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Suppose I have a qubit and the ability to act a Pauli $Z$ gate on it. This is a black box that does the phase flip and I don't know how it works on the inside. Can I use this black box to implement a Pauli $Z$ measurement? The measurement projects the qubit onto $\{\frac{1+Z}{2}, \frac{1-Z}{2}\}$. If yes, how can I do it?

Conversely, I know how to do a measurement in the sense of Stern-Gerlach experiment. We send a particle through the SG appartus and obtain two possible outcomes and label this as the projection onto $\vert 0\rangle\langle 0\vert$ or $\vert 1\rangle\langle 1\vert$. This is a Z-measurement. Can I use the Stern-Gerlach setup to implement a Pauli $Z$ gate?

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A Pauli $Z$ gate (or any unitary quantum gate) acts reversibly on quantum states. Applying a $Z$ gate twice has no effect. On the contrary, a Pauli $Z$ measurement is an irreversible process: it collapses (projects) the quantum state towards an eigenstate of the $Z$ gate (either $\left|0\right\rangle$ or $\left|1\right\rangle$). There is no going back, you lose quantum information and gain classical information.

Hence, using solely one of them to implement the other is not possible.

One connection between the two is that Pauli $Z$ measurement is completly oblivious to $Z$ gate being performed. Applying a $Z$ gate just before a $Z$ measurement has no effect on the possible measurement outcomes. You can conversely disregard any $Z$ gate applied immediately after a $Z$ measurement, because your state is in an eigenstate of the $Z$ gate, so applying it will only add a meaningless global phase.

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  • $\begingroup$ Thanks! So is there any relationship between the physical implementation of a $Z$ measurement and a $Z$ gate? I understand the connection you make in your final paragraph on paper but how does an implementation of the projector $1\pm U$ relate to the rotation $U$ in a lab? $\endgroup$
    – Brendan
    Commented Apr 10 at 20:32
  • $\begingroup$ I am not knowledgeable enough to discuss physical implementation. It is probably hard to answer your question without stating how you implement your qubits. From a computation perspective, these two operations are so different that I would be surprised if there implementation where close in practice. For the SG experiment, the $Z$ rotation is maybe an electrical field that converts $|+\rangle \leftrightarrow |-\rangle$ when placed between 2 $X$-SG apparatus, while the $Z$ measurement is probably a screen where (half of) your atoms end their journey. $\endgroup$
    – AG47
    Commented Apr 11 at 8:15

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