Figure 1

Pyramid geometry for $N=3$. The ancilla kets $\mid E_k⟩$, of unit length, are the edges of the ancilla pyramid. Its shape is determined by the parameter $\lambda$ of Eq. (1), the cosine of the acute angle between any pair of edges. The height ket $\mid H⟩$ of Eq. (2) points from the tip of the pyramid to the center of its base; its length is $\sqrt{r_0}$. The kets $\mid E_k⟩-\mid H⟩$, of length $\sqrt{1-r_0}$, point from the center of the pyramid base to its corners. The SRM kets $\mid e_k⟩$ of Eq. (9), of unit length, define the SRM pyramid, which has right angles between its edge kets. The SRM pyramid is wider than, but not as high as, the ancilla pyramid.Reuse & Permissions

Figure 2

Mutual information between Alice and Eve if Eve performs the square-root measurement. The curves refer to $N=2,3,5,10,20$, and $100$, and the plot covers the range $0⩽\lambda ⩽1$ that is relevant for quantum cryptography.Reuse & Permissions

Figure 3

Mutual information for the POVM of Eq. (13) relative to that of the SRM. For $N=10$, the plot shows the ratio of $I\left(T\right)∕I^{\left(\mathrm{SRM}\right)}$ as a function of $T$ for $\lambda =0.9,0.7,0.5,0.3$ (solid lines) and for $\lambda =0.8,0.6,0.4$ (dashed lines). The left end $\left(T=0\right)$ refers to the SRM, the right end $\left(T=1\right)$ to the MUD. For $\lambda =\mathrm{0.772\hspace{0.17em}76}$ (dash-dotted line), both give the same mutual information.Reuse & Permissions

Figure 4

Ratio of the maximal mutual information $I_{\max }$ and the SRM value $I^{\left(\mathrm{SRM}\right)}$, for $N=3,5,10,20,100$, as a function of $\lambda$.Reuse & Permissions