What observable(s) does a mirror operate on?

Question

I'm reading this introduction to the Mach-Zehnder interferometer, and I'm confused about what the mirrors are doing.

Suppose we're describing the path (upper / lower) and polarization (vertical / horizontal) of a particle in the basis $|UV\rangle$, $|UH\rangle$, $|LV\rangle$, $|LH\rangle$. They describe the mirrors' operation as:

$$-\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix} = -(I_{pos}\otimes I_{pol}) = -I_{pos}\otimes I_{pol} = I_{pos}\otimes -I_{pol}$$

A couple of questions:

1. Many resources online describe a mirror as "phase shifting" the photon, but they don't give a precise explanation of what this means. From the above, I gather it's not a relative phase shift (which is what the phrase normally means in QIT) but a global one. Is this correct? (We could say that each mirror introduces a phase shift relative to the other path, but the pair introduces a global phase shift. Another way of saying this is that if there were only one path, the phase shift would not be relative to anything.)

2. It seems like a global phase shift cannot be specific to a particular observable: $-|\psi\rangle \otimes |\phi\rangle = |\phi\rangle \otimes -|\psi\rangle$ after all. Even worse, this means it isn't specific to a specific particle: it's equivalent to phase shifting the universal wave function, so it doesn't seem to make sense to ask "where" the shift got applied.

Clearly I'm misunderstanding something fundamental, and any tips would be appreciated.

Answer 1

If the full state is being reflected, then the introduced phase is not observable because it's a global phase. If, however, only part of the beam is reflected by a mirror while some other part is not, then this phase shift is not global, and can be observed for example by making the parts of the beam interfere with each other.

Answering to an observation in the comments:

And a global phase shift is not specific to any observable (if it can be said to have happened at all). Also, a waveplate introduces a relative polarization phase shift, and there's no way to make it global by moving or extending it: position eigenstates are by definition separated in space, while the polarization eigenstates aren't.

There is complete symmetry between position and polarisation in this regard.

If a photon is in a superposition of spatial modes, say something of the form $|\text{left path}\rangle+|\text{right path}\rangle$, you can introduce a phase shift between the two modes using for example a mirror, which might send this state into one of the form $|\text{left path}\rangle-|\text{right path}\rangle$.

Similarly, a photon can be in a superposition of different polarisation states, e.g. $\lvert\uparrow\rangle+\lvert\downarrow\rangle$, and a phase between the two polarisation modes can be introduced using a waveplate, which might produce something like $\lvert\uparrow\rangle-\lvert\downarrow\rangle$.

It is true that you can think of reflecting both spatial modes of the photon via a mirror, and then this would amount to an irrelevant global phase. You could then think that the analogous operation for the polarisation should be to "apply a waveplate to both polarisation states of the photon", which is not really a meaningful physical operation. That is not overly surprising though: different physical carriers of information can be handled in different ways. You could still do something similar as the "mirror on one side" operation on the polarisation by using a PBS to spatially separate the two polarisation modes, then put a mirror on one of the two paths, and use another PBS to join the paths together again.

Another difference that you might be wondering about is the following: we can understand the phase shift introduced by the waveplate as due to a "rotation" in the polarisation space. But in the case of the mirror, it's hard to see where this "rotation" is happening. The mirror is operating on a single spatial mode, there would seem to be no "auxiliary space" on which the mirror is acting. The answer is that there actually is such a space: the photon interacts with the mirror in a highly complex way, that will involve the atoms making up the mirror etc. A rotation is happening in this much larger space, but things are set up in such a way that the overall result is simply a phase shift, so you don't need to worry about the describing the whole thing, and can just use the effective description in which the mirror is modelled as simply "applying a phase shift to the state".