I also wonder how early after infection a test will show positive.
The PCR swab test is 100% negative - ie useless - on day 1, falling to 61% on day 4. You normally become symptomatic (assuming you have symptoms) late on the 4th day or on the 5th day. Still 39% useless on day of symptoms. This assumes a proper swab, taken by someone else, via the nose and into the throat. An oral swab adds an additional 20% false negative
The antibody test done in a lab on blood from the elbow has less than 2% false negative but some people do not produce antibodies. ie it correctly says 'no antibodies' even though you have had the virus. It isnt worth doing until about day 11 but antibodies will persist at least a month and we are now thinking three months