Experimental high-dimensional quantum teleportation

TL;DR

This paper demonstrates the teleportation of high-dimensional quantum states using a six-photon system, achieving nonclassical, genuine three-dimensional teleportation with potential applications in complex quantum networks.

Contribution

It presents the first experimental realization of high-dimensional quantum teleportation in a three-dimensional six-photon system, utilizing spatial modes and auxiliary entanglement for deterministic measurements.

Findings

Teleportation fidelity of 0.596±0.037

Proved nonclassical and genuine three-dimensional teleportation

Enabled remote reconstruction of complex quantum systems

Abstract

Quantum teleportation provides a way to transmit unknown quantum states from one location to another. In the quantum world, multilevel systems which enable high-dimensional systems are more prevalent. Therefore, to completely rebuild the quantum states of a single particle remotely, one needs to teleport multilevel (high-dimensional) states. Here, we demonstrate the teleportation of high-dimensional states in a three-dimensional six-photon system. We exploit the spatial mode of a single photon as the high-dimensional system, use two auxiliary entangled photons to realize a deterministic three-dimensional Bell state measurement. The fidelity of teleportation process matrix is F=0.596\pm0.037. Through this process matrix, we can prove that our teleportation is both nonclassical and genuine three dimensional. Our work paves the way to rebuild complex quantum systems remotely and to construct complex quantum networks.

Equations

$$|\phi ⟩=\alpha |0⟩+\beta |1⟩+\gamma |2⟩,$$

$$\begin{array}{r}|\xi {⟩}_{23}=(|00{⟩}_{23}+|11{⟩}_{23}+|22{⟩}_{23})/\sqrt{3}.\end{array}$$

$$\begin{array}{rl}& |{\psi }_{00}⟩=(|00⟩+|11⟩+|22⟩)/\sqrt{3},\\ & |{\psi }_{10}⟩=(|00⟩+e^{2\pi i/3}|11⟩+e^{4\pi i/3}|22⟩)/\sqrt{3},\\ & |{\psi }_{20}⟩=(|00⟩+e^{4\pi i/3}|11⟩+e^{2\pi i/3}|22⟩)/\sqrt{3},\\ & |{\psi }_{01}⟩=(|01⟩+|12⟩+|20⟩)/\sqrt{3},\\ & |{\psi }_{11}⟩=(|01⟩+e^{2\pi i/3}|12⟩+e^{4\pi i/3}|20⟩)/\sqrt{3},\\ & |{\psi }_{21}⟩=(|01⟩+e^{4\pi i/3}|12⟩+e^{2\pi i/3}|20⟩)/\sqrt{3},\\ & |{\psi }_{02}⟩=(|02⟩+|10⟩+|21⟩)/\sqrt{3},\\ & |{\psi }_{12}⟩=(|02⟩+e^{2\pi i/3}|10⟩+e^{4\pi i/3}|21⟩)/\sqrt{3},\\ & |{\psi }_{22}⟩=(|02⟩+e^{4\pi i/3}|10⟩+e^{2\pi i/3}|21⟩)/\sqrt{3}.\end{array}$$

$$\begin{array}{rl}|\xi {⟩}_{23}|\phi {⟩}_1=& \frac{1}{\sqrt{3}}(|00{⟩}_{23}+|11{⟩}_{23}+|22{⟩}_{23})(\alpha |0{⟩}_1+\beta |1{⟩}_1+\gamma |2{⟩}_1)\\ =& \frac{1}{3}(|{\psi }_{00}{⟩}_{12}\otimes U_{00}^{†}|\phi {⟩}_3+|{\psi }_{10}{⟩}_{12}\otimes U_{10}^{†}|\phi {⟩}_3+|{\psi }_{20}{⟩}_{12}\otimes U_{20}^{†}|\phi {⟩}_3\\ & +|{\psi }_{01}{⟩}_{12}\otimes U_{01}^{†}|\phi {⟩}_3+|{\psi }_{11}{⟩}_{12}\otimes U_{11}^{†}|\phi {⟩}_3+|{\psi }_{21}{⟩}_{12}\otimes U_{21}^{†}|\phi {⟩}_3\\ & +|{\psi }_{02}{⟩}_{12}\otimes U_{02}^{†}|\phi {⟩}_3+|{\psi }_{12}{⟩}_{12}\otimes U_{12}^{†}|\phi {⟩}_3+|{\psi }_{22}{⟩}_{12}\otimes U_{22}^{†}|\phi {⟩}_3).\end{array}$$

