Principle

Quantum teleportation consists in the preparation of an entanglement and the measurement in the Bell basis, resulting in post-selective transfer of the quantum state (Fig. 1a). In the current demonstration, we first prepare an entanglement between electron spin and carbon nuclear spin, and then measure photon polarization and electron spin in the Bell basis by photon absorption23 to transfer the photon polarization state into the carbon spin state (Fig. 1b, c). In the practical protocol of the one-way quantum repeater system with an NV center at each node, the photon is emitted from one node, leaving an electron entangled with the emitted photon24 (Fig. 1d). The success of the photon storage in the other node establishes the entanglement between two adjacent nodes.

Fig. 1 Schematics of the quantum state transfer. a Protocol of the transfer scheme. We first prepare an entanglement between electron spin (e) and carbon nuclear spin (13C) (bottom), and then measure photon polarization (p) and electron spin (e) in the Bell basis by photon absorption23 (middle), which announces the success of the quantum state transfer from the photon into the carbon (top). b The lattice structure of diamond contains a nitrogen-vacancy (NV) center with surrounding carbon nuclear spins. An electron locates in the vacancy. c Energy level diagram of the photon based on the right/left \((\vert { +} 1\rangle_{\mathrm{p}} / \vert {-}1\rangle_{\mathrm{p}})\) circular polarizations, the electron based on up/down \((\vert + 1\rangle_{\mathrm{e}} / \vert -1\rangle_{\mathrm{e}})\) spin polarizations, and the carbon nuclear spin based on up/down \((\vert {\uparrow} \rangle_{\mathrm{c}} / \vert{\downarrow}\rangle_{\mathrm{c}})\) nuclear spin polarizations. The nitrogen impurity \(({ \,\! }^{14}{\mathrm{N}})\) polarized in \(\vert + 1\rangle_{\mathrm{N}} \) nuclear spin polarization serves as a nanomagnet to apply a magnetic field only for the electron, allowing the preparation of entanglement between the hyperfine coupled electron and carbon spins under a zero magnetic field. The electron is then resonantly excited to a spin-orbit correlated eigenstate \((\vert A_{1} \rangle)\) by photon absorption, allowing the measurement of the photon polarization and electron spin in the Bell basis to result in quantum state transfer from the photon into the carbon. d The supposed protocol of the one-way quantum repeater system with an NV center at each node. A photon is emitted from one node (left), leaving an electron entangled with the emitted photon. The success of the photon storage in the other node (right) establishes the entanglement between two adjacent nodes Full size image

A negatively charged NV center in diamond consists of a nitrogen impurity (14N) and an adjacent vacancy (V), where the triplet state electron (e) is localized (Fig. 1b). Both the electron and the nitrogen nucleus show a spin 1 property constituting a V-type three-level system with two degenerate m s,I = ±1 states (denoted \((\vert \pm 1\rangle_{\mathrm{e,N}}),\) which constitute a logical qubit called a geometric spin qubit22,23,25,26,27,28,29, and an m s,I = 0 state (denoted |0〉 e,N ), which constitute an ancilla. These are split by a zero-field splitting of around 2.87 GHz for the electron and a nuclear quadrupole splitting of around 4.95 MHz for the nitrogen. On the other hand, a carbon nuclear spin (13C), weakly coupled to the electron via a hyperfine interaction (0.9 MHz in this demonstration), shows a spin half property constituting a two-level system with two degenerate m I = ±1/2 states (denoted \(\vert{\uparrow}\rangle_{\mathrm{C}},\vert {\downarrow} \rangle_{\mathrm{C}}\)) under a zero magnetic field (Fig. 1c). We utilize the nitrogen as a nanomagnet localized at the vacancy.

