Indistinguishability of Quantum States and its Consequences In the book Quantum Computation and Quantum Information, there is a discussion about how if states are not orthonormal then there is no quantum measurement capable of distinguishing the states.
I am interested in the consequences of this. What does this mean physically? How is this applied when manipulating quantum states?
I am sure there are other questions that stem from this that I am not even thinking of, but that would be interesting to explore. So any help in understanding the consequences of this would be greatly appreciated.
 A: Let assume two states $|x\rangle$ and $|y\rangle$ and their inner product $\langle x|y\rangle$. If $\langle x|y\rangle = 0$, then both state can be perfecly distinguished and there is no uncertainty which is which. Examples of such states are $|0\rangle$ and $|1\rangle$ or $|+\rangle$ or $|-\rangle$. As $\langle x|y\rangle$ tends to $1$, the states are more and more similar and less distinguishable. In extereme case when $\langle x|y\rangle$ is 1, it holds that $|y\rangle = \mathrm{e}^{i\theta}|x\rangle$. Such states differ in global phase only and they are absolutely indistinguishable. This all means that not only two orthogonal states are distinguishable but others as well. However, as $\langle x|y\rangle$ goes to one, the states are more and more similar.
Note that you can employ so-called swap test to calculate value $|\langle x|y\rangle|^2$ and decide how two state are similar to each other.
A: I'm not sure if this will help but a very sharp former colleague once posed this question.
You have two boxes. The first contains 1000 horizontally polarized photons and 1000 vertically polarized photons while the second contains 1000 left circularly polarized photons and 1000 right circularly polarized photons. How do you tell which is which?
Since this is sort of a riddle, I'll make the answer below not visible by default.

 Use a vertical polarization filter. If it lets exactly 1000 photons through then you opened the first box. If there's a slight imbalance then you opened the second box. Note than even though 1000 is the most likely value for the second box, binomial distributions are such that it is more likely for you to not see the most likely value :).

