Entanglement Phase: A Geometric–Topological Formalism for Emergent Spacetime

This work explores a new perspective on quantum entanglement, where the relative phases between entangled components are treated not as secondary details, but as fundamental degrees of freedom. Traditional measures of entanglement, such as entropy, capture how “mixed” a system is, but they often ignore the rich structure encoded in phase relationships. Our goal is to elevate these phases to a central role and investigate what new physics can emerge from this viewpoint. We introduce the Schmidt coherence matrix (CCC), a tool that captures both the amplitude and relative phase of each component in a bipartite quantum system. By analyzing the CCC, we can define a metric on the space of entangled states, a way to measure “distances” that arise purely from phase differences. In other words, we show that phase itself generates a geometry, and this geometry has real physical consequences. Mathematically, we construct what can be called a purification bundle, using ideas from differential geometry and gauge theory. The framework allows us to describe the evolution of phases in a way that is invariant under global transformations, ensuring that only physically meaningful quantities contribute. We derive formulas for phase-induced metrics, connecting them to known objects in quantum information theory like the Fubini–Study metric and quantum Fisher information. Beyond the formalism, this approach has deep implications for physics. When phase gradients are interpreted as fields, they can influence matter and light in a way analogous to gravity. Remarkably, in coarse-grained limits, the equations governing the phase metric reproduce familiar results from General Relativity, suggesting a link between entanglement and the structure of spacetime itself. We also explore potential experimental signatures: for example, quantum interferometry could detect tiny shifts induced by gravitational sources through phase correlations, offering a testable connection between quantum information and spacetime geometry. This work is consistent and self-contained. The entanglement metric is gauge-invariant, the phase-induced stress-energy tensor behaves as expected under variations, and semiclassical predictions align with known physics. We also discuss extensions to mixed states, using Bures–Uhlmann metrics to retain the phase-sensitive structure even in realistic, noisy systems. In short, this work shifts the focus from entropy-based descriptions to phase-based descriptions of quantum systems, offering a bridge between quantum information, geometry, and gravity. It opens new avenues for understanding how the universe’s large-scale structure could emerge from fundamental quantum processes, and provides concrete ideas for future experiments to probe these effects.