Measuring the State of Photons: Determining Mixed & Pure States

[nard]

Suppose we have a photon whose polarization state is unknown, what measurement can be done to determine the state of that photon? How many times are we supposed to make the measurement in order to confirm our results? How can we distinguish a mixed state to a pure state?

 

[Ken G]

Suppose we have a photon whose polarization state is unknown, what measurement can be done to determine the state of that photon?

First you have to prepare a whole bunch of photons in the same state, so that you can characterize what that state is. Let's say you prepare N >> 1 photons in the same state (say, using a laser). Next you send them all through a linear polarizer at some angle. If they all get through, you know that was the polarization of the state. If they don't, turn the polarizer a little and do it again. Keep doing that until they all get through, now you know their polarization (and you know how the state is prepared, so you can assume it will always work that way). If they never all get through, they must not be linearly polarized, so you bring out the circular polarizer and do it again. If that doesn't work either, they are elliptically polarized, so you maximize the circular and the linear outputs, and use that to figure out the superposition you had originally.

How many times are we supposed to make the measurement in order to confirm our results?

You need many measurements to characterize the state. However, once you understand how the preparation of the state links up with the measurements, you can "learn how it works", and not have to do it again every time.

How can we distinguish a mixed state to a pure state?

A mixed state will never let all the photons through any polarizer, and when the pure state doesn't either, you will get different results when you start using both linear and circular polarizations-- the pure state will have a different signature that will tell you all the photons are in the same state. For the mixed state, the outcomes will not be consistent with all the photons being in the same state, they have to be a mixture of different pure states.