Heading towards an Algebraic Heisenberg Cut

Mathias van den Bossche; Philippe Grangier

REVIEW 2 major objections 4 minor 1 cited by

Heading towards an Algebraic Heisenberg Cut

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims the quantum-classical borderline is an algebraic one: during an avalanche-photodiode measurement, the meter state gradually becomes orthogonal to the no-click state, so a sector parameter converges to the classical…

desk verdict The paper's central avalanche-overlap calculation is internally inconsistent, so the claimed finite-N precursors of sectorisation do not follow. read the letter →

arxiv 2412.16574 v3 pith:FKU5JKCT submitted 2024-12-21 quant-ph physics.hist-ph

Mathias Van Den Bossche , Philippe Grangier This is my paper · ORCID

classification quant-phphysics.hist-ph MSC 81P1546L10

keywords quantummeasurementHeisenbergcutinfinitetensorproductssectorisationtype-IIIvonNeumannalgebrasnon-separableHilbertspaceavalanchephotodiodecontextuality

topics The Measurement Problem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the Heisenberg cut can be located at the algebraic transition between separable and non-separable Hilbert spaces, and that this transition has visible precursors at finite particle number. The vehicle is a deliberately simplified model of a photon-polarisation measurement using avalanche photodiodes. The authors claim that as the electron avalanche grows, the diode state with a click and the diode state without a click become progressively orthogonal, and the expectation value of a sector parameter tends smoothly to the classical value |δ|²(|h|²−|v|²). If correct, this would legitimize treating macroscopic measurement devices with von Neumann's infinite tensor products and type-III algebras, turning the classical realm from an emergent phenomenon into an algebraic requirement.

What carries the argument

The central object is the sector parameter, an observable that labels macroscopic von Neumann sectors. It is built from elementary projectors |ϕ_α⟩⟨ϕ_α| averaged over N tensor factors, returning 1 on the reference product state and deviating by order M/N when M factors are modified. The avalanche state itself is generated by a unitary recursion in which each conduction electron scatters an impurity electron with amplitude η, producing a binary tree of entangled blocks |Z_k⟩; the convergence of ⟨P̂_{2n}⟩ is driven by the overlap between the avalanche state and the no-avalanche state, which the paper claims decays as a power of √(1−|η|²).

What would settle it

Compute the exact overlap ⟨Ω^P_{[A]}|Φ^P_n⟩ directly from equations (8), (13), and the definitions in Annex 3, keeping the zeroth impurity factor ⟨⊥_0|⊤_0⟩; the product includes this zero factor for every n, so the overlap never has the claimed (√(1−|η|²))^{n−1} form and the sector-parameter expectation would equal the classical value already at n=0.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that von Neumann sectorisation has finite-N precursors. In a unitary, decoherence-free model of an avalanche photodiode, the avalanche state and the no-avalanche state are claimed to have an overlap given at leading order by (√(1−|η|²))^{n−1}, so the two meter branches separate only gradually as n grows. Consequently, the expectation value of the sector parameter ⟨P̂_{2n}⟩ = |δ|²(|h|²−|v|²)(1−|⟨Φ^H_n|Ω^H_{[A]}⟩⟨Φ^V_n|Ω^V_{[A]}⟩|²) converges to the classical result. The authors take this as evidence that the full force of von Neumann sectorisation at infinity is controlled by an exponential avalanche, and they locate the Heisenberg cut where separable and non-separable Hilbert spaces become experimentally indistinguishable.

Load-bearing premise

The argument assumes the avalanche and no-avalanche meter states start with a small, nonzero overlap that shrinks by a factor √(1−|η|²) with each generation; under the paper's own state definitions that overlap is identically zero from the start, because the photoexcited electron occupies |⊤_0⟩ while the no-avalanche state has |⊥_0⟩.

Editorial extensions

If this is right

The classical predictive content of measurement emerges without invoking external decoherence, thermal baths, or Lindblad equations.
Sectorisation is claimed to be regular enough to justify the N→∞ limit in the same spirit as the thermodynamic limit in statistical physics.
The Heisenberg cut is reinterpreted as an algebraic boundary: the point where one can no longer distinguish a separable from a non-separable Hilbert space.
The destructive-measurement model extends to quantum non-demolition measurements using an ancilla entangled with the system.
Even when η=1, the sector parameter takes classical values immediately, while non-sector interferences vanish at infinity by the sectorisation theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct check of equation (14) against the definitions of the avalanche state reveals a tension: the photoexcited electron occupies |⊤_0⟩ while the no-avalanche state has |⊥_0⟩, so the overlap may be exactly zero from the start; if so, the claimed gradual sectorisation would collapse to an instantaneous one.
Because the diode bias is treated as a classical infinite resource, a fully quantum treatment of the power supply could reintroduce entanglement with the amplifier; testing the argument under a quantized bias would clarify whether the classical context is necessary for the mechanism.
The exponential growth of the avalanche is central to the claim of finite-time convergence; a linear amplifier, by contrast, would approach the classical limit only as N grows, suggesting that the rate of convergence is itself physically observable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

