Formalize Open Quantum Problem #46: Thermodynamic implementation of Gibbs-Preserving Maps

### What is the conjecture

This is problem [#46](https://oqp.iqoqi.oeaw.ac.at/thermodynamic-implementation-of-gibbs-preserving-maps) in the [Open Quantum Problems collection](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #46: “Thermodynamic implementation of Gibbs-Preserving Maps”).**
> In a one-shot thermodynamic model with physically meaningful free operations (for instance, thermal operations), determine the least extra resources needed to realize an arbitrary Gibbs-preserving channel.

A standard formalization is as a **channel-implementation / resource-cost** problem in quantum thermodynamics:

* Fix a finite-dimensional quantum system `S` with Hamiltonian `H_S` and inverse temperature `β`.
* Its Gibbs state is
`γ_S := exp(-β H_S) / Z_S`,
where `Z_S = Tr(exp(-β H_S))`.
* A **Gibbs-preserving map** is a completely positive, trace-preserving map `Φ` on states of `S` such that
`Φ(γ_S) = γ_S`.
* A **thermal operation** is a channel of the form
`T(ρ_S) = Tr_B[ U (ρ_S ⊗ γ_B) U^† ]`,
where `B` is a heat bath / ancilla with Gibbs state `γ_B = exp(-β H_B)/Z_B`, and `U` is a unitary satisfying
`[U, H_S + H_B] = 0`.

Every thermal operation is Gibbs-preserving, but not every Gibbs-preserving map is a thermal operation. In particular:

* Thermal operations are **time-covariant**, i.e.
`T(e^{-itH_S} ρ e^{itH_S}) = e^{-itH_S} T(ρ) e^{itH_S}` for all `t ∈ R`.
* A general Gibbs-preserving map need not be time-covariant and may create coherence in the energy eigenbasis.

Thus the problem can be phrased as follows:

* Fix a target Gibbs-preserving map `Φ` and an error tolerance `ε ≥ 0`.
* Allow a physically implementable free operation `T` (typically a thermal operation) acting on the input system together with auxiliary resource systems prepared in an initial state `ω_aux`.
* Ask whether one can achieve
`‖ Tr_aux ∘ T( (·) ⊗ ω_aux ) - Φ ‖_⋄ ≤ ε`.
The auxiliary systems may encode, for example, a clock / time reference, coherence resource, work battery, or other ancillas.
* The implementation should be **one-shot** and **universal**, i.e. work for all inputs, not only asymptotically for many i.i.d. uses.

Then the open problem is:

* **Main question / open resource-theoretic problem:** determine, for a general Gibbs-preserving map `Φ`, the **minimal additional resources** needed to implement `Φ` in such a one-shot framework.
* In particular, characterize or bound:
* the minimal **clock / time-reference** resource needed to break time-translation symmetry;
* the minimal **coherence** resource (for example, number of copies of `|+⟩ := (|0⟩ + |1⟩)/√2` with a suitable energy gap, or an equivalent bounded-energy reference state);
* the minimal **work / purity** resource (e.g. battery investment or pure-state ancillas);
* which Gibbs-preserving maps are implementable with **finite** resources at all, and which require resources diverging as `ε → 0`.

The OQP page also highlights that the exact “best” one-shot framework is itself part of the problem. Natural variants include:

* explicit batteries / work-storage systems;
* allowing discarded ancillas to remain interacting with the output;
* arbitrary initial resource states with bounded energy spread;
* more “inherently quantum” notions of work.

Partial results highlighted on the OQP page include:

* If `H_S = 0`, then the Gibbs state is maximally mixed, Gibbs-preserving maps are exactly **unital** maps, and thermal operations reduce to **noisy operations**.
* In that trivial-Hamiltonian case, unital maps are strictly more general than noisy operations, but any unital map can be approximately implemented in one shot with a work cost scaling like `W ~ log(f(ε))`, independently of the system dimension.
* For **time-covariant** maps acting on **time-covariant** input states, Corollary 8.3 of the arXiv version of Faist–Berta–Brandão gives a work-cost expression in terms of hypothesis-testing relative entropy and a Stinespring dilation.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #46)](https://oqp.iqoqi.oeaw.ac.at/thermodynamic-implementation-of-gibbs-preserving-maps)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)

Key related references (as listed on the OQP page):

