Step 1: The proportion of babies dying within the first four weeks of life, expressed per 1000 live births, reflects how well a state delivers obstetric and newborn services.
Step 2: At the national level Kerala leads with the smallest such figure, but Kerala is absent from the choices, so the comparison is restricted to the four listed states.
Step 3: Of those four, the southern state of Tamil Nadu posts the minimum value, ahead of Delhi, Karnataka and Maharashtra.
Step 4: Hence the correct pick is Tamil Nadu.
\[\boxed{\text{Tamil Nadu}}\]