Date: 202009
Local time of day: 0601-1200
Location: ZZZ.Airport, US
Weather elements / visibility: Haze / Smoke; Fog; 10
Light: Daylight
Aircraft: Skylane 182/RG Turbo Skylane/RG
Operator: FBO
Operating under: Part 91
Flight plan: IFR
Mission: Training
Flight phase: Descent; Cruise
Airspace: Class E ZZZ
ATC / advisory: TRACON ZZZ
Second aircraft: Small Aircraft
Reporter organization: FBO
Reporter function: Pilot Flying
Reporter qualification: Flight Crew Student; Flight Crew Private
Human factors: Confusion; Communication Breakdown
Second reporter function: Approach
Anomaly: ATC Issue All Types; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Detector: Automation Air Traffic Control; Person Air Traffic Control
When detected: In-flight
Result: Air Traffic Control Issued Advisory / Alert; Flight Crew Became Reoriented; Flight Crew FLC complied w / Automation / Advisory
Contributing factors: Human Factors
Primary problem: Human Factors

Narrative: I was on an IFR training flight in Aircraft X. I was on vectors to ZZZZZ1 over ZZZZZ2 at 6;000 ft. with Approach; expecting the RNAV XX approach at ZZZ. At a few miles from ZZZZZ2; Approach gave me the instruction; 'Aircraft X descend and maintain 4;500 ft. direct ZZZZZ.' I read back; 'Descending 4;500 ft. direct ZZZZZ Aircraft X.' NorCal didn't give a response to my readback; so I proceeded to comply with the instruction. A few moments later; NorCal gave me an altitude advisory saying I was below MVA and was told to immediately climb to 6;000 ft.


Second narrative: I had 3 aircraft setting up for the GPS XX approach at ZZZ. #1 was Aircraft Y who was 5 miles south-southeast of ZZZZZ; IAF for ZZZ. #2 was Aircraft Z who was about 20 miles south of ZZZZZ; and #3 was Aircraft X who was about 20 east of ZZZZZ. Aircraft X checked on to my frequency level at 6000 on a heading. I acknowledged Aircraft X; told the aircraft to fly heading 200 and maintain 6000; he would be #3. After a few other transmissions; I cleared Aircraft Y to maintain 4500 until established on the approach and cleared GPS XX approach ZZZ. Aircraft X read back; 'Roger maintain 4500 till established;' which I missed.I then vectored a VFR aircraft away from a TFR. Then; Aircraft Y said; 'We copied; maintain 4500 till established; cleared approach.' After a few more transmissions I recognized that Aircraft X was descending below the assigned altitude of 6000. At 5600 which was the MVA in the area; I questioned Aircraft X why he was descending and to maintain 6000; with no response. As Aircraft X was descending below 5400 and now in the MVA; I issued a low altitude alert for Aircraft X and instructed him to climb and maintain 6000 immediately; which he acknowledged and complied. After asking Aircraft X what had happened; he responded that his auto pilot had shut off; and he was apologetic. I missed the hearback/readback. However; I never would have thought that the aircraft that was on a heading 20 miles away from the IAF would take a clearance to maintain 4500 till established and cleared approach. The pilot was issued a brasher; and upon further review I missed Aircraft X taking a clearance for Aircraft Y. Be vigilant listening to call signs and instructions on hearback/readbacks.


Synopsis: C182 Student Pilot and ATC Controller reported the former taking the clearance of another aircraft and descended below the MVA as a result.