# Why can we predict any experimental outcome, given a probability density over quantum states?

There is a very interesting answer given by Peter Shor in this website here. However, I admit I don't fully understand it.

In particular, I don't understand:

1. If we have a probability density μv on quantum states v, we can predict any experimental outcome from the density operator

What is the formula given? Is it $$\operatorname{trace}{\left(\rho A\right)}$$ for an observable $$A$$?

2. a probability distribution on quantum states is an overly specified distribution, and it is quite cumbersome to work with

How can we work with the density operator if, from an experimental point of view, all we can measure is the probability distribution?

• Welcome to SE.Physics! Could you think of a more descriptive title for this question? Ideally, something that someone seeing this question on the list can understand the physical subject matter from reading.
– Nat
Oct 28, 2018 at 2:21
• Shor is not a physicist, and he is using totally nonstandard terminology there.
– Buzz
Oct 28, 2018 at 3:07
• @Buzz: Note that the OP for the previous question was a mathematician, and not a physicist. But I'm curious—what terminology do you think I should have used? Oct 28, 2018 at 23:48
• @PeterShor You make repeated references to "probability distributions on quantum states," defined in a way that is never used. Moreover, you call such distributions "overly specified," which makes it sound like the "probability distributions on quantum states" formalism is adequate but unnecessarily complicated. However, that is not the case. The "probability distributions on quantum states" as you write them down are not overly specified but underspecified—I would even say wrongly specified—because they leave no room for phase information; and that is why they are never used.
– Buzz
Oct 28, 2018 at 23:51
• @Buzz Definitely the phase information is there inside the density matrix. Or the probability is just a deduction of the geometrical structure of quantum mechanics, the probability is just a distance, right?
– XXDD
Oct 29, 2018 at 2:30

For your first question, the expectation valuable of an observable $$A$$ on a density matrix $$\rho$$ is indeed $$\mathrm{Tr}\, (\rho A)$$.