On the EPR paradox in systems with finite number of levels
Henryk Gzyl
TL;DR
The paper reexamines the EPR paradox in composite quantum systems with a finite number of levels by focusing on measurements and conditional probabilities. It develops a joint-system formalism with ${\boldsymbol S}=\boldsymbol{A}_1+\boldsymbol{A}_2$, decomposing the state into entangled post-measurement components $|\psi_{s}\rangle$ via projectors ${\Pi}_k$ and probabilities $p(s_k)$, and derives conditional expectations $\langle\psi_s|f({\boldsymbol A}_1)|\psi_s\rangle$ that depend on the observed $S$. The analysis shows that, when $S$ is fixed, the prediction variances satisfy ${\Delta}_\psi({\boldsymbol A}_1)={\Delta}_\psi({\boldsymbol A}_2)$ and that sequential measurements (first $S$, then ${\boldsymbol A}_1$) lead to deterministic predictions for ${\boldsymbol A}_2$ in each branch, removing the EPR paradox in finite dimensions. A concrete two-qubit Pauli example confirms that conditional predictions respect the uncertainty principle and that no spooky action at a distance arises. Overall, the work clarifies how conditional predictions derived from quantum measurements in finite-level systems resolve apparent nonlocal paradoxes and refine our understanding of measurement-induced state updating.
Abstract
In this work we reexamine the EPR paradox for composite systems with a finite number of levels. The analysis emphasizes the connection between measurements and conditional probabilities. This connection implies, on the one hand, that when a measurement is performed, the new quantum state and the probability distribution becomes a function of the observable being measured. On the other hand, this becomes important when making predictions about the properties of the subsystems, since the predictions are implicitly a function of the observable that was measured. Systems with finitely many levels are simpler to describe because the analysis is not encumbered by the mathematical technicalities of the continuous case, and the underlying physical interpretations are the same.