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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?

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johani
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    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 '18 at 02:21
  • Shor is not a physicist, and he is using totally nonstandard terminology there. – Buzz Oct 28 '18 at 03:07
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    @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? – Peter Shor Oct 28 '18 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 '18 at 23:51
  • @Buzz: the OP asked why he couldn't define probability distributions on quantum states using his "ignorant definition." I don't see how I could have answered the question adequately without referring to his (admittedly nonstandard) definition. – Peter Shor Oct 28 '18 at 23:57
  • @PeterShor But you seem to use the "probability distributions on quantum states" in the definition of the density matrix and state that with the probabilities of states known, you can predict the outcome of any experiment—which is not true, because of the lack of phase information. Moreover, "overly specified" just seems to be wrong. – Buzz Oct 29 '18 at 00:05
  • I don't think you understand the OP's "ignorant definition", which is a classical probability distribution on an ensemble of quantum states (where these states include phase information). – Peter Shor Oct 29 '18 at 00:18
  • Maybe I didn't explain my answer very well, but you can find the same material explained by a real physicist in Section 2.3 of John Preskill's lecture notes. – Peter Shor Oct 29 '18 at 00:19
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    @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 '18 at 02:30
  • Please consider writing descriptive question titles with appropriate punctuation, grammar, and formatting. See this meta post: How do we write good question titles?. –  Oct 30 '18 at 14:10
  • I tried to edit the title to a more descriptive/useful one. Feel free to revert the edit if you think the current title does not properly reflect your question – glS Nov 05 '18 at 13:56

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From your question, it sounds like you've learned about observables but maybe not von Neumann measurements and POVMs. In my previous answer, when I mentioned "experimental outcome", I was referring to these types of measurement. If you want to understand my previous answer completely, you probably should learn about them.

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

For the second question, if you can prepare a mixed quantum state (represented by a density matrix) repeatedly, you can measure the density matrix by doing quantum state tomography on it. If you only have one instance of a quantum state, you can't even get a probability distribution by measuring it ... you just get one measurement outcome.

The probability distribution I referred to in my previous answer is a probability distribution that the OP was asking about. This is related to how the state was prepared, and you can't measure this probability distribution even if you can prepare arbitrarily many copies of the quantum state.

Peter Shor
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  • Then how come we can predict the future using classical physics? Eg. How does this certainty arise? I can predict, in the next 24 hours xx minutes xx seconds, the earth will make a full rotation. How does one get these predictive quantities from a non-predictive probability distribution (we have to measure it, we can't predict). – Aditya Shrivastava Sep 12 '20 at 02:13