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:1503.09176 · Physical Review A

Conditions for coherence transformations under incoherent operations

Shuanping Du, Zhaofang Bai, Yu Jing Guo

Published May 27, 2015• 206 citations
arXivHTMLPDF

Abstract

We build in this paper the counterpart of the celebrated Nielsen theorem for coherence manipulation. This offers an affirmative answer to the open question: whether, given two states $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, either $\ensuremath{\rho}$ can be transformed into $\ensuremath{\sigma}$ or vice versa under incoherent operations [Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014)]. As a consequence, we find that there exist essentially different types of coherence. Moreover, incoherent operations can be enhanced in the presence of certain coherent states. These extra states are coherent catalysts: they allow uncertain incoherent operations to be realized without being consumed in any way. Our main result also sheds light on the construction of coherence measures.

DOI
10.1103/physreva.91.052120

Related papers

A partial order on coherent states introduced by incoherent operations

Shuanping Du, Zhaofang Bai, Yu Chen Guo

We build the counterpart of the celebrated Nielsen's theorem for coherence manipulation in this paper. This offers an affirmative answer to the open question: whether, given two states $\rho$ and $\sigma$, either $\rho$ can be transformed into $\sigma$ or vice versa under incoherent operations [Phys. Rev. Lett. \textbf{113}, 140401(2014)]. As a consequence, we find that there exist essentially different types of coherence. Moreover, incoherent operations can be enhanced in the presence of certain coherent states. These extra states are coherent catalysts: they allow uncertain incoherent operations to be realized, without being consumed in any way. Our main result also sheds a new light on the construction of coherence measures.

•Mar 31, 2015•0 citations

Catalytic coherence transformations

Kaifeng Bu, Uttam Singh, Junde Wu

Catalytic coherence transformations allow the otherwise impossible state transformations using only incoherent operations with the aid of an auxiliary system with finite coherence that is not being consumed in any way. Here we find the necessary and sufficient conditions for the deterministic and stochastic catalytic coherence transformations between a pair of pure quantum states. In particular, we show that the simultaneous decrease of a family of R\'enyi entropies of the diagonal parts of the states under consideration is a necessary and sufficient condition for the deterministic catalytic coherence transformations. Similarly, for stochastic catalytic coherence transformations we find the necessary and sufficient conditions for achieving a higher optimal probability of conversion. We thus completely characterize the coherence transformations among pure quantum states under incoherent operations. We give numerous examples to elaborate our results. We also explore the possibility of the same system acting as a catalyst for itself and find that indeed self-catalysis is possible. Further, for the cases where no catalytic coherence transformation is possible we provide entanglement-assisted coherence transformations and find the necessary and sufficient conditions for such transformations.

arXiv:1511.08071•Physical Review A•Apr 18, 2016•60 citations

Catalytic transformations in coherence theory

Priyabrata Char, Dipayan Chakraborty, Amit Bhar +2 more

In 2016, Winter and Yang [A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] provided an operational meaning to relative entropy of coherence and coherence of formation by introducing coherence distillation and dilation protocols in an asymptotic setup. Here we have provided relative entropy of coherence [via incoherent operations (IO)] and coherence of formation (via IO) and quantum incoherent relative entropy [via local quantum incoherent operations with classical communications (LQICC)] as a measure using the concept of a catalyst. We have proved an existential correspondence between asymptotic and catalytic state transformation using IO, local incoherent operations with classical communications, and LQICC. We have also discussed two very important protocols, assisted distillation and quantum incoherent state merging using a catalyst. Monotone properties of relative entropy of coherence, coherence of formation, and quantum incoherent relative entropy under the catalytic transformation are also discussed here.

arXiv:2111.14645•Physical Review A•Jan 3, 2023•13 citations

Coherence manipulation with stochastic incoherent operation

Ping Li, Yongming Li

An important problem in the coherent resource theory is the convertibility of coherent states by free operations. In this paper we consider such a problem from a mixed coherent state into a pure one by using both stochastic incoherent operations and incoherent operations. We prove that stochastic incoherent operations can transform a mixed coherent state into a pure coherent state if and only if the density matrix of the initial state contains a singular principal submatrix. Then we provide two sufficient conditions with explicit expressions for the probabilistic transformation and the deterministic transformation. These two sufficient conditions facilitate the construction of concrete operators of stochastic incoherent operations and incoherent operations. These results demonstrate that incoherent operations are strictly more powerful than strictly incoherent operations within the probabilistic transformation as well as the deterministic transformation.

