Abstract
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.
Highlights
As a measure of the distinguishability between quantum states[1,2,3], plays an important role in many areas of quantum information science, for example it is related to the precision limit in quantum metrology [4], serves as a measure of entanglement preservation through noisy quantum channels [5], and a measure of entanglement preservation in quantum memory [6]; it has been used as a characterization method for quantum phase transitions [7], and a criterion for successful transmission in formulating quantum channel capacities [8]
This fidelity function on quantum channels lead to various distance measures defined directly on quantum channels, in particular we show the Bures angle and the Bures distance can be extended to quantum channels
We show the distance between quantum channels leads naturally to a new Fisher information on quantum channels which quantifies the ultimate precision limit in quantum metrology
Summary
As a measure of the distinguishability between quantum states[1,2,3], plays an important role in many areas of quantum information science, for example it is related to the precision limit in quantum metrology [4], serves as a measure of entanglement preservation through noisy quantum channels [5], and a measure of entanglement preservation in quantum memory [6]; it has been used as a characterization method for quantum phase transitions [7], and a criterion for successful transmission in formulating quantum channel capacities [8]. Here X 1 = T r X†X, ρSA denotes a state on system+ancilla, and IA denotes the identity operator on the ancillary system), is induced by the trace distance on quantum states ρ1 − ρ2 1; another measure on quantum channels which is defined as arccos Fmin(K1, K2) = arccos minρSA FS[K1 ⊗IA(ρSA), K2 ⊗IA(ρSA)][12, 13], is induced by the fidelity on quantum states FS(ρ1, ρ2) = T r ρ12 ρ2ρ12 These induced measures through quantum states lack a direct connection to the properties of quantum channels, which severely restrict the insights that can be gained from these measures. We show that this fidelity function provides a unified framework for perfect quantum channel discrimination and quantum metrology, in particular we show 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
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