Resource Theory of Engines

• Resource Theory of Engines is a framework that defines engine operation through constrained state transformations using specified free states and free operations.
• It extends thermal operations to quantum multi-bath and non-equilibrium settings, enabling analysis of cyclic transformations under finite-size, catalytic, or memory constraints.
• The approach quantifies performance trade-offs, using monotones and generalized free energies to balance work yield, efficiency, and robustness in nanoscale systems.

Resource Theory of Engines denotes a class of resource-theoretic frameworks in which engine operation is formulated through constrained state transformations, specified free states and free operations, and monotones that quantify work, heat, advantage, or other operational gains. In the quantum setting, these frameworks extend single-bath resource theories of athermality to two-bath and multi-constraint settings; in broader thermodynamic settings, they also encompass finite-state information ratchets that exploit temporal correlations and microscopic devices driven by structured nonequilibrium noise. Across these variants, the central question is not only which state transitions are possible, but which cyclic transformations can be implemented, at what work yield, with what efficiency, and under which finite-size, catalytic, or memory constraints (Brandão et al., 2011, Wojewódka-Ściążko et al., 2023, Bera et al., 2019, Boyd et al., 2016, Czartowski et al., 8 Jul 2025).

1. Conceptual foundations and scope

A standard point of departure is the resource theory of states out of thermal equilibrium. For a system with Hamiltonian $H$ at inverse temperature $\beta$, the Gibbs state $\gamma=e^{-\beta H}/Z$ is free, thermal operations are free, and any state $\rho\neq\gamma$ is a resource. In that setting, relative entropy to the Gibbs state yields the familiar nonequilibrium free energy, and asymptotic work extraction is quantified by $\Delta F(\rho)=k_B T D(\rho\|\gamma)$ (Brandão et al., 2011). The chapter on thermal operations and second laws refines this picture by emphasizing that thermodynamic state conversion is governed not only by a single free-energy inequality, but by a family of generalized free energies $F_\alpha$, which collapse to the ordinary free energy in the i.i.d. limit (Ng et al., 2018).

Resource engines generalize this single-constraint picture by fusing two resource theories. In the formulation of resource engines, a finite-dimensional system is passed back and forth between two agents, Alice and Bob, each with their own free states and free operations. The implementable operations are the alternating compositions generated by $\mathcal O_A\cup\mathcal O_B$, and the free states of the fused theory are the convex hull of states reachable from $F_A\cup F_B$ under such engine cycles. Thermodynamic two-bath engines arise as the special case in which Alice and Bob possess thermal operations at two different temperatures (Wojewódka-Ściążko et al., 2023).

Czartowski and Bistroń formalize this synthesis as an engine theory $\mathfrak E(\mathcal R_1,\mathcal R_2)$, where $$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$, the free-state set $\beta$0 is convex and contains $\beta$1, and the free-operation set $\beta$2 is convex, composition-closed, and contains $\beta$3. One admissible choice is

$\beta$4

with free states obtained as mixtures of images of $\beta$5 under $\beta$6 (Czartowski et al., 8 Jul 2025).

The scope of the framework is not restricted to thermal resource theories. In the coherence-engine example, Alice’s free operations are the diagonal unitaries $\beta$7 in one basis and Bob’s are $\beta$8 in another basis. Under the support-matrix conditions labeled $\beta$9 and $\gamma=e^{-\beta H}/Z$0, the generated subgroup is all of $\gamma=e^{-\beta H}/Z$1, so the engine becomes universal for unitary synthesis. The same analysis yields explicit stroke bounds, including the lower bound

$\gamma=e^{-\beta H}/Z$2

and, for even $\gamma=e^{-\beta H}/Z$3, the upper bound

$\gamma=e^{-\beta H}/Z$4

This establishes that the engine picture is a general mechanism for fusing constraints, rather than a thermal construction alone (Wojewódka-Ściążko et al., 2023).

