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It is said if we have been given the state $|0\rangle$, the measurement will yield $0$ with probability $1$ in Nielsen's book.

So here, the measurement will yield $0$ refers to we will get state $|0\rangle$ or eigenvalue $0$? I think under computational basis it should be $(|0\rangle\langle{0})|0\rangle=|0\rangle$, so the measurement will get $|0\rangle$ with probability $1$, and eigenvalue $1$.

The same case: Given a state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, when we measure $|\psi\rangle$, we get the result $0$ with probability $|\alpha|^2$ or the result $1$ with probability $|\beta|^2$. Here, the result $0$ refers to what?

But in many cases, we said under the Pauli basis, we can get $1$ or $-1$, in this case, are we saying the eigenvalues?

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This is indeed a common notational inconsistency (I'm writing a textbook myself, and am struggling with being clear and consistent about this).

At least in your case, there's no ambiguity: a 0 means the measurement outcome corresponds to the state $|0\rangle$ (and the arbitrary labels that are being assigned to measurement outcomes will be 0/1). It's only when you get a 1 do you have to ask yourself "does that mean $|1\rangle$, with measurement outcomes being labelled 0/1, or $|0\rangle$ with outcomes being labelled $\pm 1$?". These are the two standard conventions. Hopefully whatever text you're using is being explicit, or consistent within context. That's the only way that you can resolve it because, as I say, measurement outcomes are just two completely arbitrary labels used as a shorthand to describe which of two events happened. If nobody tells you want labels they're using, you have no hope!

For example, if the text says "measure with respect to the $Z$ basis", what that should mean is take $Z$ to be an observable, and associate the eigenvalues with the measurement labels, and the corresponding eigenvectors are the output states. But if it talks about 0/1 results, it must be talking about the computational basis.

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    $\begingroup$ What's the book you're writing and what topics would it contain? 👀 $\endgroup$
    – FDGod
    Oct 25, 2023 at 8:22
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    $\begingroup$ @FDGod It's basically a print version of qubit.guide with a lot more exercises and extension topics. $\endgroup$
    – DaftWullie
    Oct 25, 2023 at 8:52
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The labels used to denote outcomes, especially for things like many-qubit systems, are fundamentally a matter of convention. You can call the two outcomes "$0$ and $1$", or "$+1$ and $-1$", or "$\text{soup}$ and $\text{biscuits}$" for all that matters. It is standard to denote the two possible outcomes with $0,1$ because it is reminiscent of the way we call the possible states of a classical bits. But in other contexts, for example "measuring in the eigenbasis of a Pauli observable" it is more natural to call them $+1$ and $-1$, because those are the eigenvalues of the operators written in the standard form. Note that measuring in for example the eigenbasis of $Z$ is completely equivalent to measuring in the eigenbasis of $(I+Z)/2$, which would more naturally again give outcomes $0,1$ (you see it looking at the eigenvalues).

There is however one other thing to mention: when we talk about "measuring an observable", what we mean is really to measure in the eigenbasis of the observable, and then attach the associated eigenvalue to each of the possible outcomes. So you can think of the eigenvalues of the observables as not just conventional labels, but rather as a type of post-processing, meaning that you attach those particular number (for whatever reason which will depend on the context) to those particular outcomes.

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  • $\begingroup$ If you get the outcome $\text{soup}$, you conclude your post-measurement state is $|0\rangle$ and if you get the outcome $\text{biscuits}$, you conclude your post-measruement state is $|1\rangle$, lol :-) $\endgroup$
    – FDGod
    Oct 25, 2023 at 8:49

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