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I'm looking for some intuition behind how a cNOT gate works. I think I understand the mathematics; but, I'm having trouble imagining how two electrons would interact to produce the predicted result.

If I understand correctly, the input to a cNOT gate is two unentangled electrons. I picture these as two Bloch spheres with vectors pointing in arbitrary directions.

To implement a cNOT gate, we would (1) bring these two electrons close enough to become entangled, (2) prevent electron x from rotating in the z direction (thereby preserving its state in that direction), and (3) allow electron y to rotate (which changes the probably of measuring it in the up state with respect to the z axis).

My understanding is that entanglement is a result of electrons rotating to align their spin axes into a lower energy state. If that's the case, why is electron y affected more when electron x's axis is pointed up than when it's pointed down? It would make sense if the effect was symmetric (large when the angle is close to either 0 or $\pi$, and small when it is close to $\frac{\pi}{2}$), but given that y can be oriented either up or down, I don't see how the effect of x can be small with oriented up and large when oriented down.

This post addresses the implementation of cNOT, but from a perspective that seems more mathematical than intuitive. How is CNOT operation realized physically?

This video gives the type of answer I was looking for: https://youtu.be/EjdIMBOWCWo?t=30 (I would have added this as the answer; but, I can't while the post is marked as a duplicate.)

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    $\begingroup$ Sounds like a better title to your question might be "Implementation of quantum cNOT gates?". I think that that's a good question. I've read through some material on quantum cNOT gates but the material that I've read never actually describes how such a gate would be physically implemented. $\endgroup$
    – user93237
    Commented Mar 30, 2019 at 17:32
  • $\begingroup$ I avoided that title, because I wanted something more intuitive than mathematical. I found this answer yesterday; but, it's a bit over my head: physics.stackexchange.com/q/173776 $\endgroup$
    – Zack
    Commented Mar 30, 2019 at 17:41
  • $\begingroup$ the selected answer says the key is to switch the interaction on and off in a controllable way, and I agree. Imagine that you can flip between states $|0>$ and $|1>$ using microwaves, but state $|1>$ is an interacting state (e.g. its scattering length is large so that contact interaction becomes strong when two atoms in $|1>$ are on top of each other). Prepare two atoms in $|0>$ in optical tweezers. Now flip control atom to $|1>$ and overlap the tweezers. Because of interaction, the microwave resonance is shifted and you won't see spin flips when you shine the microwaves. $\endgroup$
    – wcc
    Commented Mar 30, 2019 at 17:54
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    $\begingroup$ Possible duplicate of How is CNOT operation realized physically? $\endgroup$
    – Norbert Schuch
    Commented Mar 30, 2019 at 23:19

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