$$\begin{split}&U_{00}=\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\\ \end{array}\right),\ U_{01}=\left(\begin{array}[]{ccc}0&0&1\\ 1&0&0\\ 0&1&0\\ \end{array}\right),\ U_{02}=\left(\begin{array}[]{ccc}0&1&0\\ 0&0&1\\ 1&0&0\\ \end{array}\right),\\ &U_{10}=\left(\begin{array}[]{ccc}1&0&0\\ 0&e^{\frac{2\pi i}{3}}&0\\ 0&0&e^{\frac{4\pi i}{3}}\\ \end{array}\right),\ U_{11}=\left(\begin{array}[]{ccc}0&0&e^{\frac{4\pi i}{3}}\\ 1&0&0\\ 0&e^{\frac{2\pi i}{3}}&0\\ \end{array}\right),\ U_{12}=\left(\begin{array}[]{ccc}0&e^{\frac{2\pi i}{3}}&0\\ 0&0&e^{\frac{4\pi i}{3}}\\ 1&0&0\\ \end{array}\right),\\ &U_{20}=\left(\begin{array}[]{ccc}1&0&0\\ 0&e^{\frac{4\pi i}{3}}&0\\ 0&0&e^{\frac{2\pi i}{3}}\\ \end{array}\right),\ U_{21}=\left(\begin{array}[]{ccc}0&0&e^{\frac{2\pi i}{3}}\\ 1&0&0\\ 0&e^{\frac{4\pi i}{3}}&0\\ \end{array}\right),\ U_{22}=\left(\begin{array}[]{ccc}0&e^{\frac{4\pi i}{3}}&0\\ 0&0&e^{\frac{2\pi i}{3}}\\ 1&0&0\\ \end{array}\right).\\ \end{split}$$

$$\begin{array}{rl}& |{\psi }_{00}⟩=(|00⟩+|11⟩+|22⟩)/\sqrt{3},\\ & |{\psi }_{10}⟩=(|00⟩+e^{2\pi i/3}|11⟩+e^{4\pi i/3}|22⟩)/\sqrt{3},\\ & |{\psi }_{20}⟩=(|00⟩+e^{4\pi i/3}|11⟩+e^{2\pi i/3}|22⟩)/\sqrt{3},\\ & |{\psi }_{01}⟩=(|01⟩+|12⟩+|20⟩)/\sqrt{3},\\ & |{\psi }_{11}⟩=(|01⟩+e^{2\pi i/3}|12⟩+e^{4\pi i/3}|20⟩)/\sqrt{3},\\ & |{\psi }_{21}⟩=(|01⟩+e^{4\pi i/3}|12⟩+e^{2\pi i/3}|20⟩)/\sqrt{3},\\ & |{\psi }_{02}⟩=(|02⟩+|10⟩+|21⟩)/\sqrt{3},\\ & |{\psi }_{12}⟩=(|02⟩+e^{2\pi i/3}|10⟩+e^{4\pi i/3}|21⟩)/\sqrt{3},\\ & |{\psi }_{22}⟩=(|02⟩+e^{4\pi i/3}|10⟩+e^{2\pi i/3}|21⟩)/\sqrt{3}.\end{array}$$

$$\begin{array}{rl}|\phi {⟩}_1\otimes |\xi {⟩}_{23}\otimes |H{⟩}_t=& (\alpha |H_0⟩+\beta |V_1⟩+\gamma |H_2⟩)\otimes (\frac{2}{3}|H_0⟩|H_0⟩+\frac{2}{3}|V_1⟩|V_1⟩+\frac{1}{3}|H_2⟩|H_2⟩)\otimes |H{⟩}_t\\ =& (\frac{2}{3}\alpha |H_0⟩|H_0⟩|H_0⟩+\frac{2}{3}\alpha |H_0⟩|V_1⟩|V_1⟩+\frac{1}{3}\alpha |H_0⟩|H_2⟩|H_2⟩\\ & +\frac{2}{3}\beta |V_1⟩|H_0⟩|H_0⟩+\frac{2}{3}\beta |V_1⟩|V_1⟩|V_1⟩+\frac{1}{3}\beta |V_1⟩|H_2⟩|H_2⟩\\ & +\frac{2}{3}\gamma |H_2⟩|H_0⟩|H_0⟩+\frac{2}{3}\gamma |H_2⟩|V_1⟩|V_1⟩+\frac{1}{3}\gamma |H_2⟩|H_2⟩|H_2⟩)\otimes |H{⟩}_t.\end{array}$$

$$\begin{array}{r}(\frac{2}{3}\alpha |H_0{⟩}_a|H_0{⟩}_b|H_0⟩+\frac{1}{3}\alpha |H_0{⟩}_a|H_2{⟩}_b|H_2⟩+\frac{2}{3}\beta |V_1{⟩}_a|V_1{⟩}_b|V_1⟩+\frac{2}{3}\gamma |H_2{⟩}_a|H_0{⟩}_b|H_0⟩+\frac{1}{3}\gamma |H_2{⟩}_a|H_2{⟩}_b|H_2⟩)\otimes |H{⟩}_t.\end{array}$$

