Fig. 1: Appearance of NLWE.
When \({\theta }_{0}\,\ne \,\frac{\pi }{4}\), if θ1 < θ0, both singular rates γ− and γ+ are valid. However, when \({\theta }_{1}\ge \frac{\pi }{2}-{\theta }_{0}\), both γ− and γ+ are not valid. If \({\theta }_{0}\le {\theta }_{1}<\frac{\pi }{2}-{\theta }_{0}\), γ+ is valid but γ− is not. When the ratio r becomes γ− (blue) or γ+ (red), the global minimum-error probability of a quantum ensemble \({\mathcal{E}}\) can be achieved by a finite-round LOCC, otherwise it cannot be obtained by finite-round LOCC. Furthermore, if \(r\,\notin \,{\mathcal{G}}\) (yellow), the global minimum-error probability of the quantum ensemble \({\mathcal{E}}\) can be achieved by means of a separable measurement. Meanwhile, when \(r\;\in \;{\mathcal{G}}\) (green), the global minimum-error probability of a quantum state ensemble \({\mathcal{E}}\) cannot be obtained by a separable measurement.
