Author Correction: Entanglement Availability Differentiation Service for the Quantum Internet

Correction to: Scientific Reports 10.1038/s41598-018-28801-3, published online 13 July 2018

This Article contains errors.

In Protocol 0, where

should read:

In addition, in Equation 13 and 14

$$\begin{array}{cc}\hfill {\sigma }_{\mathit{ABC}}=& |\phi (t)⟩{⟨\phi (t)|}_{\mathit{ABC}}=U{\rho }_0{U}^{†}\hfill \\ \hfill =& \frac{1}{2}(U_{\mathit{AC}}|{\psi }_{+}⟩⟨{\psi }_{+}||+⟩⟨+|U_{\mathit{AC}}^{†})+\frac{1}{2}(U_{\mathit{AC}}|{\varphi }_{+}⟩⟨{\varphi }_{+}||-⟩⟨-|U_{\mathit{AC}}^{†}),\hfill \end{array}$$

where ${|\phi (t)⟩}_{\mathit{ABC}}$ at time t is evaluated as

$$\begin{array}{cc}\hfill {|\phi (t)⟩}_{\mathit{ABC}}=& \frac{1}{\sqrt{2}}(cos(t)(|{\psi }_{+}⟩|+⟩)-isin(t)(|{\varphi }_{+}⟩|+⟩))\hfill \\ \hfill & +\frac{1}{\sqrt{2}}(cos(t)(|{\varphi }_{+}⟩|-⟩)+isin(t)(|{\psi }_{+}⟩|-⟩))\hfill \\ \hfill =& \frac{1}{\sqrt{2}}(cos(t)(|{\psi }_{+}⟩)-isin(t)(|{\varphi }_{+}⟩))|+⟩\hfill \\ \hfill & +\frac{1}{\sqrt{2}}(cos(t)(|{\varphi }_{+}⟩)+isin(t)(|{\psi }_{+}⟩))|-⟩\hfill \end{array}$$

should read:

$${\sigma }_{\mathit{ABC}}=\frac{1}{2}\left(U_{\mathit{AC}}(t)|{\psi }_{+}⟩⟨{\psi }_{+}||+⟩⟨+|U_{\mathit{AC}}^{†}(t)\right)+\frac{1}{2}\left(U_{\mathit{AC}}(t)|{\varphi }_{+}⟩⟨{\varphi }_{+}||-⟩⟨-|U_{\mathit{AC}}^{†}(t)\right),$$

while ${|\phi (t)⟩}_{\mathit{ABC}}$ at time t is defined as

$$\begin{array}{cc}\hfill |& {\phi (t)⟩}_{\mathit{ABC}}\hfill \\ \hfill & =\frac{\text{1}}{\sqrt{\text{2}}}(U_{\mathit{AC}}(t)(|{\psi }_{+}⟩|+⟩)+U_{\mathit{AC}}(t)(|{\varphi }_{+}⟩|-⟩))\hfill \\ \hfill & =\frac{\text{1}}{\sqrt{\text{2}}}(cos(t)(|{\psi }_{+}⟩|+⟩)-isin(t)(|{\varphi }_{+}⟩|+⟩))+\frac{\text{1}}{\sqrt{\text{2}}}(cos(t)(|{\varphi }_{+}⟩|-⟩)+isin(t)(|{\psi }_{+}⟩|-⟩))\hfill \\ \hfill & =\frac{\text{1}}{\sqrt{\text{2}}}(cos(t)(|{\psi }_{+}⟩)-isin(t)(|{\varphi }_{+}⟩))|+⟩+\frac{\text{1}}{\sqrt{\text{2}}}(cos(t)(|{\varphi }_{+}⟩)+isin(t)(|{\psi }_{+}⟩))|-⟩\hfill \\ \hfill \end{array}$$

Furthermore, in Equation 22 and 23

$$\begin{array}{cc}\hfill {\sigma }_{\mathit{ABC}}=& |\phi (t)⟩{⟨\phi (t)|}_{\mathit{ABC}}=U{\rho }_0{U}^{†}\hfill \\ \hfill =& \frac{1}{2}(U_{\mathit{AC}}|{\psi }_{+}⟩⟨{\psi }_{+}||+⟩⟨+|U_{\mathit{AC}}^{†})+\frac{1}{2}(U_{\mathit{AC}}|{\varphi }_{+}⟩⟨{\varphi }_{+}||-⟩⟨-|U_{\mathit{AC}}^{†}),\hfill \end{array}$$