$$\begin{array}{r}(\frac{2}{3}\alpha |V_0{⟩}_a|V_0{⟩}_b|H_0⟩+\frac{1}{3}\alpha |V_0{⟩}_a|H_1{⟩}_b|H_2⟩+\frac{2}{3}\beta |H_0{⟩}_a|H_0{⟩}_b|V_0⟩+\frac{2}{3}\gamma |H_1{⟩}_a|V_0{⟩}_b|H_0⟩+\frac{1}{3}\gamma |H_1{⟩}_a|H_1{⟩}_b|H_1⟩)\otimes |H{⟩}_t.\end{array}$$

$$\begin{array}{r}(\frac{2}{3}\alpha \frac{1}{\sqrt{2}}|H_0-V_0{⟩}_a\frac{1}{\sqrt{2}}|H_0-V_0{⟩}_b|H_0⟩+\frac{1}{3}\alpha \frac{1}{\sqrt{2}}|H_0-V_0{⟩}_a|H_1{⟩}_b|H_2⟩+\frac{2}{3}\beta \frac{1}{\sqrt{2}}|H_0+V_0{⟩}_a\frac{1}{\sqrt{2}}|H_0+V_0{⟩}_b|V_1⟩\\ +\frac{2}{3}\gamma |H_1{⟩}_a\frac{1}{\sqrt{2}}|H_0-V_0{⟩}_b|H_0⟩+\frac{1}{3}\gamma |H_1{⟩}_a|H_1{⟩}_b|H_2⟩)\otimes |H{⟩}_t.\end{array}$$

$$\begin{array}{r}(\frac{1}{3}\alpha |H_0{⟩}_a|H_0{⟩}_b|H_0⟩+\frac{\sqrt{2}}{6}\alpha |H_0{⟩}_a|V_0{⟩}_b|H_2⟩+\frac{1}{3}\beta |H_0{⟩}_a|H_0{⟩}_b|V_1⟩+\frac{\sqrt{2}}{3}\gamma |V_0{⟩}_a|H_0{⟩}_b|H_0⟩+\frac{1}{3}\gamma |V_0{⟩}_a|V_0{⟩}_b|H_2⟩)\otimes |H{⟩}_t.\end{array}$$

$$\begin{array}{rl}& \frac{\sqrt{2}}{6}(\alpha |H{⟩}_a|H{⟩}_b|H{⟩}_c|H{⟩}_d|H_0⟩+\beta |H{⟩}_a|H{⟩}_b|H{⟩}_c|H{⟩}_d|V_1⟩+\gamma |V{⟩}_a|V{⟩}_b|V{⟩}_c|V{⟩}_d|H_2⟩)\otimes |H{⟩}_t.\end{array}$$

$$\begin{array}{rl}(& \frac{1}{4}(|HHHH⟩+|HHVV⟩+|HVHV⟩+|HVVH⟩\\ & +|VHHV⟩+|VHVH⟩+|VVHH⟩+|VVVV⟩)\frac{\sqrt{2}}{6}(\alpha |H_0⟩+\beta |V_1⟩+\gamma |H_2⟩)\\ & +\frac{1}{4}(|HHHV⟩+|HHVH⟩+|HVHH⟩+|HVVV⟩\\ & +|VHHH⟩+|VHVV⟩+|VVHV⟩+|VVVH⟩)\frac{\sqrt{2}}{6}(\alpha |H_0⟩+\beta |V_1⟩-\gamma |H_2⟩))\otimes |H{⟩}_t.\end{array}$$

$${|{\psi }_{nm}⟩}_{12}=\frac{1}{\sqrt{d}}\sum _{j=0}^{d-1}{e}^{j2\pi jn/d}|j⟩\otimes |(j+m)\mathrm{mod}d⟩.$$

$$U_{nm}=\sum _{k=0}^{d-1}{e}^{i2\pi kn/d}|k⟩\otimes ⟨(k+m)\mathrm{mod}d|.$$

$$\rho_{1}=\left(\begin{array}[]{ccc}0.745+0.000i&0.110-0.109i&0.075-0.053i\\ 0.110+0.103i&0.142+0.000i&0.043-0.050i\\ 0.075+0.053i&0.043+0.050i&0.114+0.000i\\ \end{array}\right),$$

$$\rho_{2}=\left(\begin{array}[]{ccc}0.151+0.000i&-0.033+0.051i&0.051-0.003i\\ -0.033-0.051i&0.715+0.000i&-0.078-0.072i\\ 0.051+0.003i&-0.078+0.072i&0.133+0.000i\\ \end{array}\right),$$

