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arXiv:2605.13739

Quantum selective measurement as a quasilinear evolution

Jakub Rembieliński, Karol Ławniczak

quant-phSubmitted May 13, 2026• Announced May 14, 2026• 0 citations
v1 · May 13, 2026v2 · May 18, 2026
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Abstract

We propose replacing the instantaneous state reduction in von Neumann selective measurement with continuous nonlinear evolution. Despite its nonlinearity, this evolution preserves the equivalence of quantum ensembles and hence obeys the no-signaling principle. Its final states coincide with those produced by the von Neumann projection. The defining features of rank-one projective measurement are retained: convergence to the eigenstate of the observable associated with the selected outcome, independence of this final state from the initial state, and consistent action on entangled states.

Comments
8 pages, 5 figures

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We propose replacing the instantaneous state reduction in von Neumann selective measurement with continuous nonlinear evolution. Despite its nonlinearity, this evolution preserves the equivalence of quantum ensembles and hence obeys the no-signaling principle. Its final states coincide with those produced by the von Neumann projection. The defining features of rank-one projective measurement are retained: convergence to the eigenstate of the observable associated with the selected outcome, independence of this final state from the initial state, and consistent action on entangled states.
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.
Unitary evolution and projective measurement are fundamental axioms of quantum mechanics. Even though projective measurement yields one of the eigenstates of the measured operator as the outcome, there is no theory that predicts which eigenstate will be observed in which experimental run. There exists only an ensemble description, which predicts probabilities of various outcomes over many experimental runs. We propose a dynamical evolution equation for the projective collapse of the quantum state in individual experimental runs, which is consistent with the well-established framework of quantum mechanics. In case of gradual weak measurements, its predictions for ensemble evolution are different from those of the Born rule. It is an open question whether or not suitably designed experiments can observe this alternate evolution.
Unitary evolution and projective measurement are fundamental axioms of quantum mechanics. Even though projective measurement yields one of the eigenstates of the measured operator as the outcome, there is no theory that predicts which eigenstate will be observed in which experimental run. There exists only an ensemble description, which predicts probabilities of various outcomes over many experimental runs. We propose a dynamical evolution equation for the projective collapse of the quantum state in individual experimental runs, which is consistent with the well-established framework of quantum mechanics. In case of gradual weak measurements, its predictions for ensemble evolution are different from those of the Born rule. It is an open question whether or not suitably designed experiments can observe this alternate evolution.
Unitary evolution and projective measurement are fundamental axioms of quantum mechanics.Even though projective measurement yields one of the eigenstates of the measured operator as the outcome, there is no theory that predicts which eigenstate will be observed in which experimental run.There exists only an ensemble description, which predicts probabilities of various outcomes over many experimental runs.We propose a dynamical evolution equation for the projective collapse of the quantum state in individual experimental runs, which is consistent with the well-established framework of quantum mechanics.In case of gradual weak measurements, its predictions for ensemble evolution are different from those of the Born rule.It is an open question whether or not suitably designed experiments can observe this alternate evolution.
We study the problem of driving a known initial quantum state onto a known pure state without using a unitary evolution. This task can be achieved by means of von Neumann measurement processes, introducing N observables which are consecutively measured in order to approach the state of the system to the target state. We proved that the probability of projecting onto the target state can be increased meaningfully by adding suitable observables to the process, that is, it converges to 1 when N increases. We also discuss a physical implementation of this scheme.
The Projection Postulate from Standard Quantum Mechanics relies fundamentally on measurements. But measurements implicitly suggest the existence of anthropocentric notions like measuring devices, which should rather emerge from the theory. This article proposes an alternative formulation of the Standard Quantum Mechanics, in which the Projection Postulate is replaced with a version in which measurements and observations are not assumed as fundamental. More precisely, the Wigner functions representing the quantum states on the phase space are required to be tightly constrained to regions of the classical coarse graining of the phase space. This ensures that states are quasiclassical at the macrolevel. Within a coarse-graining region, the time evolution of the Wigner functions representing the quantum system is required to obey the Liouville-von Neumann equation, the phase-space equivalent of the Schr\"odinger equation. The projection is postulated to happen when the system transitions from a coarse-graining region to others, by selecting one of them according to the Born rule, but without reference to a measurements. The connection with the standard formulation of Quantum Mechanics is explained, as well as the problems that the present formulation solves, in particular the Wigner's friend type of paradoxes. Experimental consequences and open problems of the proposed formulation are discussed.