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Usually, while conducting a measurement on a qubit we are using two projectors, namely $P_0 = |0\rangle \langle 0|$ and $P_1 = |1\rangle \langle 1 |$.

For the case of $P_0$ we have two possible eigenvalues:

  • 0, with eigenvector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$,
  • 1, with eigenvector $\begin{pmatrix} 1 \\ 0 \end{pmatrix} = |0\rangle$.

In the case of $P_1$ we have simmilar eigenvalues:

  • 0, with eigenvector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$,
  • 1, with eigenvector $\begin{pmatrix} 0 \\ 1 \end{pmatrix} = |1\rangle$.

I guess, that due to the noralization condition we cannot obtain the $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ states. So the only possible outcomes of a measurement are states $|0\rangle$ and $|1\rangle$ both with eigenvalue 1.

So what is the interpretation of this eigenvalue? Is it something like the "amount of information" stored in a qubit?

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The eigenvalues are just labels for the outcomes of the measurement. They don't necessarily have any physical meaning.

See this answer for a more detailed understanding.

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Projectors have eigenvalues 0 and 1 by definition. There's nothing more to interpret here...

(Note that the projectors have eigenvectors of 0 eigenvalue which are non-trivial, which are orthogonal to the eigenvectors with eigenvalue 1.)

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If you think of a projector $P$ as representing a kind of "yes/no" question you can ask to a state, you can understand the $1$s as marking the eigenstates (more precisely, the eigenspace) corresponding to the "yes" answer. What I mean by this is that any (orthogonal) projector represents a measurement with two possible outcomes: either you find your state to be in the $+1$ eigenspace of the projector (i.e. $\ker(P-I)$), or you don't.

I would note that from a purely informational perspective, the value of these eigenvalues doesn't make any difference. If you were to perform a measurement corresponding to some other observable with some $\lambda$s and $\mu$s where $P$ has $1$s and $0$s, you would collapse the state in exactly the same way. The only difference is in the numbers/labels you attach to the measurement results ($0$ and $1$ for a projector, some other $\lambda\neq\mu$ for another operator). The only thing that really matters (at least as far as the way the state is collapsed by the measurement is concerned) is the way the operator splits the Hilbert space in different eigenspaces.

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