$$\rho_{3}=\left(\begin{array}[]{ccc}0.154+0.000i&0.026+0.014i&-0.008-0.092i\\ 0.026-0.014i&0.1369+0.000i&0.059+0.006i\\ -0.008+0.092i&0.059-0.006i&0.708+0.000i\\ \end{array}\right),$$

$$\rho_{4}=\left(\begin{array}[]{ccc}0.425+0.000i&0.316+0.156i&0.018+0.029i\\ 0.316-0.156i&0.430+0.000i&-0.079+0.069i\\ 0.018-0.029i&-0.079-0.069i&0.143+0.000i\\ \end{array}\right),$$

$$\rho_{5}=\left(\begin{array}[]{ccc}0.468+0.000i&-0.273-0.252i&-0.080+0.059i\\ -0.273+0.252i&0.411+0.000i&-0.010+0.000i\\ -0.080-0.059i&-0.010-0.000i&0.119+0.000i\\ \end{array}\right),$$

$$\rho_{6}=\left(\begin{array}[]{ccc}0.464+0.000i&-0.041+0.065i&0.232+0.148i\\ -0.041-0.065i&0.142+0.000i&-0.035-0.017i\\ 0.232-0.148i&-0.035+0.017i&0.392+0.000i\\ \end{array}\right),$$

$$\rho_{7}=\left(\begin{array}[]{ccc}0.418+0.000i&0.009+0.049i&-0.221-0.192i\\ 0.009-0.049i&0.133+0.000i&-0.064-0.014i\\ -0.221+0.192i&-0.064+0.014i&0.448+0.000i\\ \end{array}\right),$$

$$\rho_{8}=\left(\begin{array}[]{ccc}0.158+0.000i&0.000+0.051i&-0.053+0.043i\\ 0.000-0.051i&0.362+0.000i&0.247+0.140i\\ -0.053-0.043i&0.247-0.140i&0.479+0.000i\\ \end{array}\right),$$

$$\rho_{9}=\left(\begin{array}[]{ccc}0.159+0.000i&0.005+0.065i&-0.074-0.008i\\ 0.005-0.065i&0.390+0.000i&-0.079-0.244i\\ -0.074+0.008i&-0.079+0.244i&0.450+0.000i\\ \end{array}\right),$$

$$\rho_{10}=\left(\begin{array}[]{ccc}0.321+0.000i&0.111+0.047i&0.133-0.054i\\ 0.111-0.047i&0.386+0.000i&0.225-0.002i\\ 0.133+0.054i&0.225+0.002i&0.292+0.000i\\ \end{array}\right).$$

$$\begin{split}&\sigma_{1}=\left(\begin{array}[]{ccc}0&1&0\\ 1&0&0\\ 0&0&0\\ \end{array}\right),\ \sigma_{2}=\left(\begin{array}[]{ccc}0&-i&0\\ i&0&0\\ 0&0&0\\ \end{array}\right),\ \sigma_{3}=\left(\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&0\\ \end{array}\right),\\ &\sigma_{4}=\left(\begin{array}[]{ccc}0&0&1\\ 0&0&0\\ 1&0&0\\ \end{array}\right),\ \sigma_{5}=\left(\begin{array}[]{ccc}0&0&-i\\ 1&0&0\\ i&0&0\\ \end{array}\right),\ \sigma_{6}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&1&0\\ \end{array}\right),\\ &\sigma_{7}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&-i\\ 0&i&0\\ \end{array}\right),\ \sigma_{8}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&-2\\ \end{array}\right).\\ \end{split}$$

$$\begin{split}&\chi=\left(\begin{array}[]{ccccccccc}0.596+0.000i&-0.018-0.003i&-0.024+0.018i&0.002+0.064i&-0.009+0.012i&-0.012-0.006i&-0.029-0.013i&0.012+0.009i&0.020-0.001i\\ -0.018+0.003i&0.059+0.000i&-0.045-0.007i&0.000-0.016i&0.010-0.006i&-0.008+0.011i&0.006+0.003i&0.000-0.011i&-0.001+0.022i\\ -0.024-0.018i&-0.045+0.007i&0.046-0.000i&0.008+0.005i&-0.010+0.002i&0.004-0.010i&0.005+0.008i&-0.005+0.002i&0.003-0.021i\\ 0.002-0.064i&0.000+0.016i&0.008-0.005i&0.030+0.000i&-0.001-0.013i&-0.005+0.012i&-0.006-0.002i&0.008+0.008i&0.011+0.004i\\ -0.009-0.012i&0.010+0.006i&-0.010-0.002i&-0.001+0.013i&0.054-0.000i&-0.041-0.006i&-0.000+0.008i&-0.011-0.003i&0.022-0.014i\\ -0.012+0.006i&-0.008-0.011i&0.004+0.010i&-0.005-0.012i&-0.041+0.006i&0.041+0.000i&-0.014-0.004i&0.018-0.001i&-0.013+0.015i\\ -0.029+0.013i&0.006-0.003i&0.005-0.008i&-0.006+0.002i&-0.000-0.008i&-0.014+0.004i&0.052+0.000i&-0.037+0.001i&-0.029-0.002i\\ 0.012-0.009i&0.000+0.011i&-0.005-0.002i&0.008-0.008i&-0.011+0.003i&0.018+0.001i&-0.037-0.001i&0.032-0.000i&0.009+0.006i\\ 0.020+0.001i&-0.001-0.022i&0.003+0.021i&0.011-0.004i&0.022+0.014i&-0.013-0.015i&-0.029+0.002i&0.009-0.006i&0.087+0.000i\\ \end{array}\right).\\ \end{split}$$

