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arXiv:2608.11872

Limitations on Joint Coherence Transfer in Quantum Thermodynamics

Piotr Ćwikliński, Michał Studziński

quant-phmath-phSubmitted Aug 12, 2026• Announced Aug 13, 2026• 0 citations
v1 · Aug 12, 2026v2 · Sep 15, 2026v3 · Oct 26, 2026
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Abstract

Coherences between different energy levels are strongly constrained by thermodynamics. Here we ask a related question: if several coherence transfers can be optimized separately, can they also be optimized simultaneously by the same thermodynamic process? We show that this is in general a compatibility problem. Using an energy-resolved Stinespring representation of thermal operations, we express elementary coherence-transfer amplitudes as inner products of bath-weighted vectors associated with total-energy shells. Saturation of the corresponding Cauchy--Schwarz bounds requires these vectors to be collinear. Since all transfers have to arise from the same energy-preserving system--bath unitary, such saturation conditions cannot in general be imposed independently. Trace and Gibbs preservation lead to additional phase-closure conditions and to polygon inequalities that can rule out simultaneous saturation for fixed population-transfer data. We complement these analytic results with a rigorous computer-assisted semidefinite-program analysis. For the mixed-degenerate Hamiltonian $H=\operatorname{diag}(0,1,1,3)$, machine-certified primal and dual bounds show that no covariant Gibbs-preserving channel can simultaneously attain three individually optimal directional coherence transfers. Since every thermal operation is covariant and Gibbs preserving, the same triple of optimal values cannot be attained by a thermal operation. We further relate this incompatibility to range-inclusion constraints in Choi-sector Gram matrices for mixed-degenerate energy levels. Our results show that coherence transfers which are optimal when considered separately need not be compatible with one physical thermodynamic process. A full characterization of this incompatibility directly within thermal operations remains open.

