Thermal Operations Preview

Thermal operations are a resource-theoretic way to ask which state transformations are possible using a heat bath, energy-conserving dynamics, and no hidden nonequilibrium resource. They are not a replacement for master equations or fluctuation theorems. They are a complementary language for finite systems, one-shot transformations, catalysts, coherence, and work storage.

This page is a preview. It introduces the core definitions and warnings used by open-system thermodynamics. Resource Theories owns the cross-theory operational grammar, while this page retains thermal operations, Gibbs preservation, thermo-majorization, coherence constraints, and work-storage boundaries.

Basic Idea

Fix a system Hamiltonian $H_S$ and a bath temperature $T$, with

$$\beta = \frac{1}{k_B T}.$$

The system Gibbs state is

$$\tau_S = \frac{e^{-\beta H_S}} {Z_S}, \qquad Z_S=\operatorname{Tr}(e^{-\beta H_S}).$$

In resource-theoretic thermodynamics, $\tau_S$ is a free state: it is what can be obtained from a heat bath at the same temperature without spending work or another nonequilibrium resource.

States that differ from $\tau_S$ contain thermodynamic resources. These resources may include:

• population imbalance relative to the Gibbs distribution;
• coherence between different energy eigenspaces;
• correlations with another system;
• access to a work storage device;
• catalysts that enable transformations while being returned unchanged.

Thermal Operation

A thermal operation on system $S$ has the form

$$\Phi(\rho_S) = \operatorname{Tr}_B \left[ U \left( \rho_S\otimes\tau_B \right) U^\dagger \right],$$

where

$$\tau_B = \frac{e^{-\beta H_B}} {Z_B}$$

is a bath Gibbs state and the joint unitary obeys energy conservation:

$$\left[ U, H_S+H_B \right] = 0.$$

The allowed ingredients are therefore:

• add a bath system in a thermal state at temperature $T$;
• apply a unitary that conserves total energy;
• discard bath degrees of freedom.

No time-dependent external drive is hidden inside $U$. If a drive changes the Hamiltonian or supplies work, that drive is an additional resource and must be modeled explicitly.

Gibbs Preservation

Every thermal operation preserves the Gibbs state:

$$\Phi(\tau_S) = \tau_S.$$

The reason is simple. The product state $\tau_S\otimes\tau_B$ is the Gibbs state of $H_S+H_B$:

$$\tau_S\otimes\tau_B = \frac{e^{-\beta(H_S+H_B)}} {Z_SZ_B}.$$

Because $U$ commutes with $H_S+H_B$, it leaves this global Gibbs state invariant. Tracing out the bath gives back $\tau_S$.

The converse is not generally true. A map that preserves the Gibbs state is called Gibbs-preserving, but the set of Gibbs-preserving maps is larger than the set of thermal operations. This distinction matters in finite-dimensional and coherence-sensitive settings.

Free-Energy Monotone

For a fixed Hamiltonian and temperature, the nonequilibrium free energy can be written as

$$F(\rho) = \operatorname{Tr}(\rho H_S) - T S(\rho).$$

Its excess over the thermal free energy is

$$F(\rho)-F(\tau_S) = k_B T\, D(\rho\Vert\tau_S),$$

where

$$D(\rho\Vert\sigma) = \operatorname{Tr} \left[ \rho(\ln\rho-\ln\sigma) \right]$$

is quantum relative entropy. Since thermal operations preserve $\tau_S$ and relative entropy is contractive under quantum channels,

$$F(\Phi(\rho)) \le F(\rho).$$

Thus thermal operations cannot increase nonequilibrium free energy. In the many-copy macroscopic limit, this recovers the familiar role of free energy. In one-shot and catalytic settings, additional monotones can appear.

Thermo-Majorization

For states diagonal in the energy basis, thermal operations are closely related to thermo-majorization. Roughly, a state can be converted to another diagonal state when its population vector is sufficiently more ordered relative to the Gibbs weights.

The important lesson is qualitative:

• ordinary majorization applies when the Hamiltonian is effectively degenerate;
• thermo-majorization weights populations by $e^{-\beta E_n}$;
• transitions are constrained by both probability and energy;
• average free energy is not the only constraint for single-copy transformations.

This preview does not develop thermo-majorization diagrams. The point for open-system readers is that finite thermodynamic transformations can have stricter rules than the average second law suggests.

Coherence Constraints

Coherence between different energy eigenspaces is a distinct resource. Energy-conserving thermal operations cannot freely create arbitrary superpositions of different energies. If

$$\rho$$

has off-diagonal elements between different energy eigenspaces, then state transformations are constrained not only by populations but also by time-translation symmetry.

This is why “free energy decreases” is not a complete transformation criterion in coherent quantum thermodynamics. Coherence may require:

• a phase reference;
• a coherent work storage system;
• explicit time-dependent control;
• catalytic coherence;
• accounting for asymmetry under time translations.

In open-system language, coherence can decay under thermalization, but preserving, consuming, or converting it requires more structure than a population-only balance.

Relation to Master Equations

A thermal master equation describes continuous-time reduced dynamics under approximations such as weak coupling, Markovianity, and often secularization. A thermal operation describes an allowed state transformation under a resource-theory rule.