You should know that the central quantitative claim of this paper does not survive contact with its own definitions. The paper argues that the avalanche state becomes gradually orthogonal to the no-avalanche state, with overlap (√(1−|η|²))^(n−1), and that this drives finite-N precursors of von Neumann sectorisation. But under Eqs. (8) and (13), the no-avalanche state |Ω^P_[A]⟩ contains |⊥_0⟩, while every avalanche state |Φ^P_n⟩ contains |⊤_0⟩. The overlap is identically zero for all n. The formula in the paper comes from Annex 3, which actually computes the overlap with |Ω′⟩ = |⊤_0⟩⊗|Ω(1,...,A)⟩ — that is the post-absorption state, not the no-avalanche state. So Eq. (14) is a mislabeled result, and Eq. (17) collapses: the sector parameter reaches its classical value immediately, with no gradual orthogonalisation. This is a load-bearing error, and it voids the paper's main evidence for finite-N precursors of sectorisation.

I want to be fair. The conceptual program is serious and the paper does something genuinely new: it applies von Neumann's ITP sectorisation to an exponential avalanche in an APD, and it tries to quantify how the limit is approached. The attempt to connect type-III algebras to laboratory devices is thought-provoking, and the authors are honest about the limitations of taking infinite limits. They also correctly acknowledge Hepp, Emch, Araki, and Bub. The model itself, aside from the overlap calculation, is a clean unitary toy model with no environmental decoherence.

But the soft spot is not minor. The state-dependent definition of the sector parameter in Eq. (11) is also self-referential: the observable is constructed from the very states whose expectation value is computed. That makes the result less impressive even if the overlap issue were fixed.

Bottom line: the paper sketches an interesting research direction, but this particular calculation does not support it. A serious referee could catch the issue quickly. I would accept it for peer review in the sense that it deserves a careful look, but the manuscript as it stands has a fatal flaw in its central equation.

Who is this for? People working on quantum foundations and algebraic quantum theory might get some value from the conceptual discussion, but they should not rely on the avalanche model.

Referee Report

2 major / 4 minor

Summary. The paper proposes a model of an avalanche photodiode (APD) measurement of photon polarization to argue that von Neumann sectorisation at infinite tensor products has finite-N precursors. It defines a 'sector parameter' P̂_M and computes its expectation in the pure avalanche state, claiming that as the avalanche grows the expectation value converges as |δ|²(|h|²−|v|²) with corrections governed by (√(1−|η|²))^{n−1}. The claimed mechanism is the gradual orthogonalisation of the avalanche meter state |Φ^P_n⟩ to the no-avalanche state |Ω^P_[A]⟩, converting reversible interference into irreversible probabilities before the infinite limit.

Significance. The broader algebraic programme—using von Neumann's infinite tensor products to locate the Heisenberg cut—is potentially significant for quantum foundations, and the paper is clearly written. However, the specific quantitative claim that supports the programme in this manuscript is invalid: the overlap in Eq. (14) is identically zero under the paper's own definitions, so the 'gradual sectorisation' and the finite-N precursor effect do not follow from the model. The expectation value in Eq. (17) collapses to its classical value at n=0. As written, the model calculation cannot serve as evidence for the announced conclusions.

major comments (2)

