In what contexts are different notations used for indicating measurement outcomes?

I have seen a few different notations for denoting measurement outcomes. Does anyone know of which notation is more widely used in various contexts?

For instance, I like referring to this Wikipedia page to remember which notation corresponds to what. The eigenvectors listed are the states that are collapsed to when measuring along the $$X, Y$$ and $$Z$$ bases. I like to take the corresponding eigenvalues ($$+1$$ and $$-1$$) to denote the measurement outcomes. So for example, if I am talking about the measurement outcome for the qubit state $$|0 \rangle$$ when measured along the $$Z$$ basis, I will always get a measurement result of $$+1$$. When measuring $$|1 \rangle$$ the same way, I get a result of $$-1$$. I know some people prefer to instead denote these measurement results as $$0$$ and $$1$$ respectively.

Also: what notation should be used to denote an abstraction of a measurement outcome? For instance, the measurement outcome on qubit $$i$$ may be denoted as $$b_i$$. Example: "When conducting our experiment, measuring qubit at position $$i$$ gave the measurement result $$b_i = +1$$ with $$\frac{2}{3}$$ probability".

Anyone care to elaborate on the matter?

• Your questions are not very clear. As you said measurements are said as their eigenvalues. In the specific case of Z it match 0 and 1 . Z measurement is the moat common, because usually in the hardware we measure any other with Z, using a rotation before and after the Z measurment Commented Jun 11, 2022 at 6:11
• I think the confusion here comes from thinking of "measurement outcomes" as necessarily associated with observables. It is often more natural to think of a measurement as specified by a measurement basis, ie some collection of orthogonal projections $\{ P_i\}_i$. How you call the labels is then completely inconsequential. You can use numbers, people's names, or whatever you like. What matters is that you distinguish between the measurement outcomes and consider the probabilities associated to each one.
– glS
Commented Jun 11, 2022 at 8:21

It's an interesting question as to how to the bijection between the measurement eigenvalues $$\{+1,-1\}$$ and the bits $$\{0,1\}$$ or the qubits $$\{|0\rangle,|1\rangle\}$$ are reflected or intuited in the mind's eye. Briefly perhaps theoretical computer scientists most often think in terms of $$\{0,1\}$$ while physicists intuit the measurement as $$\{\pm 1\}$$.
Indeed imagine understanding Shor's algorithm by considering $$Z$$-basis measurements of the period as being the eigenvalues of the $$Z$$-operator. I for one would find that explanation awkward.