# What are the matrices in the POVM for measuring the first $m$ qubits?

Suppose you have a quantum state $$|w\rangle$$ consisting of $$m + n$$ qubits, and you set up a measurement that measures the first $$m$$ qubits in the standard basis. What are the matrices in the corresponding POVM?

Well, since these are projective measurements on the subspace of the first $$m$$ qubits, we can just list all projectors on the computational basis of this first subspace and 'pad' them with $$I$$'s on the second subspace:
$$P_{j} = |j\rangle\langle j|_{m} \otimes I_{|n|},\,\,\, \forall j \in \{0,1\}^{m},$$ which gives exactly $$|\{0,1\}^{m}| = 2^{m}$$ different operators for the POVM. (In fact, all operators are projectors here, so they actually form a PVM).
If you identify distinct measurement outcomes with every operator, say $$\lambda_{j} = j_{d}$$ (e.g. $$j$$ in decimal form), you can easily write down a measurement observable as well:
$$M = \sum_{j} \lambda_{j}P_{j} = \sum_{j} j_{d}|j\rangle \langle j \otimes I_{n}|$$
See also, for instance, this nice answer by Daftwullie for a different measurement operator. Note that that answer omits the extra subspace of $$n$$, but you can just treat that by padding with $$I$$'s again.