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1. Quantum and Classical Ergotropy from Relative Entropies

   https://doi.org/10.3390/e23091107  

   Sone, Akira; Deffner, Sebastian (September 2021, Entropy)

   null (Ed.)

   The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. Given that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state, we define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both quantum as well as classical scenarios is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem. 

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2. Complexity-Constrained Quantum Thermodynamics

   https://doi.org/10.1103/PRXQuantum.6.010346  

   Munson, Anthony; Kothakonda, Naga_Bhavya Teja; Haferkamp, Jonas; Yunger_Halpern, Nicole; Eisert, Jens; Faist, Philippe (March 2025, PRX Quantum)

   Quantum complexity measures the difficulty of realizing a quantum process, such as preparing a state or implementing a unitary. We present an approach to quantifying the thermodynamic resources required to implement a process if the process’s complexity is restricted. We focus on the prototypical task of information erasure, or Landauer erasure, wherein an $n$-qubit memory is reset to the all-zero state. We show that the minimum thermodynamic work required to reset an arbitrary state in our model, via a complexity-constrained process, is quantified by the state’s . The complexity entropy therefore quantifies a trade-off between the work cost and complexity cost of resetting a state. If the qubits have a nontrivial (but product) Hamiltonian, the optimal work cost is determined by the . The complexity entropy quantifies the amount of randomness a system appears to have to a computationally limited observer. Similarly, the complexity relative entropy quantifies such an observer’s ability to distinguish two states. We prove elementary properties of the complexity (relative) entropy. In a random circuit—a simple model for quantum chaotic dynamics—the complexity entropy transitions from zero to its maximal value around the time corresponding to the observer’s computational-power limit. Also, we identify information-theoretic applications of the complexity entropy. The complexity entropy quantifies the resources required for data compression if the compression algorithm must use a restricted number of gates. We further introduce a , which arises naturally in a complexity-constrained variant of information-theoretic decoupling. Assuming that this entropy obeys a conjectured chain rule, we show that the entropy bounds the number of qubits that one can decouple from a reference system, as judged by a computationally bounded referee. Overall, our framework extends the resource-theoretic approach to thermodynamics to integrate a notion of , as quantified by . Published by the American Physical Society2025 

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3. Statistical Efficiency of Score Matching: The View from Isoperimetry

   Koehler, F; Heckett, A; Risteski, A. (May 2023, International Conference on Learning Representations (ICLR), 2024)

   Deep generative models parametrized up to a normalizing constant (e.g. energy-based models) are difficult to train by maximizing the likelihood of the data because the likelihood and/or gradients thereof cannot be explicitly or efficiently written down. Score matching is a training method, whereby instead of fitting the likelihood for the training data, we instead fit the score function, obviating the need to evaluate the partition function. Though this estimator is known to be consistent, it's unclear whether (and when) its statistical efficiency is comparable to that of maximum likelihood---which is known to be (asymptotically) optimal. We initiate this line of inquiry in this paper and show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated---i.e. the Poincar\'e, log-Sobolev and isoperimetric constant---quantities which govern the mixing time of Markov processes like Langevin dynamics. Roughly, we show that the score matching estimator is statistically comparable to the maximum likelihood when the distribution has a small isoperimetric constant. Conversely, if the distribution has a large isoperimetric constant---even for simple families of distributions like exponential families with rich enough sufficient statistics---score matching will be substantially less efficient than maximum likelihood. We suitably formalize these results both in the finite sample regime, and in the asymptotic regime. Finally, we identify a direct parallel in the discrete setting, where we connect the statistical properties of pseudolikelihood estimation with approximate tensorization of entropy and the Glauber dynamics. 

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4. Spectral bounds on the entropy flow rate and Lyapunov exponents in differentiable dynamical systems

   https://doi.org/10.1088/1751-8121/ad8f06  

   Das, Swetamber; Green, Jason_R (January 2025, Journal of Physics A: Mathematical and Theoretical)

   Abstract Some microscopic dynamics are also macroscopically irreversible, dissipating energy and producing entropy. For many-particle systems interacting with deterministic thermostats, the rate of thermodynamic entropy dissipated to the environment is the average rate at which phase space contracts. Here, we use this identity and the properties of a classical density matrix to derive upper and lower bounds on the entropy flow rate from the spectral properties of the local stability matrix. These bounds are an extension of more fundamental bounds on the Lyapunov exponents and phase space contraction rate of continuous-time dynamical systems. They are maximal and minimal rates of entropy production, heat transfer, and transport coefficients set by the underlying dynamics of the system and deterministic thermostat. Because these limits on the macroscopic dissipation derive from the density matrix and the local stability matrix, they are numerically computable from the molecular dynamics. As an illustration, we show that these bounds are on the electrical conductivity for a system of charged particles subject to an electric field. 

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5. Thermodynamic speed limits for mechanical work

   https://doi.org/10.1088/1751-8121/acb5d6  

   Aghion, Erez; Green, Jason R. (February 2023, Journal of Physics A: Mathematical and Theoretical)

   Abstract Thermodynamic speed limits are a set of classical uncertainty relations that, so far, place global bounds on the stochastic dissipation of energy as heat and the production of entropy. Here, instead of constraints on these thermodynamic costs, we derive integral speed limits that are upper and lower bounds on a thermodynamic benefit—the minimum time for an amount of mechanical work to be done on or by a system. In the short time limit, we show how this extrinsic timescale relates to an intrinsic timescale for work, recovering the intrinsic timescales in differential speed limits from these integral speed limits and turning the first law of stochastic thermodynamics into a first law of speeds. As physical examples, we consider the work done by a flashing Brownian ratchet and the work done on a particle in a potential well subject to external driving. 

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