Help in understanding the usage of eigenvalues in the definition of the projective measurement

Question

Recently I was reading about the projective measurement in "Quantum Computation and Quantum Information" by Nielsen & Chuang, where they describe the projective measurement as follows:

Projective measurements: 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 decoomposition,

$$ M = \sum_m m P_m, $$

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. [...]

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 $\begin{pmatrix} 0 \\ 1 \end{pmatrix} = |1\rangle$.

Because qubit has to be normalized, I guess the eigenstates associated to the 0 eigenvalue cannot be obtained during a measurement. But if we would like to stick to the definition $M = \sum_m m P_m$ for $m = \{0,1\}$, we would get

$$ M = 0 \cdot P_0 + 1 \cdot P_1 = P_1,$$

which I think is incorrect.

Because of that, isn't the formulation of an observable as $M = \sum_m m P_m$ a little bit misleading? Shouldn't it be defined as

$$ M = \sum_m \lambda_m P_m,$$

where $\lambda_m$ is the eigenvalue associated to the appropriate state? Then we would get

$$ M = \lambda_0 P_0 + \lambda_1 P_1 = 1 \cdot P_0 + 1 \cdot P_1 = \mathbf{I}. $$

Answer 1

From the perspective you're coming at it from, yes your definition sounds more reasonable. However, you're not supposed to be starting from the projectors and making an observable. You're supposed to be starting from the observable and using it to specify the projectors. For example, let's take $M=Z$, the standard Pauli matrix. $Z$ has eigenvalues $\pm 1$, so $$ Z=P_+-P_-, $$ and we associate $$ P_+=|0\rangle\langle 0|,\qquad P_-=|1\rangle\langle 1|. $$ Now it is true that later you might choose to change the labels on your projectors. In this specific case, it is more common to see them written as $P_0$ and $P_1$. However, what the operator is essentially telling you is "your measurement device will give you an answer either +1 or -1, and these are the corresponding projectors" and so the meaning is sensible and consistent across all observables you might choose.