Petrovcic2x

2022-06-23

Show that measuring the observable $|1⟩⟨1|$ is equivalent to measuring the observable Z up to a relabelling of the measurement outcomes.
What does it mean by relabelling of the measurement outcomes and how do I prove it? Could anyone help me?

arhaitategr

Measuring the observable Z means that you collapse to either $|0⟩⟨0|$ or to $|1⟩⟨1|$, this is exactly the same as measuring the observable $|1⟩⟨1|$, in which case you get that outcome or the opposite outcome $1-|1⟩⟨1|=|0⟩⟨0|$.