A TRACON Controller had to issue a terrain alert to an aircraft that descended below its assigned altitude. When the aircraft recovered from the low altitude alert it climbed above its assigned altitude into confliction with another aircraft.

Narrative

Aircraft X was IFR landing at ZZZ airport. Came from ZZZ1 approach at 5000 ft. When Aircraft X got into my airspace he was told to descend to 3000 ft. And expect a visual approach for Runway XX at ZZZ. ZZZ1 arrivals landing Runway XY. Aircraft Y was being vectored to ILS XY to land. Crossing IAF at 4000 ft. Aircraft X started to descend lower then 3000 ft. When I got a low altitude alert. I told Aircraft X about the alert and told him to check his altitude immediately and told him the altimeter setting. Aircraft X was cleared for the visual approach Runway XX to ZZZ and issued traffic landing RWY XY at ZZZ1.Aircraft X then began to climb out of 2700 ft. To 3400 ft. With traffic closer the 3 miles. I issued traffic to Aircraft X about Aircraft Y on final for RWY XY. Aircraft X called the traffic in sight but IFR separation was lost. Aircraft X got 2.7 miles away from Aircraft Y with 800 feet. I didn't give Aircraft X brasher for pilot deviation. Suggestion: Aircraft X had a pilot deviation and climbed to an altitude I didn't assign. Being cleared for a visual approach aircraft cannot climb higher then the altitude I gave him at 3000 ft.