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

[MarioKrenn6240]

What is the conjecture

This is problem #46 in the Open Quantum Problems collection.

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)
• Open Quantum Problems master list (for numbering/metadata)

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

• 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
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