How wide can a sliding door be open, before it becomes a security risk?

Given that some spelunkers (cavers) manage to squeeze through 6-7" cracks, 4" seems about right (and make sure that an arm, perhaps longer than your own, or holding a tool such as @Matthew's wire, can't reach in far enough to defeat the lock - which would be one reason to go even smaller.)

You also need to be careful about creating a trap for yourself - if, in fact, you have some sort of deadbolt on the door, and you wake up to find a fire and a desperate need to leave the room, will you be able to open it, or will you die trying?

An alternate approach might be something like a pressure-pad alarm on the outside of the door.

If its open its going to be easier to break in whether its 1cm or 4inches.

A security gate always did wonders for us in South Africa, pretty sure in many other countries too.

TIP: When you install the security gate, make sure that it cannot be screwed out from its sockets, form the OUTSIDE. Or tampered with in any other way from the outside.

I have an aluminum dog door frame that fits from top to bottom and has channels so the door can be locked. One of these could be modified with hardware wire mesh instead of plexiglass.