Quantum Thermodynamics

Quantum thermodynamics asks how the laws of thermodynamics are expressed for small quantum systems: driven atoms, qubits, oscillators, mesoscopic conductors, nanomechanical modes, and engineered reservoirs. In this regime the state is a density operator, measurements can disturb the system, fluctuations are not negligible, and the distinction between heat and work depends on the physical protocol used to control and observe the system.

The subject is not a replacement for equilibrium statistical mechanics. Its central task is more operational: given a Hamiltonian, a state, a driving protocol, one or more reservoirs, and perhaps a measurement record, what can be said consistently about energy exchange, entropy production, irreversibility, and information?

The equilibrium operator, its trace-class assumptions, spectral form, and temperature limits are developed in Thermal Density Operators. This chapter uses that state to study operational and dynamical questions.

This chapter is the canonical orientation page for those questions inside open quantum systems. Detailed pages refine the definitions of heat and work, two-point measurement work distributions, fluctuation relations, entropy production, information engines, and strong-coupling caveats.

Scope and Conventions

The simplest thermodynamic setting is a weakly coupled system with Hamiltonian $H(t)$, state $\rho(t)$, and thermal reservoirs whose temperatures are externally specified. The internal energy is usually defined as

$$E(t) = \operatorname{Tr}\!\left[\rho(t)H(t)\right].$$

Differentiating gives the bookkeeping identity

$$\frac{dE}{dt} = \operatorname{Tr}\!\left[\dot\rho(t)H(t)\right] + \operatorname{Tr}\!\left[\rho(t)\dot H(t)\right].$$

In the standard weak-coupling convention,

$$\dot Q = \operatorname{Tr}\!\left[\dot\rho(t)H(t)\right], \qquad \dot W = \operatorname{Tr}\!\left[\rho(t)\dot H(t)\right],$$

where $\dot Q$ is heat current into the system and $\dot W$ is power supplied by the external drive. This split is useful, but it is not merely algebra: it assumes that state changes at fixed Hamiltonian are caused by reservoir exchange, while explicit Hamiltonian changes are controlled work.

The split becomes more subtle when the drive changes the system-bath coupling, when the interaction energy is not negligible, when the bath is finite, or when the reduced system is not described by a thermally consistent master equation. In those cases one must specify the enlarged energy budget rather than quoting $\dot Q$ and $\dot W$ as if they were protocol-independent observables.

Heat Currents from Master Equations

For a Markovian equation with several dissipative channels,

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \sum_\alpha \mathcal L_\alpha(\rho),$$

a common heat-current definition is

$$\dot Q_\alpha = \operatorname{Tr}\!\left[ H\,\mathcal L_\alpha(\rho) \right].$$

This expression is thermodynamic only when $\mathcal L_\alpha$ genuinely represents a reservoir at temperature $T_\alpha$ for the Hamiltonian $H$ being used in the energy balance. It is safest for weak-coupling, secular, global thermal master equations whose rates obey detailed balance. See Thermal Master Equations and Detailed Balance.

For a single undriven bath, thermal consistency requires the Gibbs state

$$\rho_\beta = \frac{e^{-\beta H}} {\operatorname{Tr}(e^{-\beta H})}, \qquad \beta=\frac{1}{k_B T},$$

to be stationary, possibly after including the same Lamb-shift and renormalized Hamiltonian used in the generator. A Lindblad equation with positive rates is not automatically a thermal master equation; the rates must encode the bath temperature through relations such as the Kubo–Martin–Schwinger condition.

For multiple baths, each current $\dot Q_\alpha$ should be associated with a separately modeled reservoir. Adding local dissipators to an interacting many-part system can be a useful phenomenological approximation, but it may violate thermodynamic consistency unless the local approximation is justified. The Approximation Checklist is the practical place to record those checks.

Entropy and Irreversibility

The quantum entropy of a state is the von Neumann entropy

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

For weakly coupled Markovian dynamics with heat currents into the system, the standard entropy-production rate is

$$\dot\Sigma = \frac{dS}{dt} - \sum_\alpha \frac{\dot Q_\alpha}{T_\alpha}.$$

The sign convention matters. If $\dot Q_\alpha$ is heat entering the system from bath $\alpha$, then the bath entropy change is $-\dot Q_\alpha/T_\alpha$. The total entropy-production rate is system entropy change plus bath entropy change.

In the setting of quantum dynamical semigroups with thermal stationary states, Spohn’s inequality gives a precise route to

$$\dot\Sigma\ge0.$$

One useful form is

$$-k_B \operatorname{Tr} \left[ \mathcal L(\rho) \left( \ln\rho-\ln\rho_\beta \right) \right] \ge0,$$

when $\mathcal L(\rho_\beta)=0$ and the generator satisfies the required positivity assumptions. This is a second-law statement for a particular model class, not a proof that every phenomenological equation has a sound thermodynamic interpretation.

