The nonequilibrium cost of accurate information processing

The paper addresses how accurately information can be processed given limited resources away from thermal equilibrium. Many natural and technological processes require high-fidelity information processing (e.g., DNA replication with extremely low error rates). Fundamentally, information must deviate from thermal equilibrium, requiring initial nonequilibrium resources (e.g., clean bits). The central question is to determine the minimum nonequilibrium necessary to achieve a target accuracy for general information-processing tasks, including those where perfect implementation is impossible (such as quantum cloning). The authors develop a framework that links abstract information-processing accuracy to thermodynamic resource costs, aiming to establish universal efficiency limits valid for all quantum-mechanically allowed machines.

Recent work at the interface of quantum information and thermodynamics has developed resource-theoretic approaches to nonequilibrium and work cost beyond macroscopic limits, characterizing allowed state transitions, logical work costs, and quantum erasure/work extraction. Stochastic thermodynamics has also related accuracy and entropy production in specific classical Markovian models. However, prior results often focus on specific channels or steady states and may not identify the minimal nonequilibrium cost for a desired accuracy across all implementations of a task, especially when perfect implementation is impossible. The present work builds on Gibbs-preserving and thermal operations frameworks, single-shot entropies (min/max), and time-reversal concepts (including Petz recovery) to derive universal, task-dependent limits.

The authors formulate information-processing tasks via operational performance tests: prepare input states (possibly entangled with a reference), apply the machine locally, optionally act on the reference, and perform a joint measurement yielding a score. The worst-case expected score defines the accuracy F_T(M) for a channel M. Tasks are represented by positive semidefinite performance operators (Ω_x) acting on input-output Hilbert spaces in the Choi representation, giving F_T(M) = min_x Tr[M Ω_x]. The nonequilibrium resource is quantified using an information battery of clean qubits within a Gibbs-preserving map framework; the nonequilibrium cost c(M, Π_A) is the minimal number of pure qubits required to implement M on the input subspace Π_A (commuting with the input Hamiltonian). The task-dependent cost at target accuracy F is c_T(F) := min{c(M, Π_A) | F_T(M) ≥ F}. An exact expression for c_T(F) is provided via a semidefinite program, and a computable lower bound is obtained by fixing certain dual variables. A key construct is the time-reversed task T^rev, defined through a Crooks/Petz-style time reversal that exchanges roles of states and observables relative to the Gibbs states of input/output systems. The reverse accuracy F_T^rev is the maximum over channels of the corresponding reverse test, and the reverse entropy K_T := -log F_T^rev is expressed equivalently as K_T = max_p H_min(A|B)_ω, where ω is built from convex combinations of performance operators and H_min is the conditional min-entropy. Bounds on K_T for specific task structures are derived using min/max relative entropies D_min and D_max. A smoothed variant of reverse entropy is defined for approximate/asymptotic settings, leading to an asymptotic rate lower bounded by differences of quantum relative entropies S(·||·). For restricted classes of channels (entanglement binding, including entanglement-breaking/classical machines), a stronger limit is proven using partial transpose symmetry, yielding a stricter nonequilibrium-accuracy tradeoff than the general quantum case.

