Kinematic budget of quantum correlations

TL;DR

This paper introduces a geometric framework for understanding quantum correlations using second moments, revealing structural links between classical and quantum regimes, and providing a scalable way to analyze quantum resources without full-state tomography.

Contribution

It develops a novel, topology-driven geometric model that maps quantum correlations onto compact manifolds, linking purity, symmetry, and resource activation in a unified framework.

Findings

Quantum correlations collapse to a unified geometric structure at the second-moment level.

Exceeding classical limits activates time-asymmetric generators, indicating non-positive partial transpose entanglement.

The model simplifies analysis of quantum resources, bypassing exponential state tomography complexity.

Abstract

The diversity of quantum correlations -- discord, entanglement, steering, and Bell nonlocality -- disappears at the observable second-moment kinematic level. By treating state purity as a finite resource, we introduce a local-unitary-invariant budget split of symmetrised second moments into local and nonlocal sectors that maps quantum systems onto compact, two-dimensional, hole-free manifolds. The topology of these manifolds is governed by state purity and time-reversal symmetry. This dimensional reduction reveals a deep structural link: exceeding classical capacity limits forces the activation of time-asymmetric generators, guaranteeing non-positive partial transpose entanglement. For two qubits, the geometry is analytically solvable. A single boundary elegantly isolates classical correlations, while nested regions physically dictate entanglement, steering, Bell nonlocality, and bounds on non-stabiliser magic. Beyond two qubits, dimensional capacity bottlenecks enforce these universal kinematic limits on correlation structures. Because this macroscopic representation is completely determined by global and marginal purities, it bypasses the exponential scaling of full-state tomography. By coarse-graining over gauge-like first moments, the budget geometry acts as a thermodynamic phase diagram, exposing both the static hierarchy of quantum resources and their dynamic redistribution under decoherence.