Gibbs Preserving Superchannels

• Gibbs preserving superchannels are higher-order quantum processes that keep the thermal (Gibbs) state invariant while transforming quantum channels.
• They unify principles from quantum thermodynamics, quantum information, and open systems by extending Gibbs preserving maps to process-level transformations.
• They enable new operations like coherence creation beyond standard thermal operations, though often at the cost of infinite coherence resources in practical implementations.

A Gibbs preserving superchannel is a higher-order quantum process that transforms quantum channels in such a way that the Gibbs (thermal equilibrium) state remains invariant under the transformed channel, at a specified inverse temperature and Hamiltonian. This notion unifies structural properties from resource theories in quantum thermodynamics, quantum information, and open quantum systems. Gibbs preserving superchannels extend the well-known class of Gibbs preserving maps by allowing process-level transformations, entailing operational constraints and opportunities that do not exist in strictly state-based frameworks.

1. Definition and Formal Properties

A quantum superchannel is a completely positive map acting at the channel level: for quantum CPTP maps $\Phi$, a superchannel $\Theta$ yields another CPTP map $\Theta[\Phi]$. The Gibbs preserving constraint requires that for every $\Phi$ that preserves a Gibbs state $\gamma$ (i.e., $\Phi(\gamma)=\gamma$), the image channel $\Theta[\Phi]$ also preserves (possibly the same, possibly an appropriately matched) Gibbs state: $\Theta[\Phi](\gamma)=\gamma$.

This framework generalizes the notion in the single-system case (Gibbs preserving map: $\mathcal{E}$ with $\mathcal{E}(\gamma)=\gamma$) to transformations that act on channels themselves. In thermodynamic resource theories, such superchannels must preserve not only the complete positivity and trace preservation but also the thermal equilibrium structure and, in generalizations, respect energy conservation or covariance conditions.

2. Comparison to Thermal Operations and Resource-Theoretic Significance

Classically, Gibbs preserving maps and thermal operations are equally powerful: every Gibbs-preserving operation can be realized by a thermal operation, defined as an energy-conserving unitary interaction with a heat bath initialized in the Gibbs state (Faist et al., 2014). However, in the fully quantum regime, this equivalence fails. Gibbs preserving superchannels can, for example, create coherence between energy eigenstates (off-diagonal terms in the Hamiltonian basis), while thermal operations, constrained by strict energy conservation and covariance, cannot.

For instance, let $$\gamma = p_0 |0\rangle\langle0| + p_1 |1\rangle\langle1|$$ represent the Gibbs state of a two-level system. Any map $\Phi$ with $\Phi(\gamma)=\gamma$ can be considered Gibbs preserving. The construction $$\Phi(\cdot) = \langle 0|\cdot|0\rangle \sigma + \langle 1|\cdot|1\rangle \rho$$ (with $\sigma = (\gamma - p_1 \rho)/p_0$) produces, for $|1\rangle$, an arbitrary $\rho$—which may contain coherence. The transformation $|1\rangle \to (|0\rangle+|1\rangle)/\sqrt{2}$ is allowed for Gibbs-preserving maps but forbidden for thermal operations since the latter cannot introduce coherence (Faist et al., 2014). This operational gap implies that resource monotones and "second laws" based solely on thermo-majorization or classical criteria must be re-evaluated in quantum scenarios.

3. Majorization Structures and Reachability

Gibbs-preserving superchannels are naturally characterized by generalized majorization relations. Classical majorization is generalized to $d$-majorization for vectors (based on a strictly positive vector $d$ and $d$-stochastic matrices) and further to $D$-majorization for matrices (Ende, 2020). For Gibbs-preserving maps, $A \prec_D B$ means there exists a CPTP map $T$ with $T(B)=A$ and $T(D)=D$, where $D$ is the Gibbs state.

In the qubit case, $D$-majorization admits a trace norm characterization: $A \prec_D B$ iff $\|A-tD\|_1 \leq \|B-tD\|_1$ for all $t \in \mathbb{R}$. However, this trace-norm criterion does not generalize to higher dimensions due to counterexamples and the loss of permutation invariance. Such topological and order-theoretic properties (e.g., existence of unique minimal and maximal elements, convexity, star-shapedness of reachable sets with respect to $D$) are foundational for analyzing accessible states and transformations in open quantum systems and in the context of control under thermodynamic constraints.

