Table 1 Concrete protocols for verifying the Bell state in an untrusted quantum network.
Protocol | Threshold γ* | γ(C) (pure state) | \(\hat{\gamma }(C)\) (mixed state) | v(C = 0) |
|---|---|---|---|---|
XY | \(\frac{1}{2}+\frac{1}{2\sqrt{2}}\approx 0.854\) | \(\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1+{C}^{2}}{2}}\) | \(\frac{1}{4}[2+\sqrt{2}+(2-\sqrt{2})C\ ]\) | \(\frac{1}{\sqrt{2}}{(1,1,0)}^{{{{\rm{T}}}}}\) |
XYZ | \(\frac{1}{2}+\frac{1}{2\sqrt{3}}\approx 0.789\) | \(\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1+2{C}^{2}}{3}}\) | \(\frac{1}{6}[3+\sqrt{3}+(3-\sqrt{3})C\ ]\) | \(\frac{1}{\sqrt{3}}{(1,1,1)}^{{{{\rm{T}}}}}\) |
Isotropic | \(\frac{3}{4}=0.75\) | \(\frac{3}{4}+\frac{{C}^{2}{{{\rm{arcsinh}}}}\left(\frac{\sqrt{1-{C}^{2}}}{C}\right)}{4\sqrt{1-{C}^{2}}}\) | \(\frac{3+C}{4}\) | any direction |
Equator | \(\frac{1}{2}+\frac{1}{\pi }\approx 0.818\) | \(\frac{1}{2}+\frac{1}{\pi }K(\sqrt{1-{C}^{2}}\ )\) | \(\frac{1}{2\pi }[\pi +2+(\pi -2)C\ ]\) | any direction in the xy-plane |
Polygon(3) | \(\frac{5}{6}\approx 0.833\) | \(\frac{4+\sqrt{1+3{C}^{2}}}{6}\) | \(\frac{5+C}{6}\) | any vertex direction |
Equator +Z | \(\frac{1}{2}+\frac{1}{\sqrt{4+{\pi }^{2}}}\approx 0.769\) | − | \(\frac{1+C}{2}+\frac{1-C}{\sqrt{4+{\pi }^{2}}}\) | \(\frac{1}{\sqrt{4+{\pi }^{2}}}{(\pi ,0,2)}^{{{{\rm{T}}}}}\) |
Polygon(3)+Z | \(\frac{1}{2}+\frac{1}{\sqrt{13}}\approx 0.777\) | − | \(\frac{1}{2}+\frac{1}{\sqrt{13}}+\left(\frac{1}{2}-\frac{1}{\sqrt{13}}\right)C\) | \(\frac{1}{\sqrt{13}}{(3,0,2)}^{{{{\rm{T}}}}}\) |
