Unraveling-paired dynamical maps recover the input of quantum channels

Abstract

We explore algebraic and dynamical consequences of unraveling general time-local master equations. We show that the ‘influence martingale’, the paramount ingredient of a recently discovered unraveling framework, pairs any time-local master equation with a one parameter family of Lindblad–Gorini–Kossakowski–Sudarshan master equations. At any instant of time, the variance of the influence martingale provides an upper bound on the Hilbert–Schmidt distance between solutions of paired master equations. Finding the lowest upper bound on the variance of the influence martingale yields an explicit criterion of ‘optimal pairing’. The criterion independently retrieves the measure of isotropic noise necessary for the structural physical approximation of the flow the time-local master equation with a completely positive flow. The optimal pairing also allows us to invoke a general result on linear maps on operators (the ‘commutant representation’) to embed the flow of a general master equation in the off-diagonal corner of a completely positive semi-group which in turn solves a time-local master equation that we explicitly determine. We use the embedding to reverse a completely positive evolution, a quantum channel, to its initial condition thereby providing a protocol to preserve quantum memory against decoherence. We thus arrive at a model of continuous-time error correction by a quantum channel.