Exploring Thermodynamic Quantumness

• Thermodynamic quantumness is a framework describing how specific quantum features generate or modify thermodynamic behavior, evidenced by nonclassical temporal correlations and quantum fluctuations.
• It operationally defines quantum signatures in engines via Leggett–Garg tests and fluctuation regimes where coherence and entanglement alter work, heat, and entropy.
• The concept bridges microscopic quantum spectral properties with macroscopic thermodynamic variables, offering insights into emergent quantities like pressure, temperature, and entropy.

Thermodynamic quantumness denotes the set of regimes, criteria, and structures by which thermodynamic behavior is altered, generated, or operationally witnessed by specifically quantum features rather than by classical coarse-grained statistical mechanics alone. In the literature, the term does not have a single universal definition. It may refer to temporal nonclassicality in a working cycle detected by Leggett–Garg inequality violation, to the regime in which fluctuations are quantum rather than thermal and work ceases to be an ordinary observable, or to the derivation of pressure, heat, entropy, and temperature directly from quantum spectra, eigenstates, and reduced states in chaotic few-body systems (Friedenberger et al., 2015, Millen et al., 2015, Holm et al., 3 Feb 2026).

1. Conceptual meanings of thermodynamic quantumness

Several distinct but overlapping meanings recur in the literature.

Strand	Defining object	Representative criterion
Temporal nonclassicality	Time correlations in a thermal machine	$K_3>1$ in a Leggett–Garg test
Quantum fluctuation regime	Unitary dynamics, measurement back-action, coherence, entanglement	Fluctuations are quantum rather than thermal
Microscopic emergence	Spectra, eigenstates, reduced states	$P=-\partial E/\partial \lambda$, $\delta Q=T\,dS$ from quantum structure

A central clarification is that thermodynamic quantumness is not exhausted by the statement that a device is microscopic or described by a quantum Hamiltonian. In the two-level Otto-engine analysis, a heat engine is called genuinely quantum only when its evolution exhibits temporal nonclassicality, operationally identified by violation of the Leggett–Garg inequality during the cycle (Friedenberger et al., 2015). In the broader perspective literature, the quantum regime is instead the regime in which fluctuations are no longer just thermal in their origin but quantum, so that coherence, entanglement, measurement disturbance, and noncommutativity reshape the interpretation of heat, work, entropy, and equilibration (Millen et al., 2015).

This plurality suggests that thermodynamic quantumness is best treated as a family resemblance concept. Across the field, the common element is not a single resource or observable, but the claim that thermodynamic state variables, performance bounds, or irreversibility acquire a specifically quantum origin, operational meaning, or structural reformulation.

2. Operational diagnostics in cyclic thermal machines

The most explicit operational definition is given for a single two-level Otto engine with Hamiltonian

$H(t)=\frac{\omega_t}{2}\sigma_z,$

with four strokes: heating, expansion, cooling, and compression. The average energy is

$$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$

so that

$dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$

Work is associated with changing $\omega$, and heat with changing the polarization $P$. During reservoir contact, the open dynamics is modeled by a Markovian Heisenberg-picture master equation, while the unitary strokes correspond to $\gamma_0=0$ (Friedenberger et al., 2015).

Quantumness is assessed through the Leggett–Garg inequality for the dichotomic observable $Q=\sigma_x$. With

$P=-\partial E/\partial \lambda$0

any violation $P=-\partial E/\partial \lambda$1 signals nonclassical temporal correlations. The symmetrized two-time correlation function is

$P=-\partial E/\partial \lambda$2

and for equally spaced times

$P=-\partial E/\partial \lambda$3

The relevant comparison is between the heating time $P=-\partial E/\partial \lambda$4 required by thermodynamic constraints and the largest interval $P=-\partial E/\partial \lambda$5 over which LGI violation is possible,

$P=-\partial E/\partial \lambda$6

The engine is nonclassical only if $P=-\partial E/\partial \lambda$7 (Friedenberger et al., 2015).

