Fidelity and Fisher information on quantum channels

The fidelity function for quantum states has been widely used in quantum information science and frequently arises in the quantification of optimal performances for the estimation and distinguishing of quantum states. A fidelity function for quantum channels is expected to have the same wide applications in quantum information science. In this paper we propose a fidelity function for quantum channels and show that various distance measures on quantum channels can be obtained from this fidelity function; for example, the Bures angle and the Bures distance can be extended to quantum channels via this fidelity function. We then show that the distances between quantum channels lead naturally to a quantum channel Fisher information which quantifies the ultimate precision limit in quantum metrology; the ultimate precision limit can thus be seen as a manifestation of the distances between quantum channels. We also show that the fidelity of quantum channels provides a unified framework for perfect quantum channel discrimination and quantum metrology. In particular, we show that the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.