To prepare the spin entanglement between the electron and carbon nuclear spin, we first initialize them into \(\vert0\rangle _{\mathrm{e}}\vert {\uparrow} \rangle_{\mathrm{C}}\). Although it is hard to initialize the carbon nuclear spin under a zero magnetic field, the nuclear quadrupole splitting of the nitrogen nuclear spin with the help of the polarized electron spin enables the polarization into \({\vert} {+} 1\rangle_{\mathrm{{N}}}\), which is used as a nanomagnet to apply a local magnetic field on the electron to initialize the carbon nuclear spin (Fig. 1c) with the following sequence (see Supplementary Note 1). The coherent population trapping (CPT) using a red light resonantly exciting the electron into a spin-orbit correlated eigenstate \(| A_{2} \rangle = \frac{1}{{\sqrt 2 }}(|{ + 1, - 1} \rangle _{l,{\mathrm{e}}} + |{{ - 1, + 1}\rangle }_{l,{\mathrm{e}}})\)23,24 (l and e, respectively denote orbital and spin angular momentums of an electron) with the right circular polarization |+1〉 p first polarizes the electron into \({\vert} {+} 1\rangle_{\mathrm{{e}}}\)23, which is then transferred into the nitrogen nuclear spin to polarize into \({\vert} {+} 1\rangle_{\mathrm{{N}}}\) (purple line in Fig. 2a). The degeneracy of the electron spin is therefore lifted via the hyperfine interaction with the nitrogen, allowing selective flip of the carbon nuclear spin \({\vert} \!\downarrow \rangle_{\mathrm{{C}}}\) to polarize into \({\vert}\! \uparrow \rangle_{\mathrm{{C}}}\) (green line in Fig. 2a). Initialization processes of the nitrogen and carbon nuclear spins are shown in Fig. 2b. The electron is again initialized into |0〉 e with the red light resonant to the |A 1 〉 state. The electron and carbon are then manipulated with a microwave optimized by GRAPE (GRadient Ascent Pulse Engineering)30 and a radiowave to create entanglement between them into \(| {{{\mathrm{\Phi }}^ + } \rangle }_{{\mathrm{e,C}}} = \frac{1}{{\sqrt 2 }}( {| { { + 1, \uparrow }\rangle } _{{\mathrm{e,C}}} + | { { - 1, \downarrow }\rangle }_{\mathrm{e,C}}} )\), which is one of four Bell states, along with the quantum circuit shown in Fig. 2c, where we define the SU(3) X gate that applies the X gate in the partial space spanned by \({\vert} a \rangle\) and \({\vert} b \rangle\) while retaining \({\vert} d \rangle\) as \(X_{(\pi )}^{\left( {\left. {\left| a \right.} \right\rangle ,\left| {\left. b \right\rangle } \right.} \right)} = \left| {\left. a \right\rangle } \right.\left\langle b \right.| + \left| {\left. b \right\rangle } \right.\left. {\left\langle a \right.} \right| + \left| {\left. d \right\rangle } \right.\left. {\left\langle d \right.} \right|\). The Y gate applied to the spin-half carbon nuclear spin stands for Pauli operator σ y .

Fig. 2 Detailed scheme of the quantum state transfer. a Optically detected magnetic resonance (ODMR) spectra before and after the initialization process. The red, purple and green lines respectively show the spectrum before the initialization, after the initialization of only the nitrogen, and after initialization of both the nitrogen and carbon. The splittings of 2.2 MHz and 0.9 MHz are caused by the hyperfine interaction of the electron with the nitrogen and carbon, respectively. The achieved initialization fidelities are 94% for the nitrogen and 90% for the carbon. Error bars defined as the standard deviation of photon count are within point size. b Level diagrams showing populations for the nitrogen and the carbon nuclear spin corresponding to three steps in a. c Quantum circuit representing the transfer scheme. The controlled NOT(CNOT)-like symbols represent two or three qubit SU(3) gates, where the gates shown in the box are operated on the target qubit conditioned by the control qubit if the state is as given in the black circle or not as given in the white circle (double-sign corresponds). Below the circuit, energy level diagrams show the gate-induced transitions of the population in the ideally initialized case. Microwave (MW) 1 performs Toffoli-like π-gate to introduce the electron-carbon interaction, radio frequency (RF) performs controlled π/2 gate to generate a superposition of the carbon nuclear spin, MW2 performs controlled π-gate to eliminate the electron–carbon interaction, and MW3 again performs Toffoli-like π-gate to convert the superposition into an electron–carbon entanglement. Detection in the |0〉 e state relaxed after resonant excitation into |A 1 〉 post selects the photon polarization and electron spin states on a certain Bell state23 Full size image

