Yes, you file the k-1 on the 1040, but you do not file a state income tax return. THAT IS THE WHOLE POINT OF A COMPOSITE RETURN. Since the partner has no other connection to that particular state, the only income is the k1, then you do not need to file a personal state return in that state.

Attach a copy of the composite form showing the tax paid for the individual to the resident state claiming the credit.