  We study a generalisation of thermodynamic geometry to degenerate quantum
ground states at zero temperatures exemplified by charged extremal black holes
in type II string theories. Several examples of extremal charged black holes
with non degenerate thermodynamic geometries and finite but non zero state
space scalar curvatures are established. These include black holes described by
D1-D5-P and D2-D6-NS5-P brane systems and also two charged small black holes in
Type II string theories. We also explore the modifications to the state space
geometry and the scalar curvature due to the higher derivative contributions
and string loop corrections as well as an exact entropy expression from quantum
information theory. Our construction describes state space geometries arising
out of a possible limiting thermodynamic characterisation of degenerate quantum
ground states at zero temperatures.
