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I am following Nielsen and Chuang, section 2.2.5:

A projective measurement is described by an observable, $M$, a Hermitian operator on the state space of the system being observed. The observable has a spectral decomposition, $$M = \sum_m mP_m$$ where $P_m$ is the projector onto the eigenspace of $M$ with eigenvalue $m$. The possible outcomes of the measurement correspond to the eigenvalues, $m$, of the observable.

My question is how can we view a the projective measurement of two qubits in the standard (Z) basis using this formalism. In this measurement the projectors are $$\left|00\right\rangle \left\langle 00\right|, \left|01\right\rangle \left\langle 01\right|, \left|10\right\rangle \left\langle 10\right|, \left|11\right\rangle \left\langle 11\right|$$

But how can I find them if I only know of the observable, which in this case is $Z\otimes Z$ if I understand correctly?

This observable has only two eigenvalues: $+1$ and $-1$. The eigenvalue $+1$ corresponds both to $\left|00\right\rangle$ and $\left|11\right\rangle$, so it seems to me I need a "stronger" observable, which can differentiate those two states, or I'm missing a something elementary in the definition of projective measurements.

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$Z\otimes Z$ is an observable which specifies a measurement but, as you have realised, it's not specific enough to fix it as being the measurement you want. There's no unique way of doing this, but you could use something like $$ 2Z\otimes I+I\otimes Z $$ If you work this out, you'll see it's diagonal, with distinct elements on each diagonal, which is what you need.

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An observable having degenerate eigenvalues means that it doesn't completely collapse the system. For example, $Z\otimes Z$ would describe a type of measurement in which you are only asking whether the state is in the span of $\{|00\rangle,|11\rangle\}$ or in the span of $\{|01\rangle,|10\rangle\}$. This means that, for example, this type of measurement would leave a state such as $|00\rangle+|11\rangle$ unperturbed, but make a state like $|00|\rangle+|01\rangle$ collapse in either $|00\rangle$ or $|01\rangle$.

If you want to model a measurement which distinguished between all elements of the computational basis, you can use any non-degenerate observable. The specific eigenvalues will only depend on what you want to do with your observations.

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