By Jinqiao Duan, Shunlong Luo, Caishi Wang
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Extra info for Recent Development in Stochastic Dynamics and Stochastic Analysis (Interdisciplinary Mathematical Sciences, 8)
Consider the task of sending the system state through a noisy channel in order to transmit the Decoherent Information 25 initial correlations (not just the initial system state). Since the auxiliary system is isolated and left undisturbed, the joint operation on the initial puriﬁed state is the tensor product of the system evolution E and the identity operation on the auxiliary system, and the ﬁnal bipartite state will usually have less correlations. The purpose is to maintain correlations between the ﬁnal system state and the auxiliary system as many as possible.
44 46 48 49 50 50 51 53 54 59 60 60 60 63 63 64 44 Tom´ as Caraballo and Peter E. Kloeden 1. Introduction Stabilization by noise has a long history dating back, in engineering practice, to the early days of the industrial revolution. Attempts to understand the phenomenon mathematically in terms of stochastic diﬀerential equations (SDEs) began in the 1960s, at ﬁrst mainly for ﬁnite dimensional systems and in the past decade for inﬁnite dimensional systems, that is for evolution equations, by which is meant mainly partial diﬀerential equations (PDEs), but also includes delay diﬀerential equations (DDEs).
To establish Eq. 7), note that S(ρa ) = S(ρb ) = S(ρa ), and since ρa b c is pure, we also have S(ρa b ) = S(ρc ) and S(ρb c ) = S(ρa ). Now by the deﬁnition of the decoherent information, we have D(ρb , E b ) = I(ρab ) − I(ρa b ) = 2S(ρb ) − (S(ρa ) + S(ρb ) − S(ρa b )) = S(ρb ) − S(ρb ) + S(ρc ), and similarly, D(ρb , Ecb ) = 2S(ρb ) − I(ρa c ) = 2S(ρb ) − (S(ρa ) + S(ρc ) − S(ρa c )) = S(ρb ) − S(ρc ) + S(ρb ). Summing up the above equations yields the desired Eq. 7). A particular interesting instance of Eq.