ZAB Controller described a restricted area incursion by a PHX Air Carrier departure indicating that the vector issued by the PHX Departure Controller most likely caused the infraction.

Narrative

I was working Sector 90 and Outlaw and Jackal MOA/ATCAA's had gone active about 5-10 minutes previous before the incident. Sector 46; the sector that works departures off of PHX; to the southeast; yelled across the room and said watch that guy when the sent a PVD of a B737 to my scope. PHX Approach had given the aircraft a vector out the departure gate; when they were supposed to have give the aircraft a full route clearance via TFD VOR because the gate was close due to the MOA/ATCAA's being active. I immediately got on Guard and called the aircraft that were working in OUTLAW and told them to maintain the altitudes of FL220B290 in OUTLAW; and to acknowledge with an IDENT. The aircraft working in the military airspace never IDENTED but instead came up on my frequency and asked why and I told them the reason and to maintain the assigned altitudes. By this time the B737 had already deviated the MOA/ATCAA airspace before being turned out by the Sector 46 Controller. I am not sure the procedures within PHX Approach airspace that notifies controllers that the departure gate is closed; but maybe something that makes the controllers more aware that the military airspace that is in their airspace is active would be more helpful.