is correct versus the measurement of $B$? (where the invariant is the net momentum)
Is there an algorithm to count how many "eigenvalues" / quantify how much information $A$ and $B$ disagree upon?