Quantum Mathematics Learning Guide 48: Quantum Thermodynamics, Gibbs States, Free Energy, Work and Resource Theories

Quantum thermodynamics asks what energy, entropy, heat and work mean when the working system is small enough that coherence, entanglement, fluctuations and individual quantum levels matter. The equilibrium formulas remain familiar, but the operational questions become sharper.

A macroscopic thermodynamic statement such as “free energy cannot increase spontaneously” hides many assumptions about system size, baths, control and averaging. At the quantum scale, one can ask a more precise question: given a state ρ, a Hamiltonian H, a heat bath at temperature T and a declared set of allowed operations, exactly which state transformations are possible and how much deterministic or average work can be extracted?

This guide develops Gibbs states, partition functions, thermodynamic identities, relative-entropy free energy, passivity, Landauer erasure, fluctuation relations, thermal operations, thermo-majorisation and single-shot free energies. Guide 32 owns channel capacities; Guide 48 owns energetic resource conversion.

Hamiltonian + temperature → Gibbs reference state → compare nonequilibrium state → quantify free energy and fluctuations → constrain allowed thermal transformations → account for work.

1. Gibbs state

For Hamiltonian H in contact with a thermal bath at temperature T, define inverse temperature

β=1/(k_BT).

The canonical Gibbs state is

τ=e^{-βH}/Z

with partition function

Z=Tr(e^{-βH}).

Z normalises the state and also generates thermodynamic derivatives.

2. Two-level Gibbs state

Take levels E₀=0 and E₁=Δ.

Then

Z=1+e^{-βΔ}

and excited-state probability

p_1=e^{-βΔ}/(1+e^{-βΔ})=1/(1+e^{βΔ}).

At low temperature βΔ≫1, p₁≈e^−βΔ. At infinite temperature β→0, both levels approach probability 1/2.

3. Internal energy from the partition function

The equilibrium internal energy is

U=Tr(τH).

Differentiating Z gives

U=−∂lnZ/∂β.

For the two-level example,

U=Δ/(1+e^{βΔ}).

Temperature changes occupation, not the energy eigenvalues themselves unless H is also externally changed.

4. Heat capacity is an energy fluctuation

For the canonical ensemble,

C=dU/dT.

A standard identity gives

C=k_Bβ²(⟨H²⟩−⟨H⟩²).

Heat capacity is therefore proportional to energy variance in equilibrium.

Thermodynamic response and microscopic fluctuation are mathematically linked.

5. Entropy

Quantum thermodynamic entropy is von Neumann entropy

S(ρ)=−k_BTr(ρlnρ).

If ρ is diagonal in an energy basis with probabilities p_i, this reduces to the classical Shannon/Gibbs entropy

−k_BΣ_i p_i ln p_i.

A pure state has zero von Neumann entropy even if it is a coherent superposition of many energy eigenstates.

6. Helmholtz free energy

For state ρ with Hamiltonian H at temperature T, define nonequilibrium Helmholtz free energy

F_T(ρ)=Tr(ρH)−TS(ρ).

For the equilibrium Gibbs state,

F_T(τ)=−k_BT ln Z.

The Gibbs state minimises F_T among all states on the same Hamiltonian at fixed temperature.

7. Relative entropy measures excess free energy

Quantum relative entropy is

D(ρ||τ)=Tr[ρ(lnρ−lnτ)].

Using lnτ=−βH−lnZ,

F_T(ρ)−F_T(τ)=k_BT D(ρ||τ).

Since relative entropy is nonnegative, F_T(ρ)≥F_T(τ).

Nonequilibrium free energy is therefore a distinguishability-from-thermal-equilibrium measure multiplied by k_BT.

8. Worked pure excited-state free energy

For a two-level Hamiltonian {0,Δ}, take pure excited state |1⟩.

Its entropy is zero and mean energy is Δ, so

F_T(|1⟩)=Δ.

The Gibbs free energy is

F_T(τ)=−k_BTln(1+e^{-βΔ}).

The excess free energy is positive and represents thermodynamic resource relative to the bath, subject to the permitted operations and work-storage model.

9. Work and heat depend on the process

For an infinitesimal change of state and Hamiltonian,

dU=dTr(ρH)=Tr(Hdρ)+Tr(ρdH).

A common quantum-thermodynamic bookkeeping convention identifies

• δQ=Tr(Hdρ) as heat-like change from redistribution among energy levels;
• δW=Tr(ρdH) as work-like change from controlled Hamiltonian variation.

This split is useful for driven weakly coupled systems but is not uniquely universal under strong coupling, non-Markovianity or autonomous work-storage models.

