Complete Passivity, KMS Structure, and Resource Conversion

A state can be passive for one system and still release work when several independent copies are controlled jointly. Complete passivity excludes this activation at every finite copy number. Under the Pusz–Woronowicz hypotheses, that stronger operational stability characterizes equilibrium: the completely passive states are KMS states at one inverse temperature, together with ground states. Turning this theorem into a resource-conversion statement requires additional choices—what counts as a copy, which operations are free, how work is stored, and which energy and approximation bounds survive the field-theory limit.

Required background. Passivity, work, and information fixes the cyclic-work sign convention, admissible energy domain, and distinction between subsystem energy transfer and useful work.

Helpful background. Modular KMS relations explain the analytic KMS condition without assuming a trace-class Gibbs density operator.

For the chapter-wide dictionary, see Enter this chapter; for the hypotheses attached to each claim, see the claim-validity summary; and for controls that invalidate tempting conclusions, see failure controls.

Tensor-power passivity

Let $(\mathcal A,\alpha_t,\omega)$ be a $C^*$-dynamical system. For each positive integer $n$, form an independent composite with algebra $\mathcal A^{\otimes n}$, product dynamics $\alpha_t^{\otimes n}$, and product state $\omega^{\otimes n}$. A tensor-product completion must be fixed when it is not unique. The state $\omega$ is completely passive when $\omega^{\otimes n}$ is passive for every finite $n$:

$$-i\,\omega^{\otimes n}\!\left(U^*\delta^{(n)}(U)\right)\ge0 \quad \text{for every admissible }U\text{ and every }n\ge1,$$

where

$$\delta^{(n)} =\delta\otimes\mathbf1\otimes\cdots +\cdots+ \mathbf1\otimes\cdots\otimes\delta$$

is the generator of the product dynamics on its natural differentiable domain. The collective unitary $U$ need not factor across copies. If only $U_1\otimes\cdots\otimes U_n$ were allowed, the test would merely repeat one-copy passivity and could never detect activation.

The central theorem is:

A completely passive state of a $C^*$-dynamical system is either a $\beta$-KMS state for some $\beta\ge0$ or a ground state; conversely, KMS and ground states are completely passive.

Here $\beta=0$ denotes a tracial KMS state when one exists. “Ground state” is an algebraic positive-energy condition, not necessarily a unique vacuum vector. The theorem is Pusz and Woronowicz 1978, Theorems 1.2 and 1.4, pp. 276–278. It assumes the stated dynamics and the differentiable cyclic processes used in their definition; it does not say that every passive state is thermal.

For a finite Hamiltonian $H$ with Gibbs state $\gamma_\beta=e^{-\beta H}/Z$, the easy direction is transparent. The $n$-copy state is

$$\gamma_\beta^{\otimes n} =\frac{e^{-\beta H^{(n)}}}{Z^n}, \qquad H^{(n)}=\sum_{j=1}^nH_j,$$

so it is Gibbs at the same $\beta$ and therefore passive for every $n$. The hard direction is that stability against all finite tensor powers forces KMS analyticity or the ground-state alternative. No finite list of work-extraction experiments proves that universal statement.

Exact activation in three truncated oscillator copies

Consider the first three levels of a regulated oscillator,

$$H=\epsilon\bigl(|1\rangle\!\langle1|+2|2\rangle\!\langle2|\bigr), \qquad \rho=0.5|0\rangle\!\langle0| +0.4|1\rangle\!\langle1| +0.1|2\rangle\!\langle2|.$$

The populations decrease with energy, so one copy is passive. It is not Gibbs: adjacent ratios are $p_1/p_0=0.8$ and $p_2/p_1=0.25$, which cannot both equal $e^{-\beta\epsilon}$. Two copies remain passive. Indeed, the individual basis-state probabilities at total energies $0,1,2,3,4$ are respectively

$$0.25; \quad 0.20; \quad 0.16\text{ or }0.05; \quad 0.04; \quad 0.01,$$

and no probability at a higher energy exceeds a probability at a lower energy.

