Complete Passivity, KMS Structure, and Resource Conversion

A passive state may hide work that becomes accessible when several copies are operated on jointly. Complete passivity excludes this activation at every finite copy number. Under the standard $C^*$-dynamical hypotheses, it singles out ground states and KMS equilibrium states rather than arbitrary decreasing energy populations.

Required background. Passivity, work, and information defines the Hamiltonian reference, admissible cyclic operations, and finite-energy domain used here.

Helpful background. KMS correlators connect the analytic KMS condition with modular flow.

From one copy to every copy

If $(\mathcal A,\alpha_t,\omega)$ is a dynamical system, the $n$-copy system has state $\omega^{\otimes n}$ and product dynamics $\alpha_t^{\otimes n}$. Complete passivity means that every $\omega^{\otimes n}$ is passive against every admissible cyclic perturbation of the composite:

$$W_{\rm ext}^{(n)}(U,\omega^{\otimes n})\le0, \qquad n=1,2,\ldots .$$

The collective unitary need not factor across copies. That point is essential: restricting to $U_1\otimes\cdots\otimes U_n$ would merely repeat the one-copy test and could not reveal activation.

Pusz and Woronowicz 1978, Theorems 1.1 and 1.4, pp. 275–281 show, subject to their dynamical assumptions, that completely passive states are KMS states at a common inverse temperature $\beta\ge0$ or ground states. KMS analyticity is therefore not inferred from a single successful work-extraction test; it follows from stability against all finite-copy cyclic processes.

A passive nonthermal population

Consider one three-level system with energies $(0,\epsilon,2\epsilon)$ and probabilities $p_0\ge p_1\ge p_2$. The state is passive. It is thermal only if

$$\frac{p_1}{p_0}=\frac{p_2}{p_1}=e^{-\beta\epsilon}$$

for one $\beta$. If these ratios differ, tensor products contain pairs of composite levels whose population order conflicts with total energy order. At some finite copy number a collective permutation lowers the energy. The resulting work is a resource-conversion witness for athermality, not a violation of one-copy passivity.

For a regulated field, use a finite set of modes and an energy cutoff, locate an inverted pair in $\rho^{\otimes n}$, and then increase the cutoff. A conclusion about the continuum requires a uniform finite-energy construction; an activation that recedes to infinite energy does not define an admissible QFT protocol.

Complete passivity strengthens the first branch by allowing collective cyclic operations on every finite tensor power. Its KMS conclusion does not derive the separate stress-energy or entropy bounds. The diagram is schematic.

Composition, catalysts, and resources

Several qualifications prevent overstatement:

• Copies must share a specified dynamics and temperature scale. Combining unrelated Hamiltonian normalizations can manufacture a spurious gradient.
• A catalyst must return in its full state, including correlations if the resource theory requires them. Returning only its marginal can leave hidden resources in correlations.
• An unbounded-energy catalyst can invalidate finite-energy work accounting. State an energy constraint and approximation topology.
• Conserved charges replace $H$ by a generalized potential such as $H-\mu Q$ only when the charge sector and admissible operations are fixed.

The multi-copy theorem is vulnerable to changed composition rules, uncontrolled sectors, and catalysts without a finite-energy return condition. These are domain failures, not exceptions to the theorem. The map is schematic.

Exercises

Show that a Gibbs state $\rho_\beta=e^{-\beta H}/Z$ remains Gibbs for the sum Hamiltonian on $n$ noninteracting copies.

Solution

Because the copy Hamiltonians commute, $e^{-\beta\sum_i H_i}=\bigotimes_i e^{-\beta H_i}$ and the partition function is $Z^n$. Hence $\rho_\beta^{\otimes n}$ is Gibbs at the same $\beta$ and is passive for every $n$.

References

• Pusz, Władysław, and Stanisław L. Woronowicz. “Passive States and KMS States for General Quantum Systems.” Communications in Mathematical Physics 58 (1978): 273–290. DOI.