where ${|\phi (t)⟩}_{\mathit{ABC}}$ at t is evaluated as

$$\begin{array}{cc}\hfill {|\phi (t)⟩}_{\mathit{ABC}}=& \frac{1}{\sqrt{2}}(U_{\mathit{AC}}(|{\psi }_{+}⟩|+⟩)+U_{\mathit{AC}}(|{\varphi }_{+}⟩|-⟩))\hfill \\ \hfill =& \frac{1}{\sqrt{2}}\left(cos,(t),\left(\frac{1}{2},(|010⟩,+,|011⟩,+,|100⟩,+,|101⟩)\right),+,\mathit{cos},(t),\left(\frac{1}{2},(|000⟩,-,|001⟩,+,|110⟩,-,|111⟩)\right)\right)\hfill \\ \hfill & -isin(t)\left(\frac{1}{2},(|111⟩,+,|110⟩,+,|001⟩,+,|000⟩)\right)\hfill \\ \hfill & \left(+,,i,,sin,(t),\left(\frac{1}{2},\left(|,101,⟩,-,|,100,⟩,+,|,011,⟩,-,|,010,⟩\right)\right)\right)\hfill \\ \hfill =& \frac{1}{\sqrt{2}}(cos(t)(|{\psi }_{+}⟩|+⟩+|{\varphi }_{+}⟩|-⟩)-isin(t)(|{\varphi }_{+}⟩|+⟩-|{\psi }_{+}⟩|-⟩))\hfill \\ \hfill =& \frac{1}{\sqrt{2}}((cos(t)(|{\psi }_{+}⟩)-isin(t)(|{\varphi }_{+}⟩))|+⟩+(cos(t)(|{\varphi }_{+}⟩)+isin(t)(|{\psi }_{+}⟩))|-⟩),\hfill \end{array}$$

should read:

$${\sigma }_{\mathit{ABC}}=\frac{\text{1}}{\text{2}}\left(U_{\mathit{AC}}(t)|{\psi }_{+}⟩⟨{\psi }_{+}||+⟩⟨+|U_{\mathit{AC}}^{†}(t)\right)+\frac{\text{1}}{\text{2}}\left(U_{\mathit{AC}}(t)|{\varphi }_{+}⟩⟨{\varphi }_{+}||-⟩⟨-|U_{\mathit{AC}}^{†}(t)\right),$$

and ${|\phi (t)⟩}_{\mathit{ABC}}$ at t is defined as

$$\begin{array}{cc}& {|\phi (t)⟩}_{\mathit{ABC}}\hfill \\ \hfill & =\frac{\text{1}}{\sqrt{\text{2}}}(U_{\mathit{AC}}(t)(|{\psi }_{+}⟩|+⟩)+U_{\mathit{AC}}(t)(|{\varphi }_{+}⟩|-⟩))\hfill \\ \hfill & =\frac{\text{1}}{\sqrt{\text{2}}}\left(cos(t)\left(\frac{\text{1}}{\text{2}}(|\text{010}⟩+|\text{011}⟩+|\text{100}⟩+|\text{101}⟩)\right)\right)+\text{cos}(t)\left(\frac{\text{1}}{\text{2}}(|000⟩-|\text{001}⟩+|\text{110}⟩-|\text{111}⟩)\right)\hfill \\ \hfill & -isin(t)\left(\frac{\text{1}}{\text{2}}(|\text{111}⟩+|\text{110}⟩+|\text{001}⟩+|000⟩)\right)\left(+isin(t)\left(\frac{\text{1}}{\text{2}}(|\text{101}⟩-|\text{100}⟩+|\text{011}⟩-|\text{010}⟩)\right)\right)\hfill \\ \hfill & =\frac{\text{1}}{\sqrt{\text{2}}}(cos(t)(|{\psi }_{+}⟩|+⟩+|{\varphi }_{+}⟩|-⟩)-isin(t)(|{\varphi }_{+}⟩|+⟩-|{\psi }_{+}⟩|-⟩))\hfill \\ \hfill & =\frac{\text{1}}{\sqrt{\text{2}}}((cos(t)(|{\psi }_{+}⟩)-isin(t)(|{\varphi }_{+}⟩))|+⟩+(cos(t)(|{\varphi }_{+}⟩)+isin(t)(|{\psi }_{+}⟩))|-⟩),\hfill \\ \hfill \end{array}$$

Finally, in equation A.3 and A.4 in the Supplementary Information file,

should read: