Temperature is one of the central concepts of thermodynamics, yet its meaning far from equilibrium remains unclear. The problem is especially challenging in isolated quantum many-body systems, whose states evolve unitarily, may remain far from equilibrium, and retain energy coherence, a genuinely quantum feature with no direct classical counterpart. Specifically, energy fluctuations in a nonstationary quantum state have two distinct origins. Part of them comes from uncertainty in the energy populations and has the usual thermodynamic meaning. The rest comes from quantum coherence between energy sectors and is responsible for the state's time dependence. We propose that, even away from equilibrium, temperature identifies the state within the family of regular states sharing the same energy-coherence structure. This provides a natural definition of temperature for a broad class of nonequilibrium quantum states. The resulting inverse temperature is not, in general, obtained by differentiating entropy with respect to energy. The usual maximum-entropy principle is instead replaced by a principle of minimum discrimination information, which selects the least distinguishable state compatible with the prescribed energy and coherence structure. We also extend the construction to subsystems and show that, although their inverse temperature is not determined by the reduced state alone, its instantaneous rate of change is a local quantity, determined by the thermodynamic structure induced on the subsystem at that time.
Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness $\unicode{x2013}$ the resource enabling universal quantum computation $\unicode{x2013}$ is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer R\'enyi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.
Soumyadeep Sarma, Tobias Haug, John Preskill et al.· 2 citations
The study of quantum systems far from equilibrium has become a cornerstone in the development of emerging quantum technologies. Real-world quantum devices, such as quantum computers, simulators, and sensors, inevitably operate under conditions of driving, dissipation, and decoherence. Consequently, non-equilibrium dynamics are not an exception but a defining feature of these systems. A fundamental understanding of how quantum systems relax, thermalize, or avoid thermalization is essential for stabilizing coherence, extending memory lifetimes, and protecting entanglement. Experimental observations of turbulent Bose–Einstein condensates (BECs) have demonstrated the existence of distinct relaxation pathways. Depending on system conditions, a turbulent BEC may evolve toward a recondensed coherent state or decay into a thermal, non-degenerate quantum fluid. This evolution proceeds through well-defined dynamical stages, characterized by complex vortex interactions, energy cascades, and the redistribution of coherence across scales. Understanding these pathways provides critical insight into the mechanisms governing quantum turbulence and the emergence of order from strongly non-equilibrium states, offering new opportunities to control and harness quantum coherence in realistic environments.
S. Sab, Michelle Moreno-Amijos, A. D. Garc'ia-Orozco et al.· Photonics for Quantum· 0 citations
Thermodynamics is operational in the sense that the very concept of thermal equilibrium depends crucially on the choice of observables, while the amount of extractable work is defined relative to the allowed operations. Here we show that isolated quantum many-body states admit a thermodynamic hierarchical structure of observables and allowed operations, where the same state can be thermal at one level of the hierarchy and athermal at another. Enlarging the set of allowed operations therefore renders such hidden athermality a potential resource for work extraction; the resulting work gain is bounded in terms of the difference between the entropy densities of the two levels. This entropy difference takes the form of the mutual-information density. We establish the bounds for three extensions of operational access: increased spatial resolution, increased duration of control, and nonlocal connectivity, which provide access to position-state Holevo information, correlations between neighboring regions and spatially nonlocal correlations, respectively. Thus the difference between entropies at different levels of the thermodynamic hierarchy governs the bound on the work gain associated with moving to a less restricted level.
Characterizing the time evolution of generic quantum many-body systems is a fundamental challenge, as representing the exact state requires exponentially scaling computational resources. While hydrodynamics and statistical mechanics successfully simplify this task by predicting the expectation values of local observables, these macroscopic frameworks provide no information about nonlinear characteristics of the quantum state. In this work, we demonstrate that for systems exhibiting a timescale separation, with dynamics governed by the transport of conserved charges, this lost information can be systematically recovered. By constraining the maximum-entropy Scrooge ensemble solely by the system's slow modes, we accurately reconstruct complex, nonlinear quantum properties of the global time-evolved state, including the half-chain entanglement entropy and the participation entropy in the computational basis. Our results generalize the paradigm of canonical typicality, revealing that a hydrodynamically bottlenecked system behaves as a typical pure state within a dynamically restricted submanifold of the Hilbert space - a phenomenon we term global restricted typicality.
Konrad Pawlik, P. Sierant, Jakub Zakrzewski· 0 citations
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.
Coherences between different energy levels are strongly constrained by thermodynamics. Here we ask a related question: if several coherence transfers can be optimized separately, can they also be optimized simultaneously by the same thermodynamic process? We show that this is in general a compatibility problem. Using an energy-resolved Stinespring representation of thermal operations, we express elementary coherence-transfer amplitudes as inner products of bath-weighted vectors associated with total-energy shells. Saturation of the corresponding Cauchy--Schwarz bounds requires these vectors to be collinear. Since all transfers have to arise from the same energy-preserving system--bath unitary, such saturation conditions cannot in general be imposed independently. Trace and Gibbs preservation lead to additional phase-closure conditions and to polygon inequalities that can rule out simultaneous saturation for fixed population-transfer data. We complement these analytic results with a rigorous computer-assisted semidefinite-program analysis. For the mixed-degenerate Hamiltonian $H=\operatorname{diag}(0,1,1,3)$, machine-certified primal and dual bounds show that no covariant Gibbs-preserving channel can simultaneously attain three individually optimal directional coherence transfers. Since every thermal operation is covariant and Gibbs preserving, the same triple of optimal values cannot be attained by a thermal operation. We further relate this incompatibility to range-inclusion constraints in Choi-sector Gram matrices for mixed-degenerate energy levels. Our results show that coherence transfers which are optimal when considered separately need not be compatible with one physical thermodynamic process. A full characterization of this incompatibility directly within thermal operations remains open.
Piotr Ćwikliński, Michał Studziński· 0 citations
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