Kempe-Invariant Ordering of Three-Qubit Pairwise Entanglement under Local Depolarization

Abstract

Pure three-qubit states with the same pairwise concurrences and three-tangle can still have different Kempe invariants. We show that the Kempe invariant orders the entanglement dynamics of all three pairs under identical, independent local depolarizing channels, with smaller values giving greater robustness. The same ordering also holds for the sum of squared pairwise entanglements of formation and the sudden-death thresholds of pairwise entanglement.

For pure three-qubit states, fixing the three pairwise concurrences and the three-tangle generally leaves a continuous LU-invariant degree of freedom. This paper asks whether that remaining coordinate, the Kempe invariant $\kappa$, determines an ordering of pairwise-entanglement robustness under identical, independent local depolarization, $\Lambda_\eta(\rho)=\eta\rho+(1-\eta)I/2$. It proves that, among states with the same four initial entanglement coordinates, $\kappa_1<\kappa_2$ implies $C_{ij,\eta}(\kappa_1)\ge C_{ij,\eta}(\kappa_2)$ simultaneously for all three pairs and every $0\le\eta\le1$. For intermediate noise strengths, the decrease with $\kappa$ is strict wherever concurrence is positive. Consequently, the lowest and highest allowed Kempe values respectively maximize and minimize all three concurrence trajectories. The ordering extends to coordinatewise nondecreasing functions of the concurrences, including the sum of squared entanglements of formation. For initially entangled pairs, smaller $\kappa$ also gives a strictly larger sudden-death threshold in noise strength $p=1-\eta$. The proof constructs a quartic polynomial whose smallest root in the entangled region is squared concurrence. Pure-state invariant identities determine its deformation along the fixed-entanglement fiber; sign bounds and implicit differentiation establish monotonicity, with continuity handling boundaries and zero concurrence. The stated scope is pure initial states and equal local depolarizing strengths. Unequal strengths, mixed inputs, other noise channels, and analogous orderings for Bell nonlocality or steering remain open. Non-LU-covariant channels additionally require specifying local orientations.