Three copies reveal an inversion. Compare the two product vectors

$$|0,0,2\rangle:\quad E=2\epsilon,\quad p=0.5^2(0.1)=0.025,$$ $$|1,1,1\rangle:\quad E=3\epsilon,\quad p=0.4^3=0.064.$$

A unitary that swaps only these two vectors lowers the mean energy by

$$W_{\rm ext}^{(3)} =(3\epsilon-2\epsilon)(0.064-0.025) =0.039\epsilon.$$

Thus the state is one- and two-copy passive but not three-copy passive. Every number is a product of displayed populations, so the benchmark can be reproduced without diagonalizing a large matrix. As a control, replace $\rho$ by the Gibbs distribution with the same mean energy $0.6\epsilon$. Writing $q=e^{-\beta\epsilon}$, the condition

$$\frac{q+2q^2}{1+q+q^2}=0.6$$

gives

$$q=\frac{\sqrt{22}-1}{7}=0.5272023, \qquad \beta\epsilon=0.6401710,$$

and probabilities $(0.5539723,0.2920554,0.1539723)$. Their product is exactly ordered by total energy at every copy number, so the activating inversion disappears.

This is an energy-truncated oscillator, not yet a continuum QFT theorem. A field implementation must specify a finite family of wavepacket modes, a cutoff, and a collective unitary whose work and energy-domain estimates remain controlled as the cutoff is lifted. An activation that migrates to arbitrarily high energy does not define a finite-energy protocol in the limit.

What KMS structure adds

A single passive finite system can have any decreasing population list. Complete passivity removes that freedom because inconsistent logarithmic slopes become population inversions in a sufficiently large tensor power. For a nondegenerate finite Hamiltonian, define the local inverse slopes

$$\beta_{mn}=\frac{\log p_m-\log p_n}{E_n-E_m} \qquad (E_m\lt E_n).$$

Gibbs populations have one common $\beta_{mn}=\beta$. If two slopes differ, integer combinations of the corresponding level gaps can eventually reverse the product-population ordering; the three-copy calculation above is a concrete instance. In an infinite system, the KMS boundary condition replaces this elementary logarithmic-slope argument and remains meaningful when no global Gibbs trace exists.

The word “copy” deserves special care in QFT. Two spacelike regions of one vacuum state are not automatically independent tensor copies: local algebras can be type III, the vacuum is entangled, and a canonical Hilbert-space tensor factorization need not exist. Independent laboratories may be modeled by separate QFT systems, by a split inclusion with an explicitly chosen type-I intermediary, or by a regulator. Changing among these composition rules changes the complete-passivity question.

From athermality to resource conversion

Complete passivity identifies equilibrium relative to cyclic work extraction, but it does not by itself define a full resource theory. One standard finite-dimensional choice is a thermal operation: append a bath in its Gibbs state at inverse temperature $\beta$, apply a unitary commuting with the total Hamiltonian, account for a work-storage system, and discard an allowed subsystem. The system Gibbs state $\gamma_\beta$ is then free.

For any state $\rho$ with finite energy and finite relative entropy to $\gamma_\beta$,

$$D(\rho\Vert\gamma_\beta) =\beta\left[F_\beta(\rho)-F_\beta(\gamma_\beta)\right], \qquad F_\beta(\rho)=\operatorname{tr}(\rho H)-\beta^{-1}S(\rho).$$

This follows by inserting $\log\gamma_\beta=-\beta H-\log Z$. In the independent-and-identically-distributed asymptotic regime, with vanishing conversion error and a sublinear coherent reference, the reversible conversion rate is governed by this relative-entropy free energy; for nonfree $\rho$ and $\sigma$ on the declared systems,

$$R(\rho\to\sigma) =\frac{D(\rho\Vert\gamma_\beta)} {D(\sigma\Vert\gamma_\beta)}.$$

The operation class and coherence assistance are part of this result, as emphasized by Brandão et al. 2013, main theorem and discussion, pp. 2–4. It should not be transplanted unchanged to a type-III local algebra or to a one-shot protocol.

For the passive nonthermal benchmark, the Gibbs reference was chosen to have the same mean energy. Direct substitution gives

$$D(\rho\Vert\gamma_\beta)=0.03139485, \qquad F_\beta(\rho)-F_\beta(\gamma_\beta) =0.04904134\,\epsilon.$$

The three-copy swap extracts only $0.039\epsilon$ from one selected inversion, so it need not attain the asymptotic free-energy value. At small copy number, thermo-majorization and coherence constraints replace a single scalar free energy; Horodecki and Oppenheim 2013, “Thermal Operations” and “Work of formation” give the finite-system formulation.

