Continuity of quantum entropic quantities via almost convexity
arXiv:2208.00922 · doi:10.1109/TIT.2023.3277892
Abstract
Based on the proofs of the continuity of the conditional entropy by Alicki, Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF) method. This method allows us to prove a great variety of continuity bounds for the derived entropic quantities. First, we apply the ALAFF method to the Umegaki relative entropy. This way, we recover known almost tight bounds, but also some new continuity bounds for the relative entropy. Subsequently, we apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This yields novel explicit bounds in particular for the BS-conditional entropy, the BS-mutual and BS-conditional mutual information. On the way, we prove almost concavity for the Umegaki relative entropy and the BS-entropy, which might be of independent interest. We conclude by showing some applications of these continuity bounds in various contexts within quantum information theory.
arXiv comment68 pages, 6 figures. This is a long version of the paper arXiv:2305.10140. v3: We corrected a previous flaw in Lemma 5.8
References in corpus (62)
- Quantifying Entanglement
- Entanglement Measures and Purification Procedures
- The Resource Theory of Quantum States Out of Thermal Equilibrium
- From Classical to Quantum Shannon Theory
- Structure of states which satisfy strong subadditivity of quantum entropy with equality
- Quantum Information Processing with Finite Resources -- Mathematical Foundations
- An improved bound on distillable entanglement
- A Sharp Fannes-type Inequality for the von Neumann Entropy
- Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
- Quantum conditional mutual information and approximate Markov chains
- Continuity of quantum conditional information
- Quantum f-divergences and error correction
- Monotonicity of quantum relative entropy revisited
- On the quantum Renyi relative entropies and related capacity formulas
- alpha-z-relative Renyi entropies
- Universal recovery maps and approximate sufficiency of quantum relative entropy
- Different quantum f-divergences and the reversibility of quantum operations
- Continuity bounds for entanglement
- Continuity of quantum channel capacities
- Multivariate Trace Inequalities
- Continuity bounds on the quantum relative entropy
- Strong converse rates for quantum communication
- Continuity bounds on the quantum relative entropy - II
- Quantum Skew Divergence
- On asymptotic continuity of functions of quantum states
- Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of quantum channels
- Security of quantum key distribution from generalised entropy accumulation
- Rapid thermalization of spin chain commuting Hamiltonians
- Quantum Approximate Markov Chains are Thermal
- Geometric distinguishability measures limit quantum channel estimation and discrimination
- Remainder Terms for Some Quantum Entropy Inequalities
- Role of Quantum Coherence in Thermodynamics
- Generalised entropy accumulation
- Entropy and relative entropy from information-theoretic principles
- Exponential decay of mutual information for Gibbs states of local Hamiltonians
- Reverse Test and Characterization of Quantum Relative Entropy
- Defining quantum divergences via convex optimization
- Optimal Extensions of Resource Measures and their Applications
- Quantum conditional relative entropy and quasi-factorization of the relative entropy
- Entropy decay for Davies semigroups of a one dimensional quantum lattice
- Computable Rényi mutual information: Area laws and correlations
- On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems
- Continuity bounds for information characteristics of quantum channels depending on input dimension
- Maximum and minimum entropy states yielding local continuity bounds
- Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy
- The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions
- Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its use
- Superadditivity of quantum relative entropy for general states
- Eventually entanglement breaking Markovian dynamics: structure and characteristic times
- From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
- Upper continuity bounds on the relative -entropy for
- Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its use
- Quantum Brascamp-Lieb Dualities
- A strengthened data processing inequality for the Belavkin-Staszewski relative entropy
- Continuity of characteristics of composite quantum systems
- Necessary criterion for approximate recoverability
- Decay of quantum conditional mutual information for purely generated finitely correlated states
- Modulus of convexity for operator convex functions
- Bounds on the Concavity of Quantum Entropy
- General Continuity Bounds for Quantum Relative Entropies
- Weak quasi-factorization for the Belavkin-Staszewski relative entropy
- Upper continuity bound on the quantum quasi-relative entropy
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- Quantum Rényi and -divergences from integral representations
- Continuity of characteristics of composite quantum systems
- Continuity bounds on observational entropy and measured relative entropies
- General Continuity Bounds for Quantum Relative Entropies
- Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki-Fannes-Winter technique
- Quantum processes as thermodynamic resources: the role of non-Markovianity
- Observational entropy with general quantum priors
- Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
- Continuity bounds for quantum entropies arising from a fundamental entropic inequality
- Unified framework for continuity of sandwiched Rényi divergences
- Transcendental properties of entropy-constrained sets: Part II
- Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
- Asymptotically Optimal Tests for One- and Two-Sample Problems
- Lower semicontinuity of the relative entropy disturbance and its corollaries
- New monotonicity property of the quantum relative entropy