How to normalise when the probability of measurement is zero?

Question

In one of the answers to this question on measuring one qubit it is explained that given a general two-qubit state $$ |\psi\rangle = \begin{bmatrix} \alpha_{00} \\ \alpha_{01} \\ \alpha_{10} \\ \alpha_{11} \end{bmatrix} = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle $$ one measures the most-significant (leftmost) qubit in the computational basis as follows:

• the probability that the measured qubit collapses to $|0\rangle$ is $$ P\left[|0\rangle\right] = |\alpha_{00}|^2 + |\alpha_{01}|^2 $$

• the normalized state after the measurement is $$ |\psi\rangle = \frac{\alpha_{00}|00\rangle + \alpha_{01}|01\rangle}{\sqrt{|\alpha_{00}|^2 + |\alpha_{01}|^2}}. $$

My question is: How does one normalize in the case the probability $|\alpha_{00}|^2 + |\alpha_{01}|^2$ is zero? (This happens, of course, if $\alpha_{00} = \alpha_{01} = 0$.)

Answer 1

The projection operator can be writen as $\hat{P}=|0\rangle\langle 0|$, the probability of collapse as $\langle \psi|\hat{P}|\psi\rangle$, and the measurement update rule as $$|\psi\rangle\to \frac{\hat{P}|\psi\rangle}{\sqrt{\langle \psi|\hat{P}|\psi\rangle}}.$$ Both the numerator and the denominator vanish, so one could consider taking limits to see what happens.

Suppose we take $|\psi\rangle=|0\rangle+\epsilon|\phi\rangle$ for some small $\epsilon$. We find $\hat{P}|\psi\rangle=\epsilon|\phi\rangle$ and $\langle\psi|\hat{P}|\psi\rangle=\epsilon^2 \langle \phi|\hat{P}|\phi\rangle$ such that the measurement update rule becomes $$|\psi\rangle\to\frac{\epsilon|\phi\rangle}{\sqrt{\epsilon^2\langle \phi|\hat{P}|\phi\rangle}}=\frac{|\phi\rangle}{\sqrt{\langle \phi|\hat{P}|\phi\rangle}}.$$ This is a properly normalized state, no matter how small we take $\epsilon$ to be! So the case of $\epsilon=0$ must be dealt with prescriptively, as opposed to continuing with the earlier mathematical manipulations, by saying that the update rule is only defined with normalization for results with nonzero success probability. An impossible measurement result need not have a physically well-defined state.

In sum: It's easier if you consider the whole process to be a two-step procedure: find the state, then normalize. If the state is 0 (not $|0⟩$, but 0), because it is orthogonal to the measurement device, then it is not a state, so you don't have to normalize it. It's like asking "what is the resultant state when $1=2$" - it doesn't matter what you say as an answer, no matter how absurd, because that state never has any relevance.