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arXiv:2312.03572

Generalized ααα-Observational Entropy

Shivam Sinha, Majumbdar, Nripendra, Aravinda, S.

Published Dec 6, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2312.03572

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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.
Observational entropy captures both the intrinsic uncertainty of a thermodynamic state and the lack of knowledge due to coarse-graining. We demonstrate two interpretations of observational entropy, one as the statistical deficiency resulting from a measurement, the other as the difficulty of inferring the input state from the measurement statistics by quantum Bayesian retrodiction. These interpretations show that the observational entropy implicitly includes a uniform reference prior. Since the uniform prior cannot be used when the system is infinite-dimensional or otherwise energy-constrained, we propose generalizations by replacing the uniform prior with arbitrary quantum states that may not even commute with the state of the system. We propose three candidates for this generalization, discuss their properties, and show that one of them gives a unified expression that relates both interpretations.
The notion of the thermodynamic entropy in the context of quantum mechanics is a controversial topic. Although there were proposals to refer von Neumann entropy as the thermodynamic entropy, it has its own limitations. The observational entropy has been developed as a generalization of Boltzmann entropy, and it is presently one of the most promising candidates to provide a clear and well-defined understanding of the thermodynamic entropy in quantum mechanics. In this paper, we study the behavior of the observational entropy in the context of localization-delocalization transition for one-dimensional Aubrey-Andr\'e (AA) model. We find that for the typical midspectrum states, in the delocalized phase the observation entropy grows rapidly with coarse-grain size and saturates to the maximal value, whereas in the localized phase the growth is logarithmic. Moreover, for a given coarse-graining, it increases logarithmically with system size in the delocalized phase, and obeys area law in the localized phase. We also find the increase in the observational entropy followed by the quantum quench is logarithmic in time in the delocalized phase as well as at the transition point, whereas in the localized phase it oscillates. Finally, we also venture the self-dual property of the AA model using momentum space coarse graining.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
The goal of this thesis is to define the concept of coarse-graining in quantum systems through measurements and to develop a unified mathematical framework for macroscopic states, operations, correlations, and entropy.Within this framework, irreversible phenomena arise from the loss of statistical information that is accessible to macroscopic observers.To achieve this goal, we rigorously define macroscopicity.This thesis investigates observational entropy, a concept that unifies Boltzmann entropy, Gibbs entropy, von Neumann's macroscopic entropy, and diagonal entropy.Our approach to explaining how irreversibility arises in isolated quantum systems is based on this unified perspective.When the initial state is "macroscopic," we use Petz's theory of statistical sufficiency to show that observational entropy strictly increases with unitary time evolution.Furthermore, we show that, for any initial state, the probability of the system's observational entropy increasing and rapidly reaching its maximum value under random unitary time evolution is extremely high.Specifically, we demonstrate that when measurements on a quantum system are sufficiently coarse-grained relative to its size, its initial state (whether pure or mixed) becomes indistinguishable from the maximally mixed state through random time evolution.We show that this holds not only for random time evolutions sampled according to the Haar measure, but also for approximate 2-designs, which are considered more physically realistic models.Next, we extend this framework by introducing positive operator-valued measures (POVMs) and quantum prior states.This allows us to construct a unified inferential framework for macroscopic states.A macroscopic state can be characterized using equivalent algebraic and constructive formulations based on quantum statistical sufficiency.This approach is based on the concept of an observational deficit, which quantifies the degree to which a state is "irretrodictable" relative to prior and measurement.This structure is formulated as an "inferential reference frame" that encapsulates the observer's informational perspective.Furthermore, we treat macroscopic states as free states and construct the resource theory of microscopicity.This theory unifies existing theories of coherence, athermality and asymmetry.Finally, we extend this framework to the study of quantum correlations under observational constraints.This enables us to show that quantum correlations, such as entanglement and quantum discord, are context-dependent resources that depend on the observer's reference frame.I would like to express my deepest gratitude to my supervisor,
Coarse-grained measurements offer a scalable alternative to full state tomography for characterizing complex quantum dynamics. We show that observational entropy (OE), an information-theoretic entropy defined directly from finite-resolution measurement outcomes, provides a unified and experimentally accessible framework for quantifying chaos and probing criticality. From probing the insulator-metal crossover in the Aubry-Andre model to tracking the gradual destruction of Kolmogorov-Arnold-Moser tori in the Kicked Rotor, derivatives of OE provide an accurate and unified diagnostic of probing these transitions. In both cases, the critical points extracted from dynamical evolution and eigenstate analyses converge to the exact theoretical values once the observational resolution exceeds a finite threshold. In the chaotic limit, OE exhibits a linear behavior within the Ehrenfest time regime, and its slope defines an observable Lyapunov exponent. Using a Pretty Good Measurement correction to the Husimi phase-space distribution, this entropy-production rate quantitatively reproduces the classical Lyapunov exponent in both the standard and singular kicked rotors. Our results establish OE as a compact information-theoretic bridge between classical instability, quantum criticality, and realistic finite-resolution measurements.