On spooky action at a distance and conditional probabilities

TL;DR

The paper investigates how post-measurement predictions in quantum mechanics can be understood through conditional probabilities, drawing a precise parallel with classical probability updates. It analyzes a two-component quantum system in the entangled state $\eta= a\psi\otimes\phi + b\phi\otimes\psi$ and derives quantum joint and conditional probabilities for commuting projections, showing $\mathcal{P}_\eta({\boldsymbol P}|{\boldsymbol Q})=\mathcal{P}_\eta({\boldsymbol P}{\boldsymbol Q})/\mathcal{P}_\eta({\boldsymbol Q})$ and the post-measurement state $\eta_{\boldsymbol Q}=\frac{{\mathbb I}_1\otimes{\boldsymbol Q}\eta}{\|{\boldsymbol Q}\eta\|}$. A classical counterpart on $\Omega=\{0,1\}^2$ demonstrates the same conditional-update structure via sample-space reduction, clarifying that projection-like updates do not require nonlocal influence when projections commute. The work further discusses when independence holds (product states) versus entanglement, and notes that uncorrelated observables need not be independent, connecting density-matrix formulations to the Heisenberg picture. Overall, it provides a clear bridge between quantum measurement theory and classical conditional prediction with implications for quantum information processing.

Abstract

The aim of this exposé is to make explicit the analogy between the classical notion of non-independent probability distribution and the quantum notion of entangled state. To bring that analogy forth, we consider a classical systems with two dependent random variables and a quantum system with two components. In the classical case, afet observing one of the random variables, the underlying sample space and the probability distribution change. In the quantum case, when and event pertaining to one of the components is observed, the post-measurement state captures, both, the change in the state of the system and implicitly the new probability distribution. The predictions after a measurement in the classical case and in the quantum case, have to be computed with the conditional distribution given the value of the observed variable.