Physical Review A•Oct 16, 2023•4 citations

Quantum coherence versus quantum uncertainty

Shunlong Luo, Yuan Sun

The notion of measurement is of both foundational and instrumental significance in quantum mechanics, and coherence destroyed by measurements (decoherence) lies at the very heart of quantum to classical transition. Qualitative aspects of this spirit have been widely recognized and analyzed ever since the inception of quantum theory. However, axiomatic and quantitative investigations of coherence are attracting great interest only recently with several figures of merit for coherence introduced [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. While these resource theoretic approaches have many appealing and intuitive features, they rely crucially on various notions of incoherent operations which are sophisticated, subtle, and not uniquely defined, as have been critically assessed [Chitambar and Gour, Phys. Rev. Lett. 117, 030401 (2016)]. In this paper, we elaborate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate, and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of a measurement (with respect to a state). Consequently, coherence measures may be set into correspondence with measures of quantum uncertainty. In particular, we take average quantum Fisher information as a measure of quantum uncertainty, and introduce the corresponding measure of coherence, which is demonstrated to exhibit desirable properties. Implications for interpreting quantum purity as maximal coherence, and quantum discord as minimal coherence, are illustrated.

Physical Review A•Aug 25, 2017•140 citations

Resource theory of coherence: Beyond states

Khaled Ben Dana, María García Díaz, Mohamed Mejatty +1 more

We generalize the recently proposed resource theory of coherence (or superposition) [T. Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014); A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] to the setting where not just the free (``incoherent'') resources, but also the manipulated objects, are quantum operations rather than states. In particular, we discuss an information theoretic notion of the coherence capacity of a quantum channel and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel and prove achievability results for unitaries and general channels acting on a $d$-dimensional system; we show that a maximally coherent state of rank $d$ is always sufficient as a resource if incoherent operations are allowed, and one of rank ${d}^{2}$ for ``strictly incoherent'' operations. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e., maps with nonzero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.

