How do expected values of observables depend on the current state?

Question

I'm currently looking into quantum computing following the book The Nature of Computation. On pages 835/836 they define observables and the expected value of an observable corresponding to a hermitian linear operator $M$. If I'm reading this right, they're defining it as

$$ \mathbb{E}[M] = \langle v|M|v \rangle. $$

What strikes me as odd is that the state of the system $v$ shows up on the right side but not on the left. Different states should yield different expected values, yet the left side doesn't convey that at all.
Furthermore, later on the page there's the claim that

$$ \mathbb{E}[Z_A M_B] = \frac{1}{\sqrt{2}}. $$

Where $Z_A M_B$ is some operator. Yet if I use the right hand side from earlier and choose different values of $v$ I can get pretty much any value I like for this expected value. What am I misunderstanding?

(Note: I'm not that experienced in physics and am currently only beginning to understand the finite dimensional setting required for quantum computing.)

Answer 1

What the notaton $\mathbb{E}[M]$ means here is "expectation value of the operator $M$ in the state $|\nu\rangle$". You could write the state as an index on the left, like $\mathbb{E}[M]_\nu$, but this is tedious and in most cases unnecessary, as it is clear from the context which state we mean.

As for your second equation, I don't have this book, but I'm sure they also mean the expectation value in some previously defined state.