This is an exercise from p.52 of An Introduction to Quantum Computing by P. Kaye, R. Laflamme and M. Mosca. The oberservable Z denotes Pauli Z matrix. What does it mean by relabelling of the measurement outcomes and how do I prove it? Could anyone help me?
Measuring the observable $Z$ means that you collapse to either $\lvert 0\rangle\langle 0\rvert$ or to $\lvert 1\rangle\langle 1\rvert$, this is exactly the same as measuring the observable $\lvert 1\rangle\langle 1\rvert$, in which case you get that outcome or the opposite outcome $1-\lvert 1\rangle\langle 1\rvert = \lvert 0\rangle\langle 0\rvert$.