Bures distance for completely positive maps

D. Bures defined a metric on states of a $C$-algebra as the infimum of the distance between associated vectors in common GNS representations. Now there are modifications and extensions of this notion to completely positive maps. Our approach is through Paschke’s Hilbert $C^*$-module version of Stinespring’s theorem. We present some recent results in the area. This is based on joint works with K. Sumesh and Mithun Mukherjee.