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Communications in Theoretical Physics

Comment on “Compare Quantum Operation Sensitivity for Different Distance Measures"

Gao Yun-feng, Jian Qiang Feng

Published Nov 1, 2011• 0 citations
Publisher

Abstract

We make comments on the conclusions of the paper: “Compare Quantum Operation Sensitivity for Different Distance Measures" [ Commun. Theor. Phys. 53 (2010) 635 ], and give a physical example to describe the non-equivalence of the fidelity and the trace distance. We prove that the fidelity is not more sensitive to a quantum operation than the trace distance.

DOI
10.1088/0253-6102/56/5/06

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We make comments on the conclusions of the paper: “Compare Quantum Operation Sensitivity for Different Distance Measures" [ Commun. Theor. Phys. 53 (2010) 635 ], and give a physical example to describe the non-equivalence of the fidelity and the trace distance. We prove that the fidelity is not more sensitive to a quantum operation than the trace distance.
We investigate the coherence measures induced by fidelity and trace norm, based on the coherence quantification recently proposed by Baumgratz et al. [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We show that the fidelity of coherence does not in general satisfy the monotonicity requirement as a measure of coherence under the subselection of the measurement condition. We find that the trace norm of coherence can act as a measure of coherence for qubits and some special class of qutrits with some restrictions on the incoherent operators, while the general case needs to be explored further.
The problem of quantifying the difference between evolutions of an open quantum system (in particular, between the actual evolution of an open system and the ideal target operation on the corresponding closed system) is important in quantum control, especially in control of quantum information processing. Motivated by this problem, we develop a measure for evaluating the distance between unitary evolution operators of a composite quantum system that consists of a sub-system of interest (e.g. a quantum information processor) and environment. The main characteristic of this measure is the invariance with respect to the effect of the evolution operator on the environment, which follows from an equivalence relation that exists between unitary operators acting on the composite system, when the effect on only the sub-system of interest is considered. The invariance to the environment's transformation makes it possible to quantitatively compare the evolution of an open quantum system and its closed counterpart. The distance measure also determines the fidelity bounds of a general quantum channel (a completely positive and trace-preserving map acting on the sub-system of interest) with respect to a unitary target transformation. This measure is also independent of the initial state of the system and straightforward to numerically calculate. As an example, the measure is used in numerical simulations to evaluate fidelities of optimally controlled quantum gate operations (for one- and two-qubit systems), in the presence of a decohering environment. This example illustrates the utility of this measure for optimal control of quantum operations in the realistic case of open-system dynamics.