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It depends a bit on the context, but the idea is that if two observables commute that means that measuring one will not affect measurement results of the other, and you can therefore think of the values … This means that you can characterize a state by the measurement outcomes you will get if you measure it in the different bases. …

A matrix/operator is unitary if and only if it sends orthonormal bases into orthonormal bases. Therefore, to check whether $U$ is unitary, it is enough to check that it sends a subset of a basis (i.e …

Now, if you are asking for a projective measurement basis, then you require a set of operators which describes every possible outcome of your state. …

You can frame this question as follows: define an estimator which gives an estimate for the expectation value you seek for each possible measurement outcome. … For example, using Hoeffding's inequality, denoting with $\overline o_N$ the standard mean of the estimates obtained from $N$ measurement rounds, $\overline o_N\equiv \frac{1}{N}\sum_{k=1}^N \hat o(b_k …

All this is to say that the statement "derivation X cannot be applied to Y due to interpretations of the measurement process" is nonsense. … Note that you can also model measurement processes as (non-unitary) maps applied to the state. …

To "measure $X$" means physically to apply a measurement which makes the state collapse into one of the eigenstates of $X$. …

The way I'd put it is that the state after the measurement is just $|v_m\rangle$, regardless of what's the phase of $\langle v_m|\psi\rangle$. …

Note that here by "computational basis" I simply mean the measurement basis under consideration. … You model this situation by using a projector $P$ that projects onto the required basis, and the associated measurement probability is then given by $\|P|\psi\rangle\|^2$. …

What I mean by this is that any (orthogonal) projector represents a measurement with two possible outcomes: either you find your state to be in the $+1$ eigenspace of the projector (i.e. … The only difference is in the numbers/labels you attach to the measurement results ($0$ and $1$ for a projector, some other $\lambda\neq\mu$ for another operator). …

First of all, arguably the most natural kind of measurement in QM consists in choosing a basis and having the state collapse in that basis. …

Fix a finite number of states $\sigma_k$, and consider a channel of the form $$\Phi(X)=\sum_k c_{k}(X)\sigma_k.$$ For $\Phi$ to be linear and trace-preserving we must have: $$c_k(X+X') = c_k(X) + c_k( …

At a formal level, this amounts to applying an Hadamard gate before the measurement: the Hadamard implements the operations $|0\rangle\mapsto|+\rangle$ and $|1\rangle\mapsto|-\rangle$, and thus measuring … Then, to measure in the $|\pm\rangle$ simply means to rotate the polarisation before said measurement, e.g. using a quarter waveplate. …

As discussed e.g. in this post, given two states $\rho$ and $\sigma$, the measurement that allows to optimally discriminate between them (i.e. the measurement providing the highest average probability … \langle+|$, and computing the measurement optimally discriminating $\rho$ from $\sigma_p=p |0\rangle\!\langle0| + (1-p) |1\rangle\! …

The whole point is that you do not want, nor need, to "look" how the computation is going. You can ensure that the input is what it should be by a variety of means. The simplest case being that you m …

The focus is on that measurement probabilities, nothing more and nothing less. Generalized measurements also deal with a possible post-measurement state. … Aside from the fact that in many situations you do not have a post-measurement state, the fact of the matter is that you might not care about post-measurement states for whatever you are studying. …