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arXiv:2404.03479 · Physical Review Letters

Gibbs-Preserving Operations Requiring Infinite Amount of Quantum Coherence

Hiroyasu Tajima, Ryuji Takagi

Published Apr 29, 2025• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.134.170201

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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.
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.
In this brief paper, we compare two frameworks for characterizing possible operations in quantum thermodynamics. One framework considers thermal operations—unitaries which conserve energy. The other framework considers all maps which preserve the Gibbs state at a given temperature. Thermal operations preserve the Gibbs state; hence a natural question which arises is whether the two frameworks are equivalent. Classically, this is true—Gibbs-preserving maps are no more powerful than thermal operations. Here, we show that this no longer holds in the quantum regime: a Gibbs-preserving map can generate coherent superpositions of energy levels while thermal operations cannot. This gap has an impact on clarifying a mathematical framework for quantum thermodynamics.
Coherences between different energy levels are strongly constrained by thermodynamics. Here we ask whether several coherence transfers that can be optimized separately can also be optimized simultaneously by the same thermodynamic process. Using an energy-resolved Stinespring representation of thermal operations, we express elementary transfer amplitudes as inner products of bath-weighted vectors. Joint saturation imposes collinearity and phase-closure conditions on these vectors, which must all arise from one energy-preserving system--bath unitary. This gives necessary compatibility conditions for arbitrary finite-dimensional Hamiltonians. We show that joint incompatibility persists in a system with both degenerate and nondegenerate energy levels: rigorous computer-assisted semidefinite-program bounds, combined with explicit thermal constructions, prove that three individual thermal optima cannot be approached simultaneously. We then determine how well different coherences can be protected when their optima are incompatible. For a four-level family at infinite temperature, we obtain the complete closed retention region and the cost of balancing the coherences relative to their individual optima. At positive temperature, covariance and Gibbs preservation permit joint retention that no thermal operation can approach. Finite-bath constructions and certified semidefinite relaxations bound the balanced thermal performance. When two coherences carry independent phases, these bounds describe simultaneous preservation of interference visibility and phase Fisher information.
In this brief paper, we compare two frameworks for characterizing possible operations in quantum thermodynamics. One framework considers thermal operations—unitaries which conserve energy. The other framework considers all maps which preserve the Gibbs state at a given temperature. Thermal operations preserve the Gibbs state; hence a natural question which arises is whether the two frameworks are equivalent. Classically, this is true—Gibbs-preserving maps are no more powerful than thermal operations. Here, we show that this no longer holds in the quantum regime: a Gibbs-preserving map can generate coherent superpositions of energy levels while thermal operations cannot. This gap has an impact on clarifying a mathematical framework for quantum thermodynamics.
The minimum amount of thermodynamic work required in order to implement a quantum computation or a quantum state transformation can be quantified using frameworks based on the resource theory of thermodynamics, deeply rooted in the works of Landauer and Bennett. For instance, the work we need to invest in order to implement nnn independent and identically distributed (i.i.d.) copies of a quantum channel is quantified by the thermodynamic capacity of the channel when we require the implementation's accuracy to be guaranteed in diamond norm over the nnn-system input. Recent work showed that work extraction can be implemented universally, meaning the same implementation works for a large class of input states, while achieving a variable work cost that is optimal for each individual i.i.d. input state. Here, we revisit some techniques leading to derivation of the thermodynamic capacity, and leverage them to construct a thermodynamic implementation of nnn i.i.d. copies of any time-covariant quantum channel, up to some process decoherence that is necessary because the implementation reveals the amount of consumed work. The protocol uses so-called thermal operations and achieves the optimal per-input work cost for any i.i.d. input state; it relies on the conditional erasure protocol in our earlier work, adjusted to yield variable work. We discuss the effect of the work-cost decoherence. While it can significantly corrupt the correlations between the output state and any reference system, we show that for any time-covariant i.i.d. input state, the state on the output system faithfully reproduces that of the desired process to be implemented. As an immediate consequence of our results, we recover recent results for optimal work extraction from i.i.d. states up to the error scaling and implementation specifics, and propose an optimal preparation protocol for time-covariant i.i.d. states.
We consider physical situations where the resource theories of coherence and thermodynamics play competing roles. In particular, we study the creation of quantum coherence using unitary operations with limited thermodynamic resources. We find the maximal coherence that can be created under unitary operations starting from a thermal state and find explicitly the unitary transformation that creates the maximal coherence. Since coherence is created by unitary operations starting from a thermal state, it requires some amount of energy. This motivates us to explore the trade-off between the amount of coherence that can be created and the energy cost of the unitary process. We also find the maximal achievable coherence under the constraint on the available energy. Additionally, we compare the maximal coherence and the maximal total correlation that can be created under unitary transformations with the same available energy at our disposal. We find that when maximal coherence is created with limited energy, the total correlation created in the process is upper bounded by the maximal coherence, and vice versa. For two-qubit systems we show that no unitary transformation exists that creates the maximal coherence and maximal total correlation simultaneously with a limited energy cost.