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.
Comments
27 pages, v2: some clarifications about reasons and sources of joint obstruction in coherences transfer made, v3: slightly rewritten version
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.
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.
Abstract Whether genuine quantum effects, particularly quantum coherences, can offer an advantage to quantum devices is a topic of much interest. Here, we study a minimal model, the three-level V system coupled to two heat baths, and investigate the role of quantum coherences in heat transport in both the transient regime and in the nonequilibrium steady-state. In our model, energy is exchanged between the baths through two parallel pathways, which can be made distinct through the nondegeneracy of excited levels (energy splitting Δ) and a control parameter α , which adjusts the strength of one of the arms. Using a nonsecular quantum master equation of Redfield form, we succeed in deriving closed-form expressions for the quantum coherences and the heat current in the steady state limit for closely degenerate excited levels. By including three ingredients in our analysis: nonequilibrium baths, nondegeneracy of levels, and asymmetry of pathways, we show that quantum coherences are generated and sustained in the V model in the steady-state limit if three conditions, conjoining thermal and coherent effects are simultaneously met: (i) the two baths are held at different temperatures. (ii) Bath-induced pathways do not interfere destructively. (iii) Thermal rates do not mingle with the control parameter α to destroy interference through an effective local equilibrium condition. Particularly, we find that coherences are maximized when the heat current is suppressed. Although we mainly focus on analytical results in the steady state limit, numerical simulations reveal that the transient behavior of coherences contrasts the steady-state limit: large long-lived transient coherences vanish at steady state, while weak short-lived transient coherences survive, suggesting that different mechanisms are at play in these two regimes. Enhancing either the lifetime of transient coherences or their magnitude at steady state thus requires the control and optimization of different physical parameters.
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.
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.