Next, we allow the electron to absorb an incoming photon with arbitrary polarization \(\vert \psi\rangle_{\mathrm{p}} = \alpha \vert +1\rangle_{\mathrm{p}} + \beta\vert-1\rangle_{\mathrm{p}}\), which excites the electron into another spin-orbit correlated eigenstate \(| A_{1} \rangle = \frac{1}{{\sqrt 2 }}( | {{ + 1, - 1}\rangle } _{l,{\mathrm{e}}} - {| { - 1, + 1}}\rangle _{l,{\mathrm{e}}})\)23. The photon absorption projects the polarization state of the photon and the spin state of the electron into one of the Bell states as demonstrated in Kosaka et al.23 Although the projection is probabilistic and allows only partial Bell state measurement, we repeat the transfer process until absorption into the \({\vert} A_{1} \rangle\) state stops after non-radiative relaxation into \({\vert} 0 \rangle_{\mathrm{e}}\) of the orbital ground state. The projection of the prepared state composed of an arbitrary photon polarization \({\vert}\psi\rangle_{\mathrm{p}}= \alpha\vert+ 1\rangle_{\mathrm{p}}+ \beta\vert -1\rangle_{\mathrm{p}}\) and the electron-carbon entangled state \(\left| {\left. {{\mathrm{\Phi }}^ + } \right\rangle } \right._{\mathrm{e,C}} = \frac{1}{{\sqrt 2 }}( {\left| {\left. { + 1, \uparrow } \right\rangle } \right._{{\mathrm{e,C}}} + \left| {\left. { - 1, \downarrow } \right\rangle } \right._{{\mathrm{e,C}}}} )\) into the |A 1 〉 state is described as

$$\langle A_1|\psi \rangle _l\left| {\left. {{\mathrm{\Phi }}^ + } \right\rangle } \right._{{\mathrm{e,C}}} \, = \, \frac{1}{2}\left( {\langle + 1, - 1|_{l,{\mathrm{e}}} - \langle - 1, + 1|_{l,{\mathrm{e}}}} \right)\left( {\alpha \left| {\left. { + 1} \right\rangle } \right._l + \beta \left| {\left. { - 1} \right\rangle } \right._l} \right)\\ \hskip 12pt\left( {\left| {\left. { + 1, \uparrow } \right\rangle } \right._{{\mathrm{e,C}}} + \left| {\left. { - 1, \downarrow } \right\rangle } \right._{{\mathrm{e,C}}}} \right)\\ = \frac{1}{2}\left( {\beta \left| {\left. \uparrow \right\rangle } \right._{\mathrm{C}} - \alpha \left| {\left. \downarrow \right\rangle } \right._{\mathrm{C}}} \right) = \frac{i}{2}\sigma _y\left| {\left. \psi \right\rangle } \right._{\mathrm{C}},$$ (1)

where the photon polarization state \(\vert \psi\rangle_{\mathrm{p}}\) was substituted by the electron orbital state \(\vert \psi\rangle_{l}= \alpha \vert +1\rangle_{l} + \beta \vert -1\rangle_{l}\) since the photon polarization state p corresponds to the electron orbital state l via the angular momentum conservation or the polarization selection rule. The coefficient \(\frac{i}{2}\) implies that success probability of the partial Bell state measurement is \(\left| {\frac{i}{2}} \right|^2 = \frac{1}{4}\). However, the norm of the transferred nuclear spin becomes unity, which means that the nuclear spin state is in principle purified, after the post-selection of the electron spin state. The resulting nuclear spin state corresponds to the photon polarization state with the additional unitary operation \(\sigma_{y}\).

Experiments

We first measure the phase correlation between the input photon and the transferred carbon to show that the transfer operation conserves the quantum coherence. In principle, the photon polarization state \(\left| {\left. \psi \right\rangle } \right._{\mathrm{p}} = \frac{1}{{\sqrt 2 }}\left( {\left| {\left. { + 1} \right\rangle } \right._{\mathrm{p}} + e^{i\phi }\left| {\left. { - 1} \right\rangle } \right._{\mathrm{p}}} \right)\) should be transferred into the carbon nuclear spin polarization state \(\left| {\left. \psi \right\rangle } \right._{\mathrm{C}} = \frac{1}{{\sqrt 2 }}\left( {\left| {\left. \uparrow \right\rangle } \right._{\mathrm{C}} + e^{i\left( {\pi - \phi } \right)}\left| {\left. \downarrow \right\rangle } \right._{\mathrm{C}}} \right)\) with the additional unitary transformation σ y determined by Eq. (1). Figure 3a shows the photon polarization dependence of the carbon nuclear spin population measured in \(\vert+\rangle_{\mathrm{C}}- \vert -\rangle_{\mathrm{C}}\) axis \(\left( {\left| {\left. \pm \right\rangle } \right._{\mathrm{C}} = \frac{1}{{\sqrt 2 }}\left( {\left| {\left. \uparrow \right\rangle _{\mathrm{C}}} \right. \pm \left| {\left. \downarrow \right\rangle _{\mathrm{C}}} \right.} \right)} \right)\), which is obtained by measuring the photon count after the irradiation of a radiowave and a microwave followed by the red light resonant to the \(\vert E_{x}\rangle\) state (see Supplementary Fig. 5). Strong anti-phase correlation is observed as expected, indicating the quantum nature of the transfer.