[Sec. 4.2 and Annex 3, Eq. (14)] Eq. (14) and Annex 3: the no-avalanche overlap is identically zero. Section 2 defines [N]={0,...,N}, so [A] includes 0, and Eq. (8) has |Ω^P_[A]⟩=⊗_{n∈[A]}|⊥^P_n⟩, which contains |⊥_0⟩. Equations (9), (13), and Annex 3 define |Φ^P_n⟩ to contain |Z^P_0(i^{n+1}_0)⟩=|⊤^P_0⟩ for every n. Therefore ⟨Ω^P_[A]|Φ^P_n⟩=⟨⊥_0|⊤_0⟩×...=0 for all n, not a decaying overlap. Annex 3's computation 'Avalanche to no-avalanche overlap' actually uses |Ω′⟩=|⊤_0⟩⊗|Ω(1,...,A)⟩, which is the post-absorption state |Φ_0⟩, not the no-avalanche state. Consequently Eq. (14) describes ⟨Φ_0|Φ_n⟩, and the factor (1−|⟨Φ^H_n|Ω^H_[A]⟩⟨Φ^V_n|Ω^V_[A]⟩|²) in Eq. (17) is identically 1. The central claim of gradual orthogonalisation and finite-N precursors collapses.
[Sec. 4.3, Eq. (11) and Eq. (17)] Eq. (11) and Eq. (17): the sector parameter P̂_M is defined by projecting onto the very avalanche states |Φ^H_n⟩ and |Φ^V_n⟩ that appear in the state |Ψ_n⟩. The expectation value in Eq. (17) therefore largely measures the overlap of the state with its own defining projectors. This state-dependent definition gives the computation a self-referential character; to support the measurement interpretation, the observable should be specified independently of the avalanche generation n, or at least its n-dependence should be justified physically.

minor comments (4)

[Annex 3, Eq. (24)] Eq. (24) in Annex 3 writes '=o(√(1−|η|²))' for the overlap, but the leading term is exactly √(1−|η|²), so the little-o notation is incorrect and should be replaced by an equality with the leading term plus corrections.
[Eq. (13)] Eq. (13): the notation in |Z^P_k([i^n_{2k-1}:i^n_{2^k-1}])⟩ appears to have inconsistent subscripts; from Annex 3 the intended interval is [i^n_{2k-1}:i^n_{2k-1}], so the upper endpoint in the main-text formula should be corrected for readability.
[Sec. 4.3] The cross-reference to 'Annex 3.C' in Section 4.3 does not correspond to any numbered subsection in Annex 3; update the reference.
[Annex 3, 'Unitarity considerations'] The completion of the scattering operator introduced in the 'Unitarity considerations' paragraph is not explicitly used in the main text; clarify whether it affects the overlap computation or is included only to justify unitarity.

Circularity Check

2 steps flagged · score 6.0 of 10

The claimed gradual meter orthogonalization reduces to a misdefined 'no-avalanche' state, and the sector-parameter expectation is a projection onto the branch states themselves.

self definitional [Annex 3 ('Avalanche to no-avalanche overlap') with Sections 3 and 4.2, Eqs. (8), (13), (14)]
"Let |Ω′⟩ := |⊤_0⟩⊗|Ω(1,...,A)⟩ be the state where no avalanche occurs. ... ⟨Ω′|Φ_n⟩ ∼ (√(1−|η|²))^{n−1}. ... |Ω^P_[A]⟩ :=⊗_{n∈[A]}|⊥^P_n⟩ ... |Z_0(i)⟩ := |⊤_i⟩."

Under the paper's own definitions, the true no-avalanche state |Ω^P_[A]⟩ contains |⊥_0⟩, while every avalanche state |Φ^P_n⟩ contains |Z_0⟩=|⊤_0⟩ by Eq. (13). Hence ⟨Ω^P_[A]|Φ^P_n⟩ is exactly zero for all n, not the decaying overlap claimed in Eq. (14). The state |Ω′⟩ used in Annex 3 is not the no-avalanche meter state; it is exactly the post-absorption state |Φ_0⟩=|⊤_0⟩⊗|Ω([1:A])⟩ defined in the same annex. Thus Eq. (14) computes ⟨Φ_0|Φ_n⟩, an overlap between successive avalanche generations, and the gradual orthogonalization is built into the choice of |Ω′⟩ rather than derived for the actual meter states.
self definitional [Section 4.3, Eqs. (11), (16) and (17)]
"P̂_M := |0γ⟩⊗| Φ^H_n⟩⊗| Ω^V_[A]⟩⟨0γ|⊗⟨ Φ^H_n|⊗⟨ Ω^V_[A]| − |0γ⟩⊗| Φ^V_n⟩⊗| Ω^H_[A]⟩⟨0γ|⊗⟨ Φ^V_n|⊗⟨ Ω^H_[A]| ... after some elementary algebra yields ⟨P̂_{2n}⟩ = |δ|²(|h|²−|v|²)(1−|⟨ Φ^H_n|Ω^H_[A]⟩⟨Φ^V_n|Ω^V_[A]⟩|²)."

