Quantum mechanics has two seemingly competing rules. Firstly, a system evolving without measurement follows a continuous, deterministic evolution governed by the Schrödinger equation, with dynamics determined by a Hamiltonian. Secondly, when a measurement occurs, the wavefunction collapses, producing a sudden, discontinuous change that is not derived from a Hamiltonian. Several approaches attempt to reconcile these behaviours, including the Copenhagen interpretation (which does not explain the mechanism of collapse), decoherence theory (which does not provide a single definite outcome), stochastic collapse models, and continuous measurement theory.

In this work, measurement is not treated as fundamentally different. Instead, it is described using stochastic (random) Hamiltonians that generate continuous evolution of the quantum state. In this picture, collapse emerges from noisy dynamics. The authors show that these dynamics can be understood as double-bracket gradient flows, where the system is driven to align with a measured observable, steadily reducing uncertainty until it reaches a definite outcome. Thus, wavefunction collapse can be viewed as coarse-grained continuous dynamics that minimise the variance of the observable. By interpreting this as a gradient flow, the same mechanism can be exploited using feedback to drive a system into desired states, including entangled ones.

This approach provides a continuous and physically interpretable picture of wavefunction collapse. Compared to decoherence theory, it explains the emergence of a single outcome but does not specify when measurement dynamics begin. More broadly, it replaces the notion of collapse with a dynamical process, making the theory more internally consistent, while also offering practical tools for controlling quantum systems, which is important for quantum computing and experiments.

“These geometric connections between Hamiltonian dynamics and quantum measurements open the door to exciting new approaches to quantum algorithm design.” – Aaron Villanueva, Radboud University

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Genuine quantum correlations in quantum many-body systems: a review of recent progress by Gabriele De Chiara and Anna Sanpera (2018)