Landauer's principle sets a lower bound, kBTln⁡2k_B T \ln 2 per erased bit, on the heat dissipated by logically irreversible operations, but the bound is attained only in the quasistatic limit with an infinite Markovian thermal bath. This problem asks for a general characterization of when the Landauer bound is attainable or tight for the erasure of a quantum memory once these idealizations are lifted: finite operation time, finite-size heat baths, and non-Markovian or nonequilibrium environments in which fluctuations, quantum coherence, and system-bath correlations can no longer be ignored. Existing results supply regime-specific bounds (quasistatic finite-bath corrections, Markovian finite-time bounds with coherence terms, slow-driving strong-coupling corrections, collision-model non-Markovian conditions, and a tight classical finite-time bound), but no model-independent tight quantum bound or necessary-and-sufficient attainability conditions are known. The problem is stated at the natural generality of the source review (Rep. Prog. Phys. 88, 086001 (2025)) and remains open as of 2026.

Background

A logically irreversible operation is one whose output does not uniquely determine its input; resetting a memory to a fixed state (erasure) is the canonical example. Landauer's principle states that any such operation dissipates into the environment a heat QQ obeying the Landauer bound

β Q≥ΔS\beta \, Q \ge \Delta S

where β=1/(kBT)\beta = 1/(k_B T) is the inverse temperature of the bath, kBk_B is Boltzmann's constant, and ΔS\Delta S is the entropy decrease of the memory (for erasing one fully random bit, Q≥kBTln⁡2Q \ge k_B T \ln 2). The bound follows from the second law under strong idealizing assumptions: the erasure is performed quasistatically (infinitely slowly), the bath is infinitely large and remains in equilibrium, the system-bath coupling is weak, and the bath dynamics is Markovian, i.e. it has no memory of past interactions with the system. Under these assumptions the bound is attainable: a reversible isothermal protocol can approach kBTln⁡2k_B T \ln 2 arbitrarily closely, and experiments with classical mesoscopic memories have saturated it to within a few percent.

In the quantum regime several of these assumptions fail simultaneously. First, a finite operation time τ\tau forces the protocol away from the quasistatic limit, and the driving can generate quantum coherence in the energy eigenbasis (off-diagonal density-matrix elements), which changes the full statistics of the dissipated heat: coherence contributes non-negatively to every cumulant and produces rare, strongly dissipative events (Miller et al. 2020), and the dissipated heat acquires corrections that scale inversely with τ\tau (Vu and Saito 2022). Second, a finite-size bath is perturbed by the erasure itself: system-bath correlations, quantified by quantum mutual information (a measure of the total correlation, both classical and quantum, between two systems), and heat fluctuations enter the balance. Reeb and Wolf (2014) proved an improved bound with an explicit finite-size correction,

βQ≥ΔS+M(ΔS/N)>ΔS\beta Q \ge \Delta S + M(\Delta S/N) > \Delta S

for a bath of NN constituents, showing the naive bound is not tight for any finite bath; equality conditions and the tightness of the correction itself remain only partially understood. Third, non-Markovian environments, in which information and energy flow back from the bath to the system (information backflow), and nonequilibrium or non-thermal reservoirs (e.g. squeezed baths, whose fluctuations are reduced below the thermal level along one quadrature of the relevant oscillator, or coherence-bearing baths) can transiently violate the naive inequality; generalized bounds with correlation and interaction-energy terms restore consistency, but the known statements are tied to specific models such as collision models, in which the system interacts sequentially with auxiliary subunits, or to perturbative or slow-driving regimes. At strong system-bath coupling even the definitions of heat and work become ambiguous, since the interaction energy and correlated equilibrium states must be included.

For orientation, the state of the art splits by regime. Quasistatic, weak coupling: Landauer's bound, tight. Finite bath: Reeb-Wolf-type corrections, valid but not known tight in general. Nonequilibrium erasure with finite baths: bounds by Goold, Paternostro and Modi (2015), comparative tightness studies (Campbell et al. 2017), and a one-parameter bound built on the full counting statistics of the dissipated heat — its complete probability distribution rather than the average alone — which can be made arbitrarily tight in a parametric sense but is not an attained optimum (Guarnieri et al. 2017). Classical finite time: a provably tight finite-time Landauer bound with saturating optimal dynamics was derived by Lee, Lee, Kwon and Park (2022) from classical speed limits, which bound how fast a system can be driven in terms of its statistical distance and dissipation. Quantum finite time: only lower bounds are known — bounds for Markovian dynamics governed by a Lindblad master equation (the standard memoryless evolution equation for an open quantum system), carrying 1/τ1/\tau and coherence terms (Vu and Saito 2022), and an exact strong-coupling, non-Markovian finite-time correction for a single fermionic mode restricted to the slow-driving regime via a geometric approach to thermodynamics, which expresses dissipation as a metric distance between successively driven states (Rolandi and Perarnau-Llobet 2023). Experiments (a trapped-ion quantum Landauer equality by Yan et al. 2018; a molecular nanomagnet by Gaudenzi et al. 2018) have verified the principle but not saturation in the quantum regime. The 2025 review by Chattopadhyay, Misra, Pandit and Paul in Reports on Progress in Physics surveys exactly this landscape, reviewing the finite-time, nonequilibrium, and non-Markovian erasing processes as separate threads and identifying open quests such as a bound that accounts for the fluctuations of heat — more informative than the Clausius inequality — and generalizations to initially correlated system-environment states.

Problem Statement

Under what conditions is the Landauer erasure bound attainable or tight for quantum systems? The question is kept at the natural generality of the source: erasure of information stored in a quantum memory by any physical process, with the dissipative cost compared against the bound βQ≥ΔS\beta Q \ge \Delta S (per bit, Q≥kBTln⁡2Q \ge k_B T \ln 2). The characterization is sought along the three resource axes along which the idealized derivation fails — which the source review treats as separate open threads — asking how each modifies the bound and the excess dissipation of real erasure processes:

(1) Finite operation time: determine the provably tight excess dissipation above the Landauer cost as a function of the protocol duration, the memory's spectrum and coupling, and the quantum coherence generated in the energy eigenbasis, together with the protocols or dynamical conditions under which that excess is minimal (the quantum counterpart of the tight classical finite-time bound of Lee et al. 2022).

(2) Finite-size heat baths: determine how bath back-action, system-bath correlations (quantum mutual information), and heat fluctuations modify the bound beyond the Reeb-Wolf finite-size correction, and under what conditions on bath size, spectrum, and initial state the corrected bound is tight and attainable.

(3) Non-Markovian and nonequilibrium or non-thermal environments: determine under what conditions on the environment's memory (information backflow), coupling strength, and state the standard inequality holds, fails, or must be replaced by a generalized bound including correlation and interaction-energy terms, and which of these generalized bounds are tight rather than merely valid.

These three axes are the source's separate open threads, and the question asks how each modifies the bound and the excess dissipation of real erasure processes. New results are expected to be consistent with the established quasistatic, weak-coupling Markovian, slow-driving, and classical limits in the corresponding regimes. The three axes are linked in practice: fluctuations, coherence, and system-bath correlations all enter the same quantity, the dissipated heat of erasure.

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