$$f_{\text{stateave}}=\frac{f_{\text{process }}d+1}{d+1},$$

$$\begin{array}{rl}& |{\psi }_1⟩=|0⟩,\\ & |{\psi }_2⟩=|1⟩,\\ & |{\psi }_3⟩=|2⟩,\\ & |{\psi }_4⟩=\frac{1}{\sqrt{3}}(|0⟩+|1⟩+|2⟩),\\ & |{\psi }_5⟩=\frac{1}{\sqrt{3}}(|0⟩+\omega |1⟩+{\omega }^2|2⟩),\\ & |{\psi }_6⟩=\frac{1}{\sqrt{3}}(|0⟩+{\omega }^2|1⟩+\omega |2⟩),\\ & |{\psi }_7⟩=\frac{1}{\sqrt{3}}(\omega |0⟩+|1⟩+|2⟩),\\ & |{\psi }_8⟩=\frac{1}{\sqrt{3}}(|0⟩+\omega |1⟩+|2⟩),\\ & |{\psi }_9⟩=\frac{1}{\sqrt{3}}(|0⟩+|1⟩+\omega |2⟩),\\ & |{\psi }_{10}⟩=\frac{1}{\sqrt{3}}({\omega }^2|0⟩+|1⟩+|2⟩),\\ & |{\psi }_{11}⟩=\frac{1}{\sqrt{3}}(|0⟩+{\omega }^2|1⟩+|2⟩),\\ & |{\psi }_{12}⟩=\frac{1}{\sqrt{3}}(|0⟩+|1⟩+{\omega }^2|2⟩),\end{array}$$

$$\rho_{qutrit}=\frac{1}{3}\left(\begin{array}[]{ccc}1+\frac{\sqrt{3}}{2}(\langle\lambda_{8}\rangle+\sqrt{3}\langle\lambda_{3}\rangle)&\frac{3}{2}(\langle\lambda_{1}\rangle-i\langle\lambda_{2}\rangle)&\frac{3}{2}(\langle\lambda_{4}\rangle-i\langle\lambda_{5}\rangle)\vspace{1ex}\\ \frac{3}{2}(\langle\lambda_{1}\rangle+i\langle\lambda_{2}\rangle)&1+\frac{\sqrt{3}}{2}(\langle\lambda_{8}\rangle-\sqrt{3}\langle\lambda_{3}\rangle)&\frac{3}{2}(\langle\lambda_{6}\rangle-i\langle\lambda_{7}\rangle)\vspace{1ex}\\ \frac{3}{2}(\langle\lambda_{4}\rangle-i\langle\lambda_{5}\rangle)&\frac{3}{2}(\langle\lambda_{6}\rangle+i\langle\lambda_{7}\rangle)&1-\sqrt{3}\langle\lambda_{8}\rangle\end{array}\right),$$

