Quantum Measurements are descrivbed by a collection of measurement operators with .
If then we are measuring in a computational basis.
Non-distinguishability of arbitrary states
We cannot distinguish any two arbitraryquantum states.
and be two non-orthogonal states.: Let
Then such that and similarly then and .
Thus, and let where and are orthogonal.
Hence, proved by contradiction.
We can use projective measurement formalism for any general measurement too.
In case of projective measurements, and .
Here, is expectation and is standard deviation or the square root of variance.
POVM Measurements are a formalism where only measurement statistics matters.
Here, each of are hermitian.
Global Phase doesn't matter
We say but why? Because, .
However, be aware that the global phase is quite different from the relative phase.