Fig. 2: Proof of Theorem 1 using tensor network calculations.
From: Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps
A matrix is represented using a box with left and right legs, where the left legs represent row indices and the right legs represent column indices. The connection of legs indicates the contraction between indices. In a, we illustrate the tensor network representation of the Choi matrix \({\Lambda }_{{\mathcal{N}}}\) and how it can be used to represent the action of a linear map \({\mathcal{N}}\). In b, we use the exchange of the left and right legs to represent the transposition operation. Consequently, it becomes evident that in order to ensure \({{\rm{Tr}}}_{1}[H(\rho \otimes \sigma )]={\mathcal{N}}(\rho )\sigma\), we must have \(H={\Lambda }_{{\mathcal{N}}}^{{{\rm{T}}}_{1}}\), which is highlighted by the red dashed box. In c, we graphically demonstrate the validation of \({H}_{P}={\Phi }_{A}^{+}\otimes {S}_{B}\) for realizing the evolution of \({e}^{-i{\rho }^{{{\rm{T}}}_{A}}t}\), where \({\rho }^{{{\rm{T}}}_{A}}\) is the partial transpose of ρ on system A. We use blue and purple legs to represent the indices of subsystems A and B, respectively.
