Conditional probability framework for entanglement and its decoupling from tensor product structure

TL;DR

This paper develops a conditional probability framework to clarify the foundations of quantum entanglement, focusing on perfect conditional correlations and decoupling entanglement from the notion of compound systems.

Contribution

It introduces a new perspective on entanglement as knowledge knotting via conditional probabilities, independent of the physical system's structure.

Findings

Defines a class of entangled states based on perfect conditional correlations.

Decouples the concept of entanglement from multi-partite systems.

Links entanglement to knowledge and measurement rather than physical separation.

Abstract

Our aim is to make a step towards clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In Schr\"odinger's words, this is entanglement of knowledge which can be extracted via conditional measurements. In particular, quantum probabilities are interpreted as conditional ones (as, e.g., by Ballentine). We restrict considerations to perfect conditional correlations (PCC) induced by measurements ("EPR entanglement"). Such entanglement is coupled to the pairs of observables with the projection type state update as the back action of measurement. In this way, we determine a special class of entangled states. One of our aims is to decouple the notion of entanglement from the compound systems. The rigid association of entanglement with the state of a few body systems stimulated its linking with quantum nonlocality ("spooky action at a distance"). However, already by Schr\"odinger entanglement was presented as knotting of knowledge (about statistics) for one observable A with knowledge about another observable B.