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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?

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    $\begingroup$ 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)? $\endgroup$ – DaftWullie Oct 29 '19 at 9:39
  • $\begingroup$ @DaftWullie Yes. Yes. Yes. Both should be allowed (arbitrary and particular) depends on how you choose to answer. $\endgroup$ – More Anonymous Oct 29 '19 at 9:45
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    $\begingroup$ 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 $\endgroup$ – glS Oct 30 '19 at 15:37
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There are a few strategies/arguments one might make.

Firstly, density matrices often represent an individual's state of knowledge rather than some absolute state. To that end, if two experimentalists aim to produce the same state, and have the same level of uncertainty in their equipment: random number generators, basis choice etc., then their descriptions of their outcomes are identical, and those would be the best descriptions anyone could give them.

You may not feel entirely satisfied with that answer, so I have another idea which may be a little more convincing: tell each experiemntalist to make their own maximally entangled state of two systems. Then take one of those systems as their density matrix. It should be a maximally entangled state. This has certain advantages about being basis independent, and not involving any randomness. However, there's still the issue of experimental inaccuracy and not quite producing a maximally entangled state. I think there will always be something like that that you can't really deal with if you don't accept my first comment.

One possible route would be to consider random states. If you create a large enough random state, then subsystems of it are almost certainly maximally entangled (that's a very crude paraphrasing. It's a sort of result that Andreas Winter made a lot of use of, so that's probably where you can find a better statement.) Random states, and that sort of result, are likely to be quite robust against experimental imperfections.

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  • $\begingroup$ I was hoping to use this to quantify "experimental inaccuracy" in some kind of information theoretic sense. Is that impossible? $\endgroup$ – More Anonymous Oct 29 '19 at 16:01

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