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

Uncertainty and Entropy

Harold Falk

American Journal of PhysicsMay 1, 19721 citation

Continuity bounds on observational entropy and measured relative entropies

Joseph Schindler, Andreas Winter

We derive a measurement-independent asymptotic continuity bound on the observational entropy for general positive operator valued measures measurements, making essential use of its property of bounded concavity. The same insight is used to obtain continuity bounds for other entropic quantities, including the measured relative entropy distance to a convex set of states under a general set of measurements. As a special case, we define and study conditional observational entropy, which is an observational entropy in one (measured) subsystem conditioned on the quantum state in another (unmeasured) subsystem. We also study continuity of relative entropy with respect to a jointly applied channel, finding that observational entropy is uniformly continuous as a function of the measurement. But we show by means of an example that this continuity under measurements cannot have the form of a concrete asymptotic bound.We derive a measurement-independent asymptotic continuity bound on the observational entropy for general positive operator valued measures measurements, making essential use of its property of bounded concavity. The same insight is used to obtain continuity bounds for other entropic quantities, including the measured relative entropy distance to a convex set of states under a general set of measurements. As a special case, we define and study conditional observational entropy, which is an observational entropy in one (measured) subsystem conditioned on the quantum state in another (unmeasured) subsystem. We also study continuity of relative entropy with respect to a jointly applied channel, finding that observational entropy is uniformly continuous as a function of the measurement. But we show by means of an example that this continuity under measurements cannot have the form of a concrete asymptotic bound.

arXiv:2302.00400Journal of Mathematical PhysicsSep 1, 202310 citations

A Brief Introduction to Observational Entropy

Dominik Šafránek, Anthony N. Aguirre, Joseph Schindler +1 more

arXiv:2008.04409Foundations of PhysicsOct 1, 202147 citations

Witnessing quantum chaos using observational entropy

Sreeram PG, Ranjan Modak, S. Aravinda

We study observation entropy (OE) for the quantum kicked top model, whose classical counterpart possesses different phases: regular, mixed, or chaotic, depending on the strength of the kicking parameter. We show that OE grows logarithmically with coarse-graining length beyond a critical value in the regular phase, while OE growth is much faster in the chaotic regime. In the dynamics, we demonstrate that the short-time growth rate of OE acts as a measure of the chaoticity in the system, and we compare our results with out-of-time-ordered correlators (OTOC). Moreover, we show that in the deep quantum regime, the results obtained from OE are much more robust compared to OTOC results. Finally, we also investigate the long-time behavior of OE to distinguish between saddle-point scrambling and true chaos, where the former shows large persistent fluctuations compared to the latter.We study observation entropy (OE) for the quantum kicked top model, whose classical counterpart possesses different phases: regular, mixed, or chaotic, depending on the strength of the kicking parameter. We show that OE grows logarithmically with coarse-graining length beyond a critical value in the regular phase, while OE growth is much faster in the chaotic regime. In the dynamics, we demonstrate that the short-time growth rate of OE acts as a measure of the chaoticity in the system, and we compare our results with out-of-time-ordered correlators (OTOC). Moreover, we show that in the deep quantum regime, the results obtained from OE are much more robust compared to OTOC results. Finally, we also investigate the long-time behavior of OE to distinguish between saddle-point scrambling and true chaos, where the former shows large persistent fluctuations compared to the latter.

arXiv:2212.01585Physical review. EJun 6, 202313 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.