Comments
28 pages

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Coherences between different energy levels are strongly constrained by thermodynamics. Here we ask a related question: if several coherence transfers can be optimized separately, can they also be optimized simultaneously by the same thermodynamic process? We show that this is in general a compatibility problem. Using an energy-resolved Stinespring representation of thermal operations, we express elementary coherence-transfer amplitudes as inner products of bath-weighted vectors associated with total-energy shells. Saturation of the corresponding Cauchy--Schwarz bounds requires these vectors to be collinear. Since all transfers have to arise from the same energy-preserving system--bath unitary, such saturation conditions cannot in general be imposed independently. Trace and Gibbs preservation lead to additional phase-closure conditions and to polygon inequalities that can rule out simultaneous saturation for fixed population-transfer data.
We complement these analytic results with a rigorous computer-assisted semidefinite-program analysis. For the mixed-degenerate Hamiltonian H=diag⁡(0,1,1,3)H=\operatorname{diag}(0,1,1,3)H=diag(0,1,1,3), machine-certified primal and dual bounds show that no covariant Gibbs-preserving channel can simultaneously attain three individually optimal directional coherence transfers. Since every thermal operation is covariant and Gibbs preserving, the same triple of optimal values cannot be attained by a thermal operation. We further relate this incompatibility to range-inclusion constraints in Choi-sector Gram matrices for mixed-degenerate energy levels. Our results show that coherence transfers which are optimal when considered separately need not be compatible with one physical thermodynamic process. A full characterization of this incompatibility directly within thermal operations remains open.
Thermal operations are an operational model of non-equilibrium quantum thermodynamics. In the absence of coherence between energy levels, exact state transition conditions under thermal operations are known in terms of a mathematical relation called thermo-majorization. But incorporating coherence has turned out to be challenging, even under the relatively tractable model wherein all Gibbs state-preserving quantum channels are included. Here we find a mathematical generalization of thermal operations at low temperatures, 'cooling maps', for which we derive the necessary and sufficient state transition condition. Cooling maps that saturate recently discovered bounds on coherence transfer are realizable as thermal operations, motivating us to conjecture that all cooling maps are thermal operations. Cooling maps, though a less-conservative generalization to thermal operations, are more tractable than Gibbs-preserving operations, suggesting that cooling map-like models at general temperatures could be of use in gaining insight about thermal operations.
Gibbs-preserving operations have been studied as one of the standard free processes in quantum thermodynamics. Although they admit a simple mathematical structure, their operational significance has been unclear due to the potential hidden cost to implement them using an operationally motivated class of operations, such as thermal operations. Here, we show that this hidden cost can be infinite-we present a family of Gibbs-preserving operations that cannot be implemented by thermal operations aided by any finite amount of quantum coherence. Our result implies that there are uncountably many Gibbs-preserving operations that require unbounded thermodynamic resources to implement, raising a question about employing Gibbs-preserving operations as available thermodynamics processes. This finding is a consequence of the general lower bounds we provide for the coherence cost of approximately implementing a certain class of Gibbs-preserving operations with a desired accuracy. We find that our lower bound is almost tight, identifying a quantity-related to the energy change caused by the channel to implement-as a fundamental quantifier characterizing the coherence cost for the approximate implementation of Gibbs-preserving operations.
We study the limitations on coherence evolutions under the constraints of thermodynamic laws, and focus on the optimal thermal operations (TO) reaching the bounds. For the qubit case, we prove that the TO that preserves maximal coherence can be reproduced by thermal operations involving only a single-mode reservoir, which we call single-mode thermal operations (STO). For higher dimensional systems, we derive tight limitations on coherence merging which are allowed by TO, and find that these bounds can also be reached by STO. By applying the coherence merging bounds to a two-qubit system, we observe that for some states, it is impossible to erase correlations while preserving the marginal states by TO. Our results reveal the strong ability of STO in coherence processing. Because interactions between quantum systems and single-mode reservoirs are realizable under recent techniques, our theoretical results are helpful to experimentally demonstrate the limitations of thermodynamic laws.
For finite-dimensional bipartite systems with additive endpoint Hamiltonians, the ergotropic gap measures the work advantage of global over independent local cyclic unitary control. We express this gap through mutual information and the Gibbs-relative entropies of the local and global passive states. The balance retains local passive-state corrections for arbitrary marginals and reduces to the established locally thermal ergotropy identity when those corrections vanish. Dropping the global correction gives a separability-independent upper bound whose exact slack is the global passive-state free-energy excess. We show that optimizing the reference temperature is equivalent to matching the Gibbs entropy to the joint-state entropy. The optimized result is precisely the equal-entropy Gibbs bound on global ergotropy minus the local ergotropies, rather than a new thermodynamic bound. We characterize saturation and temperature endpoints, including degenerate ground spaces. Werner states yield a unique implicit crossover with a separability-based majorization bound and a small-correlation bound-to-gap ratio approaching 3/2. Structured and random two-qubit comparisons illustrate complementary performance. We also specify pointwise quantum-battery applicability and the obstruction from persistent interactions.
The second law of thermodynamics places a limitation into which states a system can evolve into. For systems in contact with a heat bath, it can be combined with the law of energy conservation, and it says that a system can only evolve into another if the free energy goes down. Recently, it's been shown that there are actually many second laws, and that it is only for large macroscopic systems that they all become equivalent to the ordinary one. These additional second laws also hold for quantum systems, and are, in fact, often more relevant in this regime. They place a restriction on how the probabilities of energy levels can evolve. Here, we consider additional restrictions on how the coherences between energy levels can evolve. Coherences can only go down, and we provide a set of restrictions which limit the extent to which they can be maintained. We find that coherences over energy levels must decay at rates that are suitably adapted to the transition rates between energy levels. We show that the limitations are matched in the case of a single qubit, in which case we obtain the full characterization of state-to-state transformations. For higher dimensions, we conjecture that more severe constraints exist. We also introduce a new class of thermodynamical operations which allow for greater manipulation of coherences and study its power with respect to a class of operations known as thermal operations.
This work determines the sharp finite-dimensional onset of incompatibility for finite families of U(1)-covariant phase observables. In dimension two, every covariant phase observable is shown to be a convolutional postprocessing of a single canonical qubit phase observable. Consequently, every finite family of qubit covariant phase observables, including arbitrary finite coarse-grainings, is jointly measurable. In dimension three this universal compatibility breaks down. Four rank-two qutrit covariant phase observables are constructed whose binary interval coarse-grainings form a rigorously certified four-Specker family. Triple compatibility is established using explicit parent POVMs with exact-rational positivity tests and interval-bounded affine corrections, while four-way incompatibility is certified by independent dual semidefinite witnesses with exact rational normalization and interval evaluation. The work therefore establishes the sharp threshold d_min = 3 for finite-family incompatibility within this phase-covariant class. It also characterizes the qutrit canonical-smearing sector, proves robustness on a full-dimensional 28-parameter neighborhood, and provides a common-commutant certificate excluding a hidden qubit reduction. Supplementary material contains detailed analytic derivations and certificates. The accompanying reproducibility package provides verification code, exact/interval certificates, numerical summaries, and machine-readable outputs.