arxiv: 2605.13756 · v1 · submitted 2026-05-13 · 🪐 quant-ph

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Quasilinear evolution versus von Neumann selective measurement

Jakub Rembieli\'nski , Karol {\L}awniczak

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Pith reviewed 2026-05-14 17:52 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum measurementselective measurementquasilinear evolutionnonlinear von Neumann equationno-signaling principlestate reductiontwo-level systemsStern-Gerlach experiment

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The pith

A nonlinear generalization of the von Neumann equation replaces instantaneous projection with continuous quasilinear evolution in selective quantum measurements.

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

The paper proposes a new description of selective measurement in which the discontinuous von Neumann projection is replaced by a continuous quasilinear evolution driven by a nonlinear generalization of the von Neumann equation. This evolution is constructed so that quantum ensembles remain equivalent and the no-signaling condition holds, keeping the framework consistent with standard quantum mechanics and relativistic causality. The stochastic outcomes and Born-rule probabilities are left unchanged, while the state reduction is treated as ordinary dynamical evolution without reference to an apparatus state. Numerical solutions for two-level systems largely reproduce the results of standard projection, except in narrow parameter intervals where small deviations appear and could be tested experimentally. The approach is illustrated with an analytical treatment of the Stern-Gerlach setup.

Core claim

The central claim is that a nonlinear generalization of the von Neumann equation can govern quasilinear evolution that performs selective measurement, replacing the projection postulate while preserving the equivalence of quantum ensembles and satisfying the no-signaling principle, so that the stochastic character of measurement and the Born rule remain intact.

What carries the argument

The quasilinear evolution equation, a nonlinear generalization of the von Neumann equation, that continuously drives the post-measurement state reduction.

If this is right

Selective measurements are described as continuous dynamical processes rather than instantaneous collapses.
The no-signaling principle remains satisfied, preserving consistency with relativity.
Born-rule probabilities and the stochastic nature of outcomes are unchanged.
Numerical solutions for two-level systems agree with von Neumann projections except in narrow parameter regions.
The Stern-Gerlach experiment admits an analytical solution within the quasilinear framework.

Where Pith is reading between the lines

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

The continuous description might allow measurement to be embedded in a single dynamical law without hybrid unitary-plus-collapse rules.
Narrow instability regions could be searched for in precision spin or photon experiments to distinguish the model from standard projection.
Similar nonlinear extensions could be examined for compatibility with other measurement scenarios such as sequential or non-projective measurements.

Load-bearing premise

The chosen nonlinear generalization of the von Neumann equation is assumed to preserve ensemble equivalence and the no-signaling condition without extra constraints or post-selection.

What would settle it

An experimental observation of signaling between separated ensembles or a clear violation of ensemble equivalence under the quasilinear dynamics in a two-level system would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.13756 by Jakub Rembieli\'nski, Karol {\L}awniczak.

**Figure 1.** Figure 1: ). The points (α, θ, Θ) belonging to the tetrahedron, together with the angle β, determine the relative configurations of the observable Ω(ω) and the generator G(g). The parameter space depicted in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Cross-sections of the parameter space from Fig. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: The inverted Morse potential gIM(t) is of the form gIM(t) = g0 1 − [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. State evolution according to Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Rate of change of the Bloch vector under quasilinear evolution starting from three different initial states [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. State dynamics for the scale [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. State dynamics for two substantially different angles [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. State dynamics for two different forms of the function [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. State dynamics on both sides of the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. State dynamics in proximity of the [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. State dynamics for an even smaller [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

**Figure 12.** Figure 12: tends to a large constant value. As is evident from [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Idealized Stern–Gerlach arrangement considered in [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14. State evolution in the Stern–Gerlach experiment according to the quasilinear measurement dynamics. Note that [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗

**Figure 15.** Figure 15: FIG. 15. The evolution of the initially polarized state in the Stern–Gerlach system according to quasilinear dynamics. [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗

**Figure 16.** Figure 16: FIG. 16. The rate of change [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗

read the original abstract

In this article, we introduce a new form of quantum selective measurement in which the von Neumann projection postulate is replaced by quasilinear evolution, governed by a nonlinear generalization of the von Neumann equation. We demonstrate that this equation preserves the equivalence of quantum ensembles and, consequently, satisfies the no-signalling principle, ensuring consistency with both quantum mechanics and Einstein causality. Our approach eliminates the need for instantaneous, discontinuous state collapse and provides a unified description of the postmeasurement quantum state reduction as a form of quantum state evolution. Notably, it does not require invoking concepts such as the quantum state assigned to a classical apparatus. At the same time, the stochastic character of selective measurement and the Born rule remain unchanged. We present several numerical solutions of the evolution equation for quasilinear selective measurement in two-level quantum systems and compare them with the standard von Neumann projection. The results demonstrate agreement between the two measurement schemes in their fundamental properties. Furthermore, we investigate phenomena associated with the structural instability of the evolution equation and identify very narrow parameter regions in which the outcomes deviate from those predicted by the von Neumann projection. These regions may offer opportunities to test the proposed approach experimentally. Finally, using specific analytical solutions, we discuss the Stern-Gerlach experiment within the framework of quasilinear measurement.