2. Free states and free operations

For thermal operations at fixed inverse temperature $\gamma=e^{-\beta H}/Z$5, the free states are Gibbs states

$\gamma=e^{-\beta H}/Z$6

and the free operations are channels of the form

$\gamma=e^{-\beta H}/Z$7

where the ancillary system $\gamma=e^{-\beta H}/Z$8 is thermal and the unitary conserves total energy. On energy-incoherent states, this reduces to a Gibbs-preserving stochastic map. This TO structure is the basic free-operation set for the single-bath resource theory of athermality (Ng et al., 2018).

Two-bath engine theories require a broader free-operation set. In the semilocal thermal operations framework, a bipartite system $\gamma=e^{-\beta H}/Z$9 interacts with local baths at $\rho\neq\gamma$0 and $\rho\neq\gamma$1, and the global unitary must satisfy both ordinary energy conservation and conservation of the weighted energy $\rho\neq\gamma$2. On diagonal states, the criterion becomes thermo-majorization relative to $\rho\neq\gamma$3. Local thermal operations with classical communication enlarge this further by allowing rounds of local thermal operations, measurements, classical messages, and conditional local actions; in the diagonal setting, a two-round parallel stroke is represented by a four-index Gibbs-preserving tensor decomposition (Bera et al., 2019, Czartowski et al., 8 Jul 2025).

The corresponding free states depend on the engine model. In the one-shot two-bath theory based on semilocal thermal operations, the free states are semi-Gibbs states $\rho\neq\gamma$4 at the two bath temperatures (Bera et al., 2019). In the abstract engine framework, a natural free-state set for a cold–hot engine is

$\rho\neq\gamma$5

and in the two-qubit case the exact characterization of $\rho\neq\gamma$6 is sufficiently difficult that analytic inner approximations are introduced in the form of tree-states, obtained by repeated extremal two-level thermal protocols along a spanning tree of the joint energy graph (Czartowski et al., 8 Jul 2025).

A distinct line of work removes background temperature altogether. In that resource theory, no states are free. The only free operations are energy-preserving unitaries on many copies, optionally assisted by an ancilla of sublinear size and sublinear Hamiltonian spectrum. This places work and heat on equal footing and shifts the theory from a Gibbs-preserving paradigm to a purely kinematic classification by entropy and average energy (Sparaciari et al., 2016).

3. Monotones, second laws, and state-conversion criteria

In the single-bath theory of athermality, relative entropy to the Gibbs state is monotonic under thermal operations: $\rho\neq\gamma$7 Using

$\rho\neq\gamma$8

the monotone becomes the nonequilibrium free energy

$\rho\neq\gamma$9

Asymptotically, the maximal average work extractable per copy tends to $\Delta F(\rho)=k_B T D(\rho\|\gamma)$0, and the reversible interconversion rate between resource states is

$\Delta F(\rho)=k_B T D(\rho\|\gamma)$1

The construction requires a sublinear amount of coherent superposition over energy levels for reversible formation (Brandão et al., 2011).

Thermal operations also admit a full family of second-law constraints. For block-diagonal states, the generalized free energies are

$\Delta F(\rho)=k_B T D(\rho\|\gamma)$2

A transition $\Delta F(\rho)=k_B T D(\rho\|\gamma)$3 is possible by thermal operations, even with an uncorrelated catalyst, iff

$\Delta F(\rho)=k_B T D(\rho\|\gamma)$4

In the many-copy limit, after smoothing and taking $\Delta F(\rho)=k_B T D(\rho\|\gamma)$5, all $\Delta F(\rho)=k_B T D(\rho\|\gamma)$6 converge per copy to the ordinary free energy $\Delta F(\rho)=k_B T D(\rho\|\gamma)$7, recovering the standard thermodynamic second law as the sole nontrivial constraint in the i.i.d. regime (Ng et al., 2018).

