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Information Erasure Inside Action Quanta

Brian Mulloy

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Quantum states are operationally indistinguishable within finite-support Heisenberg cells. By coupling these de Gosson ``blobs'' into conjugate pairs of action quanta, we show that Landauer information erasure exhausts the available time and energy, causing the classical continuum to collapse into discrete configurations. Numerical results show that at low action, eigenstates are dense, but as action increases, they become sparse---consistent with the experimental difficulty in isolating pure states---stabilizing into the classical expectations of the correspondence principle. This model establishes an information-action constraint where finite capacity forces least action coarse-graining and the emergence of quantum mechanics.

DOI
10.17605/osf.io/f7bck

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Quantum states are operationally indistinguishable within finite-support Heisenberg cells. By coupling these de Gosson ``blobs'' into conjugate pairs of action quanta, we show that Landauer information erasure exhausts the available time and energy, causing the classical continuum to collapse into discrete configurations. Numerical results show that at low action, eigenstates are dense, but as action increases, they become sparse---consistent with the experimental difficulty in isolating pure states---stabilizing into the classical expectations of the correspondence principle. This model establishes an information-action constraint where finite capacity forces least action coarse-graining and the emergence of quantum mechanics.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
Irreversibility has a fundamental thermodynamic cost, erasing information inevitably generates heat. This connection is quantified by the Landauer bound, which gives the minimum dissipation needed to erase a single bit of information. While this bound applies in both classical and quantum settings, it is saturated only in idealised limits of infinite resources. Here, we provide a unified first principles description of finite-resource erasure in both classical and quantum systems. We begin by proving the communal folklore that in the idealised regime the erasure cost of a bit encoded in a quantum or classical system is the same. Despite this, we show that their practical implementation differs substantially: achieving comparable erasure quality in quantum systems requires more control, larger accessible energy gaps and longer operation times. Classical protocols can achieve the erasure of a comparable quantum protocol under far weaker constraints which we expose in trade-off relations. Our results explain why practical erasure schemes fall short of Landauer's bound and show that classical systems enjoy several fundamental thermodynamic advantages.
One of the outstanding challenges to information processing is the eloquent suppression of energy consumption in the execution of logic operations. The Landauer principle sets an energy constraint in deletion of a classical bit of information. Although some attempts have been made to experimentally approach the fundamental limit restricted by this principle, exploring the Landauer principle in a purely quantum mechanical fashion is still an open question. Employing a trapped ultracold ion, we experimentally demonstrate a quantum version of the Landauer principle, i.e., an equality associated with the energy cost of information erasure in conjunction with the entropy change of the associated quantized environment. Our experimental investigation substantiates an intimate link between information thermodynamics and quantum candidate systems for information processing.
Abstract Finite-dimensional local Hilbert spaces are widely used as microscopic regulators and have also been proposed as a fundamental information-capacity property of space-time. A finite local dimension, however, is not equivalent to a spatial lattice spacing, and neither is equivalent to a coarse-graining scale. We formulate a regulator-resolved framework that keeps these three notions separate and defines finite-information deviations operationally as the failure of microscopic and effective time evolution to commute with a specified coarse-graining channel. The resulting deviation is a state- and observable-dependent quantity in a common effective state space, rather than a formal subtraction of generators acting on different domains. We derive a decomposition into capacity, spatial-discretization, and coarse-graining contributions; a finite-time capacity bound controlled by discarded state weight and leakage through the cutoff boundary; observable error bounds; and a sharp continuity bound for subsystem entropies. We further show that block entropy and averaged one-cell entropy differ by a relative-entropy measure of total correlation and therefore cannot be interchanged. An information-geometric expansion of neighboring reduced states yields a nonlinear sigma-model effective action on the manifold of coarse-grained states. A single entropy field follows only under an additional one-dimensional closure condition. Finally, we test the framework in an exactly evolved periodic Klein--Gordon chain with independently variable lattice spacing and local number-state cutoff. Spectral peaks are extracted from dynamical matrix elements rather than imposed. Ground-state, dispersion, reduced-state, and entropy errors converge rapidly with local capacity, while the remaining lattice--continuum discrepancy persists at fixed spacing. The results provide a controlled mathematical and numerical foundation for finite-information models without claiming a unique microscopic theory of quantum gravity.
Quantum state engineering and quantum computation rely on information erasure procedures that, up to some fidelity, prepare a quantum object in a pure state. Such processes occur within Landauer's framework if they rely on an interaction between the object and a thermal reservoir. Landauer's principle dictates that this must dissipate a minimum quantity of heat, proportional to the entropy reduction that is incurred by the object, to the thermal reservoir. However, this lower bound is only reachable for some specific physical situations, and it is not necessarily achievable for any given reservoir. The main task of our work can be stated as the minimisation of heat dissipation given probabilistic information erasure, i.e., minimising the amount of energy transferred to the thermal reservoir as heat if we require that the probability of preparing the object in a specific pure state | φ 1 ⟩ be no smaller than p φ 1 max − δ . Here p φ 1 max is the maximum probability of information erasure that is permissible by the physical context, and δ ⩾ 0 the error. To determine the achievable minimal heat dissipation of quantum information erasure within a given physical context, we explicitly optimise over all possible unitary operators that act on the composite system of object and reservoir. Specifically, we characterise the equivalence class of such optimal unitary operators, using tools from majorisation theory, when we are restricted to finite-dimensional Hilbert spaces. Furthermore, we discuss how pure state preparation processes could be achieved with a smaller heat cost than Landauer's limit, by operating outside of Landauer's framework.