Gibbs-Preserving Operations in Quantum Thermodynamics

• Gibbs-preserving operations are completely positive trace-preserving maps that leave the Gibbs state unchanged, serving as a foundational concept in quantum thermodynamics.
• They bridge classical thermo-majorization and quantum coherence, generalizing thermal operations through relaxed symmetry constraints.
• They reveal the operational contrast between energy-diagonal processes and genuinely quantum coherent transformations, often involving unbounded coherence costs.

Searching arXiv for papers on Gibbs-preserving operations and related frameworks. Gibbs-preserving operations, also called Gibbs-preserving maps, are completely positive trace-preserving maps that leave a thermal Gibbs state invariant. For a finite-dimensional system with Hamiltonian $H$ at inverse temperature $\beta$, the Gibbs state is

$$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$

and a channel $\Phi$ is Gibbs-preserving when $\Phi(\gamma_\beta)=\gamma_\beta$. In quantum thermodynamics, this condition abstracts the equilibrium-preservation aspect of the second law and defines a broad class of free processes. The subject has developed into a resource-theoretic framework connecting thermal operations, thermo-majorization, $D$-majorization, covariant dynamics, and the operational role of coherence. A central structural fact is that Gibbs-preserving operations coincide with thermal operations only in the energy-diagonal regime; in the genuinely quantum regime they are strictly more powerful (Faist et al., 2014).

1. Formal setting and basic definitions

The standard setting fixes a finite-dimensional quantum system and a temperature. In the single-system formulation, a Gibbs-preserving operation is a CPTP map $\Phi$ satisfying $\Phi(D)=D$ for a full-rank Gibbs state

$$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$

In the more general input-output formulation, a channel $\Lambda:S\to S'$ is Gibbs-preserving if $\beta$0, where $\beta$1 denotes the Gibbs state of system $\beta$2 (Ende, 2020, Tajima et al., 2024).

At infinite temperature, $\beta$3, the Gibbs state becomes maximally mixed, and Gibbs preservation reduces to unitality. In this sense, Gibbs-preserving operations generalize unital channels from ordinary majorization theory to a thermodynamic setting with a nontrivial fixed point $\beta$4 (Ende, 2020).

In the quasi-classical or energy-diagonal regime, a Gibbs-preserving operation is represented by a Gibbs-stochastic matrix $\beta$5 acting on probability vectors, with nonnegative entries, column sums equal to one, and

$\beta$6

where $\beta$7 is the Gibbs distribution. In the unnormalized convention $\beta$8, the condition becomes $\beta$9 (Ende et al., 2022).

2. Relation to thermal operations and covariance

Thermal operations are defined microscopically by coupling the system to a thermal environment, applying an energy-conserving unitary, and tracing out an environment subsystem: $$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$0 They are automatically Gibbs-preserving, so

$$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$1

A more restrictive intermediate class is given by covariant Gibbs-preserving operations, also called enhanced thermal operations, which satisfy both Gibbs preservation and time-translation covariance (Faist et al., 2014, Shiraishi, 2024).

Class	Defining condition	Inclusion
GPO/GPM	$$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$2	Broadest class here
CGPO / EnTO	Gibbs-preserving and covariant	TO $$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$3 CGPO $$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$4 GPO
TO	Thermal ancilla + energy-conserving unitary + discard	Physically motivated subclass

The crucial dynamical distinction is covariance. Thermal operations satisfy

$$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$5

for all $$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$6. Consequently, for nondegenerate spectra, they cannot create coherence between distinct energy eigenspaces from an energy-diagonal input. An energy eigenstate is invariant under the system’s free evolution, and covariance forces the output to be time-invariant as well; for nondegenerate Hamiltonians, that means energy-diagonal. Gibbs-preserving operations impose no such covariance requirement: they need only fix the Gibbs state (Faist et al., 2014).

This separation is not merely formal. The 2014 note established that thermal operations are strictly contained in Gibbs-preserving maps in the quantum regime, while the 2024 coherence-cost analysis showed that some Gibbs-preserving operations remain unattainable even if thermal operations are assisted by any finite amount of quantum coherence (Faist et al., 2014, Tajima et al., 2024).

3. Classical regime: $$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$7-majorization and thermo-majorization

For matrices or states $$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$8, the preorder induced by Gibbs-preserving CPTP maps is

$$\gamma_\beta=\frac{e^{-\beta H}}{Z},\qquad Z=\operatorname{Tr}[e^{-\beta H}],$$9

In the diagonal case $\Phi$0, $\Phi$1, $\Phi$2, this reduces to vector $\Phi$3-majorization, written $\Phi$4 (Ende, 2020).

