Recoverability in quantum information theory

Abstract

The fact that the quantum relative entropy is non-increasing with respect to quantum physical evolutions lies at the core of many optimality theorems in quantum information theory and has applications in other areas of physics. In this work, we establish improvements of this entropy inequality in the form of physically meaningful remainder terms. One of the main results can be summarized informally as follows: if the decrease in quantum relative entropy between two quantum states after a quantum physical evolution is relatively small, then it is possible to perform a recovery operation, such that one can perfectly recover one state while approximately recovering the other. This can be interpreted as quantifying how well one can reverse a quantum physical evolution. Our proof method is elementary, relying on the method of complex interpolation, basic linear algebra, and the recently introduced Renyi generalization of a relative entropy difference. The theorem has a number of applications in quantum information theory, which have to do with providing physically meaningful improvements to many known entropy inequalities.

v5: 26 pages, generalized lower bounds to apply when supp(rho) is contained in supp(sigma)

References in corpus (8)

• No-local-broadcasting theorem for quantum correlations
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Renyi generalizations of the conditional quantum mutual information
• An additive and operational entanglement measure: conditional entanglement of mutual information
• The Fidelity of Recovery is Multiplicative
• Rényi generalizations of quantum information measures
• Multipartite quantum correlations and local recoverability
• Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures

Cited by in corpus (78)

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• Multivariate Trace Inequalities
• On Variational Expressions for Quantum Relative Entropies
• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
• Entropic uncertainty and measurement reversibility
• Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map
• Quantum algorithm for Petz recovery channels and pretty good measurements
• The Structure of Quantum Stochastic Processes with Finite Markov Order
• Approximate reversibility in the context of entropy gain, information gain, and complete positivity
• Universal recovery map for approximate Markov chains
• From time-reversal symmetry to quantum Bayes' rules
• Fundamental work cost of quantum processes
• The Fidelity of Recovery is Multiplicative
• Fundamental limits on key rates in device-independent quantum key distribution
• Reversing Lindblad Dynamics via Continuous Petz Recovery Map
• Measures of quantum correlations in infinite-dimensional systems
• Axioms for retrodiction: achieving time-reversal symmetry with a prior
• On Composite Quantum Hypothesis Testing
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
• Learnability transitions in monitored quantum dynamics via eavesdropper's classical shadows
• Dynamical maps, quantum detailed balance and Petz recovery map
• Approximate reversal of quantum Gaussian dynamics
• Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures
• Information-theoretic limitations on approximate quantum cloning and broadcasting
• Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
• Approximate quantum Markov chains
• Dependence of integrated, instantaneous, and fluctuating entropy production on the initial state in quantum and classical processes
• Operational meaning of quantum measures of recovery
• Recoverability for optimized quantum $f$-divergences
• Recovery and the Data Processing Inequality for quasi-entropies
• Uniqueness of quantum state over time function
• Hierarchy of efficiently computable and faithful lower bounds to quantum discord
• Memory Effects in Quantum Processes
• Approximating Invertible Maps by Recovery Channels: Optimality and an Application to Non-Markovian Dynamics
• Monotonic multi-state quantum $f$-divergences
• Recovery Map for Fermionic Gaussian Channels
• Recoverability of quantum channels via hypothesis testing
• Necessary criterion for approximate recoverability
• Quantum Bayesian Inference in Quasiprobability Representations
• Eigenstate Thermalization Hypothesis and Approximate Quantum Error Correction
• Entanglement monogamy via multivariate trace inequalities
• Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings
• Inverses, disintegrations, and Bayesian inversion in quantum Markov categories
• Recoverability for Holevo's just-as-good fidelity
• Bayesian inversion and the Tomita-Takesaki modular group
• On multivariate trace inequalities of Sutter, Berta and Tomamichel
• On quantum quasi-relative entropy
• Role of Dilations in Reversing Physical Processes: Tabletop Reversibility and Generalized Thermal Operations
• Trace- and improved data processing inequalities for von Neumann algebras
• Entanglement and thermalization in open fermion systems
• Incoherent Gaussian equivalence of $m-$mode Gaussian states
• The Cost of Randomness for Converting a Tripartite Quantum State to be Approximately Recoverable
• Petz map recovery for long-range entangled quantum many-body states
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Recovery With Incomplete Knowledge: Fundamental Bounds on Real-Time Quantum Memories
• Towards a functorial description of quantum relative entropy
• Approximate recoverability and relative entropy II: 2-positive channels of general v. Neumann algebras
• Pretty good measurement for bosonic Gaussian ensembles
• Fundamental limitations on the recoverability of quantum processes
• Reduced quantum dynamics with initial system-environment correlations characterized by pure Markov states
• Chain rules for quantum channels
• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence
• Reality, Causality, and Quantum Theory
• Recoverability from direct quantum correlations
• Uhlmann's theorem for relative entropies
• Virtual Quantum Markov Chains
• Proper and improper mixed states serve as different prior beliefs for quantum state retrodiction
• Bipartite quantum states admitting a causal explanation
• Quantum state merging with bound entanglement
• Lower semicontinuity of the relative entropy disturbance and its corollaries
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter
• Information-Theoretic Analysis of Weak Measurements and Their Reversal
• Quantifying Irreversibility via Bayesian Subjectivity for Classical & Quantum Linear Maps
• New monotonicity property of the quantum relative entropy