On the fidelity of two pure states

The fidelity of two pure states (also known as transition probability) is a symmetric function of two operators, and well founded operationally as an event probability in a certain preparation-test pair. Motivated by the idea that the fidelity is the continuous quantum extension of the combinatorial equality function, we enquire whether there exists a symmetric operational way of obtaining the fidelity. It is shown that this is impossible. Finally, we discuss the optimal universal approximation by a quantum operation.