Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds

Abstract

Observational entropy provides a general notion of quantum entropy that appropriately interpolates between Boltzmann's and Gibbs' entropies, and has recently been argued to provide a useful measure of out-of-equilibrium thermodynamic entropy. Here we study the mathematical properties of observational entropy from an information-theoretic viewpoint, making use of recently strengthened forms of the monotonicity property of quantum relative entropy. We present new bounds on observational entropy applying in general, as well as bounds and identities related to sequential and post-processed measurements. A central role in this work is played by what we call the ``coarse-grained'' state, which emerges from the measurement's statistics by Bayesian retrodiction, without presuming any knowledge about the ``true'' underlying state being measured. The degree of distinguishability between such a coarse-grained state and the true (but generally unobservable) one is shown to provide upper and lower bounds on the difference between observational and von Neumann entropies.

18 pages, 1 figure. v2 Greatly revised and restructured, adds new results. Questions and comments welcome

References in corpus (9)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Quantum Mechanics as Quantum Information (and only a little more)
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics
• Clean Positive Operator Valued Measures
• Quantum Dynamics as an analog of Conditional Probability
• Work extraction from unknown quantum sources
• Axioms for retrodiction: achieving time-reversal symmetry with a prior
• Quantum operator entropies under unitary evolution

Cited by in corpus (15)

• From time-reversal symmetry to quantum Bayes' rules
• Witnessing quantum chaos using observational entropy
• Continuity bounds on observational entropy and measured relative entropies
• Emergence of a second law of thermodynamics in isolated quantum systems
• On the generic increase of observational entropy in isolated systems
• Universal validity of the second law of information thermodynamics
• Quantum Bayes' rule and Petz transpose map from the minimum change principle
• Measuring energy by measuring any other observable
• Observational entropy with general quantum priors
• Modeling the Past Hypothesis: A Mechanical Cosmology
• Entropy Production from Maximum Entropy Principle: a Unifying Approach
• Periodicity of dynamical signatures of chaos in quantum kicked top
• Entropic partial orderings of quantum measurements
• Coarse-grained quantum thermodynamics: Observation-dependent quantities, observation-independent laws
• Macroscopicity and observational deficit in states, operations, and correlations