About production and disagreements between density matrices

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)

Question

Is there an algorithm to count how many "eigenvalues" / quantify how much information $$A$$ and $$B$$ disagree upon?

• Can I just clarify a few things? Are you assuming that both experimentalists produce only one copy of their density matrix? And that they cannot come together and compare their density matrices in some desctructive way (such as a SWAP test)? Does this have to be for any arbitrary density matrix, or are we allowed to specifiy a particular density matrix (such as the maximally mixed state)? Oct 29 '19 at 9:39
• @DaftWullie Yes. Yes. Yes. Both should be allowed (arbitrary and particular) depends on how you choose to answer. Oct 29 '19 at 9:45
• in practice you would just measure the state produced by your setup many times to check that it is what you want it to be, until you can trust that the setup is producing what it should. If $A$ and $B$ want to make sure they are producing the same state, they would do something of this sort. You do a bunch of measurements beforehand to check that the setups are doing what they are intendend to do, and then you just trust that they will keep on doing that
– glS
Oct 30 '19 at 15:37