10. Work is not a Hermitian observable in one universal sense

Energy has an operator H. Work describes energy transferred during a process between an initial and final Hamiltonian/control condition.

In the two-projective-measurement framework, measure initial energy E_n, drive the system, then measure final energy E’_m. Define stochastic work

W=E'_m−E_n.

This gives a legitimate work distribution but destroys initial energy coherence through the first projective measurement. Other quantum work frameworks preserve or explicitly account for coherence differently.

11. Jarzynski equality

For a system initially in thermal equilibrium and driven by a protocol satisfying the standard two-point-measurement assumptions, Jarzynski’s equality is

⟨e^{-βW}⟩=e^{-βΔF}.

Here W is work done on the system and ΔF is the equilibrium free-energy difference between final and initial Hamiltonians.

Jensen’s inequality gives

⟨W⟩≥ΔF,

recovering the ordinary second-law work bound on average.

12. Worked Jarzynski two-outcome example

Suppose β=1 in energy units and a protocol has W=0 with probability 0.75 and W=2 with probability 0.25.

Then

⟨e^{-W}⟩=0.75+0.25e^{-2}≈0.78383.

If Jarzynski applies,

ΔF=−ln(0.78383)≈0.2435.

Mean work is ⟨W⟩=0.5, which indeed exceeds ΔF.

13. Crooks fluctuation relation

Crooks’ theorem compares forward and time-reversed work distributions:

P_F(W)/P_R(−W)=e^{β(W−ΔF)}

under corresponding microscopic reversibility assumptions.

The crossing point P_F(W)=P_R(−W) occurs at W=ΔF.

Fluctuation relations quantify rare apparent second-law violations while preserving the global statistical asymmetry imposed by free energy.

14. Landauer erasure

Resetting one unknown classical bit from a maximally mixed memory state to a fixed pure logical state reduces its entropy by

ΔS=−k_Bln2.

For a thermodynamically reversible reset at temperature T, at least

Q≥k_BTln2

of heat must be dumped into the environment in the ideal degenerate-memory model.

Landauer’s principle concerns logically irreversible erasure and a complete thermodynamic accounting. It does not say every digital gate always dissipates exactly k_BTln2.

15. Work extraction from a qubit

Take a qubit with energies 0 and Δ in pure excited state |1⟩.

A controlled unitary can swap |1⟩ to |0⟩ while transferring energy Δ to an ideal work-storage system, provided the global operation conserves energy with the battery included.

The excited state is therefore active: useful work can be extracted.

A Gibbs state is completely passive: no work can be extracted from any number of copies by cyclic unitaries when the Hamiltonian returns to itself.

16. Passive states

A state is passive if no unitary cyclic process can lower its mean energy.

For a nondegenerate Hamiltonian, a passive state must be diagonal in energy and have populations that do not increase with energy.

Some passive states become active when several copies are available. Gibbs states are special because they are completely passive for all copy numbers.

This gives an operational characterisation of thermal equilibrium beyond “maximum entropy”.

17. Ergotropy

Ergotropy is the maximum work extractable from state ρ by cyclic unitary transformations of the system Hamiltonian.

Construct the passive state π_ρ by assigning the largest eigenvalue of ρ to the lowest energy, the next largest to the next energy, and so on.

Then

W_erg=Tr(ρH)−Tr(π_ρH).

Ergotropy differs from bath-assisted free-energy work extraction because it permits only cyclic unitaries on the system, not entropy exchange with a heat bath.

18. Resource theory asks: which operations are free?

A thermodynamic resource theory begins by declaring:

• free states, usually Gibbs states at a fixed temperature;
• free operations, such as thermal operations;
• valuable resources, such as nonequilibrium free energy, coherence or a raised weight/battery state.

Once the free-operation set is fixed, state-conversion questions become mathematical order relations.

Different free-operation choices can yield different transformation criteria. “Thermodynamically allowed” is incomplete unless the operational model is specified.

19. Thermal operations

A thermal operation at inverse temperature β can be modelled by:

• attach a bath system B in Gibbs state τ_B=e^{-βH_B}/Z_B;
• apply a global unitary U satisfying strict energy conservation [U,H_S+H_B]=0;
• discard part or all of the bath.

The resulting channel is

𝒯(ρ)=Tr_B[U(ρ⊗τ_B)U†].

Thermal operations preserve the Gibbs state of the system.

20. Gibbs-preserving maps are broader

A Gibbs-preserving channel satisfies

𝒢(τ)=τ.

Every thermal operation is Gibbs preserving, but not every Gibbs-preserving quantum channel is necessarily implementable as a thermal operation under the stricter energy-conserving dilation.

