Complete and fidelity-robust hyperentangled-state analysis of photon systems with single-sided quantum-dot-cavity systems under the balance condition

Abstract

Hyperentangled Bell-state analysis (HBSA) for two-photon systems and hyperentangled Greenberger–Horne–Zeilinger-state analysis (HGSA) for multi-photon systems play significant roles in quantum information processing. In this paper, we propose a complete and fidelity-robust spatial-polarization two-photon HBSA scheme and generalize it to unambiguous multi-photon HGSA based on the interaction between single photons and singly charged quantum dots (QDs) in optical microcavities under the balance condition. Under the balance condition, the requirement for side-leakage rate and coupling strength for the QD-cavity system can be relaxed and the noise brought on by the unbalanced reflectance of coupled and uncoupled QD-cavity systems is effectively suppressed, raising the fidelity of our schemes to unity in theory. When generalizing to the multi-photon HGSA, our scheme can effectively suppress the decrease in efficiency resulting from the increase in the number of photons. In addition, our schemes simplify the discrimination process and reduce the required light–matter interaction by using self-assisted mechanism. These advantages make our schemes more universal and feasible for high-capacity quantum communications and quantum networks based on hyperentanglement with currently available techniques.

Appendix A Detailed deduction of the state transformation in our HBSA scheme

This part gives detailed deduction of how the composite state transforms in our HBSA scheme. While illustrating the deduction, realistic $\vert r\vert$ is taken into account, and the structure of the system is the same as that in Fig. 2. We take the initial hyperentangled Bell state ${\vert \Omega ^{-}_{00}\rangle }^{1,2}_{P}\otimes {\vert \Omega ^{+}_{01}\rangle }^{1,2}_{S}$ as an example. $p_1$ is injected into the quantum circuit from the left. After $p_1$ passing through HWP$_1$/WFC$_1$, the composite state of $p_1,p_2$ and the electron spin in QD$_1$ will transform into

$$\begin{aligned} \begin{aligned}&\vert \alpha _{1}\beta _{2}\rangle \otimes r\ \frac{-\vert RR\rangle \vert \uparrow \rangle _1+\vert LL\rangle \vert \uparrow \rangle _1+\vert RR\rangle \vert \downarrow \rangle _1-\vert LL\rangle \vert \downarrow \rangle _1}{2} \\&\quad +\vert \beta _{1}\alpha _{2}\rangle \otimes r\ \frac{\vert RR\rangle \vert \uparrow \rangle _1-\vert LL\rangle \vert \uparrow \rangle _1+\vert RR\rangle \vert \downarrow \rangle _1-\vert LL\rangle \vert \downarrow \rangle _1}{2}. \end{aligned} \end{aligned}$$

(41)

Then, $p_2$ is injected into the quantum circuit. After $p_2$ passing through HWP$_2$/WFC$_2$, the composite state of $p_1,p_2$ and the electron spin in QD$_1$ will transform into

$$\begin{aligned} \begin{aligned}&\vert \alpha _{1}\beta _{2}\rangle \otimes r^2\ \frac{-\vert RR\rangle \vert \uparrow \rangle _1+\vert LL\rangle \vert \uparrow \rangle _1+\vert RR\rangle \vert \downarrow \rangle _1-\vert LL\rangle \vert \downarrow \rangle _1}{2} \\&\quad +\vert \beta _{1}\alpha _{2}\rangle \otimes r^2\ \frac{-\vert RR\rangle \vert \uparrow \rangle _1+\vert LL\rangle \vert \uparrow \rangle _1+\vert RR\rangle \vert \downarrow \rangle _1-\vert LL\rangle \vert \downarrow \rangle _1}{2}. \end{aligned} \end{aligned}$$

(42)

(42) is equivalent to

$$\begin{aligned} r^2\ {\vert \Omega ^{-}_{00}\rangle }^{1,2}_{P}\otimes {\vert \Omega ^{+}_{01}\rangle }^{1,2}_{S}\otimes \vert -\rangle _1. \end{aligned}$$

(43)

Next, after $p_1,p_2$ passing through BS$_1$ and BS$_2$, the composite state of $p_1,p_2$ will transform into

$$\begin{aligned} r^2\ {\vert \Omega ^{-}_{00}\rangle }^{1,2}_{P}\otimes {\vert \Omega ^{-}_{00}\rangle }^{1,2}_{S}. \end{aligned}$$

(44)

After $p_1$ passing through HWP$_3$/WFC$_3$ (no operation performed on $p_2$ in this process), the composite state of $p_1,p_2$ and the electron spin in QD$_2$ will transform into

$$\begin{aligned} \begin{aligned}&\vert \alpha _{1}\alpha _{2}\rangle \otimes r^3\ \frac{-\vert RR\rangle \vert \uparrow \rangle _2+\vert LL\rangle \vert \uparrow \rangle _2+\vert RR\rangle \vert \downarrow \rangle _2-\vert LL\rangle \vert \downarrow \rangle _2}{2} \\&\quad -\vert \beta _{1}\beta _{2}\rangle \otimes r^3\ \frac{\vert RR\rangle \vert \uparrow \rangle _2-\vert LL\rangle \vert \uparrow \rangle _2+\vert RR\rangle \vert \downarrow \rangle _2-\vert LL\rangle \vert \downarrow \rangle _2}{2}. \end{aligned} \end{aligned}$$

(45)

After $p_2$ passing through HWP$_4$/WFC$_4$ (no operation performed on $p_1$ in this process), the composite state of $p_1,p_2$ and the electron spin in QD$_2$ will transform into

$$\begin{aligned} \begin{aligned}&\vert \alpha _{1}\alpha _{2}\rangle \otimes r^4\ \frac{\vert RR\rangle \vert \uparrow \rangle _2-\vert LL\rangle \vert \uparrow \rangle _2+\vert RR\rangle \vert \downarrow \rangle _2-\vert LL\rangle \vert \downarrow \rangle _2}{2} \\&\quad -\vert \beta _{1}\beta _{2}\rangle \otimes r^4\ \frac{\vert RR\rangle \vert \uparrow \rangle _2-\vert LL\rangle \vert \uparrow \rangle _2+\vert RR\rangle \vert \downarrow \rangle _2-\vert LL\rangle \vert \downarrow \rangle _2}{2}. \end{aligned} \end{aligned}$$

(46)

(46) is equivalent to

$$\begin{aligned} r^4\ {\vert \Omega ^{-}_{00}\rangle }^{1,2}_{P}\otimes {\vert \Omega ^{-}_{00}\rangle }^{1,2}_{S}\otimes \vert +\rangle _2. \end{aligned}$$

(47)

At last, after $p_1,p_2$ passing through BS$_3$ and BS$_4$, the composite state of $p_1,p_2$ will revert to the initial status, with the global coefficient $r^4$

$$\begin{aligned} r^4\ {\vert \Omega ^{-}_{00}\rangle }^{1,2}_{P}\otimes {\vert \Omega ^{+}_{01}\rangle }^{1,2}_{S}. \end{aligned}$$

(48)

The succeeding process SPBSM has been formulated in detail in Sect. 3.