Entanglement Phase: A Geometric–Topological Formalism for Emergent Spacetime

We propose a mathematical and operational formalism for the “entanglement phase,” a quantity neglected in traditional quantum mechanics and entanglement measures. While conventional metrics such as entropy, concurrence, or negativity capture only the probability distribution across Schmidt branches, they ignore the relative phase between components. Our framework introduces the Schmidt coherence matrix (C), encoding both probability amplitudes and relative phase information. From (C), we construct measurable invariants, including trace functionals (\mathrm{Tr} C^k) and holonomies on the purification bundle, enabling the distinction of states with identical entropies but different phase structures. A central element is the definition of an emergent entanglement metric (g_{\mu\nu}^{\rm ent}), derived from gradients of phase variations. This allows a radical reinterpretation of spatial distance: two systems with stable relative phases ((\Delta \phi \approx 0)) occupy the same “topological locus,” independent of laboratory-frame separation, while accumulated phase shifts (dephasing) correspond to increased topological distance. To render the framework dynamical, we propose a phase–energy coupling: local mass and energy act as phase shifters or local decoherence sources, curving the Schmidt bundle and modifying the phase metric. Using a variational or thermodynamic approach, we derive field-equation–like relations connecting phase dynamics to the local stress–energy tensor. We also outline a route to an emergent Lorentzian signature via modular flow (Tomita–Takesaki), providing a theoretical grounding for time as an emergent property of entanglement structure. The framework includes concrete experimental proposals: SPDC entangled photons with local interferometers and shared phase references; Measurement of Uhlmann holonomy along parameter cycles; Tests in topological systems and anyonic qubits; A unique falsifiable prediction: entanglement-phase shifts induced by nearby massive bodies, measurable in satellite-based quantum communication links. Additionally, we present toy-model calculations, numerical simulations under decoherence, and a critical analysis of technical issues such as local gauge dependence, mixed states, and operational measurability. In summary, this work outlines a theory in which the entanglement phase serves as a fundamental degree of freedom, from which spacetime geometry emerges. It bridges quantum information, topology, and gravitational dynamics, providing experimentally accessible predictions and a roadmap for validation.