Medium Transport; Low Wing; 2 Turbojet Eng
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 6
Communication Breakdown
Conflict Airborne Conflict; Deviation - Altitude Excursion From Assigned Altitude; Inflight Event / Encounter Fuel Issue
Air Traffic Control Issued Advisory / Alert; Air Traffic Control Issued New Clearance; Air Traffic Control Provided Assistance
Procedure
Aircraft Y (FL270) was above Aircraft X (FL250).  Aircraft Y keyed up and said he was declaring a fuel [shortage] and would like to divert to ZZZ.  My Trainee keyed up and assigned heading 080 vectors for descent.  I did not like that heading so I reassigned heading 110.  Aircraft Y read that back; then we proceeded to gather more info.  The next update; Aircraft Y had descended to FL262 and the next update we lost separation.  He was not assigned a descent; nor did he report out of FL270.  As soon as we noticed the altitude change; I advised Aircraft X of the traffic who at this point was diverging and not a factor.  I believe the closest the two aircraft got was 300 feet and 3 miles diverging.Pilot needs to be more informed on what he is responsible to say if he needs an altitude change; or decides he needs to change altitude without assignment.
Center Controller reported that an aircraft declaring Minimum Fuel departed his assigned altitude without notification resulting in a conflict with another aircraft.