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:2301.08402

Entropy Uncertainty Relations and Strong Sub-additivity of Quantum Channels

Gao, Li, Marius Junge, Nicholas D. LaRacuente

Published Jan 20, 2023• 0 citations
arXivHTMLPDF

Abstract

We prove an entropic uncertainty relation for two quantum channels, extending the work of Frank and Lieb for quantum measurements. This is obtained via a generalized strong super-additivity (SSA) of quantum entropy. Motivated by Petz's algebraic SSA inequality, we also obtain a generalized SSA for quantum relative entropy. As a special case, it gives an improved data processing inequality.

DOI
10.48550/arxiv.2301.08402

Related papers

Entropic uncertainty relation and strong subadditvity of quantum channels

Gao Li, Marius Junge, Nicholas D. LaRacuente

We extend the entropic uncertainty principle from quantum measurements to quantum channels. This is obtained via a generalized strong subadditivity (SSA) of quantum entropy. Motivated by Petz's algebraic SSA inequality, we also obtain a generalized strong subadditivity of quantum relative entropy. The English version of this article is available at arXiv:2301.08402.

Scientia Sinica Mathematica•Jun 26, 2023•0 citations

Strong Subadditivity Inequality and Entropic Uncertainty Relations

Li Gao, Marius Junge, Nicholas D. LaRacuente

We prove a generalization of the strong-subadditivity (SSA) inequality in the context of finite dimensional von Neumann algebras. Our generalized SSA inequality subsumes various entropic uncertainty relations, such as the Maassen-Uffink inequality in the presence of quantum memory. Motivated by the generalization of SSA, we introduce an analog of conditional mutual information and discuss superadditivity and monotonicity under certain operations. Our proof uses complex interpolation and noncommutative mixed $L_p$ spaces.

•Oct 27, 2017•8 citations

Quantum Brascamp–Lieb Dualities

Mario Berta, David Sutter, Michael Walter

Abstract Brascamp–Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory—including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of ‘geometric’ Brascamp–Lieb inequalities.

arXiv:1909.02383•Communications in Mathematical Physics•Mar 9, 2023•12 citations

On the nature and decay of quantum relative entropy

Nicholas D. LaRacuente

Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features. An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty. We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity. Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels.

•Jun 2, 2020•0 citations

Uncertainty Principle for Quantum Channels

Li Gao, Marius Junge, Nicholas D. LaRacuente

We extend the quantum entropic uncertainty principle beyond measurements, to a pair of finite-dimensional quantum channels acting on the same input system. This also generalizes strong super-additivity inequality by giving a lower bound on generalizations of conditional mutual information. Moreover we show that the corresponding “squashed” conditional mutual information is additive in this general setting.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2018•11 citations

Complete Entropic Inequalities for Quantum Markov Chains

Li Gao, Cambyse Rouzé

Abstract We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or $$L_2$$ L 2 -estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.

arXiv:2102.04146•Archive for Rational Mechanics and Analysis•May 13, 2022•39 citations
We prove an entropic uncertainty relation for two quantum channels, extending the work of Frank and Lieb for quantum measurements. This is obtained via a generalized strong super-additivity (SSA) of quantum entropy. Motivated by Petz's algebraic SSA inequality, we also obtain a generalized SSA for quantum relative entropy. As a special case, it gives an improved data processing inequality.
We prove a generalization of the strong-subadditivity (SSA) inequality in the context of finite dimensional von Neumann algebras. Our generalized SSA inequality subsumes various entropic uncertainty relations, such as the Maassen-Uffink inequality in the presence of quantum memory. Motivated by the generalization of SSA, we introduce an analog of conditional mutual information and discuss superadditivity and monotonicity under certain operations. Our proof uses complex interpolation and noncommutative mixed LpL_pLp​ spaces.
Abstract Brascamp–Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory—including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of ‘geometric’ Brascamp–Lieb inequalities.
Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features. An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty. We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity. Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels.
Abstract We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L2L_2L2​ L 2 -estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.