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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 $?

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 $?

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|$, 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 $?

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 $?

I am currently taking a course on quantum computing in college and some parts really confuse me.

Starting from the basics, I understand that $ |\phi\rangle $ represents a state of a quantum system. 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| $?

We can perform operations on the system, and these are represented by operators, possible operations includes measurements, gates etc. It is represented by $ M|\phi\rangle $, and operating on the system leads to a change in the state of the system, $ M|\phi\rangle=|\psi\rangle $. However I came across this question:

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|$, 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 $?

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|$, 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 $?