Fig. 3 Experimental demonstration of the quantum state transfer. a Correlation between the photon phase angle (twice the polarization angle) and the carbon nuclear spin population measured in \(\vert + \rangle_{\mathrm{{C}}} - \vert -\rangle_{\mathrm{{C}}}\) axis, which is converted into the population of the electron spin ancilla state. b The Poincare sphere representation of the polarization state of the input light. c The Bloch sphere representation of the quantum state stored in the carbon nuclear spin reconstructed by the quantum state tomography. Six basis polarization states of the incoming photon are transferred into the memory carbon. The meshed frame shows the distorted Bloch sphere estimated by the process tomography measurements shown in e. d The state fidelities of the carbon for the six basis polarization states defined by \({\mathrm{Tr}}[\rho _{{\mathrm{ideal}}}^\dagger \rho _{{\mathrm{exp}}}]\). The averaged state fidelity is 78%, which well exceeds the classical limit of 67%. e Matrix representation of the quantum process tomography for the state transfer. The matrix elements χ exp are calculated based on the transferred states shown in c. The transfer process fidelity defined by \({\mathrm{Tr}}[\chi _{{\mathrm{ideal}}}^\dagger \chi _{{\mathrm{exp}}}]\) is 76%. Error bars defined as the standard deviation of photon count in a, c–e are within point size (<2%). f Simulated dependence of the transfer fidelity on the initialization fidelities for the carbon and nitrogen nuclear spins for the photon polarizations (i), \(\vert + \rangle_{\mathrm{p}}, \vert - \rangle_{\mathrm{p}}, \vert + i \rangle _{\mathrm{p}}, \vert - i \rangle_{\mathrm{p}},\) (ii), \(\vert\, + 1\, \rangle_{\mathrm{p}}\), and (iii), \(\vert - 1\rangle_{\mathrm{p}}\). Solid circles represent the ranges of the initialization errors for the demonstration. g Simulated dependence of the transfer fidelity on the two vector components of the crystal strain for the photon polarizations (i), | + 〉 p , (ii), | + i〉 p , and (iii), | + 1〉 p . Dashed circles represent the ranges of the crystal strain for the demonstration Full size image

To evaluate the fidelity of the quantum process during the state transfer, we prepare the six mutually-unbiased basis states of the photon polarization \(( {| { { + 1}\rangle } _{\mathrm{p}},| { { - 1} \rangle } _{\mathrm{p}},| { + \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( {| { { + 1} \rangle } _{\mathrm{p}} + | { { - 1} \rangle } _{\mathrm{p}}} ), | { - \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( {| { { + 1} \rangle } _{\mathrm{p}} - \!| { { - 1} \rangle } _{\mathrm{p}}} ),| { { + i} \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( {| { { + 1} \rangle } _{\mathrm{p}} + i| { { - 1} \rangle } _{\mathrm{p}}} )} \\ {,| { { - i} \rangle } _{\mathrm{p}} = \frac{1}{{\sqrt 2 }}( {| { { + 1} \rangle } _{\mathrm{p}} - i| { { - 1} \rangle } _{\mathrm{p}}} } ))\) (Fig. 3b) and estimate the carbon nuclear spin state after the transfer based on the quantum state tomography. The Bloch vectors for the carbon nuclear spin states transferred from the six photon polarizations are reconstructed as shown in Fig. 3c. The fidelities projected into the ideal state reach 78 ± 2% on average, which well exceeds the classical limit of 67% (Fig. 3d). With the reconstructed Bloch vectors, we evaluated the quantum channel of the transfer based on the quantum process tomography as shown in Fig. 3e. The fidelity of the transfer process is estimated from the χ matrix defined as \({\mathrm{Tr}}[\chi _{{\mathrm{ideal}}}^\dagger \chi _{{\mathrm{exp}}}]\) to be 76%, indicating that the transfer channel maintains the quantum coherence.