Full text

Quantum teleportationBennett93 ; Pirandola15 enables the rebuilding of arbitrary unknown quantum states without the transmission of a real particle. Previous efforts have shown the capability to rebuild qubit states and continuous variable states. Discrete variable statesBouwmeester97 ; Nielsen98 ; Fattal04 ; Barrett04 ; Riebe04 ; Sherson06 ; Olmschenk09 and continuous variable states Furusawa98 ; Takei05 ; Yonezawa07 ; Lee11 in one degree of freedom have been transported. Recent work has also demonstrated the capability of teleporting multiple degrees of freedom of a single photon Wang15 . However, to teleport quantum states of a real particle, for example, a single photon, one needs to consider not only the two-level states (polarization), but also those multilevel states. For example, the orbital angular momentum Dada11 ; Krenn14 , the temporal mode Martin17 , the frequency mode Kues17 and the spatial mode Schaeff15 ; Hu16 ; Hu20 ; Valencia20 of a single photon are all natural attributes of multilevel states, which are exploited as high-dimensional systems. However, to teleport high-dimensional quantum states is still a challenge for two reasons. One is the generation of high-quality high-dimensional entanglement feasible for quantum teleportation. There has been much work on high-dimensional entanglement generation Dada11 ; Krenn14 ; Martin17 ; Kues17 ; Schaeff15 ; Hu16 ; Hu20 ; Valencia20 , including attempts to observe interference between different high-dimensional entangled pairs Erhard18 ; Malik16 . Nevertheless, the interference visibility between different pairs is still quite low at $63.5\%$. The other concerns performing a deterministic high-dimensional Bell state measurement (HDBSM). Here, we use the spatial mode (path) to encode the three-dimensional states that has been demonstrated to extremely high fidelity Hu16 and use an auxiliary entangled photon pair to perform the HDBSM. We thereby overcome these obstacles and demonstrate the teleportation of a three-dimensional quantum state using the spatial mode of a single photon.

Suppose Alice wishes to teleport to Bob the quantum state of a single photon (photon 1, Fig. 1), encoded in the path mode as

[TABLE]

where $|0\rangle$, $|1\rangle$, and $|2\rangle$ denote the path degree of freedom (DOF). This DOF exists in an infinite dimensional space of the photonic system; here, we take only three dimensions as an example. The coefficients $\alpha$, $\beta$, and $\gamma$ are complex numbers satisfying $|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}=1$. Alice and Bob initially need to share a high-dimensional entangled photon pair (photons 2 and 3) in path

[TABLE]

Then, Alice performs a two-particle HDBSM on photons 1 and 2, which projects the two-photon state onto the basis of the nine orthogonal three-dimensional Bell states and discriminates one of them,

[TABLE]

After the HDBSM, photons 1 and 2 are projected onto the state $|\psi_{00}\rangle$ with a probability of 1/9, then photon 3 is projected onto state $|\varphi\rangle$. For instances where photons 1 and 2 are projected onto the other eight three-dimensional Bell states, Bob needs to perform a three-dimensional unitary operation on photon 3 to rotate the state of photon 3 to $|\varphi\rangle$ according to the measurement results of photons 1 and 2.

However, HDBSM is still a challenge with linear optics Calsamiglia02 ; Zhang19 . Although one can classify high-dimensional entangled states into several categories Hu182 ; Hill16 , one cannot identify any of them. The possible solution to this HDBSM is to introduce an auxiliary system. Here, we introduce a pair of assistant entangled photons to complete the HDBSM.

The Bell state measurement (BSM) of a two-dimensional polarized state is divided into two steps Liu16 . The four Bell states are first divided into two categories ($(|HH\rangle\pm|VV\rangle)/\sqrt{2}$ and $(|HV\rangle\pm|VH\rangle)/\sqrt{2}$) by a polarizing beam splitter (PBS) according to classical terms. Second, the two states are distinguished with different phases by projecting onto basis $|H\pm V\rangle/\sqrt{2}$. The structure of HDBSM in our system is similar to that of qubit polarized BSM. According to classical terms, nine three-dimensional Bell states are divided into three categories, then, the localized projection measurement is used to identify the three-dimensional Bell states.

Fig. 2 illustrates our linear optical scheme for teleporting the three-dimensional quantum states. The first step is to divide nine Bell states into three categories according to classical terms $|i,i\rangle$, $|i,i+1\rangle$, and $|i,i+2\rangle$, $i\in\{0,1,2\}$ under modulo-3 arithmetic. Photons 1 and 2 are sent to a PBS, which transmits horizontally polarized terms ($|H\rangle$) and reflects vertically polarized terms ($|V\rangle$). In the three-dimensional path state, we control the polarization of each path to satisfy ($|0\rangle\rightharpoonup|H\rangle$, $|1\rangle\rightharpoonup|V\rangle$, and $|2\rangle\rightharpoonup|H\rangle$). After the PBS, we post-select the event in which there is one and only one photon in each outport. For the nine classical terms of the three-dimensional Bell states ($|i,j\rangle$, $i,j\in\{0,1,2\})$, five of them are selected ($|i,j\rangle$ with $i+j$ even).

The second step is to use a local projection measurement to determine which Bell state is post-selected through $|\psi_{00}\rangle$, $|\psi_{10}\rangle$ and $|\psi_{20}\rangle$ (here, terms $|20\rangle$ and $|02\rangle$ are noise terms and are cancelled later). We can construct an arbitrary single qutrit basis (e.g., $(|0\rangle+|1\rangle+|2\rangle)/\sqrt{3}$) by half-wave plates (HWPs), beam displacers (BDs) and PBSs, so that we can determine whether the measured state is $|\psi_{00}\rangle$ by measuring this basis on both sides Hu16 .

