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What are some examples of measuring qubits in different basis states (it's known that preferable choices are the computation basis states $|0\rangle$, $|1\rangle$ and the Bell basis states $|+\rangle$, $|-\rangle$ on a two-qubit system)?

Context: I am an undergraduate sophomore working on quantum information with a professor so this isn't for a class. Over break, I was given the task of writing a Mathematica function that provides a numerical modeling procedure for measuring a qubit. The function's inputs are the quantum state, measurement basis, and which qubit to measure. The outputs of the function are the post-measurement state of the qubit, and what the outcome of the measurement is (being $|0\rangle$ or $|1\rangle$) by running a similar function to a density probability. That being said, I am still in the process of learning this material. Thank you.

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  • $\begingroup$ Hi, Frank. Welcome to Quantum Computing SE! Please review How to write a good question? I've edited your question this time, but it would be quite annoying for us to do it on your behalf every time. $\endgroup$ – Sanchayan Dutta Jan 2 at 9:16
  • $\begingroup$ are you asking about measuring a single qubit in different bases, or about measuring systems composed of several qubits? $\endgroup$ – glS Jan 2 at 16:39
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The way that you phrase the question makes it sound like you're given a pure state, $|\psi\rangle$ of $n$ qubits, and you are tasked with performing a single qubit projective measurement on a given qubit, $k$. That's adding in a little precision to your statement which may not be accurate, so please clarify if it's not (because you also mention a Bell basis measurement).

Probably the easiest way to specify a one-qubit measurement is to accept 3-component real vector $\underline{n}=(n_x,n_y,n_z)$ which satisfies $\underline{n}\cdot\underline{n}=1$. This corresponds to a measurement observable $M=n_xX+n_yY+n_zZ$, where $X,Y,Z$ are the standard Pauli matrices.

To calculate the probability of getting the $\pm1$ measurement outcomes on qubit $k$, you calculate $$ p_{\pm}=\frac12\pm\langle\psi|\mathbb{I}^{\otimes(k-1)}\otimes M\otimes\mathbb{I}^{\otimes(n-k)}|\psi\rangle, $$ and the post-measurement state is $$ \left(|\psi\rangle\pm\mathbb{I}^{\otimes(k-1)}\otimes M\otimes\mathbb{I}^{\otimes(n-k)}|\psi\rangle\right)/\sqrt{p_{\pm}}, $$ depending on which of the two outcomes was obtained.

There are alternative approaches to calculating the effect of the measurement. Take a look at the answers to this question.

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