The resulting phase diagram in $P=-\partial E/\partial \lambda$8, with reduced internal-friction parameter $P=-\partial E/\partial \lambda$9, contains three regimes: a low-friction regime with a single quantum–classical transition at $\delta Q=T\,dS$0, an intermediate “blurry boundary” regime with multiple transitions, and a large-friction regime in which the engine is classical for all $\delta Q=T\,dS$1. In the low-friction regime, the transition temperature is

$\delta Q=T\,dS$2

with the critical damping criterion $\delta Q=T\,dS$3. The analysis also shows that the intuition “go faster to stay coherent” fails because larger internal friction increases thermalization times and therefore bath exposure (Friedenberger et al., 2015).

A distinct operational notion appears in the finite-time harmonic Otto cycle, where quantum and classical versions are compared exactly. There, thermodynamic quantumness is identified with the combination of Bose–Einstein statistics, the commutator contribution from $\delta Q=T\,dS$4, and coherence indicators $\delta Q=T\,dS$5 and $\delta Q=T\,dS$6. In the quasistatic limit, quantumness reduces productivity and precision, whereas in finite time it can increase both. As the heat conductance $\delta Q=T\,dS$7 increases, the precision of the quantum Otto cycle can overtake that of the classical one. The conventional thermodynamic uncertainty relation is violated by both quantum and classical finite-time Otto cycles in the small-entropy-production regime, while a tighter bound is found when the entropy production is large enough (Lee et al., 2020).

3. Emergence of thermodynamic variables from quantum structure

A separate research program treats thermodynamic quantumness not as a resource added to pre-existing thermodynamics, but as the microscopic origin of thermodynamic state variables themselves. In a single-particle two-dimensional billiard, pressure is extracted from the geometry dependence of quantum levels through the Hellmann–Feynman theorem,

$\delta Q=T\,dS$8

For an integrable billiard, a consistent scalar pressure does not emerge because the force is anisotropic. For a chaotic Sinai billiard, ergodic eigenstates delocalized over the area make pressure isotropic, and the numerically extracted pressure obeys the two-dimensional analog of Boyle–Mariotte’s law,

$\delta Q=T\,dS$9

up to finite-wavelength quantum fluctuations (Holm et al., 3 Feb 2026).

The same framework extends to two interacting particles in adjacent boxes with Hamiltonian

$H(t)=\frac{\omega_t}{2}\sigma_z,$0

Because the wall is immobile, energy exchange is interpreted as heat rather than work. The net exchanged energy is

$H(t)=\frac{\omega_t}{2}\sigma_z,$1

or equivalently

$H(t)=\frac{\omega_t}{2}\sigma_z,$2

so heat is defined from the change in reduced local populations caused by interaction-induced equilibration. Local entropies are

$H(t)=\frac{\omega_t}{2}\sigma_z,$3

and temperatures follow from

$H(t)=\frac{\omega_t}{2}\sigma_z,$4

The equilibrium condition is $H(t)=\frac{\omega_t}{2}\sigma_z,$5, and the paper summarizes the first-law split as

$H(t)=\frac{\omega_t}{2}\sigma_z,$6

This construction is presented as a transparent illustration of the eigenstate thermalization hypothesis, because the relevant chaotic eigenstates already encode balanced local energies (Holm et al., 3 Feb 2026).

A broader implication is that thermodynamic quantumness can mean that the foundational objects are quantum spectral data rather than classical phase-space distributions. In that sense, classical laws appear as emergent consequences of quantum chaos and reduced-state structure rather than as assumptions imposed at the outset.

4. Nonequilibrium dynamics, strong coupling, and quantum fluctuation structure

In exact nonequilibrium thermodynamics for single-particle open quantum systems, thermodynamic quantities are defined directly from the reduced density matrix $H(t)=\frac{\omega_t}{2}\sigma_z,$7: $H(t)=\frac{\omega_t}{2}\sigma_z,$8

$H(t)=\frac{\omega_t}{2}\sigma_z,$9

with

$$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$0

In the weak-coupling regime $$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$1, classical thermodynamics emerges dynamically without an equilibrium hypothesis. In the strong-coupling regime $$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$2, a localized bound state appears, the system retains memory of its initial condition, energy can backflow to the reservoir, entropy can increase and then decrease, temperature may oscillate rather than converge, and classical thermodynamics breaks down (Ali et al., 2018).

