Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints

Abstract

We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, $E_R$, and its regularization $E_R^{\infty }$, as well as of the entanglement of formation, $E_F$. Using a novel "quantum coupling" of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, $E_C=E_F^{\infty }$. Second, analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.

12pp, lots of lemmas but no theorem. V2, v3 v4, v5, v6 have ever more references and more accurate attribution of previous results. In v6 correct proof of the asymptotic continuity of entanglement cost, which had been claimed already in v4 and been retracted in v5

References in corpus (10)

• A Sharp Fannes-type Inequality for the von Neumann Entropy
• The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography
• The private classical capacity with a symmetric side channel and its application to quantum cryptography
• The quantum capacity with symmetric side channels
• Continuity of quantum channel capacities
• An additive and operational entanglement measure: conditional entanglement of mutual information
• Approximate Degradable Quantum Channels
• Measures of quantum correlations in infinite-dimensional systems
• Squashed entanglement in infinite dimensions
• Bounds on the Concavity of Quantum Entropy

Cited by in corpus (177)

• General Benchmarks for Quantum Repeaters
• Fundamental Limits of Repeaterless Quantum Communications
• Quantum Resource Theories
• Operational Resource Theory of Coherence
• Universal recovery maps and approximate sufficiency of quantum relative entropy
• Approaches for approximate additivity of the Holevo information of quantum channels
• Resource Theory of Coherence - Beyond States
• Should Entanglement Measures be Monogamous or Faithful?
• Principles of Quantum Communication Theory: A Modern Approach
• Quantum speed limits and the maximal rate of information production
• Approximate Degradable Quantum Channels
• Entropic proofs of Singleton bounds for quantum error-correcting codes
• Advantage distillation for device-independent quantum key distribution
• Quantum optimal transport with quantum channels
• Energy-constrained diamond norms and their use in quantum information theory
• Approximate reversibility in the context of entropy gain, information gain, and complete positivity
• Quantum Channel Simulation and the Channel's Smooth Max-Information
• Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels
• Energy-constrained private and quantum capacities of quantum channels
• Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of quantum channels
• Asymptotic security of discrete-modulation protocols for continuous-variable quantum key distribution
• Useful states and entanglement distillation
• Entropy of a quantum channel
• Entanglement cost and quantum channel simulation
• Dimension Reduction in Quantum Key Distribution for Continuous- and Discrete-Variable Protocols
• Amortized entanglement of a quantum channel and approximately teleportation-simulable channels
• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
• Optimal universal programming of unitary gates
• Fundamental Limits on Correlated Catalytic State Transformations
• Entanglement and secret-key-agreement capacities of bipartite quantum interactions and read-only memory devices
• Generic emergence of objectivity of observables in infinite dimensions
• Framework for resource quantification in infinite-dimensional general probabilistic theories
• Measures of quantum correlations in infinite-dimensional systems
• Quantum Monge-Kantorovich problem and transport distance between density matrices
• Classical capacities of quantum channels with environment assistance
• Relational Proofs for Quantum Programs
• On contraction coefficients, partial orders and approximation of capacities for quantum channels
• Convexity and Operational Interpretation of the Quantum Information Bottleneck Function
• Deconstruction and conditional erasure of quantum correlations
• Energy-Consumption Advantage of Quantum Computation
• Is there a finite complete set of monotones in any quantum resource theory?
• Continuity of the quantum Fisher information
• Entanglement-assisted private communication over quantum broadcast channels
• Convergence rates for quantum evolution & entropic continuity bounds in infinite dimensions
• Hadamard quantum broadcast channels
• Secure and Robust Identification via Classical-Quantum Channels
• Assisted work distillation
• Asymptotic state transformations of continuous variable resources
• Convergence rates for the quantum central limit theorem
• Conditional quantum one-time pad
• Continuity bounds for information characteristics of quantum channels depending on input dimension
• Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy
• Universal limitations on implementing resourceful unitary evolutions
• Squashed entanglement in infinite dimensions
• Maximum and minimum entropy states yielding local continuity bounds
• Continuity of quantum entropic quantities via almost convexity
• Device-independent Certification of One-shot Distillable Entanglement
• Distributed Compression of Correlated Classical-Quantum Sources or: The Price of Ignorance
• A converse to Lieb-Robinson bounds in one dimension using index theory
• Quantum Broadcast Channels with Cooperating Decoders: An Information-Theoretic Perspective on Quantum Repeaters
• Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its use
• Witnessing Negative Conditional Entropy
• A single-qubit position verification protocol that is secure against multi-qubit attacks
• Resource theory of unextendibility and non-asymptotic quantum capacity
• Private states, quantum data hiding and the swapping of perfect secrecy
• Towards practical security of continuous-variable quantum key distribution
• Reversibility of quantum resources through probabilistic protocols
• How much entanglement can be created in a closed system?
• Relative entropy for von Neumann subalgebras
• Bosonic quantum communication across arbitrarily high loss channels
• Quantum Optimal Transport
• The variance of relative surprisal as single-shot quantifier
• Rates of multi-partite entanglement transformations and applications in quantum networks
• Fundamental limitations on the device-independent quantum conference key agreement
• Programmability of covariant quantum channels
• Eventually entanglement breaking Markovian dynamics: structure and characteristic times
• Optimal uniform continuity bound for conditional entropy of classical--quantum states
• Autonomous Quantum Devices: When Are They Realizable without Additional Thermodynamic Costs?
• Converse bounds for quantum and private communication over Holevo-Werner channels
