Hidden variables: reconsidering quantum entanglement
Quantum mechanics presents a view of reality that is often at odds with our classical intuitions. One of the most striking features is entanglement, where two or more particles can be linked in such a way that they share the same fate, no matter how far apart they are. This concept becomes particularly puzzling when considering measurements.
Imagine a scenario with two entangled photons. If we measure the polarization of the first photon and find it to be vertical, quantum mechanics tells us that the polarization of the second photon is instantly determined as well, even if it’s light-years away. This apparent instantaneous connection troubled Einstein, who believed in locality – the idea that physical influences should not travel faster than light.
Einstein questioned whether quantum mechanics provided a complete picture of reality. He proposed the idea of “hidden variables”. The core idea is that perhaps quantum mechanics is incomplete, and there are underlying properties, “hidden variables”, that we are not aware of. These variables, if known, would predetermine the outcomes of quantum measurements, removing the inherent randomness and the need for instantaneous influences.
Let’s consider entangled photons again. In a hidden variable theory, at the moment of creation, each photon pair is endowed with a hidden property, represented by \lambda. This \lambda dictates the outcomes of polarization measurements. For photon \nu_1 and polarizer (I) oriented along \mathbf a, the measurement outcome is A(\lambda, \mathbf a) = \pm 1. Similarly, for photon \nu_2 and polarizer (II) oriented along \mathbf b, the outcome is B(\lambda, \mathbf b) = \pm 1.
To account for the observed statistical distribution of measurement results, we introduce a probability distribution \rho(\lambda) for these hidden variables. The correlation between measurements on the two photons can then be expressed as an average over these hidden variables:
\mathcal C = \int \rho(\lambda) A(\lambda, \mathbf a) B(\lambda, \mathbf b) \mathrm d \lambda
This formulation suggests that the correlations we observe in quantum experiments are not due to spooky action at a distance, but rather because both photons’ properties were determined from the start by the shared hidden variable \lambda.
Hidden variable models offer a way to reconcile quantum correlations with a more classical, deterministic, and local worldview. While these models can reproduce some quantum mechanical predictions, it is important to note that they face significant challenges when confronted with the full scope of quantum phenomena.
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