State-independent purity and fidelity of quantum operations

Abstract

The purity and fidelity of quantum operations are of great importance in characterizing the quality of quantum operations. The currently available definitions of the purity and fidelity of quantum operations are based on the average over all possible input pure quantum states, i.e. they are state-dependent (SD). In this paper, without resorting to quantum states, we define the state-independent (SI) purity and fidelity of a general quantum operation (evolution) in virtue of a new density matrix formalism for quantum operations, which is extended from the quantum state level to quantum operation level. The SI purity and fidelity gain more intrinsic physical properties of quantum operations than state-dependent ones, such as the purity of a one-qubit amplitude damping channel (with damping rate 1) is 1/2, which is in line with the fact that the channel is still a nonunitary operation described by two Kraus operators rather than a unitary one. But the state-dependent Haar average purity is 1 in this case. So the SI purity and fidelity proposed here can help the experimentalists to exactly quantify the implementation quality of an operation. As a byproduct, a new measure of the operator entanglement is proposed for a quantum evolution (unitary or nonunitary) in terms of the linear entropy of its density matrix on the orthonormal operator bases (OOBs) in Hilbert–Schmidt space.