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Suppose I have a quantum state represented by a Complex vector, arbitrarily:

$[~-0.538, 0.044 - 0.323i, 0.706, 0.044 + 0.323i~]$

With each value in that vector corresponding to some classical state:

$[~ |00\rangle, |01\rangle, |10\rangle, |11\rangle ~]$

What is the correct mathematical formula to derive a probability distribution across possible classical states that could be observed in measurement from this complex vector?

E.g. how do I find individual real values for $p(|00\rangle)$, $p(|01\rangle)$ etc. in this example?

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The quantum state is a vector $v$ of unit norm. The probability of measuring outcome $x$ is $\|v_x\|^2$, the squared magnitude of the corresponding coefficient in the vector. Since $v$ has unit norm, these probabilities add up to $1$.

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  • $\begingroup$ to check I understand correctly; in my example, p(|01>) = 0.044^2 + (-0.323)^2? $\endgroup$ – Tim Atkinson Dec 16 '16 at 17:24
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    $\begingroup$ @Atkrye Exactly. $\endgroup$ – Yuval Filmus Dec 16 '16 at 20:01

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