# Is there a way to measure the degree of superposition of a quantum state?

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!

• 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. – Lucas Baldo Jul 9 at 4:40