To cancel the disturbance terms ($|20\rangle$ and $|02\rangle$), we introduce another entangled photon pair Supple . Hence, we can distinguish at least one Bell state deterministically. For Bell states $|\psi_{10}\rangle$ and $|\psi_{20}\rangle$, we need to select different local projection measurements. Finally, we have teleported a three-dimensional quantum state with a success probability of 1/54. To increase the success probability, we use the non-maximally entangled state $(2|00\rangle+2|11\rangle+|22\rangle)/3$ to replace the maximally entangled state $|\xi\rangle_{23}$, and adjust the measurement base on HDBSM correspondingly, this increases the success probability of teleportation to 1/18 while does not affect the fidelity Supple . This method of completing HDBSM in linear optical systems can be extended to higher dimensions and can be applied to different degrees of freedom such as orbital angular momentum (OAM). For d-dimensional systems, we only need $\lceil\log_{2}(d)\rceil-1$ pairs of auxiliary entangled photons Supple .

The implementation of the HDBSM requires Hong-Ou-Mandel (HOM)-type interference between indistinguishable single photons with good temporal, spatial, and spectral overlap. We use a narrow band interference filter (3 nm) and a single-mode fiber to improve the visibility of HOM interference. For photon 3 and photon t, we use a broad band interference filter (8 nm) to increase the coincidence efficiency.

The verification of the teleportation results relies on the coincidence events of six photons. To suppress the statistical error, the data collection time is tens of hours. Hence, the stability of the whole system becomes a crucial aspect for the experiment. In our system, the HOM interference between the different photons is stable enough Zhang15 ; Hu19 , whereas the interference between different spatial modes after passing through the single-mode fibers is not. Here, we use a fiber phase locking system Supple to maintain a phase-stable interferometer. The measured interference visibility remained above 0.98 in 45 hours Supple .

We prepared ten different initial states to be teleported: $|\varphi_{1}\rangle=|0\rangle$, $|\varphi_{2}\rangle=|1\rangle$, $|\varphi_{3}\rangle=|2\rangle$, $|\varphi_{4}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$, $|\varphi_{5}\rangle=(|0\rangle+i|1\rangle)/\sqrt{2}$, $|\varphi_{6}\rangle=(|0\rangle+|2\rangle)/\sqrt{2}$, $|\varphi_{7}\rangle=(|0\rangle+i|2\rangle)/\sqrt{2}$, $|\varphi_{8}\rangle=(|1\rangle+|2\rangle)/\sqrt{2}$, $|\varphi_{9}\rangle=(|1\rangle+i|2\rangle)/\sqrt{2}$, and $|\varphi_{10}\rangle=(|0\rangle+|1\rangle+|2\rangle)/\sqrt{3}$. The first nine states ($|\varphi_{1}\rangle$–$|\varphi_{9}\rangle$) constitute a complete orthogonal basis in three-dimensional space; the last state $|\varphi_{10}\rangle$ is a linear-dependent superposition of quantum states in this space. All these states are prepared by the BDs, QWPs, and HWPs.

To evaluate the performance of the high-dimensional teleportation, we reconstruct the density matrix of $|\varphi_{1}\rangle-|\varphi_{10}\rangle$ by state tomography. Conditioned on the detection of the trigger photon and the four-photon coincidence after the HDBSM, we registered the photon counts of teleported photon 1. As shown in Fig. 3, the average fidelity of states $|\varphi_{1}\rangle-|\varphi_{10}\rangle$ is $F=0.685\pm 0.027$, which is significantly higher than that of qutrit nonclassical teleportation bound ($>0.5$ Hayashi05 ; Brub99 ).

All reported data are the raw data without background subtraction. The main sources of error include double pair emission, imperfect initial states, entanglement of photons 2–3 and 4–5, two-photon interference, and phase stabilization. We note that the teleportation fidelities of the states are affected differently by errors from the various sources. The fidelity of $|\varphi_{1}\rangle$-$|\varphi_{3}\rangle$ is higher than that of $|\varphi_{4}\rangle$-$|\varphi_{10}\rangle$. The reason is that imperfect interference does not affect the first three quantum states, but the latter.

