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Physicist help needed: From my near null knowledge of quantum mechanics it seems they decrease randomness in phase and amplitude measurement by increasing uncertainty in mirrors position. So, this is interesting because the mirrors are macroscopic objects that interacted with the light in a rather simple way and conditions. On the other hand... Don't they require the mirror position in a well determined position to measure the gravitational waves?

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They are increasing the precision of the phase while decreasing the precision of the amplitude. Since the amplitude of the light determines how hard it pushes on the mirrors, this has the side-effect of also making the positions of the mirrors more uncertain. To get the most accurate measurement possible, I'd guess that they would squeeze the light until getting any additional precision in the phase was not worth the added noise in mirror position.

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