• Quantum reading capacity: General definition and bounds
• A solution of the generalised quantum Stein's lemma
• From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
• Advanced Alicki-Fannes-Winter method for energy-constrained quantum systems and its use
• State exchange with quantum side information
• One-shot Capacity bounds on the Simultaneous Transmission of Classical and Quantum Information
• Continuity of characteristics of composite quantum systems
• Bounding quantum capacities via partial orders and complementarity
• Continuous-variable quantum state designs: theory and applications
• Quantum Channel Capacities Per Unit Cost
• Capacities of Gaussian Quantum Channels with Passive Environment Assistance
• Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
• Necessary criterion for approximate recoverability
• A tight uniform continuity bound for the Arimoto-Rényi conditional entropy and its extension to classical-quantum states
• Decay of quantum conditional mutual information for purely generated finitely correlated states
• Relative entropy of steering: On its definition and properties
• Continuity bounds on observational entropy and measured relative entropies
• The black hole information puzzle and the quantum de Finetti theorem
• Quantum Earth mover's distance, No-go Quantum Kantorovich-Rubinstein theorem, and Quantum Marginal Problem
• Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
• On quantum channels and operations preserving finiteness of the von Neumann entropy
• Communication Cost for Non-Markovianity of Tripartite Quantum States: A Resource Theoretic Approach
• Complexity-constrained quantum thermodynamics
• Entanglement-Assisted Quantum Data Compression
• Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy
• Estimate distillable entanglement and quantum capacity by squeezing useless entanglement
• 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
• Entanglement-assisted capacities of compound quantum channels
• Refined diamond norm bounds on the emergence of objectivity of observables
• Uniform continuity bounds for characteristics of multipartite quantum systems
• Matrix Product Density Operators: when do they have a local parent Hamiltonian?
• Operator E-norms and their use
• Continuity of entropies via integral representations
• The dynamical $\alpha$-Rényi entropies of local Hamiltonians grow at most linearly in time
• Uniform finite-dimensional approximation of basic capacities of energy-constrained channels
• A Tight Uniform Continuity Bound for Equivocation
• Operator-valued Schatten spaces and quantum entropies
• An Extention of Entanglement Measures for Pure States
• Dimensional advantage in secure information trading via the noisy dense coding protocol
• Approximate broadcasting of quantum correlations
• Infinite-Dimensional Programmable Quantum Processors
• Hayden-Preskill Recovery in Hamiltonian Systems
• Tight uniform continuity bound for a family of entropies
• Approximating quantum channels by completely positive maps with small Kraus rank
• From Quantum Source Compression to Quantum Thermodynamics
• Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes
• Matrix product states and the decay of quantum conditional mutual information
• Universal random codes: Capacity regions of the compound quantum multiple-access channel with one classical and one quantum sender
• Correlation measures of a quantum state and information characteristics of a quantum channel
• Generalizing Entanglement
• Continuity of the relative entropy of resource
• Quantum channels that destroy negative conditional entropy
• Entanglement cost for infinite-dimensional physical systems
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• On discontinuity of information characteristics of quantum systems and channels
• Heralded Channel Holevo Superadditivity Bounds from Entanglement Monogamy
• Continuity bounds for quantum entropies arising from a fundamental entropic inequality
• Analytic Semi-device-independent Entanglement Quantification for Bipartite Quantum States
• Unifying Entanglement with Uncertainty via Symmetries of Observable Algebras
• Hybrid quantum network design against unauthorized secret-key generation, and its memory cost
• Device-independent quantum key distribution from computational assumptions
• Quantum Polarization of Qudit Channels
• Fundamental limitations to key distillation from Gaussian states with Gaussian operations
• Unified framework for continuity of sandwiched Rényi divergences
• On completion of the cone of CP linear maps with respect to the energy-constrained diamond norm
• Fundamental limitations on the recoverability of quantum processes
• Random private quantum states
• Linear gate bounds against natural functions for position-verification
• Tightening continuity bounds for entropies and bounds on quantum capacities
• Quantitative non-classicality of mediated interactions
• A Rate-Distortion Perspective on Quantum State Redistribution
• Energy preserving evolutions over Bosonic systems
• Entropy Accumulation under Post-Quantum Cryptographic Assumptions
• Single-shot entanglement manipulation of states and channels revisited
• Continuous-variable Quantum Position Verification secure against entangled attackers
• Energy-limited quantum dynamics
• On the relation between quantum Darwinism and approximate quantum Markovianity
• Quantum Chernoff divergence in advantage distillation for quantum key distribution and device-independent quantum key distribution
• Transcendental properties of entropy-constrained sets: Part II
• Entropic and operational characterizations of dynamic quantum resources
• Simple characterization of positive linear maps preserving continuity of the von Neumann entropy
• A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian Channels
• Adding Photonic Entanglement to Superradiance by Using Multilevel Atoms
• Entanglement-Assisted Capacity of Quantum Channels with Side Information
• Multipartite entanglement theory with entanglement-nonincreasing operations
• Semantic Security with Infinite Dimensional Quantum Eavesdropping Channel
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• Lipschitz continuity of quantum-classical conditional entropies with respect to angular distance and related properties
• Uniform continuity bound for sandwiched Rényi conditional entropy
• Lower bounds on distances between a given quantum channel and certain classes of channels
• Converting entanglement into ensemble basis-free coherence
• One-shot Multiple Access Channel Simulation
• What are temporal correlations
• Information storage and transmission under Markovian noise
• Matrix systems, algebras, and open maps
• Universal proofs of entropic continuity bounds via majorization flow
• Classical State Masking over a Quantum Channel