For engines coupled to two baths, the monotone structure becomes explicitly multi-temperature. Under catalytic semilocal thermal operations, the allowed transformations are characterized by the non-increase of the $\Delta F(\rho)=k_B T D(\rho\|\gamma)$8-free entropies

$\Delta F(\rho)=k_B T D(\rho\|\gamma)$9

for all $F_\alpha$0. These are equivalent to one-shot free energies $F_\alpha$1, and the stated theorem is necessary and sufficient for arbitrary state and Hamiltonian changes in the block-diagonal regime (Bera et al., 2019).

Without a background temperature, asymptotic equivalence is instead governed by the pair $F_\alpha$2. Two many-copy states are asymptotically interconvertible by energy-preserving unitaries iff they have the same average energy and von Neumann entropy. The set

$F_\alpha$3

is a closed convex region in the energy–entropy plane, bounded below by $F_\alpha$4 and above by the thermal curve $F_\alpha$5. Equivalently,

$F_\alpha$6

with $F_\alpha$7 interpreted as $F_\alpha$8-athermality. If discarding of unentangled subsystems is allowed, the preorder is determined by entropy together with all $F_\alpha$9 monotones (Sparaciari et al., 2016).

In the alternating two-stroke thermal-engine model, monotones can also be genuinely engine-specific. For an incoherent initial state $\mathcal O_A\cup\mathcal O_B$0, every achievable $\mathcal O_A\cup\mathcal O_B$1 satisfies

$\mathcal O_A\cup\mathcal O_B$2

where $\mathcal O_A\cup\mathcal O_B$3 is the hot-bath Gibbs state. This gives an explicit upper-bound monotone for the fused theory. A matching lower-bound construction is obtained from the points $\mathcal O_A\cup\mathcal O_B$4, and in the infinite-temperature limit $\mathcal O_A\cup\mathcal O_B$5 the achievable set becomes the full probability simplex (Wojewódka-Ściążko et al., 2023).

4. Work, heat, advantage, and efficiency

In the microscopic heat-engine model built from thermal operations, a cycle acts on a hot bath $\mathcal O_A\cup\mathcal O_B$6, a cold bath $\mathcal O_A\cup\mathcal O_B$7, a catalytic machine $\mathcal O_A\cup\mathcal O_B$8, and a battery with levels $\mathcal O_A\cup\mathcal O_B$9 and $F_A\cup F_B$0. Work extraction corresponds to lifting the battery from $F_A\cup F_B$1 to $F_A\cup F_B$2, with efficiency

$F_A\cup F_B$3

Because the full transformation is a thermal operation, the generalized free-energy constraints must hold jointly for the cold bath and battery. The resulting nanoscale efficiency bound is

$F_A\cup F_B$4

For a finite cold bath consisting of $F_A\cup F_B$5 qubits, the quasi-static near-perfect-work efficiency is

$F_A\cup F_B$6

showing that Carnot efficiency can be attained for $F_A\cup F_B$7 and is strictly unattainable for $F_A\cup F_B$8 (Ng et al., 2018).

The semilocal two-bath theory gives a one-shot formulation of engine work. A one-step cycle may swap the two subsystems and, optionally, their Hamiltonians: $F_A\cup F_B$9 The deterministic work in one cycle is governed by the free-entropy distance

$\mathfrak E(\mathcal R_1,\mathcal R_2)$0

For equality-saturating reversible choices of input and output energy eigenstates,

$\mathfrak E(\mathcal R_1,\mathcal R_2)$1

and hence

$\mathfrak E(\mathcal R_1,\mathcal R_2)$2

The same framework also allows work extraction exclusively from inter-system correlations (Bera et al., 2019).

