Quantum Thermodynamics

Quantum thermodynamics studies energy exchange and irreversibility when fluctuations, coherence, measurement, and finite control resources matter. Its predictions depend on a physical protocol: which Hamiltonian defines the energy, which reservoirs exchange heat, what is driven, and what is measured. Specifying that protocol comes before assigning heat, work, or entropy production.

Required background. Thermal Master Equations and Detailed Balance describe equilibrium-compatible rates; the Fluctuation–Dissipation Relation relates equilibrium noise to response. The equilibrium state itself is developed in Thermal Density Operators.

Energy balance requires a physical partition

For a weakly coupled system with controlled Hamiltonian $H(t)$, define $U=\operatorname{Tr}(\rho H)$. Then

$$\dot U=\operatorname{Tr}(\dot\rho H) +\operatorname{Tr}(\rho\dot H).$$

For thermally consistent dissipators $\mathcal L_\alpha$ and controlled driving, a common convention identifies

$$\dot Q_\alpha=\operatorname{Tr}[H\mathcal L_\alpha(\rho)], \qquad \dot W=\operatorname{Tr}(\rho\dot H),$$

with heat and work positive into the system. The Hamiltonian part of $\dot\rho$ contributes no change to its own energy expectation. An arbitrary state change at fixed $H$ is not automatically heat from a thermal reservoir: measurement, feedback, and engineered nonthermal dissipation require their own physical interpretation.

Energy, Heat, and Work develops this distinction. At strong coupling, interaction energy and the work of changing couplings can be important. Enlarging the system, for example through Reaction-Coordinate Mapping, can make those contributions explicit.

Work fluctuations depend on how they are measured

In the Two-Point Measurement Scheme, an initial energy measurement gives $E_n^0$, a drive acts, and a final energy measurement gives $E_m^\tau$. The assigned work is $W=E_m^\tau-E_n^0$ for the closed driven protocol. Work Distributions combines the outcome probabilities into a distribution.

This is a two-time experimental prescription, not generally a single one-time observable independent of the protocol. The initial measurement can remove energy-basis coherence. Special commuting cases can simplify the description; alternative measurement schemes need their own definitions rather than borrowing the same distribution automatically.

For a closed unitary drive beginning in the Gibbs state at inverse temperature $\beta$, with finite endpoint partition functions, the two-point protocol satisfies

$$\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}.$$

Jarzynski and Crooks Relations states the forward/reverse preparations and time-reversal assumptions. Fluctuation Theorems compares broader settings. Individual runs can have work below $\Delta F$; Jensen’s inequality constrains the mean, $\langle W\rangle\ge\Delta F$. Estimating the exponential average can be dominated by rare trajectories.

Entropy production includes the reservoirs

With $S=-k_B\operatorname{Tr}(\rho\ln\rho)$ and heat positive into the system, the weak-coupling expression is

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

For suitable thermal GKLS generators, Spohn’s inequality yields nonnegative entropy production. The bath terms matter: a system can become purer while exporting entropy. Positive Lindblad rates alone do not identify reservoirs or establish that this heat interpretation is valid. Entropy Production gives the assumptions and relative-entropy formulation.

Landauer Principle and Maxwell’s Demon and Information include memory, correlations, feedback, and resetting. The familiar $k_BT\ln2$ scale applies to an ideal unbiased degenerate bit reset under specified isothermal conditions. It is not a universal cost assigned to every measurement or every recorded bit regardless of its state and available side information.

Different tasks permit different operations

Task	Page and defining restriction
Transform states using an equilibrium environment	Thermal Operations specifies allowed energy-conserving interactions and resources.
Extract work from a given state by cyclic unitary control	Ergotropy and Passive States identifies the unitary work bound and its assumptions.
Operate a cycle between reservoirs	Quantum Heat Engines distinguishes work, heat, efficiency, and power.
Infer a temperature from a probe	Thermometry connects sensitivity, equilibration, and measurement constraints.

Results for one allowed-operation set need not carry over unchanged to another. A useful first study is a two-level system with thermal upward and downward rates, followed by a controlled change of its gap. The Quantum Thermodynamics Toy Models page turns those processes into calculations with explicit energy and sign conventions.

Exercises

Cooling need not increase system entropy. A qubit with excited energy $\epsilon>0$ starts in $I/2$ and relaxes to its ground state at fixed Hamiltonian. In the weak-coupling energy convention, find its energy change and heat into the system.

Solution

The initial energy is $\epsilon/2$ and the final energy zero, so $\Delta U=-\epsilon/2$. There is no Hamiltonian driving, hence $W=0$ and $Q=-\epsilon/2$. The system entropy falls from $k_B\ln2$ to zero. The environment must be included when assessing total irreversibility; the zero-temperature limit requires care with expressions involving $Q/T$.

A fluctuating second law. Explain why the Jarzynski equality allows some runs with $W<\Delta F$ but still implies $\langle W\rangle\ge\Delta F$ for positive temperature.

Solution

Convexity gives $e^{-\beta\langle W\rangle}\le\langle e^{-\beta W}\rangle=e^{-\beta\Delta F}$. Taking logarithms and dividing by $-\beta<0$ reverses the inequality. The constraint is on the ensemble mean, not each realization. Rare low-work events can contribute strongly to the exponential average.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Thermal generators and entropy production.
• Campisi, M., P. Hänggi, and P. Talkner. “Colloquium: Quantum fluctuation relations: Foundations and applications.” Reviews of Modern Physics 83, 771–791 (2011). doi:10.1103/RevModPhys.83.771.
• Goold, J., 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). Information, resources, and thermodynamic tasks.
• Spohn, H. “Entropy production for quantum dynamical semigroups.” Journal of Mathematical Physics 19, 1227–1230 (1978). doi:10.1063/1.523789.