Resource-theory statements should therefore name the allowed map class rather than treating all Gibbs-preserving operations as physically equivalent.

21. Classical majorisation at infinite temperature

If all energy levels are degenerate, the Gibbs state is uniform and thermal operations reduce, on diagonal states, to ordinary noisy operations.

A probability vector p can be transformed deterministically into q by doubly stochastic mixing exactly when

p majorises q.

This is the same majorisation mathematics used for pure-state entanglement in Guide 31, but the direction and resource interpretation differ because here ordered probability means nonequilibrium purity relative to a uniform thermal reference.

22. Thermo-majorisation

At finite temperature, energy levels have unequal Gibbs weights.

For a state diagonal in energy with probabilities p_i and energies E_i, order levels by decreasing

p_i e^{βE_i} = p_i/τ_i ×(1/Z)

where τ_i∝e^−βE_i.

This β-ordering ranks how overpopulated each level is relative to equilibrium.

A thermo-majorisation curve is then built by cumulative Gibbs weights on the horizontal axis and cumulative state probabilities on the vertical axis.

23. Thermo-majorisation criterion

For energy-diagonal states under standard thermal operations without catalysts, a transition p→q is possible if the thermo-majorisation curve of p lies nowhere below that of q.

This is a complete single-shot transition criterion in the corresponding quasiclassical setting.

Ordinary free energy is only one scalar summary. Two states can have F(p)>F(q) and still fail the detailed thermo-majorisation test.

24. Worked β-ordering

Take energies E=(0,1,2), β=1 and state probabilities p=(0.50,0.30,0.20).

Compute

• p₁e^{0}=0.50;
• p₂e^{1}≈0.815;
• p₃e^{2}≈1.478.

The β-order is therefore level 3, then 2, then 1 despite the raw probabilities being largest at low energy.

Thermo-majorisation sorts by excess population relative to thermal expectation, not by p_i alone.

25. Single-shot work

Macroscopic thermodynamics focuses on average work over many repetitions.

In a single microscopic run, work fluctuates. One may instead ask for:

• the maximum deterministic work guaranteed except with failure probability ε;
• the minimum work cost to form a target state;
• the full work distribution;
• average extractable work.

These are different operational quantities and need not coincide at finite size.

26. Family of generalised free energies

For catalytic thermal state transformations of energy-diagonal states, a family of Rényi-divergence free energies appears:

F_α(ρ)=F(τ)+k_BT D_α(ρ||τ).

Monotonicity of all relevant F_α can impose stronger constraints than the ordinary α=1 free energy alone.

In the many-copy asymptotic limit, these separate second laws collapse toward the standard Helmholtz free-energy criterion under regularity assumptions.

27. Coherence is an additional thermodynamic resource

A state with off-diagonal terms in the energy basis contains quantum coherence between different energies.

Thermal operations are covariant under time translations generated by H, which constrains how such coherence can be created or consumed.

Two states with the same energy populations and same ordinary free energy can differ operationally because one contains coherent phase information.

Thermo-majorisation alone is therefore incomplete for coherent states. Additional asymmetry/coherence monotones are required.

28. Catalysts

A catalyst C can enable a state conversion

ρ_S⊗σ_C → ρ'_S⊗σ_C

while being returned unchanged and uncorrelated in the ideal catalytic definition.

As in entanglement catalysis, the auxiliary resource changes the ordering of possible transformations without being consumed.

If approximate catalysts are allowed with unconstrained dimension and tiny trace-distance changes, “embezzling” phenomena can trivialise some resource restrictions. Catalyst accuracy and size must therefore be part of the theorem.

29. Strong coupling changes the bookkeeping

The Gibbs state e^−βH_S/Z assumes the system Hamiltonian can be separated cleanly from the bath in the equilibrium description.

At strong system–bath coupling, interaction energy is not negligible and the reduced equilibrium state can be described using a Hamiltonian of mean force rather than the bare H_S.

Heat/work splits and free-energy formulas must then be adapted. Weak-coupling textbook identities should not be applied blindly.

30. Quantum engines

Quantum heat engines cycle a working medium among Hamiltonian changes and contacts with hot/cold reservoirs.

The Carnot efficiency

η_C=1−T_c/T_h

remains the reversible upper bound for engines operating between two equilibrium baths under standard thermodynamic assumptions.

Quantum coherence can affect finite-time power, fluctuations and control cost, but it does not permit a cyclic engine to violate a correctly formulated second law.

31. Maxwell demons and information

A demon measures a system and uses the result for feedback, apparently extracting work from fluctuations.

Once the demon’s memory, correlations, measurement and eventual reset are included, information has thermodynamic value and cost.

