Rigorous minimal-work bounds for reliable information-bearing switches and memories as a function of error probability and temperature

Implementing a reliable switch or memory element — a device that moves a physical system between two ergodicity-broken states and holds information there — is heuristically known to cost work growing as $k_B T\ln(1/\epsilon)$ for error probability $\epsilon$, with stored-information lifetime $\tau\simeq\tau_0/\epsilon$. These formulas rest only on specific solvable models (Kish's classical electronic switch, Alicki's spin-boson quantum switch, in which $\epsilon=e^{-\bar W/\Theta}$ with $\Theta$ an effective noise temperature including zero-point energy) and on analogy. The open problem is to derive rigorous, model-independent bounds on the minimal work needed to implement a reliable switch or memory at temperature $T$ and error probability $\epsilon$, together with the associated lifetime-cost relation, and to determine whether such bounds are optimal or can be beaten in broader model classes — non-Gaussian, many-body, engineered, and quantum or low-temperature realizations — where the stability-accuracy cost may dominate the Landauer term. Rigorous results exist only under restricted control classes, so the question of universality remains genuinely open.

Background

Landauer's principle — erasing one bit of information in contact with a heat bath at temperature $T$ costs at least $k_B T\ln 2$ of work (R. Landauer, IBM J. Res. Dev. 5, 183 (1961)) — is the foundational statement on the thermodynamic cost of information processing. Far less is known about the cost of reliability: making a device that holds or transmits a bit with small error probability $\epsilon$.

R. Alicki approached this question through ergodicity breaking. A physical system whose phase space (or, quantum mechanically, whose Hilbert space) is decomposed by external constraints into disconnected ergodic components can carry information — one bit per two components; ergodicity breaking means the system's dynamics does not explore the full state space, so each component acts as a metastable memory state. The constraints are never perfect: they are realized by finite energy barriers, which thermal noise renders penetrable, so stored information has a finite lifetime (estimable by Kramers theory, the standard theory of thermally activated barrier crossing), and probability tails leaking into the wrong component produce an encoding error $\epsilon$. A switch is the elementary device that moves the system between two such components and back.

Two solvable switch models have been analyzed: L. B. Kish's classical electronic switch (IEE Proc.-Circ. Devices Syst. 151, 190 (2004)) and Alicki's quantum switch (arXiv:1305.4910 (2013)), a spin-boson model — a two-level system coupled to a bosonic (harmonic-oscillator) bath that realizes a double-well potential for the pointer. In both, the minimal work to switch with error $\epsilon\ll 1$ is

$W \simeq k_B T\ln\frac{1}{\epsilon},$

and the lifetime of the stored information scales as $\tau\simeq\tau_0/\epsilon$, where $\tau_0$ is the relaxation time of the system without ergodicity-breaking constraints. In the quantum model the more precise statement is $\epsilon=e^{-\bar W/\Theta}$, i.e. $\bar W=\Theta\ln(1/\epsilon)$, where $\Theta$ is an effective 'noise temperature' — the mean energy of the oscillator mode including zero-point energy — so $\Theta\simeq T$ at high temperature but $\Theta\simeq\omega_0/2$ at low temperature: the cost does not vanish as $T\to 0$ and always exceeds the Landauer value $k_B T\ln 2$. L. Brillouin (J. Appl. Phys. 24, 1152 (1953)) had obtained the same form for a photonic measurement, and a precursor trade-off between minimal dissipation and maximal error goes back to Landauer and Woo (J. Appl. Phys. 42, 2301 (1971)). Alicki's notes 'Information is not physical' (arXiv:1402.2414) and 'Information is not a thermodynamic resource' (arXiv:1801.05237) pose the cost question in this generality; the optimality of the model-derived relations is argued there only by analogy with the saturation of the Heisenberg uncertainty relation by Gaussian states — explicitly heuristic.

Subsequent stochastic thermodynamics (the theory of entropy production in fluctuating systems far from equilibrium) has produced rigorous but conditional results. Riechers, Boyd, Wimsatt, and Crutchfield (Phys. Rev. Research 2, 033524 (2020)) proved that under time-symmetric control — driving protocols invariant under time reversal — acting on metastable memories, the dissipation per nonreciprocal logical transition (one not equal to its own time reverse) is at least $k_B T\ln(1/\epsilon)$ and diverges as $\epsilon\to 0$; the same paper states that the bound does not apply to enlarged autonomous systems and poses the general error-dissipation trade-off surface and its quantum extension as open. Wimsatt et al. (J. Stat. Phys. 183 (2021)) refined this for erasure. Analogous rigorous trade-offs hold within other specific frameworks — KL-control, a control framework whose cost functional is the Kullback–Leibler divergence between the controlled process and the uncontrolled relaxation dynamics (Gopalkrishnan, Entropy 18, 165 (2016)), an optimal-bit design class (Deshpande et al., Proc. R. Soc. A 473, 20170117 (2017)), a refresh-based model in which the maintenance cost at fixed $\epsilon$ can be reduced by more frequent, cheaper refreshes (Chiuchiù et al., Phys. Rev. A 97, 052108 (2018)), and finite-time initialization with cost equal to the Landauer term plus a thermodynamic-length term, i.e. dissipation growing with the squared distance traversed in the space of equilibrium states (Ma et al., Phys. Rev. E 106, 034112 (2022)) — each relative to a stated class of devices, controls, or protocols. No result establishes, or refutes, a model-independent bound.

Problem Statement

Determine rigorous, model-independent bounds on the minimal work $W_{\min}(\epsilon,T)$ required to implement a reliable information-bearing switch or memory element — a physical device that moves a system between two ergodicity-broken (metastable) states and holds information there with total error probability $\epsilon$ — at temperature $T$, together with the associated lifetime-reliability relation (the dependence of the information storage time $\tau$ on the invested work, barrier height, and control protocol). The heuristic relations $W\simeq k_B T\ln(1/\epsilon)$, $\epsilon=e^{-\bar W/\Theta}$ with $\Theta$ the effective noise temperature including zero-point energy, and $\tau\simeq\tau_0/\epsilon$ currently rest on specific solvable models and on analogy, and are valid only for $\epsilon\ll 1$. The high-temperature restriction applies only to the $T$-form $W\simeq k_B T\ln(1/\epsilon)$; the low-temperature regime is governed by $\epsilon=e^{-\bar W/\Theta}$. All three of the following parts are jointly required:

(i) prove, or disprove, a model-independent lower bound on the minimal work of reliable switching and storage as a function of $\epsilon$ and $T$ that accounts for the finite energy barriers realizing ergodicity-breaking constraints and for their relaxation times; here 'model-independent' means the bound holds for every device in the ergodicity-breaking class described above, and 'minimal work' is the infimum over all control protocols acting on such a device, not only over a restricted control class (such as time-symmetric driving) or a single solvable model;

(ii) determine whether such bounds are optimal — i.e., tight — across broader model classes, such as non-Gaussian, many-body, and engineered realizations of switches and memories, or else construct explicit realizations that beat the $\ln(1/\epsilon)$ scaling at fixed $T>0$;

(iii) settle the quantum-regime extension, in which work and heat must be consistently defined in quantum models with dynamically induced superselection rules — that is, effective restrictions on the observable algebra that make information sectors dynamically disconnected — and the cost is governed by the noise temperature $\Theta$ including zero-point fluctuations rather than by $T$ alone.

These components jointly serve one objective: a fundamental thermodynamic limit for reliable information-processing hardware, fixing when the stability-accuracy cost of storing information dominates the Landauer term $k_B T\ln 2$.

References

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