Local Geometry of Quantum Information: A Framework for the Arrow of Time in Open Quantum Systems

We develop a geometric framework for analysing quantum dynamics through the short-time expansion of the survival probability S (t) = Tr₀ (t). The expansion S (t) = 1 - \, t - 12\, t² - 16j\, t³ + O (t⁴) defines three local coefficients — the tilt, the curvature, and the jerk j — that separate, in a model-independent way, the time-asymmetric dissipative content from the time-symmetric unitary content of the evolution. For Markovian (GKSL) dynamics with a pure initial state, 0 equals half the instantaneous purity-loss rate and serves as a geometric measure of the local arrow of time. We establish a four-fold entropy hierarchy connecting to linear entropy, von Neumann entropy Sₕ₍ (t) = -\, t t + O (t), Bures distance, and quantum Fisher information, independent of the specific dynamical model. The framework extends in three directions. First, kinematic time asymmetry is formulated through forward–backward overlap functionals controlled by. Second, for non-Markovian dynamics, (t) < 0 signals information backflow, and the monotonicity of the fidelity deficit (t) = 1 - S (t) provides an operationally accessible Markovianity witness; a spectral closure theorem provides an independent signature of memory effects. Third, for multipartite systems, the entanglement-flux decomposition Q = ₓ₎ₓ₀₋ + 12ĖL (0) shows that the local arrow of time of a subsystem decomposes exactly into global irreversibility plus entanglement production. This generalises to N-partite systems through a telescoping decomposition and an ordering-independent sum rule ᵢ ₐ㶁 = ₓ₎ₓ₀₋ + 12Ṫ₂ (0) governed by the Tsallis-2 total correlation rate. All decompositions rest on a single universal identity requiring only differentiability of the state.