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arXiv:1506.02304 · Physical Review A

Cohering and decohering power of quantum channels

Azam Mani, Vahid Karimipour

Published Sep 29, 2015• 155 citations
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Abstract

We introduce the concepts of cohering and decohering power of quantum channels. Using the axiomatic definition of the coherence measure, we show that the optimization required for calculations of these measures can be restricted to pure input states and hence greatly simplified. We then use two examples of this measure, one based on the skew information and the other based on the ${l}_{1}$ norm; we find the cohering and decohering measures of a number of one-, two-, and $n$-qubit channels. Contrary to the view at first glance, it is seen that quantum channels can have cohering power. It is also shown that a specific property of a qubit unitary map is that it has equal cohering and decohering power in any basis. Finally, we derive simple relations between cohering and decohering powers of unitary qubit gates and their tensor products, results which have physically interesting implications.

DOI
10.1103/physreva.92.032331

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We introduce the concepts of cohering and decohering power of quantum channels. Using the axiomatic definition of the coherence measure, we show that the optimization required for calculations of these measures can be restricted to pure input states and hence greatly simplified. We then use two examples of this measure, one based on the skew information and the other based on the l1{l}_{1}l1​ norm; we find the cohering and decohering measures of a number of one-, two-, and nnn-qubit channels. Contrary to the view at first glance, it is seen that quantum channels can have cohering power. It is also shown that a specific property of a qubit unitary map is that it has equal cohering and decohering power in any basis. Finally, we derive simple relations between cohering and decohering powers of unitary qubit gates and their tensor products, results which have physically interesting implications.
Having a suitable measure to quantify the coherence of quantum states, a natural task is to evaluate the power of quantum channels for creating or destroying the coherence of input quantum states. In the present paper, by introducing the maximal coherent Gaussian states based on the relative entropy measure of coherence, we propose the (generalized) cohering power and the (generalized) decohering power of Gaussian unitary operations for continuous-variable systems. Some basic properties are obtained and the cohering power and decohering power of two special kinds of Gaussian unitary operations are calculated.
Abstract In this paper, we investigate the cohering and decohering power of the one-qubit Markovian channels with respect to coherence measures based on the l 1 - norm, the Rényi α - relative entropy and the Tsallis α - relative entropy of coherence, respectively. The amplitude damping channel, phase damping channel, depolarizing channel, and flip channels are analytically calculated. It shows that the decohering power of the amplitude damping channel on the x , y , and z basis is equal to each other. The same phenomenon can be seen for the phase damping channel and the flip channels. The cohering power for the phase damping channel and the flip channels on the x , y basis also equals to that on the z basis. However, the cohering and decohering power of the depolarizing channel is independent to the reference basises. And the cohering power of the amplitude damping channel on the x , y basis is different to that on the z basis.
The coherence power of a quantum channel, that is, its maximum ability to increase the coherence of input states, is a fundamental concept within the framework of the resource theory of coherence. In this note we discuss various possible definitions of coherence power and coherence rate and their basic properties. Then we prove that the coherence power of a unitary operator acting on a qubit, computed with respect to the l1-coherence measure, can be calculated by maximizing its coherence gain over pure incoherent states. We proceed to show that this result fails in the general case, that is, the maximal coherence gain is found when acting on a state with non-vanishing coherence in the case of the l1-coherence and dimension N > 2, the relative entropy of coherence and the geometric measure of coherence.
The coherence power of a quantum channel, that is, its ability to increase the coherence of input states, is a fundamental concept within the framework of the resource theory of coherence. In this note we discuss various possible definitions of coherence power. Then we prove that the coherence power of a unitary operator acting on a qubit, computed with respect to the l1l_1l1​-coherence measure, can be calculated by maximizing its coherence gain over pure incoherent states. We proceed to show that this result fails for dimensions N>2, that is, the maximal coherence gain is found when acting on a state with non-vanishing coherence.
In this paper, we consider the cohering power of unitary operation, which is defined by the maximum coherence caused by an unitary operation acting on incoherent states. We obtain that the optimizations can be reduced to a simple discrete set that depends only on the fixed reference basis. Then, we investigate that the de-cohering power of quantum channel, which is defined by the maximum decay of coherence caused by quantum channel acting on the maximally coherent states. Based on the relative entropy, we discuss the cohering power of 1-qubit and 2-qubit unitary operations and the de-cohering power of quantum channel on 1-qubit system. Finally, we study the relation between the coherence and the discord via incoherent operation.
We study two distinct diagnostics of bipartite quantum channels: the average fidelity between a desired state and noisy output across different classes of input states, and generation of entanglement from pure product input states, quantified by well-defined entangling power. These diagnostics are generally independent, but together they provide a more complete description of bipartite channels. We show that the fidelity averaged over any local-unitary orbit with fixed Schmidt coefficients for equal local dimensions is completely determined by the average input-output fidelity and its restriction to product inputs. We also introduce concurrence- and negativity-based entangling power for two-qubit channels, prove their convexity and monotonicity under local postprocessing, and show that, unlike the previously proposed linear-entropy quantity, they vanish for all separable channels. Representative separable and nonseparable channels are investigated. Finally, we generalize our definitions of entangling power to nonproduct inputs, and provide an analytical lower bound for the concurrence-based entanglement variation, showing its effectiveness with specific examples.