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In many papers and books on quantum theory it is written that it is not possible to obtain amplitudes exactly when we have a qubit in superposition. In a course we said that actually it is possible, but with exponentially numbers of shots, to obtain a probability near to one. I couldn't find references. Can someone fill in my gaps please?

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  • $\begingroup$ Do you mean: "Given many copies of an unknown state $|\psi\rangle$, how do you determine the probability amplitudes of the state to high accuracy?" $\endgroup$
    – DaftWullie
    Jul 26, 2022 at 7:50

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We can start with a classical example. For an unfair coin, with heads probability $p$ and tails probability $1-p$, it is not possible to determine $p$ with just a single toss. With additional tosses, your estimation of $p$ will improve, and tend to perfect if tosses are unlimited.

General single-qubit state can be written as $$ |\psi\rangle = \cos\frac\theta2 |0\rangle+e^{i\phi}\sin\frac\theta2|1\rangle $$ Measuring this state in the computational basis gives $|0\rangle$ with probability $p=\cos^2\frac\theta2$ and $|1\rangle$ with probability $1-p=\sin^2\frac\theta2$. Hence, measurements in the computational basis are completely analogous to a classical coin toss in this respect. With more measurements, your estimation of $\theta$ will impove.

Note also that there is an additional phase $\phi$, which does not affect probabilities measured in the computation basis. To learn about $\phi$ one needs to perform measurement in a different basis, say $|+\rangle, |-\rangle$. Generally, the task of learning the state from measurements is known as the quantum state tomography.

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Depends on what you mean by "exactly". Amplitudes are related to outcome probabilities. Probabilities can always only be "obtained" by collecting some amount of statistics and estimating the underlying probability distribution. With finite statistics, there will always be some amount of error in the estimation, so in this sense, you can never retrieve the probability with infinite precision. But that's the same with any other measurement in physics really: you always have some uncertainty associated with a measurement, so in this sense probabilities aren't really that different.

Another interpretation of the statement is that if you perform a projective measurement in a fixed measurement basis, you won't be able to fully characterise probability amplitudes, even with infinite statistics. This is because probability amplitudes tell you more than the outcome probabilities associated with a single projective basis: they tell you what the outcome probabilities will be for any choice of measurement. So typically to estimate amplitudes you'll need to collect statistics associated with different measurement bases (see quantum state tomography for more details).

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