But they both died after sustaining group A streptococcal infections shortly after discharge.
A statement from the trust said Ms Kimmance developed fatal toxic shock on 23 December and Ms Pickett died from a sudden onset of severe pneumonia the following day.
Winchester and Eastleigh Healthcare NHS Trust said the two incidents were separate and were not likely to be linked to the clinical care received at the trust.
However maternity staff and the unit itself have been swabbed as a precaution and has begun an investigation to find out the causes of the infections.
The trust’s medical director, Kevin Stewart, said: ‘We live in an age where deaths during or after childbirth are very rare as are those from group A streptococcal infections. This makes these two deaths in a short space of time all the more shocking.’
Both babies are reported to be in good health. An inquest has been opened and adjourned into the two deaths.