Fig. 9: The LOCC operation Λ^D defined in Eq. (74).

Here we provide a visualization of the LOCC operation Λ^D for the case where n = 4. Let’s begin with (a), where we employ concentric circles to depict quantum states that are uniform mixtures of four tensor product states. Each layer, labeled by a basis state \(\left\vert t\right\rangle\) of the auxiliary system T ranging from the innermost to the outermost for t = 1, …, 4, symbolizes a component of the state. Specifically, the innermost concentric circle represents \({\rho }_{1}\otimes {{{{\rm{Tr}}}}}_{1}[{{{\mathcal{E}}}}({\rho }^{\otimes 4})]\otimes \left\vert 1\right\rangle {\left\langle 1\right\vert }_{T}/4\), where i ∈ {1, …, 4} abbreviates systems A_iB_i. The red quarter stands for state ρ, while the white part in the t-th layer denotes \({{{{\rm{Tr}}}}}_{1\cdots t}[{{{\mathcal{E}}}}({\rho }^{\otimes 4})]\). Next, let’s discuss the construction of Λ^D, which comprises three steps: First, applying \({{{\mathcal{E}}}}\) to the state with classical register \(\left\vert 4\right\rangle {\left\langle 4\right\vert }_{T}\) yields the state depicted in (b). Second, after implementing a permutation to the classical register system T: \({\left\vert t\right\rangle }_{T}\to {\left\vert t+1\right\rangle }_{T}\) for t < 4 and \({\left\vert 4\right\rangle }_{T}\to {\left\vert 1\right\rangle }_{T}\), we obtain (c). Third, a SWAP on systems 1 and t for in t-th layer leads to (d).