Effective W-state fusion strategies for electronic and photonic qubits via the quantum-dot-microcavity coupled system

We propose effective fusion schemes for stationary electronic W state and flying photonic W state, respectively, by using the quantum-dot-microcavity coupled system. The present schemes can fuse a n-qubit W state and a m-qubit W state to a (m + n − 1)-qubit W state, that is, these schemes can be used to not only create large W state with small ones, but also to prepare 3-qubit W states with Bell states. The schemes are based on the optical selection rules and the transmission and reflection rules of the cavity and can be achieved with high probability. We evaluate the effect of experimental imperfections and the feasibility of the schemes, which shows that the present schemes can be realized with high fidelity in both the weak coupling and the strong coupling regimes. These schemes may be meaningful for the large-scale solid-state-based quantum computation and the photon-qubit-based quantum communication.

Entanglement is a unique phenomenon in quantum mechanics, and it is an important quantum resource in the quantum information field. Especially, quantum entanglement plays a vital role in quantum communication and quantum information processing (QIP), such as quantum computation 1 , quantum teleportation 2 , quantum key distribution (QKD) 3 , and so on. It is known to all that bipartite entanglement is different from multipartite entanglement. Among the multipartite entangled states, W state, GHZ state and cluster state form inequivalent classes and they can't be transformed into each other by local operations and classical communication. W state is a special kind of entangled state in the multipartite system. Compared with the GHZ state, W state is highly robust against the qubits loss. Hence, W state has recently attracted considerable attention in the field of quantum computing and information science [4][5][6][7] . For example, W state has been used for the optimal universal quantum cloning machine 8 and has also been proposed as a resource for QKD 9 . The more the number of particles forming an entangled state is, the more complex the entanglement structures are. Therefore the creation of multipartite entangled states has been paid much attention.
In the last decade, expansion and fusion operations have been proposed and demonstrated as efficient ways to prepare large scale multipartite entangled states. One can get a larger entangled state from two or more qubits entangled states by fusion operation on the condition that the access is granted only to one qubit of each of the states entering the fusion operation. Nowadays, much attention has been paid to the preparation of large scale multipartite entangled states by fusion operation. Currently, many expansion and fusion proposals of multipartite entangled states have been put forward, such as using cluster states with smaller-scale qubits to prepare larger cluster states 10 , the creation of large-scale GHZ states 11 and W states [12][13][14][15][16][17][18][19][20] . Among those schemes, in 2011 Ozdemir et al. first used a simple optical fusion gate to get a W state W n+m−2 from W n and W m (n,m ≥ 3 and W x denotes a x-qubit W state) 16 . In the following years, they put forward several W states fusion schemes with the help of complex quantum gate sets. However,

