# Connection between a Pauli measurement and the corresponding Pauli gate?

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?

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.
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.
• 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? Commented Apr 10 at 20:32
• 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.