Statement Breakdown:

1. It is a safe motherhood intervention of the State Health Departments:
   • Incorrect: The Janani Suraksha Yojana (JSY) is a centrally sponsored scheme under the National Rural Health Mission (NRHM) of the Union Ministry of Health & Family Welfare, not solely managed by state health departments. While it is implemented at the state level, it is a central government initiative.
2. Its objective is to reduce maternal and neonatal mortality among poor pregnant women:
   • Correct: The main goal of JSY is indeed to reduce maternal and neonatal mortality, particularly among poor pregnant women by promoting institutional deliveries. It focuses on vulnerable groups in rural areas.
3. It aims to promote institutional delivery among poor pregnant women:
   • Correct: One of the key objectives of JSY is to encourage institutional delivery, particularly for women in rural areas, by providing financial assistance to them.
4. Its objective includes providing public health facilities to sick infants up to one year of age:
   • Incorrect: JSY focuses on safe motherhood and institutional delivery for pregnant women, not directly on providing public health services to sick infants. The scheme does not target infant care up to one year of age.

Based on the corrected analysis, statements 2 and 3 are correct, while statements 1 and 4 are incorrect.

Thus, the correct answer is Only two.