Fig. 3

These graphics compare one slice through the magic simplex^42,43,45 for different dimension \(d=3,5,7\). The state space of \(\rho (\alpha ,\beta )=\frac{1-\alpha -\beta }{d^2} {\mathbbm{1}}_d+\alpha \; \rho _1 + \beta \; \rho _2\) with \(\alpha\) on the x-axis and \(\beta\) on the y-axis is visualized. The mirrored witness pairs \(\{W_1,W_2\}\) (red/gold lines) are always “parallel” to the isotropic state \(\rho (\alpha ,\alpha )\equiv \rho _{iso}(p)=p \,P_{0,0}+(1-p)\, \frac{1}{d^2} {\mathbbm{1}}_{d^2}\) (black dotted line). In all dimensions, both witnesses detect bound entangled states. In the case of \(d=3,7\), the PPT states (blue and green) are symmetric around the isotropic state, and by that, they detect the same amount of bound entangled states. However, for \(d=5\) the PPT region (blue) is not symmetric with respect to the positivity region (green); both EWs detect bound entangled states, but not the same amount. Thus symmetry on the Hermitian but not positive space is not in one-to-one correspondence with symmetry with respect to transposition in one subsystem and positivity.