• Universal tradeoff: For any task T with time-invariant input subspace, the nonequilibrium cost satisfies c(F) ≤ K_T + log F, where K_T = −log F_T^rev is the reverse entropy determined solely by the task's time-reversed version. Equivalently, with c clean qubits, the achievable accuracy obeys F_T(c) ≤ 2^(−c). • Reverse entropy characterizations: K_T admits a conditional min-entropy form K_T = max_p H_min(A|B)ω. For deterministic classical computations y = f(x), K_T = D_max(p_f || q_B), the max Rényi divergence between the distribution of outputs produced from thermal inputs and the output Gibbs distribution. Physical constraints can make K_T > 0 even without deviations from equilibrium (e.g., transposition has K_trans = log[(d+1)/2]). • Attainability: If the bound is achievable at F_max (the physics-allowed maximal accuracy), then it is achievable for all F in [F_min, F_max], with F_min = 2^(−min K). The bound is tight for all deterministic classical computations and their quantum extensions, and for universal quantum cloning. • Work cost linkage: Work cost W(M, Π_A) under thermal operations is bounded by W ≥ kT ln 2 × c. For erasure of arbitrary states to the ground state, the exact minimum work is W_erase(F) = ΔA + kT ln F for all F in [e^(−ΔA/(kT)), 1]. • Classical cloning: For N → N′ cloning of orthonormal-basis states, K_clon = ΔN ΔA_max / (kT ln 2), giving the exact minimum nonequilibrium cost C_clon(F) = ΔN ΔA_max / (kT ln 2) + log F. For fully degenerate systems, W_clon(F) = kT ln 2 × C_clon(F). • Quantum cloning: Universal quantum cloning has the same accuracy–nonequilibrium tradeoff as classical cloning (same K), though quantum fidelity is capped below 1 by no-cloning. • Limits for classical/entanglement-binding machines: For entanglement binding channels, the cost obeys c*(F) ≥ 2 max(κ_T, κ_X) + log F, with κ_X the reverse entropy of the transpose task. For cloning, κ_clon′ ≤ κ_clon + log(d{N+N′}/d_N), implying classical machines need more nonequilibrium than optimal quantum machines. In fully degenerate cases, these become equalities. • Storage/transmission benchmark (qubits): The minimal nonequilibrium cost over all entanglement-binding (classical) machines achieving fidelity F is C_store/transmit(F) = log[ F^F / (2F − 1)^{2(1−F)} ] for F ∈ [F_min, F_max] with F_max = 2/3 and F_min = (e + 1)/(2 + e). Any setup achieving fidelity F with work below kT ln 2 × C_store/transmit(F) demonstrates a quantum thermodynamic advantage; this can certify quantum advantage even for fidelities below 2/3 when accounting for work constraints. • Asymptotics: For many-copy, smoothed tasks, the per-copy smooth reverse entropy satisfies κ_T^id ≥ max_x S(ρ_x || Γ_B) − S(ρ_x || Γ_A), coinciding with the thermodynamic capacity in settings allowing perfect transformations, showing asymptotic achievability of the universal tradeoff.

The results establish a task-dependent, easily computable reverse entropy that governs the ultimate efficiency of accurate information processing across quantum systems. The framework unifies nonequilibrium resource costs with operational accuracy tests and connects to single-shot entropies. For exact state generation and work extraction, suitable accuracy measures recover known single-shot work costs via D_max and D_min. The bounds sharpen for classical/measurement-based (entanglement-binding) machines, proving a thermodynamic advantage for genuinely quantum devices in cloning, storage, and transmission. The approach also provides practical benchmarks: for qubits, a closed-form classical cost curve as a function of fidelity enables certification of quantum advantage under work/nonequilibrium constraints, even below the usual 2/3 fidelity threshold. In the asymptotic regime, smoothing leads to relative-entropy rates (thermodynamic capacity), and the tradeoff becomes achievable for all allowed processes. Compared to thermodynamic uncertainty relations in stochastic thermodynamics, the present tradeoff addresses general information-processing accuracy beyond steady states and specific models, suggesting opportunities to link these approaches through concrete physical implementations.

The paper introduces the reverse entropy as a universal, task-dependent quantity that sets fundamental limits on the nonequilibrium resources required for accurate information processing. A general bound c(F) ≤ K_T + log F holds for all quantum tasks, is tight for broad classes including classical computations, their quantum extensions, universal quantum cloning, and erasure, and yields operational benchmarks linking fidelity and work cost. Applications demonstrate optimal tradeoffs for erasing, cloning, storage, and transmission, and reveal thermodynamic advantages of quantum over classical (measurement-based) machines. In asymptotic settings, the framework connects to thermodynamic capacity, establishing achievability of the tradeoff. This provides targets for device design and a methodology to certify quantum thermodynamic advantages experimentally. Future research may integrate these information-theoretic bounds with stochastic thermodynamics in concrete models approaching the ultimate efficiency limits.

The work is primarily theoretical and relies on resource-theoretic models (Gibbs-preserving and thermal operations) and information batteries. Exact evaluation of c_T(F) requires solving semidefinite programs, which is tractable only for low-dimensional systems; hence, universal bounds are emphasized. Attainability is proven for specific families of tasks (deterministic classical computations, their quantum extensions, universal quantum cloning, and erasure), while for other tasks the bound may be non-tight. Some results assume input subspaces commute with the Hamiltonian, and certain equalities hold exactly only for fully degenerate Hamiltonians. Experimental certification requires accurate accounting of work/nonequilibrium resources, and extensions to complex noise models may need smoothed or approximate formulations.