We discuss the interconnections between basic correlation measures of a bipartite quantum state and basic information characteristics of a quantum channel, focusing on the benefits of these interconnections for solving specific problems concerning the characteristics of both types. We describe properties of the (unoptimized and optimized) quantum discord in infinite-dimensional bipartite systems. In particular, using the generalized Koashi–Winter relation, a simple condition is obtained that guarantees that a state with zero quantum discord is quantum-classical. Two possible definitions of the quantum discord for states with infinite one-way classical correlation are proposed and analysed. The generalized versions of Koashi–Winter and Xi–Lu–Wang–Li relations are used to obtain advanced continuity bounds for the Holevo information at the outputs of a channel and its complementary channel (as functions of a channel for a given ensemble of input states), for the Holevo capacity and the unregularized private capacity of a quantum channel depending either on the input dimension or on the input energy bound. We also discuss the properties of quantum channels which are “doppelgangers” of the monotonicity of the quantum discord and the entropy reduction of a local measurement under quantum channels acting on an unmeasured subsystem.

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An example of such a state and a channel can be constructed by taking any state ω in S(HBC) such that S(ωB) = S(ωC) = +,but I(B: C)ω < +. Let ω̂ be a purification of ω in S(HABC). Then the channel Φ(σ) = TrAσ, σS(HAB) and the state ρ=ω̂AB have the required properties.

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