Approximate reversibility in the context of entropy gain, information gain, and complete positivity

Abstract

There are several inequalities in physics which limit how well we can process physical systems to achieve some intended goal, including the second law of thermodynamics, entropy bounds in quantum information theory, and the uncertainty principle of quantum mechanics. Recent results provide physically meaningful enhancements of these limiting statements, determining how well one can attempt to reverse an irreversible process. In this paper, we apply and extend these results to give strong enhancements to several entropy inequalities, having to do with entropy gain, information gain, entropic disturbance, and complete positivity of open quantum systems dynamics. Our first result is a remainder term for the entropy gain of a quantum channel. This result implies that a small increase in entropy under the action of a subunital channel is a witness to the fact that the channel's adjoint can be used as a recovery map to undo the action of the original channel. Our second result regards the information gain of a quantum measurement, both without and with quantum side information. We find here that a small information gain implies that it is possible to undo the action of the original measurement if it is efficient. The result also has operational ramifications for the information-theoretic tasks known as measurement compression without and with quantum side information. Our third result shows that the loss of Holevo information caused by the action of a noisy channel on an input ensemble of quantum states is small if and only if the noise can be approximately corrected on average. We finally establish that the reduced dynamics of a system-environment interaction are approximately completely positive and trace-preserving if and only if the data processing inequality holds approximately.

v3: 12 pages, accepted for publication in Physical Review A

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• Dynamical maps, quantum detailed balance and Petz recovery map
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• Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
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• Strong convergence of quantum channels: continuity of the Stinespring dilation and discontinuity of the unitary dilation
• Geometric inequalities from phase space translations
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• Coherence generating power of unitary transformations via probabilistic average
• Thermodynamic Constraints on Quantum Information Gain and Error Correction: A Triple Trade-Off
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• Thermodynamic Implementations of Quantum Processes
• Approximating Invertible Maps by Recovery Channels: Optimality and an Application to Non-Markovian Dynamics
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• Universal validity of the second law of information thermodynamics
• Thermodynamically free quantum measurements
• Thermodynamic Reverse Bounds for General Open Quantum Processes
• On lower semicontinuity of the quantum conditional mutual information and its corollaries
• Correlation measures of a quantum state and information characteristics of a quantum channel
• Recovery With Incomplete Knowledge: Fundamental Bounds on Real-Time Quantum Memories
• Lower semicontinuity of the entropic disturbance and its applications in quantum information theory
• Fundamental limitations on the recoverability of quantum processes
• Classical and quantum parts of conditional mutual information for open quantum systems
• On the relation between quantum Darwinism and approximate quantum Markovianity
• Fermionic duality: General symmetry of open systems with strong dissipation and memory
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• Squashed quantum non-Markovianity: a measure of genuine quantum non-Markovianity in states
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• Lower semicontinuity of the relative entropy disturbance and its corollaries