The observable is defined as a projector onto the same branch states |Φ^H_n⟩ and |Φ^V_n⟩ that already appear in the measurement state |Ψ_n⟩ with weights hδ and vδ. The expectation value in Eq. (17) therefore returns those input weights together with overlap cross-terms; with the correct zero overlap the result is |δ|²(|h|²−|v|²) at every n, independent of any avalanche dynamics. The 'convergence' of the sector parameter is thus a restatement of the branch weights encoded in the definitions of both the state and the observable, not a derived finite-N precursor of sectorisation.

full rationale

The paper's broad appeal to von Neumann's infinite-tensor-product sectorisation theorems is external and would constitute independent mathematical support if the finite-N calculation were sound; the authors' self-citations to their CSM framework shape the interpretation but are not the quantitative derivation. The quantitative derivation, however, is not self-contained. In Annex 3 the 'no-avalanche' state is defined as |Ω′⟩=|⊤_0⟩⊗|Ω(1,...,A)⟩, which is precisely the post-absorption avalanche state |Φ_0⟩ defined earlier in the same annex; it contains the excited electron |⊤_0⟩. The actual no-avalanche state defined in Eq. (8), |Ω^P_[A]⟩=⊗_{n∈[A]}|⊥^P_n⟩, contains |⊥_0⟩. Since every |Φ^P_n⟩ contains |Z_0⟩=|⊤_0⟩ by Eq. (13), the inner product ⟨Ω^P_[A]|Φ^P_n⟩ is identically zero for every n, rather than (√(1−|η|²))^{n−1} as claimed in Eq. (14). Consequently the overlap factor in Eq. (17) is identically 1 and the classical value |δ|²(|h|²−|v|²) appears already at n=0; no gradual, finite-N precursor of sectorisation is exhibited. In addition, the sector parameter in Eq. (11) is constructed as a projector onto the very branch states whose weights are inserted into |Ψ_n⟩, so the expectation value in Eq. (17) largely restates the input amplitudes rather than deriving a dynamical tendency. These two features make the central model calculation circular in the specific sense that the claimed outcome is built into the definitions used, even though the surrounding algebraic programme may still be worth studying on other grounds.

Assumptions & free parameters 2 free parameters · 5 assumptions · 3 invented entities

The central calculation depends on two hand-chosen amplitudes (δ, η) and on the CSM ontology of contexts and modalities. The mathematical backbone (von Neumann sectorisation) is external, but its physical applicability to a finite photodiode is a domain assumption.

free parameters (2)

δ
Photoexcitation amplitude in Eq. 7; it sets the detection efficiency and enters the final expectation value |δ|²(|h|²−|v|²). Chosen by hand, not fitted.
η
Avalanche collision amplitude in Eq. 12; controls the decay rate of the overlap in Eq. 14 and the speed of convergence to sectorisation. Chosen by hand, not fitted.

assumptions (5)

standard math Von Neumann's incomplete direct product (ITP) sectorisation theorems apply and yield a decomposition into uncountable sectors at N→∞
External theorem (ref. [9]); the paper relies on it to claim self-decoherence and type-III algebras.
domain assumption The infinite limit N→∞ is a legitimate and regular physical limit for the measurement device, analogous to the thermodynamic limit
Stated in §5.1; this is the paper's justification for applying ITP sectorisation to a finite APD.
domain assumption The avalanche evolves unitarily as a pure state with no decoherence from phonons, resistivity, or a continuum band
Explicit simplification in §3; the paper claims sectorisation alone gives the measurement result.
standard math Sector parameters X∞ and P∞ lie in the centre of the operator algebra and commute at the limit
Follows from von Neumann's sectorisation theorem, used to define macroscopically distinct sectors.
ad hoc to paper Contexts and modalities exist as physical realities (CSM ontology)
Postulated in Annex 1; the paper's interpretation assumes classical contexts are fundamental, not emergent.

invented entities (3)

Macroscopic context
purpose: A classical measurement device that defines the set of possible modalities; used to make the Heisenberg cut a primitive.
CSM postulate, no falsifiable handle outside the framework.
Modality
purpose: A real, certain, repeatable property of a system within a context; replaces the usual state vector ontology.
CSM postulate; probabilities between modalities are derived from Gleason and Uhlhorn theorems, but the entities themselves are not independently evidenced.
Sector parameter
purpose: Observable labelling macroscopic sectors; the paper introduces it to compute the measurement outcome value.
Mathematical construct defined using the target states; no independent physical handle beyond the paper's definitions.

