Toy model for local and deterministic wave-function collapse

A local, deterministic toy model for quantum mechanics is introduced and discussed. It is demonstrated that the model makes the same predictions as quantum mechanics when averaged over the hidden variables. In this model the dynamics depends on the settings of the measurement device at the time of detection but not on how those settings were chosen. As a consequence, the model has a future input dependence and violates statistical independence but is local. We further show that the model is neither fine tuned nor allows for superluminal signaling and discuss the causality relations.

Figure 1

Schematic illustration of the collapse described by the toy model. Shades of gray indicate absolute values of amplitudes, where darker (lighter) shades are larger (smaller) amplitudes. Just after the superposition is created (e.g., by a beam splitter), the two branches have the same amplitude, but as they approach the detector (gray bar on top), one branch becomes dominant over the other one.

Figure 2

The dynamics induced by the Lindblad terms involving the operators $L_{NM}$ with $M=1,2,...N-1$. For ${\sigma }_N>0$ each state will evolve to the state $|{\chi }_N\rangle$. For ${\sigma }_N<0$, the state $|{\chi }_N\rangle$ will evolve evenly into the states $|{\chi }_1\rangle ,...|{\chi }_{N-1}\rangle$.

Figure 3

Sketch to illustrate local causality. A state is prepared in region P and measured at two detectors ${\mathbf{D}}_1$ and ${\mathbf{D}}_2$. The hidden variables $\mathrm{\Lambda }$ are present already at P and therefore in the backward light cones of both detectors.

Figure 4

Classical example for optimization problem that cannot be expressed by Euler-Lagrange equations.