Fig. 2: The LOCC operation ΛCS defined in Eq. (22).
We use a visual example with n = 4 to illustrate the convex-split lemma. Each color represents a different state: red for ρ, white for τ, and blue for the maximally entangled state ϕ+. The circle portrays a uniform mixture of 4 concentric circles. Each concentric circle, from the innermost to the outermost, is designated as the t-th layer, with t ranging from 1 to 4, and represents a quantum state. For instance, every concentric circle in (a) stands for a tensor product state of the form ρ1 ⊗ τ2 ⊗ τ3 ⊗ τ4, where i ∈ 1, …, 4 denotes systems AiBi. In (b), we take layer 2 as an illustration, depicting τ1 ⊗ ρ2 ⊗ τ3 ⊗ τ4. Finally, (c) showcases the state \({\phi }_{1}^{+}\otimes {\tau }_{2}\otimes {\tau }_{3}\otimes {\tau }_{4}\). The symbol ≈ indicates that the purified distance between the quantum states depicted in (b) and (c) is at most \(\eta =\sqrt{{2}^{k}/4}+P(\tau ,{\phi }^{+})\).