arXiv:1704.03710•Physical Review A•Jun 20, 2017•95 citations
We build in this paper the counterpart of the celebrated Nielsen theorem for coherence manipulation. This offers an affirmative answer to the open question: whether, given two states \ensuremathρ\ensuremath{\rho}\ensuremathρ and \ensuremathσ\ensuremath{\sigma}\ensuremathσ, either \ensuremathρ\ensuremath{\rho}\ensuremathρ can be transformed into \ensuremathσ\ensuremath{\sigma}\ensuremathσ or vice versa under incoherent operations [Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014)]. As a consequence, we find that there exist essentially different types of coherence. Moreover, incoherent operations can be enhanced in the presence of certain coherent states. These extra states are coherent catalysts: they allow uncertain incoherent operations to be realized without being consumed in any way. Our main result also sheds light on the construction of coherence measures.
We build the counterpart of the celebrated Nielsen's theorem for coherence manipulation in this paper. This offers an affirmative answer to the open question: whether, given two states ρ\rhoρ and σ\sigmaσ, either ρ\rhoρ can be transformed into σ\sigmaσ or vice versa under incoherent operations [Phys. Rev. Lett. \textbf{113}, 140401(2014)]. As a consequence, we find that there exist essentially different types of coherence. Moreover, incoherent operations can be enhanced in the presence of certain coherent states. These extra states are coherent catalysts: they allow uncertain incoherent operations to be realized, without being consumed in any way. Our main result also sheds a new light on the construction of coherence measures.
Catalytic coherence transformations allow the otherwise impossible state transformations using only incoherent operations with the aid of an auxiliary system with finite coherence that is not being consumed in any way. Here we find the necessary and sufficient conditions for the deterministic and stochastic catalytic coherence transformations between a pair of pure quantum states. In particular, we show that the simultaneous decrease of a family of R\'enyi entropies of the diagonal parts of the states under consideration is a necessary and sufficient condition for the deterministic catalytic coherence transformations. Similarly, for stochastic catalytic coherence transformations we find the necessary and sufficient conditions for achieving a higher optimal probability of conversion. We thus completely characterize the coherence transformations among pure quantum states under incoherent operations. We give numerous examples to elaborate our results. We also explore the possibility of the same system acting as a catalyst for itself and find that indeed self-catalysis is possible. Further, for the cases where no catalytic coherence transformation is possible we provide entanglement-assisted coherence transformations and find the necessary and sufficient conditions for such transformations.
In 2016, Winter and Yang [A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] provided an operational meaning to relative entropy of coherence and coherence of formation by introducing coherence distillation and dilation protocols in an asymptotic setup. Here we have provided relative entropy of coherence [via incoherent operations (IO)] and coherence of formation (via IO) and quantum incoherent relative entropy [via local quantum incoherent operations with classical communications (LQICC)] as a measure using the concept of a catalyst. We have proved an existential correspondence between asymptotic and catalytic state transformation using IO, local incoherent operations with classical communications, and LQICC. We have also discussed two very important protocols, assisted distillation and quantum incoherent state merging using a catalyst. Monotone properties of relative entropy of coherence, coherence of formation, and quantum incoherent relative entropy under the catalytic transformation are also discussed here.
An important problem in the coherent resource theory is the convertibility of coherent states by free operations. In this paper we consider such a problem from a mixed coherent state into a pure one by using both stochastic incoherent operations and incoherent operations. We prove that stochastic incoherent operations can transform a mixed coherent state into a pure coherent state if and only if the density matrix of the initial state contains a singular principal submatrix. Then we provide two sufficient conditions with explicit expressions for the probabilistic transformation and the deterministic transformation. These two sufficient conditions facilitate the construction of concrete operators of stochastic incoherent operations and incoherent operations. These results demonstrate that incoherent operations are strictly more powerful than strictly incoherent operations within the probabilistic transformation as well as the deterministic transformation.
The notion of measurement is of both foundational and instrumental significance in quantum mechanics, and coherence destroyed by measurements (decoherence) lies at the very heart of quantum to classical transition. Qualitative aspects of this spirit have been widely recognized and analyzed ever since the inception of quantum theory. However, axiomatic and quantitative investigations of coherence are attracting great interest only recently with several figures of merit for coherence introduced [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. While these resource theoretic approaches have many appealing and intuitive features, they rely crucially on various notions of incoherent operations which are sophisticated, subtle, and not uniquely defined, as have been critically assessed [Chitambar and Gour, Phys. Rev. Lett. 117, 030401 (2016)]. In this paper, we elaborate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate, and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of a measurement (with respect to a state). Consequently, coherence measures may be set into correspondence with measures of quantum uncertainty. In particular, we take average quantum Fisher information as a measure of quantum uncertainty, and introduce the corresponding measure of coherence, which is demonstrated to exhibit desirable properties. Implications for interpreting quantum purity as maximal coherence, and quantum discord as minimal coherence, are illustrated.
We generalize the recently proposed resource theory of coherence (or superposition) [T. Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014); A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] to the setting where not just the free (``incoherent'') resources, but also the manipulated objects, are quantum operations rather than states. In particular, we discuss an information theoretic notion of the coherence capacity of a quantum channel and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel and prove achievability results for unitaries and general channels acting on a ddd-dimensional system; we show that a maximally coherent state of rank ddd is always sufficient as a resource if incoherent operations are allowed, and one of rank d2{d}^{2}d2 for ``strictly incoherent'' operations. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e., maps with nonzero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.