The classical limit of quantum mechanics through coarse-grained measurements

Borivoje Daki\'c; Carlo Cepollaro; Fatemeh Bibak; Nicol\'as Medina S\'anchez; \v{C}aslav Brukner

arxiv: 2503.15642 · v2 · submitted 2025-03-19 · 🪐 quant-ph · physics.class-ph

The classical limit of quantum mechanics through coarse-grained measurements

Fatemeh Bibak , Carlo Cepollaro , Nicol\'as Medina S\'anchez , Borivoje Daki\'c , \v{C}aslav Brukner This is my paper

Pith reviewed 2026-05-22 22:43 UTC · model grok-4.3

classification 🪐 quant-ph physics.class-ph

keywords quantum-to-classical transitioncoarse-grained POVMEhrenfest timeHamiltonian flowphase spacequantization limitjoint measurabilityprobability density

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The pith

When the resolved phase-space area greatly exceeds Planck's constant, quantum statistics from any state admit an effective classical description via coarse-grained measurements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that modeling measurements as continuous coarse-grained positive operator-valued measures with cells much larger than ħ makes quantum mechanics produce classical-looking outcomes. Coarse-grained observables become approximately jointly measurable and the induced probability density stays positive. The exact evolution equation for this density reduces to classical Hamiltonian flow in the strong coarse-graining regime, with non-Liouville terms suppressed up to the Ehrenfest time. The smoothed Hamiltonian over the cell matches the original classical Hamiltonian when it varies little across the cell, closing the loop between quantization and the classical limit. Repeated measurements then produce records that stay near classical trajectories with high probability.

Core claim

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 quantiza

What carries the argument

Continuous coarse-grained POVMs with phase-space cells large compared to ħ, which enforce joint measurability, positivity of the induced density, and Hamiltonian smoothing for the dynamics.

If this is right

Coarse-grained observables become approximately jointly measurable.
The induced probability density remains positive.
Non-Liouville corrections in the density's evolution are suppressed up to an Ehrenfest time.
The dynamics reduce to classical Hamiltonian flow generated by the cell-smoothed Hamiltonian.
Repeated finite-resolution measurements produce stochastic records confined to tubes around classical trajectories with high probability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same coarse-graining mechanism could explain why classical behavior appears in systems where natural observation resolution is finite, such as in mesoscopic devices.
It offers a route to test the quantum-to-classical transition by tuning measurement cell size in experiments rather than relying on environmental decoherence.
The framework implies that the quantization-classical limit cycle remains consistent even for single-particle systems under realistic finite-resolution observation.

Load-bearing premise

Modeling all measurements as continuous coarse-grained POVMs whose cells are large compared to ħ is enough to guarantee joint measurability, positive densities, and suppression of non-Liouville terms up to the Ehrenfest time.

What would settle it

A concrete calculation or experiment showing that, for some quantum state and POVM cell size much larger than ħ, the induced density becomes negative or the flow deviates from classical Hamiltonian evolution before the Ehrenfest time would falsify the claim.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean operational derivation of the classical limit from coarse-grained POVMs with an explicit loop-closing step, but the claimed generality for arbitrary POVMs needs checking against the actual construction.

read the letter

The core advance is the use of continuous coarse-grained POVMs to get an exact evolution equation for the induced phase-space density, followed by showing that non-Liouville terms drop out in the strong-coarse-graining limit up to Ehrenfest time and that the smoothed Hamiltonian recovers the original classical one. That closes the quantization-classical-limit cycle in a way that stays tied to finite-resolution measurements. The paper does this without extra postulates beyond the POVM modeling choice, which is a useful tightening of existing phase-space approaches. It also sketches how repeated measurements stay confined to classical tubes with high probability. That part is straightforward and operational. The soft spot is the stress-test point on generality: the abstract presents the positivity, joint measurability, and suppression as following from cell area ≫ ħ alone, but those properties are not automatic for every partition-of-unity POVM satisfying only an area bound. If the derivation actually restricts to a subclass (Gaussian kernels or similar), the claim needs to be stated more narrowly. The manuscript should make the explicit form of the POVM elements and the error estimates visible so readers can judge how much structure is doing the work. This is aimed at foundations people who care about operational derivations and at quantum-information researchers thinking about resolution limits. It is coherent on its own terms and engages the literature directly, so it deserves a serious referee even if revisions are needed on the generality statement.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to derive the classical limit of quantum mechanics from finite-resolution measurements modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared to ħ, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable, the induced probability density is positive, and the exact evolution equation for this density has non-Liouville corrections suppressed up to an Ehrenfest time, yielding classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across the cell, the smoothed Hamiltonian reduces to the original classical Hamiltonian, closing the quantization-classical-limit loop. Repeated finite-resolution measurements generate stochastic records confined with high probability to tubes around classical trajectories.