Feedback fluctuation theorems modify ordinary work inequalities by mutual-information terms.

Information does not abolish thermodynamics; it becomes part of the resource accounting.

32. Common misconception: entropy is always ignorance

A subsystem can have positive von Neumann entropy because it is entangled with another system even when the global state is pure and perfectly known.

33. Common misconception: free energy alone decides every microscopic transition

Ordinary free energy governs asymptotic average thermodynamics. Single-shot thermal operations can impose thermo-majorisation and families of generalised free-energy constraints, especially with finite systems and coherence.

34. Common misconception: extracting work means measuring energy

Work can be stored coherently in an explicit battery through energy-conserving dynamics. Projective energy measurements are one framework for defining work statistics, not a universal physical necessity for every engine.

35. Worked synthesis problem

A qubit has energies 0 and Δ=2k_BT, so βΔ=2.

Step 1: Partition function.

Z=1+e^{-2}≈1.135335.

Step 2: Excited probability.

p_1=e^{-2}/Z≈0.119203.

Step 3: Mean energy.

U≈0.238406 k_BT.

Step 4: Equilibrium free energy.

F(τ)=−k_BT ln Z≈−0.126928 k_BT.

Step 5: Prepare pure excited state. Its free energy is 2k_BT, so excess nonequilibrium free energy above equilibrium is approximately 2.126928 k_BT.

This excess is the average bath-assisted work scale available in a reversible asymptotic setting; deterministic single-shot work can obey stricter constraints.

36. Practice set

1. Define β and the Gibbs state.
2. What is the partition function?
3. How is internal energy obtained from ln Z?
4. Write von Neumann entropy with k_B included.
5. Define nonequilibrium Helmholtz free energy.
6. How is excess free energy related to relative entropy?
7. Why is work not one universal Hermitian observable?
8. State Jarzynski’s equality under the work-on-system convention.
9. What does Landauer’s bound say for erasing one unbiased bit?
10. What is a passive state?
11. Define a thermal operation.
12. Why can thermo-majorisation be stronger than ordinary free-energy comparison?

Answers

1. β=1/(k_BT); τ=e^{-βH}/Z.
2. Tr(e^{-βH}).
3. U=−∂lnZ/∂β.
4. S(ρ)=−k_BTr(ρlnρ).
5. Tr(ρH)−TS(ρ).
6. F_T(ρ)−F_T(τ)=k_BT D(ρ||τ).
7. Work refers to energy transfer during a process and depends on initial/final Hamiltonians and the operational measurement/storage model.
8. ⟨e^{-βW}⟩=e^{-βΔF}.
9. An ideal reversible reset must dump at least k_BTln2 heat into the environment in the standard degenerate-memory model.
10. A state from which no work can be extracted by cyclic unitaries on the system Hamiltonian.
11. Attach a Gibbs bath, apply a global energy-conserving unitary, then discard bath degrees of freedom.
12. It compares the full β-ordered population geometry rather than compressing the state to one scalar F.

Sources and further study

[1] Fernando Brandão, Michał Horodecki, Nelly Ng, Jonathan Oppenheim and Stephanie Wehner, The second laws of quantum thermodynamics. A resource-theoretic family of microscopic second laws and generalised free energies.

[2] Michał Horodecki and Jonathan Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics. Thermo-majorisation and single-shot work in thermal resource theory.

[3] Christopher Jarzynski, Nonequilibrium Equality for Free Energy Differences, Physical Review Letters 78, 2690 (1997). The fluctuation equality connecting nonequilibrium work statistics to equilibrium free energy.

[4] Rolf Landauer, Irreversibility and Heat Generation in the Computing Process, IBM Journal of Research and Development 5, 183–191 (1961). The foundational thermodynamics-of-erasure argument.

[5] Robert Alicki and Ronnie Kosloff, Introduction to Quantum Thermodynamics: History and Prospects. A broad review of quantum engines, open-system thermodynamics and foundational issues.

Batch 12 series navigation

• Guide 45: Quantum Many-Body Hamiltonians, Ground States, Correlation Functions and Phase Transitions
• Guide 46: Quantum Spin Chains, Jordan–Wigner Transformation, Free Fermions and Criticality
• Guide 47: Quantum Chaos, Random Matrices, Level Statistics and Scrambling
• Guide 48: Quantum Thermodynamics, Gibbs States, Free Energy, Work and Resource Theories
• Return to the BTT Mathematics Learning Hub

Educational note: thermodynamic work, heat and free-energy claims are operational-model dependent at microscopic scales. Strong coupling, coherence, catalysts, finite failure probability and explicit work-storage systems can change the correct formulation.