In the no-background-temperature theory, work and heat are both read off from additive monotones on the energy–entropy diagram. If $\mathfrak E(\mathcal R_1,\mathcal R_2)$3 is the effective inverse temperature associated with two finite reservoirs, then

$\mathfrak E(\mathcal R_1,\mathcal R_2)$4

and

$\mathfrak E(\mathcal R_1,\mathcal R_2)$5

For finite reservoir changes, the engine efficiency is

$\mathfrak E(\mathcal R_1,\mathcal R_2)$6

which is strictly below Carnot, while the large-reservoir limit recovers the Carnot value (Sparaciari et al., 2016).

The abstract engine-theory formalism also introduces task-based performance measures. Given a benefit function $\mathfrak E(\mathcal R_1,\mathcal R_2)$7, the relative advantage and free-state-referenced advantage are

$\mathfrak E(\mathcal R_1,\mathcal R_2)$8

From these, engine efficiency is defined by

$\mathfrak E(\mathcal R_1,\mathcal R_2)$9

For semilocal thermal-operation engines, a faithful monotone is

$$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$0

with the lemma $$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$1 under any SLTO-engine stroke (Czartowski et al., 8 Jul 2025).

5. Correlations, memory, and self-correcting information engines

The resource-theoretic engine perspective is not limited to quantum heat engines. Boyd, Mandal, and Crutchfield analyze a finite-state information engine, or ratchet, that interacts sequentially with an input tape. The ratchet has internal state $$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$2, input symbols $$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$3, output symbols $$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$4, and detailed-balanced transition kernel

$$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$5

In the stationary long-time limit, the average work per symbol is

$$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$6

where

$$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$7

A key structural result is that if the ratchet has only one internal state, then $$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$8 depends only on the single-symbol distribution of the input; temporal correlations beyond alphabet bias are invisible. Hence any engine that leverages temporal correlations must have internal memory, at least two states (Boyd et al., 2016).

The explicit example is a three-state ratchet $$\mathcal R_i=(\mathcal F_i,\mathcal O_i,\mathcal M_i)$$9 driven by an unbiased, perfectly correlated period-2 input. Because there is no single-bit bias, a memoryless engine cannot extract work and $\beta$00. The three-state design splits into two dynamical modes: a counterclockwise dissipative mode on $\beta$01 with net work $\beta$02 per bit, and a clockwise generative mode on $\beta$03 with net work $\beta$04 per bit. By allowing transitions through the synchronizing state $\beta$05, the clockwise mode becomes the unique attractor. In that asymptotic regime,

$\beta$06

so the engine is powered solely by an increase in entropy rate rather than by single-symbol bias (Boyd et al., 2016).

The same construction yields an autonomous self-correction mechanism. If the period-2 input is corrupted by phase slips with probability $\beta$07, then for optimal $\beta$08,

$\beta$09

The zero-work curve is

$\beta$10

and $\beta$11 is the critical corruption level above which no choice of $\beta$12 yields positive work. The thermodynamic-function phase diagram contains engine, eraser, and dud regions. The physical interpretation is explicitly nonergodic: the dissipative and generative phases are disconnected attractors, while the transient bridge $\beta$13 breaks the nonergodicity just enough to synchronize the ratchet to the correct phase. This is presented as a nonergodicity-based thermodynamic mechanism for autonomous error correction (Boyd et al., 2016).

6. Catalysis, trade-offs, and constructive protocols

Constructive control of engine performance often proceeds through catalytic or combinatorial protocols. In the thermal engine framework of Czartowski and Bistroń, tree-states provide analytic inner approximations to the set of free states. A spanning tree on the $\beta$14 joint energy levels selects $\beta$15 two-level couplings, and repeated application of extremal two-level thermal protocols fixes edge-population ratios to the corresponding cold–hot endpoints. In the two-qubit case, this yields closed-form lower bounds on attainable ground- and excited-state populations. The same framework also proves a catalytic advantage: with a two-qubit catalyst prepared in its local ground state, any diagonal joint state can be reached in the infinite-stroke limit, so all states are free in the SLTO-catalytic engine (Czartowski et al., 8 Jul 2025).