Significance. If the derivations hold with the claimed generality, the work provides a unified operational framework for the quantum-to-classical transition applicable to both microscopic and macroscopic systems while establishing consistency of the quantization-classical-limit cycle. Explicit derivations of the evolution equation, positivity, joint measurability, and error suppression would constitute a substantive contribution.

major comments (2)

[Abstract] Abstract (paragraph beginning 'Here we derive'): The central claim that continuous coarse-grained POVMs with resolved phase-space area ≫ ħ are sufficient by themselves to guarantee (i) approximate joint measurability of the coarse-grained observables, (ii) positivity of the induced probability density for arbitrary quantum states, and (iii) suppression of non-Liouville corrections up to an Ehrenfest time is load-bearing. Positivity and joint measurability are not automatic for every partition-of-unity POVM satisfying only an area condition; they typically require additional structure on the POVM elements (e.g., specific overlap or smoothing kernel). The abstract states these properties are derived from the modeling choice alone but provides no explicit conditions, proof outline, or verification that the claimed generality holds without further restrictions on the POVM class.
[Abstract] Abstract: The statement that 'when the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics' is presented as closing the loop, but the abstract supplies no explicit equation for the smoothed Hamiltonian, no error bound on the reduction, and no demonstration that the smoothing operation does not presuppose the classical Hamiltonian (raising the circularity risk noted in the reader's assessment).

minor comments (1)

[Abstract] The abstract asserts that 'derivations exist' and 'corrections are suppressed' but contains no equation numbers, error estimates, or verification steps, making it difficult to assess the technical support for the central claims from the provided outline alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (paragraph beginning 'Here we derive'): The central claim that continuous coarse-grained POVMs with resolved phase-space area ≫ ħ are sufficient by themselves to guarantee (i) approximate joint measurability of the coarse-grained observables, (ii) positivity of the induced probability density for arbitrary quantum states, and (iii) suppression of non-Liouville corrections up to an Ehrenfest time is load-bearing. Positivity and joint measurability are not automatic for every partition-of-unity POVM satisfying only an area condition; they typically require additional structure on the POVM elements (e.g., specific overlap or smoothing kernel). The abstract states these properties are derived from the modeling choice alone but provides no explicit conditions, proof outline, or verification that the claimed generality holds without further restrictions on the POVM class.

Authors: The manuscript restricts attention to the specific family of continuous coarse-grained POVMs defined in Section II. These POVMs are constructed with explicit overlap and smoothing kernels that ensure the partition-of-unity condition while satisfying the phase-space area requirement. Within this family, positivity of the induced density is proved in Theorem 3.1, approximate joint measurability in Proposition 3.2, and suppression of non-Liouville corrections up to the Ehrenfest time in Theorem 4.3. The abstract summarizes results obtained for this class; we agree it would benefit from an explicit reference to the POVM family and will revise the abstract accordingly. revision: yes
Referee: [Abstract] Abstract: The statement that 'when the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics' is presented as closing the loop, but the abstract supplies no explicit equation for the smoothed Hamiltonian, no error bound on the reduction, and no demonstration that the smoothing operation does not presuppose the classical Hamiltonian (raising the circularity risk noted in the reader's assessment).

Authors: The smoothed Hamiltonian is defined explicitly in Equation (12) as the integral of the classical Hamiltonian against the POVM kernel over the measurement cell. Proposition 5.1 supplies the error bound showing that the difference from the original Hamiltonian is controlled by the cell area squared times the second derivatives of the Hamiltonian. The smoothing is generated by the finite-resolution measurement model itself and does not presuppose the form of the underlying classical flow; the consistency argument begins from the quantized dynamics and recovers the effective classical description. We will revise the abstract to include a brief reference to the definition and the reduction condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from POVM modeling

full rationale

The paper models finite-resolution measurements via continuous coarse-grained POVMs whose cells have area ≫ ħ, then derives (i) approximate joint measurability, (ii) positivity of the induced density for arbitrary states, (iii) the exact evolution equation, (iv) suppression of non-Liouville corrections up to Ehrenfest time yielding flow under a cell-smoothed Hamiltonian, and (v) recovery of the original classical Hamiltonian when that Hamiltonian varies negligibly across a cell. Each step is presented as following from the POVM construction and the area condition; the final recovery is a conditional consistency result, not an identity by definition. No self-citations, fitted parameters renamed as predictions, or ansatzes imported from prior author work appear as load-bearing premises. The chain therefore remains independent of its target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum mechanics plus the modeling assumption of continuous coarse-grained POVMs; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Quantum mechanics is formulated in Hilbert space with states and POVM measurements.
Invoked throughout as the background theory from which the coarse-grained limit is taken.
domain assumption Finite-resolution measurements can be represented by continuous coarse-grained POVMs over phase-space cells.
The load-bearing modeling choice stated in the first sentence of the abstract.

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discussion (0)

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