Infinite operator-sum representation of density operator for a dissipative cavity with Kerr medium derived by virtue of entangled state representation

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By using the thermo entangled state representation we solve the master equation for a dissipative cavity with Kerr medium to obtain density operators' infinite operator-sum representationρ(t) =∑ₘ,ₙ,ₗ₌₀∞Mₘ,ₙ,ₗρ₀Mₘ,ₙ,ₗ†. It is noticeable thatMₘ,ₙ,ₗ is not hermite conjugate to Mₘ,ₙ,ₗ†, nevertheless the normalization∑ₘ,ₙ,ₗ₌₀∞Mnm,,l†Mₘ,ₙ,ₗ=1 still holds, i.e., they are trace-preserving in a general sense. This example may stimulate further studying if general superoperator theory needs modification.