PCR positive doesn't mean infected.
You'll need an high amount in PCR, i.e. low cycle threshold for the probability to be high.
Antigen tests will show if there was/is an infection.
PCR does not show infection only presence. (WHO has now put out guidance about this. They don't have to it's well known.)
If you're European this may be strange to you but I'm the U.S. we don't receive the cycle count when the RNA for the virus is detected.
Here's Fauci talking about it a little: https://youtu.be/a_Vy6fgaBPE&t=4m30s