Controls that change the conclusion

Change the composition rule. Correlated preparations, spacelike subalgebras of one field, and independent tensor copies are different resources. Replacing $\omega^{\otimes n}$ by an unspecified correlated state invalidates the complete-passivity theorem’s premise.

Allow an uncontrolled catalyst. Approximate return in trace distance alone can hide a resource in an ever-larger or ever-higher-energy catalyst. State whether the catalyst returns uncorrelated, bound its dimension or mean energy, and specify the error topology. Quantitative restrictions under dimension and energy bounds are developed by Ng et al. 2015, §§3–5.

Mix temperatures or Hamiltonian scales. A product of Gibbs states at different $\beta$ values is not a common-temperature equilibrium and can power an engine. Rescaling one copy’s Hamiltonian while keeping its populations fixed likewise changes the dynamics against which passivity is tested.

Exchange limits silently. First establish a finite-cutoff protocol, then show uniform control of its energy and error before taking the field limit. Taking infinitely many copies, removing an energy cutoff, and sending the conversion error to zero in a different order can produce a different—or undefined—rate.

Common pitfalls

Saying “passive means thermal.” One-copy passivity only orders populations. It is complete passivity, under the theorem’s tensor-power and dynamical hypotheses, that selects KMS or ground states.

Calling every Gibbs-preserving map a thermal operation. These operation classes need not coincide. State-conversion conclusions inherit the chosen free-operation set.

Treating a catalyst’s marginal return as exact return. If correlations with the output remain, the catalyst has not necessarily been restored in the operational sense required by the protocol.

Exercises

1. Verify that the two-copy benchmark state is passive even though the three-copy state is not.

Solution

At each total energy, list the smallest and largest basis-state probabilities. They are $0.25$ at $0$, $0.20$ at $\epsilon$, $0.05$ and $0.16$ at $2\epsilon$, $0.04$ at $3\epsilon$, and $0.01$ at $4\epsilon$. Every probability at a lower energy is at least every probability at a higher energy; degeneracy at $2\epsilon$ is irrelevant. Hence no two-copy permutation lowers the energy. The displayed $0.025\lt0.064$ inversion at energies $2\epsilon\lt3\epsilon$ proves failure at three copies.

2. Derive the Gibbs control distribution with mean energy $0.6\epsilon$.

Solution

With $q=e^{-\beta\epsilon}$, the probabilities are $(1,q,q^2)/(1+q+q^2)$. The mean-energy equation reduces to $7q^2+2q-3=0$. Its positive root is $q=(\sqrt{22}-1)/7$; the other root is negative and inadmissible. Normalization gives the probabilities quoted in the benchmark.

3. Show that equal-temperature Gibbs states remain Gibbs on $n$ noninteracting copies.

Solution

The copy Hamiltonians commute, so

$$e^{-\beta\sum_jH_j}=\bigotimes_je^{-\beta H_j}, \qquad Z^{(n)}=Z^n.$$

Therefore $e^{-\beta H^{(n)}}/Z^n=\gamma_\beta^{\otimes n}$. Its populations depend only on total energy and decrease exponentially, proving passivity for every finite $n$.

4. A protocol returns a catalyst’s reduced density matrix exactly but leaves it correlated with the target. Why is this not automatically exact catalysis?

Solution

The joint final state need not equal $\rho_{\rm target}\otimes\eta_{\rm cat}$. Correlations can store free energy, coherence, or information and can alter the catalyst’s usefulness in the next run. A definition that requires product return rejects the protocol; a correlated-catalytic resource theory may allow it, but then its conversion theorem and monotones must be stated separately.

References

• Brandão, Fernando G. S. L., Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, and Robert W. Spekkens. “Resource Theory of Quantum States Out of Thermal Equilibrium.” Physical Review Letters 111 (2013): 250404. DOI.
• Horodecki, Michał, and Jonathan Oppenheim. “Fundamental Limitations for Quantum and Nanoscale Thermodynamics.” Nature Communications 4 (2013): 2059. DOI.
• Ng, Nelly Huei Ying, Laura Mančinska, Cristina Cirstoiu, Jens Eisert, and Stephanie Wehner. “Limits to Catalysis in Quantum Thermodynamics.” New Journal of Physics 17, no. 8 (2015): 085004. DOI.
• Pusz, W., and S. L. Woronowicz. “Passive States and KMS States for General Quantum Systems.” Communications in Mathematical Physics 58, no. 3 (1978): 273–290. DOI.