The two languages overlap but are not identical:

Question	Thermal Master Equation	Thermal Operation
time-resolved dynamics	central	usually absent
bath model	correlation functions and rates	Gibbs ancilla at temperature $T$
energy conservation	appears through rates and detailed balance	imposed on the joint unitary
allowed transformations	generated by the equation	constrained by resource monotones
coherence	damped or rotated dynamically	constrained by symmetry and resources

For continuous-time thermal dynamics, see Thermal Master Equations. For entropy balances, see Entropy Production.

Work Storage and Batteries

To discuss work extraction, one often adds an explicit work storage system. A schematic energy-conserving unitary then acts on

$$S+B+W,$$

where $W$ is a battery or weight. The battery should not secretly provide entropy, coherence, or correlations beyond the intended work resource.

This is harder than it sounds. A good battery model should specify:

• its Hamiltonian;
• which states count as charged or discharged;
• whether it supplies coherence or only energy;
• whether its entropy changes;
• whether it is returned close to its original form.

The planned ergotropy page will focus on extractable work from a given state. This preview only flags why a work storage model is needed in resource-theoretic statements.

Simple Consequences

Thermal operations imply several useful checks:

1. A system already in $\tau_S$ cannot be turned into a nonequilibrium state for free.

2. Population inversion is not produced from a single thermal bath by energy-conserving unitaries alone.

3. Nonequilibrium free energy cannot increase under a thermal operation.

4. Coherence across energy gaps is not freely available from an ordinary thermal bath.

5. A process that uses a time-dependent Hamiltonian is not a bare thermal operation unless the work source is included as a resource.

These are sanity checks, not a full transformation theorem.

Common Mistakes

1. Treating “thermal operation” as a synonym for “any operation involving a thermal bath.”

2. Assuming every Gibbs-preserving map is physically implementable as a thermal operation.

3. Ignoring coherence constraints and applying population-only criteria to coherent states.

4. Hiding a work source inside a time-dependent Hamiltonian while claiming the operation is free.

5. Using average free energy as the only finite-size transformation criterion.

6. Confusing resource-theory reachability with the actual time required to implement a process.

Exercises

Gibbs State Is Fixed

Show that a thermal operation preserves $\tau_S$.

Solution

Start from

$$\Phi(\tau_S) = \operatorname{Tr}_B \left[ U(\tau_S\otimes\tau_B)U^\dagger \right].$$

The product state is

$$\tau_S\otimes\tau_B = \frac{e^{-\beta(H_S+H_B)}}{Z_SZ_B}.$$

If $[U,H_S+H_B]=0$, then $U$ commutes with every function of $H_S+H_B$, so

$$U(\tau_S\otimes\tau_B)U^\dagger = \tau_S\otimes\tau_B.$$

Taking the bath trace gives

$$\Phi(\tau_S) = \operatorname{Tr}_B(\tau_S\otimes\tau_B) = \tau_S.$$

Free Energy from Relative Entropy

Show that

$$F(\rho)-F(\tau_S) = k_BT\,D(\rho\Vert\tau_S).$$

Solution

Since

$$\tau_S=\frac{e^{-\beta H_S}}{Z_S},$$

we have

$$\ln\tau_S = - \beta H_S-\ln Z_S.$$

Then

$$\begin{aligned} k_BT\,D(\rho\Vert\tau_S) &= k_BT\operatorname{Tr} \left[ \rho(\ln\rho-\ln\tau_S) \right] \\ &= k_BT\operatorname{Tr}(\rho\ln\rho) + \operatorname{Tr}(\rho H_S) + k_BT\ln Z_S. \end{aligned}$$

Using $S(\rho)=-k_B\operatorname{Tr}(\rho\ln\rho)$,

$$k_BT\,D(\rho\Vert\tau_S) = \operatorname{Tr}(\rho H_S)-T S(\rho) + k_BT\ln Z_S.$$

The thermal free energy is

$$F(\tau_S) = - k_BT\ln Z_S.$$

Therefore

$$k_BT\,D(\rho\Vert\tau_S) = F(\rho)-F(\tau_S).$$

Why Time-Dependent Driving Is Not Free

Explain why a protocol using an externally controlled $H_S(t)$ is not automatically a thermal operation.

Solution

A thermal operation is built from a fixed system Hamiltonian, a thermal bath state, and a joint unitary satisfying

$$[U,H_S+H_B]=0.$$

If an external agent changes $H_S(t)$, then energy is being exchanged with a work source. That work source is not part of the free thermal bath. To treat the process resource-theoretically, one must include the drive or battery explicitly and track its energy, entropy, and coherence. Otherwise the protocol hides a resource inside the control parameter.

References

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• F. G. S. L. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, “The second laws of quantum thermodynamics,” Proceedings of the National Academy of Sciences 112, 3275, 2015.
• M. Horodecki and J. Oppenheim, “Fundamental limitations for quantum and nanoscale thermodynamics,” Nature Communications 4, 2059, 2013.
• F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, “Resource theory of quantum states out of thermal equilibrium,” Physical Review Letters 111, 250404, 2013.
• M. Lostaglio, D. Jennings, and T. Rudolph, “Description of quantum coherence in thermodynamic processes requires constraints beyond free energy,” Nature Communications 6, 6383, 2015.
• J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, “The role of quantum information in thermodynamics: a topical review,” Journal of Physics A 49, 143001, 2016.