For commuting states, this is the standard thermo-majorization criterion. One characterization uses $\Phi$5-ordered Lorenz curves: $\Phi$6 where the Gibbs weights $\Phi$7 determine the ordering. In this abelian regime, Gibbs-preserving operations and thermal operations coincide at the level of state transitions, and thermo-majorization completely characterizes reachability (Faist et al., 2014, Ende, 2020).

The order-theoretic structure is richer than the diagonal criterion alone suggests. The relation $\Phi$8 is a preorder—reflexive and transitive—but not a partial order in general. For trace-constrained Hermitian matrices, $\Phi$9 is the unique minimal element, while a rank-one projector supported on the minimal coordinate of $\Phi(\gamma_\beta)=\gamma_\beta$0 is maximal in the positive cone under suitable trace normalization. The reachable-set operator

$\Phi(\gamma_\beta)=\gamma_\beta$1

has strong regularity properties: $\Phi(\gamma_\beta)=\gamma_\beta$2 is convex; $\Phi(\gamma_\beta)=\gamma_\beta$3 is compact for compact $\Phi(\gamma_\beta)=\gamma_\beta$4; on sets of states it is star-shaped with respect to the Gibbs state $\Phi(\gamma_\beta)=\gamma_\beta$5; and on compact subsets it is non-expansive in the Hausdorff metric (Ende, 2020).

In the quasi-classical theory, these reachable sets become thermomajorization polytopes. For a Gibbs vector $\Phi(\gamma_\beta)=\gamma_\beta$6 and initial vector $\Phi(\gamma_\beta)=\gamma_\beta$7, the polytope

$\Phi(\gamma_\beta)=\gamma_\beta$8

admits both a halfspace description and an extreme-point description. The geometry depends sensitively on the Gibbs spectrum. “Stable” Gibbs states are precisely those for which global cyclic state transfers are impossible in the quasi-classical regime, while “well-structured” Gibbs states allow degeneracies of extreme points to witness equilibrium subspaces (Ende et al., 2022).

4. Quantum regime: strict separation and explicit coherence-generating maps

The decisive quantum feature is energy coherence, meaning off-diagonal matrix elements between distinct energy eigenspaces. In this regime,

$\Phi(\gamma_\beta)=\gamma_\beta$9

The strict inclusion is witnessed by explicit Gibbs-preserving channels that transform an energy eigenstate into an arbitrary target state, including coherent superpositions, while preserving the Gibbs state (Faist et al., 2014).

For a qubit with Gibbs state

$D$0

fix any target state $D$1, possibly coherent in the energy basis. Define

$D$2

with

$D$3

This channel is a measurement in the energy basis followed by conditional state preparation. It is CPTP, satisfies $D$4, and obeys $D$5. If $D$6 contains coherence across distinct energies, the transition is impossible under thermal operations by covariance, but possible under Gibbs preservation alone (Faist et al., 2014).

The construction generalizes to $D$7-level systems by taking a highest-energy level $D$8 and setting

$D$9

Again, $\Phi$0 is CPTP, maps $\Phi$1 to $\Phi$2, and preserves $\Phi$3 (Faist et al., 2014).

A common misconception is that classical equivalence implies full quantum equivalence. The literature distinguishes these statements sharply. The classical result concerns state transitions between block-diagonal states; it does not imply that the channel sets themselves coincide. Even when diagonal-state convertibility agrees, the quantum channel sets differ because thermal operations retain symmetry restrictions that Gibbs-preserving maps do not (Faist et al., 2014).

5. Monotones and free-energy formulations

The basic monotone under any Gibbs-preserving operation is the relative entropy to the Gibbs state,

$\Phi$4

If $\Phi$5, then the data-processing inequality gives

$\Phi$6

Equivalently, the nonequilibrium free energy

$\Phi$7

satisfies

$\Phi$8

so $\Phi$9 cannot increase under Gibbs-preserving maps. The same logic applies to Rényi relative entropies and the associated generalized free energies (Faist et al., 2014).

These monotones constrain both Gibbs-preserving operations and thermal operations, but they do not capture the full distinction between them. The missing ingredient is symmetry: thermal operations satisfy time-translation covariance, whereas Gibbs preservation alone does not encode the asymmetry constraints associated with coherence (Faist et al., 2014).

Recent work on covariant Gibbs-preserving operations sharpened the free-energy picture in a different direction. For finite-dimensional systems with commensurate energy gaps, if the initial state is coherent, distillable, and has shortest period $\Phi(D)=D$0, then correlated-catalytic state convertibility under covariant Gibbs-preserving operations is fully characterized by the single monotone

$\Phi(D)=D$1

Under these assumptions, $\Phi(D)=D$2 is possible arbitrarily well if and only if

$\Phi(D)=D$3

and if the inequality is strict and $\Phi(D)=D$4 is full-rank, there exists an exact correlated-catalytic implementation. In this correlated-catalytic regime, imposing covariance does not change convertibility for coherent initial states (Shiraishi, 2024).