The first nine states ($|\varphi_{1}\rangle-|\varphi_{9}\rangle$) are a set complete basis for three-dimensional tomography. The reconstructed density matrices of the teleported quantum states allow us to fully characterize the teleportation procedure by quantum process tomography Fiur2001 . We can completely describe the effect of teleportation on the input states $\rho_{ideal}$ by determining the process matrix $\chi$, defined by $\rho=\sum_{l,k=0}^{8}\chi_{lk}\sigma_{l}\rho_{ideal}\sigma_{k}$ where $\sigma_{0}-\sigma_{8}$ are the Pauli matrices for three dimension Thew2002 . The ideal process matrix of quantum teleportation $\chi_{ideal}$ has only one non-zero component $(\chi_{ideal})_{00}=1$, represents that the process of teleportation is perfect. Fig. 4 shows the real and imaginary components of $\chi$ for quantum teleportation based on our experimental results respectively. The process fidelity of our experiment was $f_{\text{process }}=\operatorname{Tr}\left(\chi_{ideal}\chi\right)=0.596\pm 0.037$.

In general, to demonstrate that the three-dimensional teleportation is nonclassical using average fidelity, one need to measure 12 states from four mutually unbiased bases settings ($|\psi_{1}\rangle-|\psi_{12}\rangle$) Ivanovic1981 ; Supple . In the three-dimensional case, the lower bound of average fidelity for nonclassical teleportation is 0.5. This condition can be converted to process fidelity Gilchrist05 and the lower bound of process fidelity is 1/3. In our experiment, the measured process fidelity is $f_{\text{process }}=0.596\pm 0.037$, which is 7 standard deviations above the fidelity of 1/3, and proves that our teleportation is nonclassical.

For high-dimensional teleportation, it is not enough to only prove that teleportation is nonclassical. Genuine d-dimensional teleportation should be distinguished from the low dimensional case, excluding the hypothesis that the teleportation can be expressed in a smaller dimension. In our case, we need to exclude qubit and make sure that we have completed the genuine three-dimensional teleportation. In reference Luo2019 , two-dimensional states are transmitted, it is found that the maximum fidelity with state $(|0\rangle+|1\rangle+|2\rangle)/\sqrt{3}$ is 2/3. Therefore, quantum states with fidelity more than 2/3 are genuine three-dimensional states. However, this is a sufficient but not a necessary condition. For some states (like $\sqrt{1/8}|0\rangle+\sqrt{1/8}|1\rangle-\sqrt{3/4}|2\rangle$), the fidelity can not reach 2/3, but they are still genuine three-dimensional coherent superposition states. If someone who can transmit all qubit states is unable to simulate a teleportation, then it is reasonable to say that the teleportation performed is genuine three dimensional. We assume that the qubit state ($\rho_{qubit}=(P_{1}\rho_{01}+P_{2}\rho_{02}+P_{3}\rho_{12}$) is incoherent at different subspaces ($\{|0\rangle,|1\rangle\},\{|0\rangle,|2\rangle\},\{|1\rangle,|2\rangle\}$). If we cannot use this qubit state to simulate the state after teleportation, then we prove that our teleportation state is in the state of three levels of coherent superposition. First, we derive a nonlinear criterion Supple , which is more powerful than the fidelity criterion. This criterion can be used to determine states like $\sqrt{1/8}|0\rangle+\sqrt{1/8}|1\rangle-\sqrt{3/4}|2\rangle$ are genuine three-dimensional states. Of course, this criterion is still not a necessary and sufficient condition. We define the robustness ($\mu$) Cavalcanti2017 of the genuine three-dimensional state. The optimal solution $\mu^{*}$ of this problem gives the minimum amount of “white noise” that has to be added to the qutrit state such that the mixture can be simulated by qubit states Supple . If $\mu>0$, we can certify that this state is a genuine three-dimensional state. On the contrary, if $\mu=0$, the state is not a genuine three-dimensional state. We choose 400 maximum coherent superposition states ($1/\sqrt{3}(|0\rangle+e^{i\varphi_{1}}|1\rangle+e^{i\varphi_{2}}|2\rangle)$,where $\varphi_{1},\varphi_{2}$ are 20 phases at equal interval in $[0,\pi)$) as the input states, and then through the evolution of $\chi$ matrix. After the semidefinite program (SDP), we find that 149 states can be simulated by qubit, while 251 states cannot be simulated. The average $\mu$ of these states that can not be simulated is $\mu=0.111\pm 0.034>0$. This means that within three standard deviation ranges, using qubit states cannot simulate our teleportation process. This result can prove that our teleportation is beyond qubit. All the errors are obtained by raw data through the Monte Carlo method, in which all the generated data have the same Poissonian error as the raw data.

In summary, we have reported the quantum teleportation of high-dimensional quantum states of a single quantum particle, demonstrating the capability to control coherently and teleport simultaneously a high-dimensional state of a single object. The generation of high-quality high-dimensional multi-photon state will stimulate the research on high-dimensional quantum information tasks, and entanglement-assisted methods for HDBSM are feasible for other high-dimensional quantum information tasks.