Error identification entanglement purification for stationary system using high-dimensional entanglement

High-fidelity entanglement plays an important role in quantum networks for guaranteeing the integrity of quantum information processing. We present an error identification entanglement purification protocol (EIEPP) for maintaining high-fidelity entanglement of stationary systems. In EIEPP, the hybrid controlled-x gate for stationary systems and high-dimensional spatial-mode state of photons has been constructed using a stationary-whispering gallery mode resonator-waveguides system, and it is used to construct an error number gate (ENG) and error position gate (EPG) for noisy entangled stationary systems. Using ENG and EPG, the stationary systems with errors are discarded according to the measurement results of high-dimensional entangled photon states, and typically high-fidelity entangled states of stationary systems can be achieved with high yield in both ideal and practical situations. This EIEPP has superior performance for stationary systems with amplitude damping noise, and it is also applicable for stationary systems with other types of noise.

Figure 1

(a) Atomic structure of ${\mathrm{SiV}}^{-}$ center in diamond. (b) The energy-level structure of ${\mathrm{SiV}}^{-}$ center and its optical transition. $|g\rangle$ and $|e\rangle$ represent two electron-spin sublevels of the lowest orbital branch in the ground state. $|g^{\prime }\rangle$ and $|e^{\prime }\rangle$ represent two electron-spin levels of the lower orbital branch in the exited state [56]. (c) Schematic diagram of a hybrid controlled-x gate constructed by ${\mathrm{SiV}}^{-}$-WGM-waveguides-HWP system. $a_0-a_3$ ($a_0^{\prime }-a_3^{\prime }$) are input (output) ports of four waveguides coupled to WGM resonator. HWP represents the half-wave plate with function $|H\rangle \to -|H\rangle$ and $|V\rangle \to |V\rangle$.

Figure 2

Schematic diagram of hybrid cx gates for the Bell states of the ${\mathrm{SiV}}^{-}$ center system and four-dimensional photon system. Photon $A$ and ${\mathrm{SiV}}^{-}$ center system ${\mathrm{Si}}_1$ belong to Alice. Photon $B$ and ${\mathrm{SiV}}^{-}$ center system ${\mathrm{Si}}_2$ belong to Bob. $a_0-a_3$ ($a_0^{\prime }-a_3^{\prime }$) are input (output) ports of the ${\mathrm{SiV}}_1^{-}\text{−}{\mathrm{WGM}}_1$-waveguides-HWP system. $b_0-b_3$ ($b_0^{\prime }-b_3^{\prime }$) are input (output) ports of the ${\mathrm{SiV}}_2^{-}\text{−}{\mathrm{WGM}}_2$-waveguides-HWP system.

Figure 3

Schematic diagram of error identification entanglement purification protocol. (a) Error number gate. (b) Error position gate. ${\mathrm{Si}}_1^1{\mathrm{Si}}_2^1-{\mathrm{Si}}_1^4{\mathrm{Si}}_2^4$ represent four ${\mathrm{SiV}}^{-}$ center systems in a subset. ${\mathrm{SiV}}^{-}$ centers ${\mathrm{Si}}_1^1-{\mathrm{Si}}_1^4$ and photons $AC$ belong to Alice. ${\mathrm{SiV}}^{-}$ centers ${\mathrm{Si}}_2^1-{\mathrm{Si}}_2^4$ and photons $BD$ belong to Bob. The schematic diagrams of ENG and EPG are shown in Fig. 4.

Figure 4

Schematic diagrams of (a) ENG and (b) EPG in Alice's side. S represents an optical switch [71], and it is used to convey photon $C$ from the output port $c_k^{y\oplus 1}$ to the corresponding input port $c_k^y$ ($y=1,2,3$). The schematic diagrams of ENG and EPG in Bob's side are similar to the ones in Alice's side by replacing ${\mathrm{Si}}_1^y$ and photons $AC$ to ${\mathrm{Si}}_2^y$ and photons $BD$.

Figure 5

The (a) fidelity and (b) efficiency of the hybrid cx gates for $|ge\rangle$ ($|{\varphi }^{+}\rangle$) and $|{\mathrm{\Psi }}_{0n}{\rangle }_{AB}$ as the function of the parameter $g/{\left(\kappa \gamma \right)}^{1/2}$.

Figure 6

(a) Average fidelity $\langle F^{\prime }\rangle$ and (b) yield ${\mathrm{Y}}^{\prime }$ as the function of the parameter $g/{\left(\kappa \gamma \right)}^{1/2}$ with initial fidelity $F=0.9$. (c) Yields under ideal and experimental conditions as the function of initial fidelity $F$, where the experimental parameters satisfy $g/{\left(\kappa \gamma \right)}^{1/2}=12.7$.