Free energy provides another compact way to organize the equilibrium statement. At fixed $H$ and $T$,

$$F(\rho) = \operatorname{Tr}(\rho H)-T S(\rho),$$

and the Gibbs state minimizes $F$ because

$$F(\rho)-F(\rho_\beta) = k_B T\,D(\rho\Vert\rho_\beta) \ge0.$$

Here $D(\rho\Vert\sigma)=\operatorname{Tr}[\rho(\ln\rho-\ln\sigma)]$ is the quantum relative entropy, defined when the support of $\rho$ lies inside the support of $\sigma$.

Work Is Protocol-Dependent

Energy at a time is represented by the Hamiltonian. Work, however, is energy transferred by a process. For a closed driven system, the standard two-point measurement scheme samples an initial energy $E_n^0$, evolves under the drive, samples a final energy $E_m^\tau$, and assigns

$$W = E_m^\tau-E_n^0.$$

The corresponding distribution has the schematic form

$$P(W) = \sum_{n,m} p_n^0\,p(m,\tau\mid n,0)\, \delta\!\left( W-E_m^\tau+E_n^0 \right).$$

This construction is central to many quantum fluctuation relations, but it is not a Hermitian “work operator” measured at one time. The initial projective measurement can destroy coherence in the energy basis, and alternative schemes are used when one wants quasiprobabilities, weak measurements, continuous monitoring, or fully inclusive system-plus-bath energy budgets.

The caution is simple: ask what experimental record or operational protocol defines the random variable. Without that, statements about “the work distribution” are incomplete.

Fluctuation Relations

Small systems fluctuate strongly, so thermodynamics is often expressed as an identity for distributions rather than only as an inequality for averages. For a closed driven system initially prepared in equilibrium, the quantum two-point measurement scheme leads to the Jarzynski equality

$$\left\langle e^{-\beta W}\right\rangle = e^{-\beta\Delta F},$$

under its standard assumptions. A related Crooks relation compares forward and reverse work distributions:

$$\frac{P_F(W)} {P_R(-W)} = e^{\beta(W-\Delta F)}.$$

These formulas are powerful because Jensen’s inequality recovers the average second-law bound

$$\langle W\rangle \ge \Delta F.$$

Their quantum content lies in the preparation, measurement scheme, driving protocol, and time-reversal convention. Later pages in this chapter separate the closed-system two-point measurement setting from open-system fluctuation theorems, counting-statistics formulations, and measurement-feedback variants.

Information, Measurement, and Feedback

Quantum thermodynamics becomes especially delicate when a controller gains information during the process. A measurement record can reduce the controller’s uncertainty, but the measurement backaction changes the system state, and storing or erasing records has thermodynamic costs.

The operational question is not whether information is “really” physical. It is how the full accounting is done:

• What system was measured?
• What outcome record was kept?
• Was feedback conditioned on the outcome?
• Which degrees of freedom store the record?
• Where is the entropy exported when the record is reset?

Pages on Quantum Instruments, Measurement Backaction, Stochastic Master Equations, and Reservoir Engineering provide the open-system language needed for feedback and information engines.

Strong-Coupling and Memory Caveats

The weak-coupling split into system energy, bath heat, and drive work is not the final word. At strong coupling, the interaction energy can be thermodynamically important. If the system is continuously dressed by the bath, an effective equilibrium state may involve the Hamiltonian of mean force rather than the bare system Hamiltonian. If the environment has memory, heat currents defined from a time-local reduced generator may fail to capture energy temporarily stored in structured modes.

Two practical strategies are common:

• enlarge the system boundary so that strongly coupled modes become explicit system or reaction-coordinate degrees of freedom;
• compute an inclusive energy balance for system plus relevant environmental degrees of freedom.

The first strategy connects directly to Reaction-Coordinate Mapping and structured bath models such as the Spin-Boson Model. The second is often needed in numerical and nonequilibrium field-theory treatments.

Modeling Checklist

Before assigning thermodynamic meaning to a quantum model, check:

• the Hamiltonian whose expectation value defines internal energy;
• whether explicit time dependence is controlled driving, modulation of coupling, or both;
• the sign convention for heat currents;
• whether each dissipator corresponds to a reservoir with a temperature or chemical potential;
• whether the dissipators satisfy detailed balance with respect to the chosen Hamiltonian;
• whether local dissipators are compatible with interactions inside the system;
• whether coherences in the energy basis affect the chosen work protocol;
• whether measurement records and feedback controllers are included in the entropy budget;
• whether strong coupling or memory requires an enlarged system boundary.

These checks are not bureaucratic. They determine whether a calculation is a thermodynamic statement or only a dynamical simulation.

Common Mistakes

• Treating work as an ordinary one-time observable without specifying a protocol.
• Calling any Lindblad dissipator a heat bath.
• Forgetting the sign convention for heat currents.
• Mixing local and global master equations without checking detailed balance.
• Using a bare Hamiltonian for energy while using a dressed Hamiltonian for rates.
• Applying Jarzynski or Crooks relations outside their preparation and measurement assumptions.
• Ignoring measurement backaction in feedback thermodynamics.
• Treating strong-coupling corrections as optional when the interaction energy is comparable to system energy scales.

Exercises

First-Law Split

Let $E(t)=\operatorname{Tr}[\rho(t)H(t)]$. Derive the standard weak-coupling split into heat current and power, and state one physical assumption behind the split.

Solution

Differentiate:

$$\frac{dE}{dt} = \operatorname{Tr}[\dot\rho H] + \operatorname{Tr}[\rho\dot H].$$

The usual convention identifies

$$\dot Q=\operatorname{Tr}[\dot\rho H], \qquad \dot W=\operatorname{Tr}[\rho\dot H].$$

The physical assumption is that changes of $\rho$ at fixed $H$ are caused by reservoir exchange, while explicit changes of $H$ are caused by an external work source. If the drive also changes the system-bath coupling or if interaction energy is significant, this split must be refined.

Entropy-Production Sign

Suppose a system absorbs heat $\dot Q>0$ from a single bath at temperature $T$, while its von Neumann entropy changes at rate $dS/dt$. With the convention that $\dot Q$ is heat into the system, what is the entropy-production rate?

Solution

The bath loses heat, so its entropy change rate is

$$\dot S_{\mathrm{bath}} = - \frac{\dot Q}{T}.$$

The total entropy-production rate is

$$\dot\Sigma = \frac{dS}{dt} + \dot S_{\mathrm{bath}} = \frac{dS}{dt} - \frac{\dot Q}{T}.$$

For a thermodynamically consistent Markovian relaxation process, this quantity is nonnegative.

Gibbs State and Free Energy

Show that the Gibbs state minimizes

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

at fixed $H$ and $T$.

Solution

Let

$$\rho_\beta = \frac{e^{-\beta H}}{Z}, \qquad Z=\operatorname{Tr}(e^{-\beta H}).$$

Then

$$\ln\rho_\beta = -\beta H-\ln Z.$$

The relative entropy is

$$D(\rho\Vert\rho_\beta) = \operatorname{Tr}(\rho\ln\rho) - \operatorname{Tr}(\rho\ln\rho_\beta).$$

Substituting $\ln\rho_\beta$ gives

$$k_B T\,D(\rho\Vert\rho_\beta) = \operatorname{Tr}(\rho H) - T S(\rho) + k_B T\ln Z.$$

Since $F(\rho_\beta)=-k_B T\ln Z$,

$$F(\rho)-F(\rho_\beta) = k_B T\,D(\rho\Vert\rho_\beta) \ge0.$$

Work Is Not a State Function

Why does the equality $W=E_m^\tau-E_n^0$ in the two-point measurement scheme not imply that work is represented by a single Hermitian operator at one time?

Solution

The quantity $W$ is defined from two measurement outcomes and a process connecting them. It depends on the initial energy measurement, the driven evolution, and the final energy measurement. A one-time Hermitian operator would assign outcomes from the state at one instant, but work records energy transferred along a protocol. In quantum mechanics the initial measurement can also remove energy-basis coherence, so the work distribution is inseparable from the measurement scheme.

Cross-Links

• Thermal Density Operators
• Applications and Experimental Platforms
• Computational Notebooks
• Reference
• Thermal Master Equations
• Detailed Balance
• Fluctuation–Dissipation Relation
• Noise Spectra
• System-Bath Hamiltonians
• Markov Approximation
• Secular Approximation
• Reservoir Engineering
• Reaction-Coordinate Mapping
• Approximation Checklist

References

• R. Alicki, “The quantum open system as a model of the heat engine,” Journal of Physics A: Mathematical and General 12, L103-L107 (1979).
• H. Spohn, “Entropy production for quantum dynamical semigroups,” Journal of Mathematical Physics 19, 1227-1230 (1978).
• H. Spohn and J. L. Lebowitz, “Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs,” Advances in Chemical Physics 38, 109-142 (1978).
• P. Talkner, E. Lutz, and P. Hanggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102(R) (2007).
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665-1702 (2009).
• M. Campisi, P. Hanggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771-791 (2011).
• R. Kosloff, “Quantum thermodynamics: A dynamical viewpoint,” Entropy 15, 2100-2128 (2013).
• 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: Mathematical and Theoretical 49, 143001 (2016).
• S. Vinjanampathy and J. Anders, “Quantum thermodynamics,” Contemporary Physics 57, 545-579 (2016).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).