Catalytic quantum thermodynamics beyond additivity and reduced-state monotones

Abstract: The generalized second laws of quantum thermodynamics are usually formulated in terms of Rényi divergences and the associated family of generalized free energies. In catalytic thermal transformations, this framework typically certifies the existence of a suitable catalyst but does not make the catalytic contribution explicit in the resulting system-level inequalities. Here we develop a complementary formulation based on non-additive divergences, whose pseudo-additive structure yields a family of generalized free energies with an explicit catalyst-dependent correction term. For uncorrelated catalytic thermal transformations, we show that this leads to non-additive second-law relations that make the catalytic contribution explicit and provide nontrivial constraints on admissible catalysts when the catalyst is returned only approximately. We also analyze correlated catalytic thermal transformations and show, through explicit finite-dimensional examples, that reduced-state data are generally insufficient to characterize thermodynamic accessibility: the thermo-majorization behavior of the joint transformation can change while the system and catalyst marginals remain fixed, and even states with identical marginals and the same mutual information can exhibit different thermo-majorization accessibility. Our results show that non-additivity can be thermodynamically informative in uncorrelated catalysis, whereas correlated catalysis generally requires a genuinely joint-state-sensitive description beyond reduced-state monotones.

• The paper develops a thermodynamically equivalent non-additive second-law framework in which pseudo-additivity exposes the catalyst’s athermality through a multiplicative free-energy correction while preserving the same state-conversion preorder and work distance.
• The paper derives finite-size trade-offs for approximately returned catalysts, showing corrections scale as O(ε) for α≥1 and O(d_M^{1−α}ε^α) for 0<α<1, with sensitivity to how errors are distributed across catalyst levels.
• The paper demonstrates with finite-dimensional examples that correlated-catalytic accessibility depends on the full joint state: identical marginals, and even similar mutual information, can correspond to different thermo-majorization outcomes.

Overview

This paper develops a complementary formulation of the generalized second laws of quantum thermodynamics based on non-additive (Tsallis-type) divergences rather than the standard Rényi divergences, and uses it to address two conceptual gaps in catalytic thermal operations (2604.21509). The first gap concerns uncorrelated catalysis: in the Rényi framework of Brandão et al., additivity causes the catalyst's contribution to cancel exactly from the system-level free-energy balance, so the resulting inequalities certify the existence of a catalyst without making its thermodynamic role explicit. The second gap concerns correlated catalysis, where the authors show—via explicit finite-dimensional thermo-majorization examples—that no family of second laws built solely from reduced-state monotones can completely characterize state accessibility.

The central technical observation is that the non-additive divergence $D_\alpha$ satisfies a pseudo-additive composition law,

$$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$

which leaves a residual cross term in the free energy of product states. Rather than treating this term as a defect, the paper retains it and interprets it as an explicit catalyst-dependent correction to the generalized second-law balance. Importantly, this reformulation does not change the underlying transition preorder: the work distance is invariant under the substitution of $D_\alpha$ for $D_\alpha^{\mathrm{R}}$, and the same state conversions are certified. What changes is the bookkeeping.

Non-additive second laws

The paper defines a family of non-additive free energies $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$, where $D_\alpha$ is the Tsallis-type divergence. Because $D_\alpha$ satisfies the data processing inequality for all $\alpha \in \mathbb{R}$ (proved via the generalized log-sum inequality), $F_\alpha$ constitutes a valid family of monotones under thermal operations. The main structural result is that the transition conditions $\Delta F_\alpha \le 0$ for all $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$0 hold if and only if the corresponding Rényi conditions hold, since $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$1 is a strictly increasing function of $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$2 at each fixed $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$3. The pseudo-additivity then implies that for exact uncorrelated catalysis $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$4, the total free-energy change factorizes as

$$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$5

so the catalyst's athermality enters multiplicatively rather than canceling. The paper also proves that the work distance $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$6 is invariant under the non-additive formulation, so the asymmetry between maximal extractable work and minimal formation cost persists: only at $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$7 do these two quantities coincide. A corollary extends Gour's order-theoretic characterization of catalytic convertibility to the non-additive divergence, showing that $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$8 catalytically relatively majorizes $$D_\alpha(p\otimes r\|q\otimes s) = D_\alpha(p\|q) + D_\alpha(r\|s) + \mathrm{sgn}(\alpha)(\alpha-1)\,D_\alpha(p\|q)\,D_\alpha(r\|s),$$9 iff both $D_\alpha$0 and $D_\alpha$1 for every $D_\alpha$2.

