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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?

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  • $\begingroup$ 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 $\endgroup$
    – Ron Cohen
    Jun 11, 2022 at 6:11
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    $\begingroup$ 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. $\endgroup$
    – glS
    Jun 11, 2022 at 8:21

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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 $\{\pm 1\}$.

Similarly consider a computer scientist and her friend the computer engineer. The computer scientist probably intuits bits 0 and 1, while the computer engineer maybe worries about the voltages on various nodes of a transistor. Clearly they can nonetheless communicate with each other, but one just has to be careful in the transliteration.

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.

That's not to say that algorithms that map 0 to +1 and 1 to -1 can't be useful - indeed, Ryan O'Donnell - a theoretical computer scientist, not a physicist - wrote a whole book on the analysis of Boolean functions, wherein thinking of the outputs of such functions as +1 and -1 (as opposed to 0 or 1) often provides the most natural and intuitive perspective. For example, this enables perhaps "the right" way to take Fourier transforms of Boolean functions, even classically in the context of various algorithms in learning theory.

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  • $\begingroup$ Thanks for the elaboration. This question is a bit more 'meta' than most. I think its important to understand where certain notations come from. In an ideal world, we would all use the same notation. $\endgroup$ Jun 12, 2022 at 17:06

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