How to initialize std::unique_ptr in constructor?

You need to initialize it through the member-initializer list:

A::A(std::string filename) :
    file(new std::ifstream(filename));
{ }

Your example was an attempt to call operator () on a unique_ptr which is not possible.

Update: BTW, C++14 has std::make_unique:

A::A(std::string filename) :
    file(std::make_unique<std::ifstream>(filename));
{ }