Universal recovery map for approximate Markov chains

Abstract

A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information $I(A:C|B)$ of a tripartite quantum state ${\rho }_{ABC}$ can be bounded from below by its distance to the closest recovered state ${\mathcal{R}}_{B\to BC}({\rho }_{AB})$, where the $C$-part is reconstructed from the $B$-part only and the recovery map ${\mathcal{R}}_{B\to BC}$ merely depends on ${\rho }_{BC}$. One particular application of this result implies the equivalence between two different approaches to define topological order in quantum systems.

v3: 31 pages, 1 figure, application to topological order of quantum systems added (Section 3). v2: 29 pages, relation to [Wilde, arXiv:1505.04661] clarified (Remark 2.5)

References in corpus (6)

• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Recoverability in quantum information theory
• Renyi generalizations of the conditional quantum mutual information
• Lower bounds on information complexity via zero-communication protocols and applications
• Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures

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• The Structure of Quantum Stochastic Processes with Finite Markov Order
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• Conditional Mutual Information of Bipartite Unitaries and Scrambling
• Measures of quantum correlations in infinite-dimensional systems
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
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• Towards holography via quantum source-channel codes
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• Approximate quantum Markov chains
• Operational meaning of quantum measures of recovery
• Hierarchy of efficiently computable and faithful lower bounds to quantum discord
• Recovery and the Data Processing Inequality for quasi-entropies
• Memory Effects in Quantum Processes
• Decay of quantum conditional mutual information for purely generated finitely correlated states
• Necessary criterion for approximate recoverability
• Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings
• On quantum quasi-relative entropy
• On multivariate trace inequalities of Sutter, Berta and Tomamichel
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• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
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• Recoverability from direct quantum correlations
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• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence
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• Proper and improper mixed states serve as different prior beliefs for quantum state retrodiction
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• Virtual Quantum Markov Chains
• Compactness criterion for families of quantum operations in the strong convergence topology and its applications
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