Um,Antibody tests are irrelevant for this. I am talking PCR. It has a non negligible false negative rate but basically non-existent false positive (not that that matters). Look at the Paul Romer link above, it is quite informative.
The PCR test is the current standard test and would be perfectly adequate for this purpose. Let’s try to do 100 million per month. Not sure why we are not already doing so. It’s not like people have anything else to do - they may as well get paid for making and administering tests.
The false-negative rate for SARS-CoV-2 RT-PCR testing is highly variable: highest within the first 5 days after exposure (up to 67%), and lowest on day 8 after exposure (21%)."
False-Negative Rate of RT-PCR SARS-CoV-2 Tests - American College of Cardiology
In the real world, testing conditions and process are far from perfect, and accuracy suffers. Researchers still don’t know what the real-world false positive rate is, but clinical sensitivity of RT-PCR tests ranges from 66% to 80%. That means nearly one in three infected people who are tested will receive false negative results.
Coronavirus tests are pretty accurate, but far from perfect