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Comparison between cohering power and de-cohering power

Kaifeng Bu, Chunhe Xiong

Published Apr 22, 2016• 1 citation
OpenAlex

Abstract

In this paper, we list the results about cohering power and de-cohering power and make a comparison between them. Moreover, we use the results about the cohering power to show that there is no monotone for $l_1$ norm of coherence under maximally incoherent operation and dephasing-covariant incoherent operation.

Related papers

A note on cohering power and de-cohering power

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Cohering power and de-cohering power have recently been proposed to quantify the ability of a quantum operation to produce and erase coherence respectively. In this paper, we investigate the properties of cohering power and de-cohering power. First, we prove the equivalence between two different kinds of cohering power for any quantum operation on single qubit systems, which implies that l1 norm of coherence is monotone under Maximally incoherent operation (MIO) and Dephasing-covariant operation (DIO) in 2-dimensional space. In higher dimensions, however, we show that the monotonicity under MIO or DIO does not hold. Besides, we compare the set of quantum operations with zero cohering power with Maximally incoherent operation (MIO) and Incoherent operation (IO). Moreover, two different types of de-cohering power are defined and we find that they are not equal in single qubit systems. Finally, we make a comparison between cohering power and de-cohering power for single qubit unitary operations and show that cohering power is always larger than de-cohering power.

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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.

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Generalized Coherence Concurrence and Coherence Number.

Seungbeom Chin

We propose a family of coherence monotones, named the \emph{generalized coherence concurrence} (or coherence $k$-concurrence), which has a close relation with the generalized entanglement concurrence. The coherence $k$-concurrence of a state is nonzero if and only if the coherence number (a recently introduced discrete coherence monotone) of the state is not smaller than $k$, and a state can be converted to a state with nonzero entanglement $k$-concurrence via incoherent operations if and only if the state has nonzero coherence $k$-concurrence. The generalized coherence concurrence can be considered as the entanglement-based coherence monotone.

•Feb 20, 2017•0 citations
In this paper, we list the results about cohering power and de-cohering power and make a comparison between them. Moreover, we use the results about the cohering power to show that there is no monotone for l1l_1l1​ norm of coherence under maximally incoherent operation and dephasing-covariant incoherent operation.
Cohering power and de-cohering power have recently been proposed to quantify the ability of a quantum operation to produce and erase coherence respectively. In this paper, we investigate the properties of cohering power and de-cohering power. First, we prove the equivalence between two different kinds of cohering power for any quantum operation on single qubit systems, which implies that l1 norm of coherence is monotone under Maximally incoherent operation (MIO) and Dephasing-covariant operation (DIO) in 2-dimensional space. In higher dimensions, however, we show that the monotonicity under MIO or DIO does not hold. Besides, we compare the set of quantum operations with zero cohering power with Maximally incoherent operation (MIO) and Incoherent operation (IO). Moreover, two different types of de-cohering power are defined and we find that they are not equal in single qubit systems. Finally, we make a comparison between cohering power and de-cohering power for single qubit unitary operations and show that cohering power is always larger than de-cohering power.
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.
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.
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.
We study the task of coherence filtration under strictly incoherent operations. The aim of this task is to transform a given state $\ensuremath{\rho}$ into another one ${\ensuremath{\rho}}^{\ensuremath{'}}$ whose fidelity with the maximally coherent state is maximal by using stochastic strictly incoherent operations. We find that the maximal fidelity between ${\ensuremath{\rho}}^{\ensuremath{'}}$ and the maximally coherent state is given by a multiple of the $\mathrm{\ensuremath{\Delta}}$ robustness of coherence, $R(\ensuremath{\rho}\ensuremath{\parallel}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}):=min{\ensuremath{\lambda}|\ensuremath{\rho}\ensuremath{\le}\ensuremath{\lambda}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}}$, which provides $R(\ensuremath{\rho}\ensuremath{\parallel}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho})$ an operational interpretation. Finally, we provide a coherence measure based on the task of coherence filtration.
We propose a family of coherence monotones, named the \emph{generalized coherence concurrence} (or coherence kkk-concurrence), which has a close relation with the generalized entanglement concurrence. The coherence kkk-concurrence of a state is nonzero if and only if the coherence number (a recently introduced discrete coherence monotone) of the state is not smaller than kkk, and a state can be converted to a state with nonzero entanglement kkk-concurrence via incoherent operations if and only if the state has nonzero coherence kkk-concurrence. The generalized coherence concurrence can be considered as the entanglement-based coherence monotone.