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 replaces von Neumann projection with a quasilinear nonlinear evolution for selective measurements, but the no-signaling claim rests mainly on two-level numerics without a general proof.

read the letter

The main new element here is a specific nonlinear generalization of the von Neumann equation that turns selective measurement into continuous quasilinear evolution instead of instantaneous projection. The authors keep the Born rule and stochastic outcomes intact while claiming the reduced state of an unmeasured subsystem stays unchanged, which would preserve no-signaling and ensemble equivalence. They back this with numerical runs on two-level systems that track the standard projection closely except in narrow parameter windows, plus an analytical Stern-Gerlach example that matches the usual result. That unification of reduction with ordinary evolution, without extra apparatus states, is the cleanest part of the work and gives a concrete dynamical alternative worth looking at. The soft spot is the no-signaling argument. The numerics are limited to two-level cases, and nothing in the presented material supplies an explicit general derivation showing that the nonlinear term cancels exactly in the partial trace over an entangled partner for arbitrary dimension and entanglement. Nonlinear dynamics can leak information across subsystems even when the effect is small on average, so the extrapolation from low-dimensional checks needs a clear analytical step rather than just agreement in special cases. The structural instability regions they flag are interesting for possible tests but also highlight where the dynamics can deviate, which makes the general claim more delicate. This is aimed at people who follow dynamical resolutions of the measurement problem. It has enough concrete equations and comparisons to justify sending it to referees who can verify the derivations and check whether the no-signaling property holds beyond the numerics shown.

Referee Report

2 major / 2 minor

Summary. The paper proposes replacing the von Neumann projection postulate with quasilinear evolution governed by a nonlinear generalization of the von Neumann equation for selective quantum measurements. It claims this preserves equivalence of quantum ensembles (hence no-signaling), maintains the Born rule and stochasticity of outcomes, eliminates discontinuous collapse, and agrees with standard QM except in narrow parameter regions of structural instability; numerical solutions for two-level systems and analytical Stern-Gerlach results are presented to support agreement with von Neumann projections.

Significance. If the no-signaling preservation holds rigorously beyond the presented cases, the approach would supply a continuous dynamical alternative to state collapse that remains consistent with relativity and ensemble equivalence, offering a potential unification of unitary evolution and measurement without invoking apparatus states. The numerical agreement and identification of testable instability regions add concrete value, though the overall significance depends on establishing the central invariance property analytically.

major comments (2)

[evolution equation and no-signaling claims] The central claim that the nonlinear generalization preserves ensemble equivalence (and thus no-signaling) for entangled states rests on numerical solutions for two-level systems and Stern-Gerlach analytics, without a general analytical derivation showing that the nonlinear term vanishes under partial trace Tr_B[ρ_AB(t)] for arbitrary dimensions and initial entanglement (see the evolution equation section and the no-signaling discussion).
[numerical solutions and instability regions] The noted structural instability in narrow parameter regions is stated to produce deviations from von Neumann predictions, but no analysis is given of whether these regions preserve or violate the ensemble equivalence property required for no-signaling (see the numerical solutions and instability discussion).

minor comments (2)

[Abstract] The abstract asserts preservation of key properties without referencing the specific equation form or the scope of the supporting derivations.
[figures and numerics] Figure captions for the numerical comparisons could more explicitly state the initial states, parameter values, and error metrics used to quantify agreement with von Neumann evolution.

Circularity Check

0 steps flagged

No circularity: preservation of ensemble equivalence shown via explicit demonstration and numerics rather than by definition or self-citation

full rationale

The paper defines a specific nonlinear generalization of the von Neumann equation for quasilinear selective measurement and states that it preserves quantum ensemble equivalence (hence no-signaling). The abstract and description indicate this is demonstrated through numerical solutions for two-level systems, analytical Stern-Gerlach comparisons, and agreement with von Neumann properties, without reducing the no-signaling claim to a fitted parameter, ansatz smuggled via self-citation, or renaming of a known result. The derivation chain supplies independent content via the explicit evolution equation and its solutions; no load-bearing step collapses to the input by construction. This is the expected honest non-finding for a proposal whose central property is asserted to be verified rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a nonlinear generalization whose properties (ensemble equivalence, no-signaling) are asserted without derivation from first principles or external benchmarks in the abstract; no free parameters or invented entities are explicitly named.

axioms (1)

domain assumption Nonlinear generalization of von Neumann equation preserves equivalence of quantum ensembles and satisfies no-signaling
Stated as demonstrated in the abstract without supporting derivation or external reference

pith-pipeline@v0.9.0 · 5522 in / 1193 out tokens · 47940 ms · 2026-05-14T17:52:40.030460+00:00 · methodology

discussion (0)

Reference graph

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Quasilinear evolution versus von Neumann selective measurement

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The following figure presents the state dy- namics under the scaleg0 much lower and much higher than in previous examples

Variation of the scaleg 0 The potentialg(t)may have a different scale, quanti- fied byg 0. The following figure presents the state dy- namics under the scaleg0 much lower and much higher than in previous examples. The remaining parameters of the evolution are the same as in Fig. 4. The initial state is taken to be one more mixed staten0 = [ 0,−1 2,−1 2 ] ...

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