The same work defines a “quantum regime” by $$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$3, or a vacuum reservoir with the system initially in a pure excited state. There the nonequilibrium temperature

$$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$4

can become negative, not in the usual equilibrium bounded-spectrum sense, but as a dynamical negative temperature arising from nonequilibrium evolution. A dynamical quantum phase transition is proposed near $$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$5, while the late-time limit $$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$6, $$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$7 is interpreted as a dynamical proof of the third law in the deep quantum regime (Ali et al., 2018).

Strong-coupling equilibrium thermodynamics has also been reformulated using the Hamiltonian of Mean Force,

$$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$8

The reduced state is

$$E=\langle H\rangle=\omega P,\qquad P=\frac{\langle \sigma_z\rangle}{2},$$9

the effective energy operator is

$dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$0

and the generalized entropy is

$dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$1

In weak coupling, specific heat reduces to the familiar variance form. In strong coupling, however,

$dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$2

so heat capacity contains both a Wigner–Yanase–Dyson skew-information term and a correction from the explicit temperature dependence of the effective energy operator. In the two-qubit model, strong coupling can yield negative specific heat, negative entropy at low temperature, nonzero ergotropy due to coherence, and enhanced entropy production; in the Jaynes–Cummings model without the rotating wave approximation, ergotropy vanishes because the HMF Gibbs state is diagonal (Pathania et al., 2024).

For Gaussian open systems, an even more microscopic basis is given in terms of the Robertson–Schrödinger uncertainty function

$dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$3

where $dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$4, $dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$5, and $dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$6. The nonequilibrium partition function becomes

$dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$7

and the effective inverse temperature is

$dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$8

Internal energy and free energy are therefore functionals of the uncertainty function. This identifies noncommutativity of canonical operators, rather than thermal randomness alone, as the microscopic source of a class of nonequilibrium thermodynamic relations (Dong et al., 2022).

5. Correlations, speed, ergotropy, and quantum thermodynamic tasks

A large part of the literature treats thermodynamic quantumness as the operational role of coherence, entanglement, discord, or non-passivity in heat, work, and metrology. One line of work derives a quantum Maxwell relation for a Hamiltonian $dE=P\,d\omega+\omega\,dP=\delta W+\delta Q.$9,

$\omega$0

leading to the isothermal entropy change

$\omega$1

and the adiabatic temperature change

$\omega$2

For an isotropic Heisenberg interaction with control parameter $\omega$3, the paper relates the spin-$\omega$4 dimer entropy change directly to the temperature derivative of the Schatten 1-norm quantum discord,

$\omega$5

Standard and inverse caloric effects are then associated with antiparallel and parallel alignment, respectively (Cruz et al., 2024).

A more dynamical resource interpretation is given by the thermodynamic speed of work extraction. For cyclic unitary evolution, the speed $\omega$6 is defined from a Hellinger-type distance between extractable energies, and the maximal quantum energy-exchange speed is

$\omega$7

For pure states under time-independent probing Hamiltonian $\omega$8,

$\omega$9

The main results are that coherence can speed up work extraction beyond all incoherent states, and genuine multipartite entanglement can speed it up beyond all biseparable states, thereby providing a thermodynamic witness of quantum resources without full state tomography. For the product coherent battery $P$0,

$P$1

and for a GHZ state

$P$2

both indicating quadratic speed-up over the corresponding benchmarks (Luo, 2024).

