How to experimentally measure a quantum gate

[confused_man]

I'm trying to figure out how you would actually measure the result of a quantum gate. For example, suppose I build a Hadamard gate by using a beam splitter.

The output of this gate creates either $$|\psi> = \frac{1}{\sqrt{2}} |0> + \frac{1}{\sqrt{2}} |1>$$ or $$|\psi> = \frac{1}{\sqrt{2}} |0> - \frac{1}{\sqrt{2}} |1>$$ depending on if the input state was $$|\psi> = |0>$$ or $$|\psi> = |1> .$$

So how would you measure this experimentally? I'm I know that you can set up photon detectors to measure photon counts going either through the beam splitter or reflected from it, but how would you actually see the effect of the minus sign?

Any insight would be greatly appreciated. Thanks!

 

[Strilanc]

If someone is handing you $|+\rangle$ or $|-\rangle$ states and you can only do Z-axis measurements then you can distinguish between the two cases by applying a Hadamard gate before the Z-axis measurement.

You could also do a $\sqrt{Y}$ operation, or various other ones. Maybe even complicated combinations of operations which together add up into something that ultimately rotate what was the qubit's X axis so that it ends up vertical.