This result does not erase the earlier separation between thermal operations and Gibbs-preserving operations. Rather, it identifies a catalytic regime in which the additional covariance constraint ceases to affect state convertibility, provided the initial coherence already supplies the relevant timing resource (Shiraishi, 2024).

6. Coherence cost and operational significance

The operational status of Gibbs-preserving operations is complicated by implementation cost. Coherence in this setting means time-translation asymmetry, and the 2024 analysis quantifies it by the quantum Fisher information

$\Phi(D)=D$5

for $\Phi(D)=D$6. The coherence cost of implementing a channel via thermal operations assisted by an ancillary state is defined as the minimum ancilla QFI required for exact or approximate realization (Tajima et al., 2024).

For pairwise reversible Gibbs-preserving operations, the key quantity is

$\Phi(D)=D$7

where $\Phi(D)=D$8 is a reversible pair. The main lower bound is

$\Phi(D)=D$9

so if $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$0, the required coherence diverges as $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$1. Exact implementation then requires an infinite amount of coherence (Tajima et al., 2024).

A concrete qubit example is the coherence-detecting channel

$$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$2

with $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$3. For a qubit input Hamiltonian $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$4 and trivial output Hamiltonian, this channel is Gibbs-preserving, the reversible pair is $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$5, and

$$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$6

Accordingly, no finite amount of coherence suffices for exact implementation by thermal operations. The same paper shows that there are uncountably many Gibbs-preserving operations with unbounded coherence cost, obtained from continuous families of coherent-basis measure-and-prepare channels (Tajima et al., 2024).

The contrast with earlier examples is important. Coherence-creating Gibbs-preserving measure-and-prepare maps, such as the qubit channel that sends an excited energy eigenstate to a coherent target state, can have finite coherence cost, whereas coherence-detecting Gibbs-preserving maps can require infinite coherence. This suggests that Gibbs preservation alone is too weak to guarantee operational affordability (Faist et al., 2014, Tajima et al., 2024).

7. Broader hierarchies, limitations, and open problems

A broader 2025 hierarchy of thermodynamically consistent quantum operations places Gibbs-preservation-like fixed-point conditions in a larger framework. In that hierarchy, Class III channels—intended to encode consistency with both the second and third laws—must be rank non-decreasing and must not perturb a strictly positive state. More precisely, any Class III system channel has at least one faithful fixed point $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$7. This generalizes the fixed-point aspect of Gibbs-preserving operations, but does not identify the fixed point with a preassigned Gibbs state unless additional structure, such as energy conservation and a thermal ancilla, is imposed. Thermal operations therefore lie inside Class III, but Class III is in general broader than Gibbs-preserving operations for a fixed $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$8 (Shahbeigi et al., 29 May 2025).

Several technical limitations recur across the literature. Most results are formulated for finite-dimensional systems and finite inverse temperature. The sharp classical equivalence between thermal operations and Gibbs-preserving operations concerns only block-diagonal state transitions. The qubit case admits special simplifications: $$D:=\frac{e^{-\beta H}}{\operatorname{Tr}[e^{-\beta H}]}.$$9-majorization is completely characterized by the family of trace-norm inequalities

$\Lambda:S\to S'$0

but this characterization fails in dimensions $\Lambda:S\to S'$1, where the transpose-based counterexample shows that simple trace-norm families are not complete (Ende, 2020).

Open problems identified in the cited works include a full characterization of $\Lambda:S\to S'$2 beyond qubits, efficient algorithmic tests for noncommuting states, a complete family of monotones for general Gibbs-preserving operations in higher dimensions, tighter relations between thermal operations, enhanced thermal operations, and Gibbs-preserving operations in catalytic regimes, and the extension of coherence-cost bounds to infinite-dimensional systems with unbounded Hamiltonians (Ende, 2020, Tajima et al., 2024, Shiraishi, 2024).

Taken together, these developments delineate a precise picture. Gibbs-preserving operations provide a mathematically natural and thermodynamically meaningful fixed-point condition. In the quasi-classical regime they reproduce thermo-majorization and the reachability theory of Gibbs-stochastic maps. In the quantum regime they strictly exceed thermal operations because Gibbs preservation does not enforce time-translation covariance. Recent work further shows that this broader power can conceal unbounded coherence requirements, so the choice between Gibbs-preserving operations, covariant Gibbs-preserving operations, and thermal operations is not merely formal; it determines which thermodynamic resources are treated as genuinely free (Faist et al., 2014, Tajima et al., 2024, Shiraishi, 2024).