A different resource-theoretic construction appears in the analysis of reversible computers sharing resources. There the relevant currency is the computational bias

$\beta$16

where $\beta$17 and $\beta$18 are bias species that earn or pay one unit of free energy. For a reaction of affinity $\beta$19, attaching $\beta$20 bias-mediating transitions requires

$\beta$21

to overcome the unfavorable direction. EP-III then constructs a sequestration-klona bank $\beta$22 that dynamically allocates the required bias to an arbitrary reaction of unknown cost $\beta$23, automatically adapting when the favorable direction changes. The overhead is quantitatively controlled: the sequestration-bank concentration is suppressed by $\beta$24 when $\beta$25, giving overhead $\beta$26, and by $\beta$27 far from equilibrium, giving overhead $\beta$28. Since a raw mona transition already runs at $\beta$29, the net time-penalty becomes $\beta$30 in the best case and $\beta$31 in the worst case. The resulting three-way frontier is among raw reversible-computation rate, free-energy cost per operation, and interaction overhead (Earley, 2020).

These constructions make explicit a recurrent theme of engine resource theories: catalytic flexibility is operationally powerful, but it typically introduces concentration, rate, or control overheads. This suggests that “free” cyclic composability and “cheap” asymptotic reversibility need not coincide once metrological precision, resource sharing, or unknown reaction costs are included.

7. Generalized thermodynamic resources

Recent work broadens the notion of an engine resource beyond temperature difference or athermality. In the nonlinear Landauer–Büttiker framework with Haldane–Wu exclusion statistics, the working medium is characterized by an exclusion parameter $\beta$32. For optimal step-function transmission, the maximum power has the universal form

$\beta$33

with

$\beta$34

Accordingly,

$\beta$35

and the bosonic bound exceeds the fermionic one by a factor of approximately $\beta$36. The derivative $\beta$37 shows strict monotonic decrease of the power bound with increasing exclusion parameter. In the resource-theoretic interpretation, free operations are those that engineer the transmission function without changing $\beta$38, while monotones include $\beta$39, the optimal bias cost $\beta$40, and efficiency-at-fixed-power curves. A concrete realization is magnon transport through a ferromagnetic spin chain, where realistic parameters achieve more than $\beta$41 of the ideal bosonic power bound and surpass the fermionic bound by about $\beta$42 (Karmakar et al., 17 Jun 2026).

Gaussian white noise has likewise been identified as a thermodynamic resource. In an overdamped ratchet with state-dependent friction $\beta$43, periodic asymmetric potential $\beta$44, equilibrium bath at temperature $\beta$45, and additive Gaussian white noise of intensity $\beta$46, the effective temperatures become

$\beta$47

The extra noise drives the system to a nonequilibrium steady state with finite current when the potential is asymmetric. Under a small opposing load $\beta$48, the extracted power is $\beta$49, the noise-injection rate is $\beta$50, and the efficiency is defined by

$\beta$51

at maximum power. In the small-$\beta$52 regime, $\beta$53 while $\beta$54, so $\beta$55; in the slow-switching limit $\beta$56, while in the fast-switching limit $\beta$57. The resource-theoretic interpretation given in the paper is that Gaussian white noise beyond the fluctuation–dissipation value acts as an athermal work resource, equivalently a weakly coupled high-temperature reservoir with $\beta$58 (Dechant et al., 2016).

Taken together, these developments show that the resource theory of engines is not a single formalism but a layered family of frameworks. In one layer, it extends the resource theory of athermality from states to cyclic engine operation; in another, it fuses distinct resource theories into a common engine theory; and in yet another, it treats temporal correlation, catalytic structure, computational bias, exclusion statistics, or nonequilibrium noise as explicit engine resources. The unifying feature is the same: engine performance is characterized by constrained transformations, complete or partial monotones, and precise trade-offs among extractable work, efficiency, memory, control, and robustness (Czartowski et al., 8 Jul 2025, Wojewódka-Ściążko et al., 2023).