[Paper Review] Universal entangleability of non-classical theories

Guillaume Aubrun, Ludovico Lami+1 more|arXiv (Cornell University)|Oct 10, 2019

Quantum Mechanics and Applications4 citations

TL;DR

This paper introduces a universal framework for entanglement in general probabilistic theories (GPTs), proving that any two non-classical theories are entangleable—meaning their composite systems admit entangled states or measurements. Using convex geometry and functional analysis, it establishes that non-classicality at the single-system level implies global entangleability, with rigorous results for 3D theories, discrete state spaces, and quantum mechanics as one subsystem.

ABSTRACT

Inspired by its fundamental importance in quantum mechanics, we define and study the notion of entanglement for abstract physical theories, investigating its profound connection with the concept of superposition. We adopt the formalism of general probabilistic theories (GPTs), encompassing all physical models whose predictive power obeys minimal requirements. Examples include classical theories, which do not exhibit superposition and whose state space has the shape of a simplex, quantum mechanics, as well as more exotic models such as Popescu-Rohrlich boxes. We call two GPTs entangleable if their composite admits either entangled states or entangled measurements, and conjecture that any two non-classical theories are in fact entangleable. We present substantial evidence towards this conjecture by proving it (1) for the simplest case of 3-dimensional theories; (2) when the local state spaces are discrete, which covers foundationally relevant cases; (3) when one of the local theories is quantum mechanics. Furthermore, (4) we envision the existence of a quantitative relation between local non-classicality and global entangleability, explicitly describing it in the geometrically natural case where the local state spaces are centrally symmetric.

Motivation & Objective

• To define and formalize entanglement in a model-independent way across all physical theories using general probabilistic theories (GPTs).
• To investigate the foundational link between single-system non-classicality (superposition) and bipartite entangleability.
• To prove that any pair of non-classical GPTs is entangleable under natural assumptions, supporting a universal role for entanglement in physical theories.
• To explore a quantitative geometric relationship between local non-classicality and global entangleability, especially in centrally symmetric state spaces.
• To resolve the foundational puzzle of why entanglement is not merely a quantum feature but a universal consequence of non-classicality.

Proposed method

• Formalizes physical theories as convex compact state spaces within finite-dimensional real vector spaces, defining GPTs as the foundational framework.
• Introduces the notion of entangleability between two GPTs as the existence of entangled states or entangled measurements in their composite system.
• Employs tools from convex geometry—such as shadow boundaries, affine hulls of boundary points, and projections—using Zamfirescu’s theorem on generic convex bodies.
• Applies functional analytic techniques, including the use of retractions and projections onto subspaces, to rule out non-trivial retracts in generic cones.
• Uses the geometric structure of centrally symmetric state spaces to explicitly describe a quantitative relation between local non-classicality and global entangleability.
• Proves results via contradiction: assuming a retract exists leads to a contradiction with strict convexity and generic boundary properties of most convex bodies.

Experimental results

Research questions

• RQ1 Can entanglement be defined universally across all physical theories, independent of the quantum formalism?
• RQ2 Is every pair of non-classical GPTs entangleable, meaning their composite system admits entangled states or measurements?
• RQ3 What is the geometric and functional-analytic mechanism linking single-system non-classicality to bipartite entangleability?
• RQ4 Can a quantitative measure of entangleability be derived from local non-classicality, particularly in symmetric state spaces?
• RQ5 Are there exceptional GPTs that are non-classical yet fail to produce entanglement in their composite systems?

Key findings

• Any two non-classical GPTs are entangleable when their local state spaces are 3-dimensional, establishing the result in the simplest non-trivial case.
• For discrete local state spaces (finite number of pure states), the conjecture of universal entangleability holds, covering foundational models like Popescu–Rohrlich boxes.
• When one local theory is quantum mechanics, the composite system always admits entangled states or measurements, confirming entangleability in a physically central case.
• In the geometrically natural case of centrally symmetric local state spaces, a quantitative relation is explicitly constructed between local non-classicality and global entangleability.
• Most convex bodies in dimension $ n-1 $ are strictly convex and satisfy the generic boundary condition $ ext{aff} ig( igcup_{x otin K} ext{bd}(K,x) ig) = n-1 $, implying that their associated cones have no $ (n-1) $-dimensional retracts.
• The absence of such retracts in generic cones proves that entanglement is not a rare or special feature, but a robust consequence of non-classicality.