Fault-tolerant quantum repeaters with minimal physical resources and implementations based on single-photon emitters

We analyze a method that uses fixed, minimal physical resources to achieve generation and nested purification of quantum entanglement for quantum communication over arbitrarily long distances and discuss its implementation using realistic photon emitters and photonic channels. In this method, we use single-photon emitters with two internal degrees of freedom formed by an electron spin and a nuclear spin to build intermediate nodes in a quantum channel. State-selective fluorescence is used for probablistic entanglement generation between electron spins in adjacent nodes. We analyze in detail several approaches which are applicable to realistic, homogeneously broadened single-photon emitters. Furthermore, the coupled electron and nuclear spins can be used to efficiently implement entanglement swapping and purification. We show that these techniques can be combined to generate high-fidelity entanglement over arbitrarily long distances. We present a specific protocol that functions in polynomial time and tolerates percent-level errors in entanglement fidelity and local operations. The scheme has the lowest requirements on physical resources of any current scheme for fully fault-tolerant quantum repeaters.

Figure 1

(Color online) (a) Generic level structure showing the state-selective optical transitions and electronic spin sublevels required for entanglement generation. (b) Setup used to create entanglement. The two emitters act as state-dependent mirrors in an interferometer. The outputs of the cavities ($a_1$ and $a_2$) are combined on a beam splitter. By proper alignment of the interferometer the photons always exit through the $(a_1+a_2)∕\sqrt{2}$ port if both centers are in the scattering state ∣0⟩. A detection of a photon in the $(a_1-a_2)∕\sqrt{2}$ mode thus leads to an entangled state. (c) Scheme for balancing the interferometer. Each node is optically excited by a laser pulse which first reflects off the other node, so that the optical path lengths for the two excitation-emission paths are identical.

Figure 2

(Color online) The relevant electronic and nuclear states of the coupled NV center and ${}^{13}\mathrm{C}$ impurity nuclear spin. The electron spin states can be coupled by ESR microwave fields near $2.88\mathrm{GHz}$, while the nuclear spin states can be addressed by NMR pulses on the $130\text{−}\mathrm{MHz}$ hyperfine transition. A laser applied on resonance with the $M_s=0$ optical transition produces strong fluorescence for the entanglement scheme, but it only weakly excites the $M_s=\pm 1$ transitions, because of the large detuning ${\Delta }_1$. The optically excited $\mid M_s=0⟩$ state decays at a rate $\gamma$ and the optical transition has an inhomogeneous broadening $\Gamma$ caused by fluctuations $\Delta \left(t\right)$ in the energy of the excited state. In our model we also assume that only the $M_s=\pm 1$ states decay to the shelving state $\mid W⟩$ at rate ${\Gamma }_S$.

Figure 3

(Color online) Entanglement propagation by swapping. To generate an entangled nuclear spin pair [black (top) circle] over the distance $L_{n+1}=2L_n+L_0$, we first generate nuclear entanglement over the first and second pairs of repeater stations which are distance $L_n$ apart. The electron spin [red (bottom) circle] is then used to generate entanglement between the middle stations, separated by distance $L_0$. Nuclear (electron) spin entanglement is illustrated by solid (dashed) lines. Entanglement swapping is performed by measuring in the Bell-state basis of the two-qubit system.

Figure 4

(Color online) Nesting scheme for generation and purification of entangled nuclear and electron spin pairs. In each node, the nuclear spin degree of freedom is represented by the upper (black) circle, while the electron degree of freedom is represented by the lower (red) circle. Entanglement between different nodes is represented by a line connecting them. Ovals represent entanglement swap steps, and rectangles represent entanglement purification steps. For $n=2$ the $B$ and $C$ pairs may be directly generated. For $n⩾3$, the first step illustrates how the $B$ pair is generated, while the remaining two steps illustrate how the $C$ pair is generated while storing the $B$ pair. The arbitrary distance algorithm works for $n⩾6$.

Figure 5

Approach to asymptotic fidelity. The solid curve shows the purified fidelity obtained from the auxiliary pair, while the dotted curve corresponds to the auxiliary pair (constructed from two smaller purified pairs) on the next nesting level. The system moves between the curves at each nesting step, and the upper intercept of the two curves gives the asymptotic fidelity. For this calculation $F_0=p=\eta =0.99$ and $\upsilon =0$.

Figure 6

(Color online) (a) Fidelity scaling with distance. Points show results using three purification steps at each nesting level, dashed lines show the fixed point $F_{\mathit{FP}}$ at each distance, and dotted lines indicate the asymptotic fidelity $F_{\infty }$. For (a) and (b), measurements and local two-qubit operations $\eta =p$ contain 0.5% errors. The initial fidelity $F_0$ is (i) 100%, (ii) 99%, (iii) 98%, (iv) 97%, and (v) 96% with phase errors only. (b) Average time scaling with distance for $m=3$, given in units of $T_0=(t_0+t_c)∕P$, the time required to generate entanglement between nearest neighbors, and $L_0$, the distance between nearest neighbors. Measurement and local operation times are neglected. Note that the axes are logarithmic, so time scales polynomially with distance.

Figure 7

(Color online) (a) Long-distance asymptote dependence on initial fidelity $F_0$ of (i) 100%, (ii) 99%, (iii) 98%, (iv) 97%, and (v) 96% with phase errors only. (b) Long-distance asymptote dependence on error type. For the calculations shown, $F_0=0.99$, and the shape parameter ranges from $\upsilon =0$ to $\upsilon =0.3$. In both (a) and (b) measurement errors are set equal to operational errors, $\eta =p$.

Figure 8

(Color online) (a) Level structure for single electron to trion transition in a single-electron doped III-V or II-VI quantum dot with external magnetic field in close to a Faraday geometry with Zeeman splitting ${\omega }_z$, heavy-hole splitting ${\omega }_z^h$ and up to four optical fields of different frequency and polarization. Dashed lines indicate weak dipole moments due to small magnetic field mixing. The triplet two-electron states are not included due to a large $(>1\mathrm{meV})$ exchange energy allowing for complete suppression of their effects. (b) Electronic (∣↑⟩, ∣↓⟩) and collective nuclear (∣0⟩, ∣1⟩) states and their transitions for singly doped quantum dots in a polarized nuclear spin lattice.

Figure 9

(Color online) Electronic level structure of atomic magnesium. The electronic ground state ${}^{1}S_0$ has vanishing spin and orbital angular momentum. In this state the nuclear spin therefore decouples from the electronic degrees of freedom. An entangling operation which is also insensitive to the nuclear degree of freedom can be achieved with lasers which are detuned much further than the hyperfine splitting in the ${}^{1}P_1$ level.