# What's the difference between observing in a given direction and operating in that same direction?

So starting with an up particle: $$\lvert \uparrow \rangle = \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$$ My understanding is that you can measure $$\lvert \uparrow \rangle$$ in $$X$$ and have a 50% chance of getting $$\lvert \leftarrow \rangle$$ and a 50% chance of getting $$\lvert \rightarrow \rangle$$ where: $$\lvert \rightarrow \rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix} \text{ and } \lvert \leftarrow \rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \end{bmatrix}$$ but you could also operate in $$X$$ which would be the equivalent of passing it though a NOT gate: $$X \cdot \lvert \uparrow \rangle = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix} = \lvert \downarrow \rangle$$

I've read that measuring qubit spin is more or less equivalent to measuring the orientation of a photon which can be done by passing it through polarized filters. If the photon is measured to be in one orientation and then is measured in a different orientation it has a certain probability of snapping to be in that other direction.

So in this example, that would be equivalent to measuring an up qubit in $$X$$. But how does operating in $$X$$ come into the equation? I understand the mathematical effect it has, but what does this mean for the physical qubit? Is it also being passed through a filter and how is that different from measuring?

Thanks!

• you can rotate spin using magnetic field; I don't remember how exactly to do it, but it should be explained in good QM courses. – kludg Oct 16 '19 at 10:49

In the particular case of $$X$$, you can actually think of it as an evolution, because $$X$$ is both Hermitian (thus an observable) and unitary (thus an evolution). What this evolution means physically depends on the implementation. If you are dealing with the polarisation of a photon, it can be representing the action of a phase shifter, which rotates the polarisation.
But it's important to distinguish between "acting on a state with the evolution $$X$$" and "measuring the state with the observable $$X$$". To "measure $$X$$" means physically to apply a measurement which makes the state collapse into one of the eigenstates of $$X$$. This involves doing something like acting on the state with a Hadamard gate and then measuring in the computational basis, so you can see how this is rather different than evolving with $$X$$.