AT1 FY21 - General Discussion + FA, page-528

So from my understanding of PCR as provided by scientists with whom I worked with when designing PCR test devices, the sample is lysed(chemically or mechanically broken apart) to exposed any target RNA strands in the sample of (sputum/nasal secretions in this case), then the target RNA is selectively amplified a number of times(30-40 typically) and tagged with a marker molecule so that the presence can be detected and the viral load can be estimated.

The RNA segments in the sample that identified the virus can therefore come from either live (viable) or dead virus and will still show up in the test due to its sensitivity. If a person has significant antibodies, has been previously shown to have recovered and test positive to PCR it is likely the virus they are shedding is not viable and therefore they are not contagious. The other less likely possibility is that the person is fighting the virus but has not yet recovered and is shedding live virus.

This is where symptoms come into the equation and the experts review the case and make a judgement on whether the case is still active or an old infection that is under control.

You may want to read further and DYOR on this.