Having the string about the thickness of a business card or a dollar bill above the fingerboard right at the nut is a good height. So that is less than a mm.
That way, the string is easy to hold down in 1/2 and 1st position and it shouldn't buzz on open strings. It'll make notes higher up on the board easier to hold down and play in tune, as well.
If the space there is too high, when you finger a note, the string bends in from both the bridge and the nut, much like the bending of a bow & arrow string. If the height is low, when you hold down a note, the string between the nut and your finger is pretty much flat against the board.
The luthier will use some round needle files to lower the grooves to the right height. This is not hard to do yourself, just be sure the front edge of the nut, where it meets the fingerboard, is the highest part of the groove. Use lots of pencil in the groove to lubricate it before you put the string back on. Put some pencil in the groove of the bridge and give the string a deep cleaning with methyl hydrate or rubbing alcohol (see #8 of
this thread) while you are at it.
Remember you can only remove wood and not put more back on, so go slowly when you are nearing the end. However, if you go too low, there are several ways to increase the height slightly, but I won't go into that right now.
If you are doing this yourself, do it one string at a time so the pressure is always on the belly of the instrument and the soundpost won't fall down.