How to compute the measurement probabilities of $|\phi\rangle=\sum_k c_k |k\rangle$ in a rotated basis $V|k\rangle$?

Question

I came across the following question and have some conceptual questions.

Consider a general quantum state $|\phi\rangle$ of dimension $N$ spanned by some standard basis $\{|k\rangle,k=0,1,...N-1\}$. Suppose we wish to measure it in a different basis $\{|\bar{0},|\bar{1},...|\overline{N-1}\rangle\}$ such that $|\bar{k}\rangle=V|k\rangle$ for some unitary $V$ for each $k=0,1,...N-1$. Write down the probability of getting outcome $|\bar{k}\rangle$ in terms of $|\phi\rangle$ and $V$.

For a question like this,I would think of it as that we need to first apply V onto the state, and then do a measurement in the new basis, which would result in something like $ MV|\phi\rangle $ where $ M=\sum^{N-1}_i|\bar i\rangle \langle \bar i|$ and is the measurement operator, but how do we go on to find the probability? Do I simply insert $ \langle \bar k| $ in front and turn it into $ |\langle \bar k|MV|\phi\rangle|^2 $?

Answer 1

Beware! If $M$ is an operator describing a measurement, it is not that the output after measurement is $M|\psi\rangle$ for initial state $|\psi\rangle$. Instead, let $\{P_i\}$ be projectors onto the different eigenspaces of $M$. The you get the outcome $i$ with probability $p_i=\langle\psi|P_i|\psi\rangle$ and the state after measurement, if you get that outcome, is $P_i|\psi\rangle/\sqrt{p_i}$.

So, in the case where $M$ only has unique eigenvalues, each of the $P_i=|\phi_i\rangle\langle\phi_i|$ and your probability is $p_i=|\langle\psi|\phi_i\rangle|^2$, giving the output state $|\phi_i\rangle$. This circles back around to your first question

I know that$ |\langle\psi|\phi\rangle|^2 $ gives a probability, but is it right to interpret it as the probability of the system in state $ > |\phi\rangle $ collapsing to state $ \langle\psi| $?

If you start with the state $|\phi\rangle$ and measure in some basis where one of the measurements is the state $|\psi\rangle$, then yes, this is the probabilty that you get that outcome.