Entanglement-Flux Decomposition of the Local Arrow of Time

This work extends the local-invariant framework for the survival probability $S(t) = 1 - \alpha\, t - \frac{1}{2}\kappa\, t^2 - \frac{1}{6}j\, t^3 + O(t^4)$ to multipartite quantum systems. Bipartite decomposition. For an initially pure product state, we derive the exact identity $$\alpha_S = \alpha_{SA} + \frac{1}{2}\,\dot{E}_L(0),$$ demonstrating that the local arrow of time of subsystem $S$ splits into a global irreversibility contribution $\alpha_{SA}$ and an entanglement-driven contribution $\frac{1}{2}\dot{E}_L(0)$, where $\dot{E}_L(0)$ is the instantaneous production rate of linear entanglement entropy. This "entanglement-flux decomposition" explains why local irreversibility can arise even when the global dynamics is microscopically reversible. $N$-partite generalization. We establish three equivalent formulations for $N$-partite systems: (i) a canonical single-cut form $\alpha_{S_i} = \alpha_{\mathrm{total}} + \frac{1}{2}[\dot{S}_{L,S_i}(0) - \dot{S}_{L,\mathrm{total}}(0)]$; (ii) a telescoping decomposition $$\alpha_{S_i} = \alpha_{\mathrm{total}} + \sum_{k=1}^{N-1}\frac{1}{2}\,\dot{E}_L^{(k)}(0),$$ which resolves the local tilt into global irreversibility plus a chain of entanglement fluxes across nested bipartitions; and (iii) an ordering-independent sum rule $\sum_{i=1}^{N}\alpha_{S_i} = N\,\alpha_{\mathrm{total}} + \frac{1}{2}\dot{C}_2(0)$, governed by the Tsallis-2 total correlation rate. Universality and obstructions. The underlying identity $\alpha = \frac{1}{2}\dot{S}_L(0)$ is proven to be universal—requiring no assumption of Markovianity, GKSL structure, or specific dynamics beyond differentiability—making all decompositions valid for arbitrary quantum dynamics. We prove that pairwise decomposition of multipartite entanglement flux is fundamentally obstructed by the failure of strong subadditivity for Tsallis-2 entropy (Petz–Virosztek, 2015), and identify the available inequality structure from subadditivity and polygon relations. Corrected examples. This revision (v2, February 2026) corrects the amplitude damping example from the original version: the survival probability is $\frac{1}{2}(1+e^{-\Gamma t/2})$ with tilt $\alpha = \Gamma/4$, consistent with Paper I v2. Additional examples include pure dephasing, unitary entangling dynamics, and a Bell state under local amplitude damping (illustrating negative entanglement flux from entanglement destruction). The decomposition links purity flow, subsystem decoherence, entanglement generation, and Tsallis-2 correlation theory within a unified differential framework, providing geometric and operational tools for analysing temporal asymmetry in composite open quantum systems.