Optimal Entanglement Recovery from Environmental Measurements

AbstractWe solve the Gregoratti–Werner optimization over environmental measurements exactly for the four-label dephasing example of Buscemi, Chiribella and D’Ariano [Phys. Rev. Lett. 95, 090501 (2005)]. The optimal deterministic entanglement fidelity under environmental measurement is cos2 (π/16), whereas coherent access permits unit fidelity. An attaining projective measurement and analytic operator certificate establish optimality over all finite POVMs. Varying one environmental state gives a family whose measurement optimization reduces to a one-dimensional concave envelope. Writing c = cos(θ/2), we prove exact projective-basis optimality throughout 0.85 ≤ c ≤ 1 and √ 1/ 2 ≤ c ≤ 0.728, using analytic contact bounds and uniform rational certificates. Inside the intervening region, a three-outcome measurement has exact optimum 153/160 at c = 3/4, with a strict, exactly quantified advantage over every projective basis. At c = 4/5, a certified four-outcome protocol exceeds every projective-simulable measurement on the environmental qubit by more than 0.00499536. The optimized measurement loss is nonmonotone. We also obtain an exact phase-only optimum for a related four-label problem and constructive correlation-matrix recovery formulas for mixed states with partial environmental access, relating the latter to conditional min-entropy and robustness of asymmetry.