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I am wondering if there is a way to calculate the amount of superposition that a quantum state is in. For example, if I have a $2$-qubit quantum system, with basis $\mathcal{B} = \{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$, and a state $|\psi\rangle$ of this system, I would assume that if such a function that computes the degree of superposition exists, it would be defined as $f : \mathcal{H} \rightarrow [0,1]$, and satisfy the following properties:

$$f(|\psi\rangle) = 0 \ \text{iff} \ |\psi\rangle \in \mathcal{B} \ \text{(up to global phase)}, \text{and}$$

$$ f(|\psi\rangle) = 1 \ \text{iff} \ |\psi\rangle = \frac{1}{\sqrt{4}}(|00\rangle + |01\rangle +|10\rangle + |11\rangle) \ \text{(up to relative phase)}.$$

As well, the following would be true: if $|\phi\rangle = \sqrt{\frac{1}{10}}|00\rangle + \sqrt{\frac{9}{10}}|01\rangle$, and $|\theta\rangle = \sqrt{\frac{2}{10}}|00\rangle + \sqrt{\frac{8}{10}}|01\rangle$, then $f(|\phi\rangle) < f(|\theta\rangle)$.

Has such a function been defined as of yet? Thanks for any help!

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    $\begingroup$ So, in other words, this function $f$ is a way of measuring how close a state is to an element of the basis? Notice that the function depends on the choice of the basis, so it would not reflect an information about an internal structure of the system, but its relation to the measurement apparatus. That said, I have never seen this done. $\endgroup$ – Lucas Baldo Jul 9 at 4:40
  • $\begingroup$ Potentially helpful reading: en.wikipedia.org/wiki/… $\endgroup$ – Noiralef Jul 9 at 8:33
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As pointed out in a comment, the notion of "degree of superposition" is basis-dependent, and thus only makes sense if there is a specific preferred basis, unlike concepts such as entanglement. Thus, in a typical quantum information scenario, where there is no preferred basis, this is not a meaningful concept.

However, there are of course scenarios where there is a preferred basis, such as a "classical basis" which is stable under interactions with the environment and in which the system will be measured, especially for "macroscopic" objects. In that case, it makes sense to study the amount of quantum superposition, which would be what allows to e.g. see interference between those objects. This has indeed been done under the name "Coherence", and has been initiated in T. Baumgratz, M. Cramer, M. B. Plenio: Quantifying Coherence.

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