4. Coherence Cost and Physical Implementability

While Gibbs-preserving superchannels are mathematically simple (preserving the Gibbs state), their physical realization within operationally motivated classes, such as thermal operations, reveals subtleties. Notably, (Tajima et al., 4 Apr 2024) demonstrates that for a broad class, the cost in quantum coherence required can be infinite. For "pairwise reversible" Gibbs-preserving operations, the energy change $$(\Lambda, P)=\|\sqrt{\rho_1}(H_S-\Lambda^\dagger(H_{S'}) )\sqrt{\rho_2}\|_2$$ quantifies the off-diagonal energy transfer. The coherence cost for implementing such an operation $\Lambda$ with approximation error $\varepsilon$ is bounded below by $$\sqrt{c^\varepsilon(\Lambda)} \ge (\Lambda, P)/\varepsilon - \Delta(H_S) - 3\Delta(H_{S'})$$, indicating divergence for vanishing error whenever $(\Lambda, P)>0$.

This result establishes that there exist uncountably many Gibbs-preserving operations that cannot be simulated by thermal operations aided by any finite coherence resource. The implication is that Gibbs-preserving superchannels are not universally operationally accessible, and their status as "free" operations in resource theory must be more judiciously considered.

5. Covariant Gibbs-Preserving Superchannels and State Convertibility

Enhanced thermal operations, or covariant Gibbs-preserving operations (CGPOs), are those that both preserve the Gibbs state and commute with the time-evolution group generated by the system Hamiltonian: $$\mathcal{E}(e^{-iHt} \rho e^{iHt}) = e^{-iHt} \mathcal{E}(\rho) e^{iHt}$$ for all $t$ (Shiraishi, 10 Jun 2024). State convertibility under CGPOs, in the presence of correlated catalysts, is fully characterized by the free energy $F(\rho) = S(\rho || \gamma)$, where $S(\cdot || \cdot)$ denotes the quantum relative entropy. If $F(\rho) \ge F(\rho')$, then $\rho$ can be converted to $\rho'$ via a covariant Gibbs-preserving operation using a correlated catalyst, regardless of the presence of quantum coherence.

This finding broadens the equivalence between Gibbs-preserving and thermal operations in the asymptotic, catalytic regime for states with full coherence and distillability, demonstrating that energetic covariance does not restrict convertibility under these operational settings.

6. Superchannel Realization, Basis Dependence, and Circuit Models

Deterministic supermaps (which include Gibbs-preserving superchannels) admit circuit-level realizations (Allen et al., 2 Oct 2024), with the transformation constructed from auxiliary spaces and channels such that for input channel $F$, the output $S(F)$ is operationally implemented via a well-defined sequence: input preparation, processing, invocation of $F$, and final output mapping. This applies not just to conventional quantum channels but also to channels involving POVMs, quantum instruments, and more. In Gibbs-preserving contexts, the Gibbs state preservation is incorporated into the design of the auxiliary operations.

The Choi–Jamiołkowski isomorphism is central to the representation of these superchannels. However, the correspondence between complete positivity of maps and positivity of the associated Choi operator is basis-dependent (Sohail et al., 2023). Only certain "canonical" bases guarantee this duality, with necessary and sufficient conditions specified by complete order isomorphisms. For analysis and physical implementation of Gibbs-preserving superchannels, it is essential to maintain the canonical basis in all Choi-type representations to ensure that all duality formulas and semidefinite programming techniques remain valid.

7. Applications and Experimental Realizations

Gibbs-preserving superchannels have potential applications in quantum thermodynamics, quantum control, noise-adapted channel simulation, and quantum error correction. Experimentally, superchannel simulation algorithms capable of realizing arbitrary superchannels—including Gibbs-preserving ones in principle—have been demonstrated in NMR systems (Li et al., 2023) using GRAPE-engineered pulses and convex decomposition of target superchannels into generalized extreme points. While such experiments do not yet impose explicit Gibbs-preserving constraints, the methodology extends naturally to impose such operational restrictions, facilitating tests of theoretical models in realistic thermal settings.

8. Summary and Future Directions

Gibbs-preserving superchannels generalize the preservation of thermal equilibrium from states to processes, providing a rigorous mathematical framework for resource theories in quantum thermodynamics. Their ability to create or manipulate coherence distinguishes them sharply from strictly energy-conserving thermal operations, particularly in the quantum regime, but exposes fundamental limits in physical implementability due to possible infinite coherence cost. The framework incorporates majorization structures, circuit-based realization theorems, and basis-dependent positivity conditions in the Choi representation.

Future research will refine resource theories to accommodate the operational constraints identified, develop monotones beyond distance to the Gibbs state, characterize the true boundary between physically implementable and idealized free processes, and pursue experimental protocols under strict thermodynamic or covariance constraints to clarify the role and reach of Gibbs-preserving superchannels in quantum technologies.