Traditional measures of entropy, like the Von Neumann entropy, while fundamental in quantum information theory, are insufficient when interpreted as thermodynamic entropy due to their invariance under unitary transformations, which contradicts observed entropy increases in isolated systems. Recognizing this limitations of existing measures for thermodynamic entropy, recent research has focused on observational entropy (OE) as a promising alternative, offering practical applicability and theoretical insights. In this work, we extend the scope of observational entropy by generalizing it to a parameterized version called $α$ -Observational entropy ( $α$ -OE). $α$ -OE is expressed in terms of the Petz-Rényi relative entropy between the states on which a quantum-to-classical channel is applied. The $α$ -OE reduces to OE under $α\rightarrow 1$ . We prove various properties of the $α$ -OE, which are the generalization of the properties of OE, including the monotonically increasing of $α$ -OE as a function of refinement of coarse-graining. We further explore the role of $α$ -OE in thermodynamic contexts, particularly for the entropy production in open and closed quantum systems and its relation with the Helmholtz free energy.Traditional measures of entropy, like the Von Neumann entropy, while fundamental in quantum information theory, are insufficient when interpreted as thermodynamic entropy due to their invariance under unitary transformations, which contradicts observed entropy increases in isolated systems. Recognizing this limitations of existing measures for thermodynamic entropy, recent research has focused on observational entropy (OE) as a promising alternative, offering practical applicability and theoretical insights. In this work, we extend the scope of observational entropy by generalizing it to a parameterized version called $α$-Observational entropy ($α$-OE). $α$-OE is expressed in terms of the Petz-Rényi relative entropy between the states on which a quantum-to-classical channel is applied. The $α$-OE reduces to OE under $α\rightarrow 1$. We prove various properties of the $α$-OE, which are the generalization of the properties of OE, including the monotonically increasing of $α$-OE as a function of refinement of coarse-graining. We further explore the role of $α$-OE in thermodynamic contexts, particularly for the entropy production in open and closed quantum systems and its relation with the Helmholtz free energy.

arXiv:2312.03572Dec 6, 20230 citations

A-entropy for spectrally absolutely continuous observables

Marian Grabowski

Reports on Mathematical PhysicsDec 1, 19788 citations

Remarks on A-entropy

Alfred Wehrl

Reports on Mathematical PhysicsDec 1, 197719 citations

Observational entropy, coarse-grained states, and the Petz recovery map: information-theoretic properties and bounds

Francesco Buscemi, Joseph Schindler, Dominik Šafránek

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.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.

arXiv:2209.03803New Journal of PhysicsApr 14, 202318 citations

α

-Observational entropy (

α

-OE).

α

-OE is expressed in terms of the Petz-Rényi relative entropy between the states on which a quantum-to-classical channel is applied. The

α

-OE reduces to OE under

α\rightarrow 1

. We prove various properties of the

α

-OE, which are the generalization of the properties of OE, including the monotonically increasing of

α

-OE as a function of refinement of coarse-graining. We further explore the role of

α

-OE in thermodynamic contexts, particularly for the entropy production in open and closed quantum systems and its relation with the Helmholtz free energy.