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arXiv:2209.10273 · Physical Review A

Observational-entropic study of Anderson localization

Ranjan Modak, S. Aravinda

Published Dec 28, 2022• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.106.062217

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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 investigate the detailed properties of observational entropy, introduced by \ifmmode \check{S}\else \v{S}\fi{}afr\'anek et al. [Phys. Rev. A 99, 010101 (2019)] as a generalization of Boltzmann entropy to quantum mechanics. This quantity can involve multiple coarse grainings, even those that do not commute with each other, without losing any of its properties. It is well-defined out of equilibrium, and for some coarse grainings it generically rises to the correct thermodynamic value even in a genuinely isolated quantum system. The quantity contains several other entropy definitions as special cases, it has interesting information-theoretic interpretations, and mathematical properties---such as extensivity and upper and lower bounds---suitable for an entropy. Here we describe and provide proofs for many of its properties, discuss its interpretation and connection to other quantities, and provide numerous simulations and analytic arguments supporting the claims of its relationship to thermodynamic entropy. This quantity may thus provide a clear and well-defined foundation on which to build a satisfactory understanding of the second thermodynamical law in quantum mechanics.
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.
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.
Nonequilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce an abrupt change in the scaling law of the bipartite entanglement entropy, thus suggesting the existence of different nonequilibrium regimes. However, our understanding of how these regimes appear and whether they survive in the thermodynamic limit is much less established. Here we investigate these questions on a one-dimensional quadratic fermionic model: this allows us to reach system sizes relevant in the thermodynamic sense. We show that local projective measurements induce a qualitative modification of the time growth of the entanglement entropy which changes from linear to logarithmic. However, in the stationary regime, the logarithmic behavior of the entanglement entropy does not survive in the thermodynamic limit and, for any finite value of the measurement rate, we numerically show the existence of a single area-law phase for the entanglement entropy. Finally, exploiting the quasiparticle picture, we further support our results by analyzing the fluctuations of the stationary entanglement entropy and its scaling behavior.
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 ααα-Observational entropy (ααα-OE). ααα-OE is expressed in terms of the Petz-Rényi relative entropy between the states on which a quantum-to-classical channel is applied. The ααα-OE reduces to OE under α→1α\rightarrow 1α→1. We prove various properties of the ααα-OE, which are the generalization of the properties of OE, including the monotonically increasing of ααα-OE as a function of refinement of coarse-graining. We further explore the role of ααα-OE in thermodynamic contexts, particularly for the entropy production in open and closed quantum systems and its relation with the Helmholtz free energy.
The authors study the metal-insulator transition in the three-dimensional Anderson model from the entanglement perspective. They show that at the transition point the entanglement entropy after a quantum quench grows logarithmically in time, while the number entropy shows a double logarithmic growth. This behavior is consistent with exact bounds. It is also the same scaling recently found in many-body localized (MBL) phases, suggesting that the MBL phase might be more akin to an extended critical regime, rather than a fully localized phase.