Finite-size implications of pseudo-additivity

For exact uncorrelated catalysis with trivial catalyst Hamiltonian, the multiplicative factor alone yields no nontrivial dimension bound—the condition reduces to an identically satisfied inequality. The finite-size content emerges only when the catalyst is returned approximately, under a trace-distance constraint $D_\alpha$3. In that regime, continuity bounds on $D_\alpha$4 yield explicit sufficient trade-offs between the target transformation, the return error, and the catalyst dimension $D_\alpha$5: for $D_\alpha$6 the correction scales as $D_\alpha$7, while for $D_\alpha$8 it scales as $D_\alpha$9.

Two benchmark constructions make the spectral-profile dependence concrete. With a uniform returned catalyst and error distributed symmetrically over half the levels, the leading pseudo-additive correction is quadratic, $D_\alpha^{\mathrm{R}}$0, independent of $D_\alpha^{\mathrm{R}}$1 at leading order. Concentrating the same trace-distance error into two levels instead produces a correction carrying an explicit factor $D_\alpha^{\mathrm{R}}$2—parametrically stronger within a fixed-$D_\alpha^{\mathrm{R}}$3 comparison—though constrained by the positivity requirement $D_\alpha^{\mathrm{R}}$4. The paper is careful here: it notes that these are sufficient but not necessary conditions, that the continuity bounds are worst-case estimates rather than sharp minimum-dimension proofs, and that the profile sensitivity is not exclusive to the non-additive formulation—an additive Rényi treatment of the same ansätze exhibits the same qualitative scaling contrast. The distinguishing feature is structural: in the pseudo-additive framework the profile dependence couples directly to the system contribution inside the monotonicity condition itself.

Correlated catalysis beyond reduced-state monotones

The second half of the paper addresses correlated catalytic transformations $D_\alpha^{\mathrm{R}}$5, where the catalyst is returned only marginally. All examples share a fixed initial product state of two local thermal states, fixed final marginals ($D_\alpha^{\mathrm{R}}$6, $D_\alpha^{\mathrm{R}}$7), and a fixed bath temperature; only the correlation structure of the final joint state varies. For the chosen parameters ($D_\alpha^{\mathrm{R}}$8, $D_\alpha^{\mathrm{R}}$9, $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$0, $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$1, $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$2), the reduced-state Rényi divergence changes are fixed at $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$3, $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$4, $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$5, entirely insensitive to correlations.

Three results follow:

• Classical correlation amount matters: for the classically correlated family $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$6, the transition is allowed at $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$7 ($$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$8) but forbidden at $$F_\alpha(\rho_S) = k_B T\, D_\alpha(\rho_S\|\gamma_S) - k_B T \ln Z_S$$9 ($D_\alpha$0), despite identical marginals.
• Correlation nature matters: a discordant state $D_\alpha$1 has the same marginals and essentially the same mutual information ($D_\alpha$2) as the allowed classically correlated state, yet its transition is forbidden by thermo-majorization.
• Consequence: no family of second laws formulated solely in terms of reduced-state functionals—even supplemented by a single scalar correlation measure such as mutual information—can provide a complete characterization of accessibility in finite-dimensional correlated catalysis.

This is consistent with earlier observations that the work cost of quantum processes depends on full system–environment correlations rather than reduced-state data.

Limitations and open questions

The paper concedes several restrictions explicitly. The entire framework assumes block-diagonal states; incorporating coherence remains impossible within this picture per prior no-go results. The approximate-catalysis trade-off relations are sufficient but not necessary, and the continuity bound for $D_\alpha$3 is a worst-case estimate not yet constituting a sharp minimum-catalyst-dimension proof. For exact uncorrelated catalysis with trivial catalyst Hamiltonian, the pseudo-additive condition yields no nontrivial restriction at all—the informative content is confined to the approximate regime. The correlated-catalysis conclusions rest on specific two-qubit numerical examples rather than a general theorem, leaving open whether the insufficiency of reduced-state monotones admits a clean axiomatic characterization or holds universally across parameter regimes. The $D_\alpha$4 non-additive quantity becomes ill-defined in the chosen parameterization. Finally, the proposed synthesis treating pseudo-additivity and correlation structure as manifestations of a common compositional principle is deferred to a companion work still in preparation.

Conclusion

The paper establishes that non-additive divergences support a fully equivalent formulation of the generalized second laws whose pseudo-additive structure makes the catalyst's athermality and spectral profile visible in the free-energy balance itself, converting approximate catalytic feasibility into an explicit finite-resource accounting problem. In parallel, it demonstrates through explicit counterexamples that correlated catalytic thermodynamics is intrinsically joint-state dependent: neither marginals nor marginal data plus mutual information suffice to determine accessibility. Together these results indicate that the limitations of standard generalized second-law formulations stem from how thermodynamic constraints behave under composition, whether through pseudo-additive residues on product states or through genuinely joint-state constraints once correlations are admitted.