A still more radical proposal replaces conventional open heat machines by few-mode coherent closed systems with nonlinear couplings. In that framework, the machine is a coherent thermal-noise filter rather than a dissipative working fluid. Linear blocks and a nonlinear core are composed as

$P$3

and cross-Kerr processing can transform passive thermal inputs into non-passive outputs with nonzero ergotropy,

$P$4

The claim is not that any coherent feature is automatically thermodynamically quantum, but that nonlinear coherent processing can turn thermal noise into a resource of work and information while respecting the global second law through entropy redistribution among modes (Kurizki et al., 6 Feb 2025).

The broader perspective literature places these results in the context of quantum demons, work extraction, and refrigeration. There, discord can enhance the performance of a quantum demon, decoherence reduces the advantage, entanglement can appear as a by-product of work extraction from thermal ensembles, and quantum refrigerators may exploit resources without direct classical analogues, as in the example where a single qutrit cools a qubit (Millen et al., 2015).

6. Reformulations of thermodynamic observables and extensions beyond standard settings

One foundational reformulation treats quantum thermodynamics as an emergent gauge theory. The mean energy

$P$5

is taken as the thermodynamic scalar, and admissible gauge transformations satisfy

$P$6

The usual Alicki work,

$P$7

is not gauge invariant, so the framework replaces it by

$P$8

with gauge potential $P$9 and covariant derivative $\gamma_0=0$0. Heat becomes

$\gamma_0=0$1

where $\gamma_0=0$2 is coherent heat produced by basis rotation. In this picture, work is tied to eigenvalue changes, while coherence contributes through the coherent-heat channel (Céleri et al., 2021).

Degeneracy produces another nonclassical reformulation. For open systems with exact degeneracies, coherences between degenerate eigenstates survive the Born–Markov–secular treatment and remain coupled to populations. The first law,

$\gamma_0=0$3

and second law,

$\gamma_0=0$4

still hold, but local eigenbasis coherences can suppress or enhance currents, reduce entropy production, and reshape full counting statistics. In a degenerate double quantum dot, a dark-state steady state can become pure and coherent, so that the von Neumann entropy vanishes while the Shannon entropy in the local basis remains finite if off-diagonal terms are ignored (Cuetara et al., 2016).

Thermodynamic quantumness has also been extended beyond Hermitian Hamiltonians. For pseudo-Hermitian systems with real spectrum, the Jarzynski equality remains valid,

$\gamma_0=0$5

and in the quasistatic limit the Carnot efficiency

$\gamma_0=0$6

is preserved even when eigenvalues are complex, provided they come in conjugate pairs with equal multiplicity. This places metric structure and generalized unitarity, rather than Hermiticity alone, at the center of the thermodynamic formulation (Gardas et al., 2015).

At an even more structural level, generalized probabilistic theories have been constrained by two thermodynamic postulates: generalized bit symmetry and the requirement that every state decompose into perfectly distinguishable pure states. These imply self-duality, projective measurements, spectral decompositions, and a consistent thermodynamic entropy constructed by a von Neumann–style thought experiment, while still allowing classical theory and certain Jordan-algebraic alternatives. The same framework links absence of third-order interference to equality of two max-entropy notions, showing that thermodynamic consistency constrains, but does not uniquely determine, the structure of quantum theory (Krumm et al., 2016).

A geometrical variant formulates quantum thermodynamics directly on the projective state manifold $\gamma_0=0$7, with canonical ensembles given by continuous densities

$\gamma_0=0$8

Heat and work are defined intrinsically from the decomposition of $\gamma_0=0$9, and Jarzynski’s relation is recovered without a two-time projective measurement scheme,

$Q=\sigma_x$0

This approach suggests that thermodynamic quantumness may reside not only in operator algebra or open-system dynamics, but also in the geometry of quantum state space itself (Anza et al., 2020).

Across these reformulations, a recurring conclusion is that thermodynamic quantumness is not a single phenomenon. It can denote temporal nonclassicality in engines, the quantum origin of fluctuations and uncertainty, the spectral construction of state variables, the role of coherence and correlations in work and heat, or the structural constraints that thermodynamics imposes on the very formulation of quantum theory.