* D. Janzing et al., “The thermodynamic cost of reliability and low temperatures: Tightening Landauer’s principle and the Second Law” (arXiv: quant-ph/0002048)
(early resource-theoretic / Gibbs-preserving perspective)
* F. G. S. L. Brandão et al., “Resource Theory of Quantum States Out of Thermal Equilibrium” (Phys. Rev. Lett. 111, 250404 (2013); arXiv: 1111.3882)
* M. Horodecki and J. Oppenheim, “Fundamental limitations for quantum and nanoscale thermodynamics” (Nat. Commun. 4, 2059 (2013); arXiv: 1111.3834)
(thermal operations and thermomajorization)
* M. Horodecki, P. Horodecki, and J. Oppenheim, “Reversible transformations from pure to mixed states and the unique measure of information” (Phys. Rev. A 67, 062104 (2003); arXiv: quant-ph/0212019)
(noisy operations, i.e. the trivial-Hamiltonian analogue of thermal operations)
* P. Faist, J. Oppenheim, and R. Renner, “Gibbs-preserving maps outperform thermal operations in the quantum regime” (New J. Phys. 17, 043003 (2015); arXiv: 1406.3618)
(shows that Gibbs-preserving maps can be strictly more powerful than thermal operations)
* L. J. Landau and R. F. Streater, “On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras” (Linear Algebra Appl. 193, 107 (1993)); see also U. Haagerup and M. Musat, “Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras” (Commun. Math. Phys. 303, 555 (2011); arXiv: 1009.0778)
(strict separation between unital maps and noisy operations)
* P. Faist et al., “The minimal work cost of information processing” (Nat. Commun. 6, 7669 (2015); arXiv: 1211.1037)
(approximate implementation in the trivial-Hamiltonian / unital special case)
* S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information” (Rev. Mod. Phys. 79, 555 (2007); arXiv: quant-ph/0610030)
(background on clocks / reference frames and breaking time-translation symmetry)
* M. P. Woods, R. Silva, and J. Oppenheim, “Autonomous Quantum Machines and Finite-Sized Clocks” (Ann. Henri Poincaré 20, 125–218 (2019); arXiv: 1607.04591)
(finite clock resources)
* I. Marvian, “Coherence distillation machines are impossible in quantum thermodynamics” (Nat. Commun. 11, 25 (2020); arXiv: 1805.01989)
(coherence as a thermodynamic resource)
* P. Faist, M. Berta, and F. G. S. L. Brandão, “Thermodynamic Implementations of Quantum Processes” (Commun. Math. Phys. 384, 1709–1750 (2021); arXiv: 1911.05563)
(work-cost formulas and universal implementations in important special cases, especially time-covariant settings)

Recent additional progress directly relevant to the OQP question:

* H. Tajima and R. Takagi, “Gibbs-Preserving Operations Requiring Infinite Amount of Quantum Coherence” (Phys. Rev. Lett. 134, 170201 (2025); arXiv: 2404.03479)
(shows that some Gibbs-preserving operations cannot be implemented by thermal operations aided by any **finite** amount of quantum coherence)

### Prerequisites needed

* Finite-dimensional quantum mechanics: density matrices, Hamiltonians, Gibbs states, partial trace, CPTP maps
* Resource-theoretic thermodynamics: thermal operations, Gibbs-preserving maps, noisy / unital operations, one-shot vs asymptotic viewpoints
* Symmetry / covariance ideas: time-translation symmetry, coherence in the energy eigenbasis, clocks / reference frames
* Linear algebra / operator theory: completely positive maps, Stinespring dilations, norms / distances on channels
* (Helpful for partial results) thermomajorization, semidefinite-program viewpoints, and basic one-shot entropic quantities such as hypothesis-testing relative entropy

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-81 (Quantum theory)
* ams-82 (Statistical mechanics, structure of matter)
* ams-47 (Operator theory)
* ams-94 (Information and communication theory)

### Choose either option

* [ ] I plan on adding this conjecture to the repository
* [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

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Start with the Open Quantum Problems page for Problem #46 and the listed references, especially the Faist–Berta–Brandão work. No repository file, test, or formalization entry point is named; define a tractable theorem scope first, and consider the work complete only when that statement is formalized in the repository.

Written by the indexing model from the issue text.