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Cite this review

Pith. "Pith review of Heading towards an Algebraic Heisenberg Cut." pith.science (2026). https://pith.science/paper/FKU5JKCT

@misc{pith2026241216574,
  author       = {Pith},
  title        = {Pith review of: Heading towards an Algebraic Heisenberg Cut},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FKU5JKCT}},
  note         = {Machine review of arXiv:2412.16574}
}

read the original abstract

In previous papers we have explained how a sequence of theorems by John von Neumann on infinite tensor products (ITP) can be understood as providing elements to support both sectorisation of the Hilbert space of large quantum systems, and a mechanism of self decoherence thereof. These two effects may help understanding the articulation of the classical and quantum realms. However, as they involve considering an infinite number of quantum degrees of freedom, legitimate concerns can be raised on their applicability. In this paper, we address explicitly the interface between both realms through the example of a simplified model of a photon polarisation measurement device. Guided by the fact that there is von Neumann sectorisation at infinity, and by the necessity of classical contexts to perform measurements, we show that this limit can be under control, and that although the full force of the sectorisation theorems requires taking the infinite limit, early signs of the macroscopic behaviour appear before infinity. In our example, this shows up in photodiodes through diverging electron avalanches that simultaneously make the system classical, localise it randomly in a macroscopic sector and provide a macroscopic signal. This lays the grounds for justifying the inclusion in quantum physics of the ITP formalism, which involves non-separable Hilbert spaces and potentially type-III von Neumann algebras. Such an approach could make sense of the quantum-classical transition as a primarily algebraic one.

Figures

Figures reproduced from arXiv: 2412.16574 by the authors.

**Figure 1.** Example – here S is a photon in state |γ⟩ and M is a polarisation beamsplitter and two photodiodes. In either APD, an avalanche that involves NP electron-hole pairs (P = V, H) might occur. These two cases correspond to two different sectors of M, so one can define a sector parameter as linked to the state of the APDs. 3 Modeling an avalanche photodiode Our goal is to spell out how sectorisation is at work in a meas… view at source ↗

**Figure 2.** Labelling of the sequence of events following the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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Contextual objectivity: a realistic interpretation of quantum mechanics

P. Grangier, “Contextual objectivity: a realistic interpretation of quantum mechanics”, European Journal of Physics23:3, 331 (2002) [arXiv:quant- ph/0012122]

arXiv 2002

[2] [2]

Contexts, Systems and Modalities: a new ontology for quantum mechanics

A. Auffèves and P. Grangier, “Contexts, Sys- tems and Modalities: a new ontology for quan- tum mechanics”, Found. Phys. 46, 121 (2016) [arXiv:1409.2120]. 8

work page Pith review arXiv 2016

[3] [3]

Extracontextuality and Extravalence in Quantum Mechanics

A. Auffèves and P. Grangier, “Extracontextu- ality and extravalence in quantum mechanics”, Phil. Trans. R. Soc. A 376, 20170311 (2018) [arXiv:1801.01398]

work page Pith review arXiv 2018

[4] [4]

Revisiting Born’s rule through Uhlhorn’s and Glea- son’s theorems

A. Auffèves and P. Grangier, “Revisiting Born’s rule through Uhlhorn’s and Glea- son’s theorems”, Entropy 24(2), 199 (2022) https://doi.org/10.3390/e24020199

work page doi:10.3390/e24020199 2022

[5] [5]

Completing the quantum formalism in a contextually objective framework

P.Grangier, “Completingthequantumformalism in a contextually objective framework”, Found. Phys. 51, 76 (2021) [arXiv:2003.03121]

work page Pith review arXiv 2021

[6] [6]

Con- textual unification of classical and quan- tum physics

M. Van Den Bossche and P. Grangier, “Con- textual unification of classical and quan- tum physics”, Found. Phys. 53, 45 (2023) [arXiv:2209.01463]

arXiv 2023

[7] [7]

Revisiting Quantum Contextuality in an Algebraic Frame- work

M. Van Den Bossche and P. Grangier, “Revisiting Quantum Contextuality in an Algebraic Frame- work”, Proceedings of the DICE 2022 Confer- ence, in J. Phys. Conference Series (2023) [ar- viv:2304.07757]

arXiv 2023

[8] [8]

Postu- lating the Unicity of the Macroscopic Physical World

M. Van Den Bossche and P. Grangier, “Postu- lating the Unicity of the Macroscopic Physical World”, Entropy25(12), 1600 (2023)

work page 2023