Universal Limits on Quantum Correlations

Quantum correlations are the singular, defining resource of quantum information science and metrology, forming the basis of every operational advantage that quantum systems hold over classical ones. Yet exact bounds on these correlations-such as the Lieb-Robinson bound on entanglement propagation and the Heisenberg limit on metrological precision-are known only in special cases and have long appeared to arise from unrelated mechanisms. Here we show that these limits share a common geometric origin. We identify a positivity invariant of the block correlation matrix, denoted $\chi$, that quantifies how far a bipartite state lies from the positivity boundary of quantum state space. For any system with a specified observable algebra and parameter-encoding map, every correlation measure determined solely by the positive correlation matrix obeys a $\chi$-dependent inequality. For systems with simple symmetry structures these inequalities take closed analytic form, reproducing the structure of the Heisenberg and Cram\'er-Rao limits and producing new results, including an exact entanglement floor and a universal Fisher-information ceiling even in all-to-all connected quantum networks. We thus demonstrate that positive geometry provides a unified framework for the attainable strength of quantum correlations, linking entanglement, metrological sensitivity, and dynamical causal structure through a single invariant.