On April 24 the World Health Organization (WHO) posted the scientific brief "Immunity passports" in the context of COVID-19. It is very readable, and I encourage people to read it.

In a nutshell, the data is not yet in regarding immunity from the SARS-CoV-2 virus. We don’t know if the antibody tests now being deployed are accurate, and even if they are accurate, we don’t yet know what the results imply.

As of 24 April 2020, no study has evaluated whether the presence of antibodies to SARS-CoV-2 confers immunity to subsequent infection by this virus in humans.
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Laboratory tests that detect antibodies to SARS-CoV-2 in people, including rapid immunodiagnostic tests, need further validation to determine their accuracy and reliability.
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At this point in the pandemic, there is not enough evidence about the effectiveness of antibody-mediated immunity to guarantee the accuracy of an “immunity passport” or “risk-free certificate.”

Not mentioned in the WHO scientific brief is what all this might mean to herd immunity.

Key to the herd immunity concept is that once you have become infected and recovered you cannot become infected again.

I suspect the purposes of this scientific brief are to guide policy makers and to adjust people’s expectations.