Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1609.08759 · Communications in Theoretical Physics

Quantum Coherence Quantifiers Based on Rényi α -Relative Entropy*

Lian-He 连合 Shao 邵, Yong-Ming 永明 Li 李, Yu 宇 Luo 罗, Zheng-Jun 政军 Xi 席

Published Jun 1, 2017• 41 citations
arXivHTMLPDF

Abstract

Abstract The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the Rényi α-relative entropy, which present in [Phys. Rev. A 94 (2016) 052336]. We show that the Rényi α-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement, which presents in [Phys. Rev. A 93 (2016) 032136]. Due to the Rényi α-relative entropy of coherence can act as a coherence monotone quantifier, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of Rényi 2-relative entropy of coherence are derived .

DOI
10.1088/0253-6102/67/6/631

Related papers

Quantum-coherence quantifiers based on the Tsallis relative α entropies

Alexey E. Rastegin

The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum level. Various measures could be proposed to quantify coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. Basic properties of the considered coherence quantifiers are derived. Tradeoff relations between coherence and mixedness are examined. The property of monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.

arXiv:1512.06652•Physical Review A•Mar 30, 2016•194 citations

Coherence measure in terms of the Tsallis relative α entropy

Haiqing Zhao, Chang‐shui Yu

Coherence is the most fundamental quantum feature of the nonclassical systems. The understanding of coherence within the resource theory has been attracting increasing interest among which the quantification of coherence is an essential ingredient. A satisfactory measure should meet certain standard criteria. It seems that the most crucial criterion should be the strong monotonicity, that is, average coherence doesn't increase under the (sub-selective) incoherent operations. Recently, the Tsallis relative α entropy has been tried to quantify the coherence. But it was shown to violate the strong monotonicity, even though it can unambiguously distinguish the coherent and the incoherent states with the monotonicity. Here we establish a family of coherence quantifiers which are closely related to the Tsallis relative α entropy. It proves that this family of quantifiers satisfy all the standard criteria and particularly cover several typical coherence measures.

Scientific Reports•Jan 4, 2018•68 citations

Remedying the strong monotonicity of the coherence measure in terms of the Tsallis relative $α$ entropy

Haiqing Zhao, Chang‐shui Yu

Coherence is the most fundamental quantum feature of the nonclassical systems. The understanding of coherence within the resource theory has been attracting increasing interest among which the quantification of coherence is an essential ingredient. A satisfactory measure should meet certain standard criteria. It seems that the most crucial criterion should be the strong monotonicity, that is, average coherence doesn't increase under the (sub-selective) incoherent operations. Recently, the Tsallis relative $α$ entropy [A. E. Rastegin, Phys. Rev. A \textbf{93}, 032136 (2016)] has been tried to quantify the coherence. But it was shown to violate the strong monotonicity, even though it can unambiguously distinguish the coherent and the incoherent states with the monotonicity. Here we establish a family of coherence quantifiers which are closely related to the Tsallis relative $α$ entropy. It proves that this family of quantifiers satisfy all the standard criteria and particularly cover several typical coherence measures.

arXiv:1704.04876•Apr 17, 2017•2 citations

Parameterized coherence measure

Meng‐Li Guo, Zhi‐Xiang Jin, Jin‐Min Liang +2 more

Quantifying coherence is an essential endeavor for both quantum mechanical foundations and quantum technologies. We present a bona fide measure of quantum coherence by utilizing the Tsallis relative operator (α,β)-entropy. We first prove that the proposed coherence measure fulfills all the criteria of a well defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. We then study the ordering of the Tsallis relative operator (α,β)-entropy of coherence, Tsallis relative α-entropies of coherence, Rényi α-entropy of coherence and l1 norm of coherence for both pure and mixed qubit states. This provides a new method for defining new coherence measure and entanglement measure, and also provides a new idea for further study of quantum coherence.

arXiv:2306.11973•Results in Physics•Jun 7, 2023•2 citations

Quantum coherence measures based on the Tsallis relative entropies

Alexey E. Rastegin

The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum world. Various measures could be proposed for quantifying the coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. It is proved that considered measures of coherence satisfy all the required properties. The only point is that monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. This extension is reduced to the standard form when the standard relative entropy is dealt. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.

