Resources for implementing Gibbs-preserving channels

Problem

What are the exact one-shot coherence and work costs of implementing an arbitrary Gibbs-preserving channel by thermal operations? Let finite systems \(S\) and \(S'\) have Hamiltonians \(H_S\) and \(H_{S'}\) at inverse temperature \(\beta\), with Gibbs states

\begin{equation} \gamma_X:=\frac{e^{-\beta H_X}}{\operatorname{Tr}(e^{-\beta H_X})}, \qquad X\in\{S,S'\}. \tag{1} \end{equation}

A channel \(\Phi:S\to S'\) is Gibbs preserving when it maps the first state in Eq. (1) to the second. For approximation error \(\varepsilon\geq0\), define its quantum-Fisher-information coherence cost by

\begin{equation} F_c^{\varepsilon}(\Phi) :=\inf_{\substack{(R,H_R),\ \eta_R\in\mathcal D(R)\\ \mathcal T:S\otimes R\to S'\ \mathrm{thermal}}} \left\{ F_Q(\eta_R,H_R): D_{\mathrm{ch}}\!\left( \Phi,\mathcal T(\,\cdot\,\otimes\eta_R) \right)\leq\varepsilon \right\}, \tag{2} \end{equation}

where the infimum ranges over finite resource systems \(R\) with Hamiltonian \(H_R\), and \(\mathcal T\) is a thermal operation implemented with Gibbs ancillas and an energy-conserving unitary; set \(\inf\varnothing:=+\infty\). The distance \(D_{\mathrm{ch}}\) is the supremum of purified distance between the two channel outputs over inputs entangled with an arbitrary reference. If \(\eta_R=\sum_j\lambda_j\lvert j\rangle\!\langle j\rvert\), the quantity in Eq. (2) is

\begin{equation} F_Q(\eta_R,H_R) :=2\!\sum_{j,k:\,\lambda_j+\lambda_k>0} \frac{(\lambda_j-\lambda_k)^2}{\lambda_j+\lambda_k} \left\lvert\langle j\rvert H_R\lvert k\rangle\right\rvert^2. \tag{3} \end{equation}

Equation (3) fixes the normalization of the coherence measure used in Eq. (2). Determine Eq. (2), up to matching bounds, for every Gibbs-preserving \(\Phi\) and every \(\varepsilon\). Give the analogous tight deterministic work cost when an ideal battery begins and ends in sharp energy states and the battery’s energy loss is charged as work. Which structural features of \(\Phi\) determine whether the exact costs are finite, and what is their sharp divergence as \(\varepsilon\downarrow0\) when they are infinite at zero error?

Source

Tajima and Takagi explicitly leave a tight channel-by-channel characterization of coherence and work costs for Gibbs-preserving operations open [TT25].

Progress

Every thermal operation is Gibbs preserving and time-translation covariant, whereas a general Gibbs-preserving channel need not be covariant and may create energy coherence. The operation classes are therefore inequivalent [FOR15].
Universal optimal work rates are known in the independent and identically distributed limit, and one-shot implementations are known for important time-covariant channels [FBB21]. These results do not determine either cost requested above for every Gibbs-preserving channel.
Some Gibbs-preserving channels cannot be implemented exactly by a thermal operation aided by any finite amount of coherence. For pairwise reversible channels, lower bounds on Eq. (2) scale as \(1/\varepsilon^2\) and are essentially attained by suitable examples [TT25]. This disproves universal finite coherence sufficiency but does not give a tight formula for arbitrary channels.

Comment

The conclusion of [TT25] explicitly leaves a complete tight characterization of coherence cost for all Gibbs-preserving operations and the corresponding work-cost problem. Universal exact implementation with finite coherence is already known to be false; the open task is the channel-by-channel cost characterization.

References

[FOR15]: P. Faist, J. Oppenheim, and R. Renner, “Gibbs-Preserving Maps Outperform Thermal Operations in the Quantum Regime,” New Journal of Physics 17, 043003 (2015).DOI arXiv
[FBB21]: P. Faist, M. Berta, and F. G. S. L. Brandão, “Thermodynamic Implementations of Quantum Processes,” Communications in Mathematical Physics 384, 1709–1750 (2021).DOI arXiv
[TT25]: H. Tajima and R. Takagi, “Gibbs-Preserving Operations Requiring Infinite Amount of Quantum Coherence,” Physical Review Letters 134, 170201 (2025).DOI arXiv

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