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 $. But this is where majority of my confusion lies:
The textbook mentions that a measurement operator is defined as $$ M=\sum^n_{i=0}\lambda_i|m_i\rangle\langle m_i| $$ and the expectation of the measurement is $$ \langle E\rangle=\langle\psi|M|\psi\rangle $$ It is thus clear that the measurement operator is a diagonal matrix, however are the above only for measurement operators? For example Hadamard gate is not a diagonal matrix yet we can still operate them on the system.
Upon applying an operator to the system, we yield a new state e.g. $ |\psi\rangle $, I understand that this shows that a measurement disturbs the system, however in physical reality, do we not yield a value from a measurement? How is that demonstrated in $ M|\phi\rangle=|\psi\rangle $?
Measurement is done with respect to a set of orthonormal basis $ \{|m_i\rangle\} $, how do we interpret this set of basis in reality? E.g. if we are measuring the momentum of the system, what would each basis $m_i$ represent?
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, which would result in something like $ MV|\phi\rangle $, 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 $?