•Dec 21, 2015•0 citations

Maximal Rényi Relative Entropy for $α>2$

Roberto Rubboli

Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for $α\in[0,1/2)$ and the sandwiched Rényi relative entropy for $α\geq 1/2$. In contrast, the maximal extension had previously been identified only for $α\in[0,2]$, where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for $α>2$, the maximal extension is given by the $α$-$z$ Rényi relative entropy with $z=α-1$. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.

quant-pharXiv:2608.01124•Aug 4, 2026•0 citations
Abstract The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the Rényi α-relative entropy, which present in [Phys. Rev. A 94 (2016) 052336]. We show that the Rényi α-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement, which presents in [Phys. Rev. A 93 (2016) 032136]. Due to the Rényi α-relative entropy of coherence can act as a coherence monotone quantifier, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of Rényi 2-relative entropy of coherence are derived .
The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum level. Various measures could be proposed to quantify coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. Basic properties of the considered coherence quantifiers are derived. Tradeoff relations between coherence and mixedness are examined. The property of monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.
Coherence is the most fundamental quantum feature of the nonclassical systems. The understanding of coherence within the resource theory has been attracting increasing interest among which the quantification of coherence is an essential ingredient. A satisfactory measure should meet certain standard criteria. It seems that the most crucial criterion should be the strong monotonicity, that is, average coherence doesn't increase under the (sub-selective) incoherent operations. Recently, the Tsallis relative α entropy has been tried to quantify the coherence. But it was shown to violate the strong monotonicity, even though it can unambiguously distinguish the coherent and the incoherent states with the monotonicity. Here we establish a family of coherence quantifiers which are closely related to the Tsallis relative α entropy. It proves that this family of quantifiers satisfy all the standard criteria and particularly cover several typical coherence measures.
Remedying the strong monotonicity of the coherence measure in terms of the Tsallis relative ααα entropy
Coherence is the most fundamental quantum feature of the nonclassical systems. The understanding of coherence within the resource theory has been attracting increasing interest among which the quantification of coherence is an essential ingredient. A satisfactory measure should meet certain standard criteria. It seems that the most crucial criterion should be the strong monotonicity, that is, average coherence doesn't increase under the (sub-selective) incoherent operations. Recently, the Tsallis relative ααα entropy [A. E. Rastegin, Phys. Rev. A \textbf{93}, 032136 (2016)] has been tried to quantify the coherence. But it was shown to violate the strong monotonicity, even though it can unambiguously distinguish the coherent and the incoherent states with the monotonicity. Here we establish a family of coherence quantifiers which are closely related to the Tsallis relative ααα entropy. It proves that this family of quantifiers satisfy all the standard criteria and particularly cover several typical coherence measures.
Quantifying coherence is an essential endeavor for both quantum mechanical foundations and quantum technologies. We present a bona fide measure of quantum coherence by utilizing the Tsallis relative operator (α,β)-entropy. We first prove that the proposed coherence measure fulfills all the criteria of a well defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. We then study the ordering of the Tsallis relative operator (α,β)-entropy of coherence, Tsallis relative α-entropies of coherence, Rényi α-entropy of coherence and l1 norm of coherence for both pure and mixed qubit states. This provides a new method for defining new coherence measure and entanglement measure, and also provides a new idea for further study of quantum coherence.
The concept of coherence is one of cornerstones in physics. The development of quantum information science has lead to renewed interest in properly approaching the coherence at the quantum world. Various measures could be proposed for quantifying the coherence of a quantum state with respect to the prescribed orthonormal basis. To be a proper measure of coherence, each candidate should enjoy certain properties. It seems that the monotonicity property plays a crucial role here. Indeed, there is known an intuitive measure of coherence that does not share this condition. We study coherence measures induced by quantum divergences of the Tsallis type. It is proved that considered measures of coherence satisfy all the required properties. The only point is that monotonicity under incoherent selective measurements has to be reformulated. The proposed formulation can naturally be treated as a parametric extension of its standard form. This extension is reduced to the standard form when the standard relative entropy is dealt. Finally, two coherence measures quadratic in moduli of matrix elements are compared from the monotonicity viewpoint.
Maximal Rényi Relative Entropy for α>2α>2α>2
Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for α∈[0,1/2)α\in[0,1/2)α∈[0,1/2) and the sandwiched Rényi relative entropy for α≥1/2α\geq 1/2α≥1/2. In contrast, the maximal extension had previously been identified only for α∈[0,2]α\in[0,2]α∈[0,2], where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for α>2α>2α>2, the maximal extension is given by the ααα-zzz Rényi relative entropy with z=α−1z=α-1z=α−1. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.