This question tests knowledge of which Indian banks are government owned, called public sector banks, and which are privately owned. A bank is called a public sector bank when the majority of its shares are held by the Government of India.
The correct option is South Indian Bank because it is the only bank among the four that has always been in the private sector and was never taken over by the government.
Let's summarize:
So South Indian Bank is the one that is not a public sector bank.