Results
Electronic W-state Fusion based on QD-microcavity coupled system. Here, we consider a singly charged GaAs/InAs QD, which has four relevant electronic levels, ↑ , ↓ , ↑ ↓ , and ↑ ↓ , as shown in Fig. 1, being embedded in a double-sided optical microcavity with both the top and bottom mirrors partially reflective. The negatively charged exciton (X − ), produced by the optical excitation of the system, contains two electrons bound in one hole. = , − represent heavy hole states with spin 3/2 and 3 2 − / components. The total spin of the two electrons in an exciton is zero, which prevents the interaction between the electron-spin and the heavy hole spin. The quantization axis for angular momentum is the z axis because the quantum dot confinement potential is much tighter in the z(growth) direction than in the transversal direction due to the quantum dot geometry. According to this feature, it has two optical transitions between the electron state and the exciton state by involving the photon whose spin is s z = + 1 ( R ↑ or L ↓ ) or s z = − 1 ( R ↓ or L ↑ ). Based on the optical selection rules and the transmission and reflection rules of the cavity for an incident circular polarization photon, the interactions between photons and electrons in the QD-microcavity coupled system can be described as follows 26,27 : where R L ( ) denotes the right(left)-circularly polarized photon state. The superscript up arrow (down arrow) denotes the propagating direction of polarized photon along (against) the z axis. Now, we introduce how to implement a (m + n − 1) qubits electronic W-state fusion scheme from m qubits W state and n qubits W state based on QD-microcavity coupled system. The schematic is depicted in Fig. 2 In this notation, a tripartite W state is written as with W 2 corresponding to the EPR pair W 2 2 = ( ↓↑ + ↑↓ )/ . For simplicity, here we have substituted a for n n 1 With the help of two ancillary photons in right-circular polarization state R , we can fuse Alice's and Bob's electronic W states to a larger W state W m n 1 + − . The initial state of the whole system is In the fusion process, only the electron spin 1 and 2 interaction with photons, the remaining electrons in modes a (b) are kept constant at Alice's (Bob's) side. Namely, only 1 1 interact with the photons, whose probabilities are P nm Firstly, photon 1 passes through the c-PBS 1 , which is polarizing beam splitter in the circular basis transmiting the right-circularly-polarized photon R and reflecting the left-circularly-polarized photon L . That is, the components R 1 and L 1 enter the cavity 1 from the top and the bottom, respectively. PS is a phase shifter that contributes a π phase shift to the photon passing through it (i.e., L L → − and R R → − ). After the interaction with cavity 1, the components R 1 and L 1 mix at c-PBS 1 again. The While the component R 1 transmits from c-PBS 2 and does not interact with the cavity. It passes c-PBS 2 → DL 1 → OWM in turn. DL is the time-delay device for matching path lengths of the two components. OWM is one-way mirror transmits photons from one side, and reflects photons from the other side without remodulating 38,39 . Hence, before the detectors click, the state will be It is obvious that when the detector D 2 detects photon, the state is In this case, we will obtain two separate W states with a smaller number of qubits, W n 1 − and W m 1 − , which can be recycled using the same fusion mechanism. However, when the detector D 1 detects photon, the state of the system is changed as φ is used to continue the fusion process. Now, let the photon 2 pass through c-PBS 1 and interact with the spin 1. After passing though c-PBS 1 again, we obtain Then, the photon 2 exits from the optical switch and goes towards c-PBS 6 . The component R 2 transmits c-PBS 6 and does not interact with the spin 2 and spin 1. However, the component L 2 passes through HWP 1 which realizes R R L 2 → ( + )/ and L R L 2 → ( − )/ , and then enters cavity 2 via c-PBS 5 to interact with the spin 2. It is worth noting that before and after the photon 2 interacts with the electron spin 2, a Hadamard operation (H e ) on the electron spin 2 is performed by a π/2 microwave pulse or an optical pulse 23 It is clearly that electron spin 1 is disentangled with other electron spins. Therefore the state of the remaining electron spins is given by Photonic W-state Fusion based on QD-microcavity coupled system. In this section, we will introduce the fusion protocol of the photonic W-state in detail. The schematic is depicted in Fig. 3. We use the similar notation W n A and W m B as electronic W states in the above section, , and d for Before and after the photon 2 passes through the second optical microcavity, Hadamard operations H p and H e are performed on photon 2 and electron 2, respectively. PS 1 realizes the transformations 2 , the state of the photon-electron system is given by Now, the photon 1 passes through the c-PBS 6 , the component R 1 does not interact with the spin 2 and spin 1 while the component L 1 passes through the optical switch S 2 and the c-PBS 2 . Then it enters cavity 2 and interacts with the spin 2. These operations (c-PBS 6 Next, the photon 1 goes through the optical switch S 1 , HWP 3 , c-PBS 3 , one by one. Before and after the photon 1 interacts with the electron spin 1, a Hadamard operation H e is performed on the electron spin 1 by using a π/2 microwave pulse or an optical pulse. Then photon 1 passes HWP 4 . After these operations (S 1 → HWP 3 → H e → c-PBS 3 → spin 1 → H e → c-PBS 3 → HWP 4 ), the state becomes ↓ , we will get two separate W states with a smaller number of qubits, W n 1 − and W m 1 − , which can be recycled by the same fusion mechanism. For the other situation we acquire 2 ψ , it will be used to continue the fusion process. At this time, the component R 1 in the DL 1 passes through the optical switch S 2 , c-PBS 2 and enters the spin 2. When the photon leaves away from the cavity 2, it passes through the switch S 1  The σ z operation on the photon 2 is needed before it passes through QWP, and the state of photons will become the same as Eq. (22). With the same operation above, we can obtain a (n + m − 1)-qubit W state in Eq. (23) with the probability (n + m − 1)/nm. So far, we have completed the W-state fusion schemes for electronic and photonic W state, respectively, based on the quantum-dot-microcavity coupled system.

Discussion
In this section, we will briefly analyze and discuss the feasibility and the success probability of the proposed schemes. When the side leakage and cavity loss are taken into account, the reflection and transmission coefficients of the coupled and the uncoupled cavities are generally different in a realistic X − -cavity system. The reflection and transmission coefficients of a double-sided optical microcavity for weak excitation limit can be described by 26 Therefore, in a realistic spin-QD-double-side-cavity unit, the rules of the optical transitions can be described as  F photonic ), respectively, which show our schemes can be achieved with high fidelities. Nevertheless, the cavity side leakage and cavity field decay have obvious impact on the fusion-scheme fidelities. Fortunately, the strong coupling of the QD-microcavity system has been observed in [42][43][44][45]  b photon = . %; even when setting κ s = 1.0κ, g = 0.4κ, (i.e. g/ (κ + κ s ) = 0.2 which is the weak coupling regime) we also can obtain F electron = 87.55%, F 72 68 a photon = . %, and F 77 26 b photon = . %. Therefore, our scheme can work well in both the weak coupling and the strong coupling regimes.
In addition, the electron spin decoherence and the exciton dephasing could also effect the fidelity. Exciton dephasing reduces the fidelity by the amount of e 1 T e − − τ , where τ is the photon life time in the cavity and T e is the exciton coherence time 26,27 . The optical dephasing reduces the fidelity only a few percent that is because in a self-assembled In(Ga)As-based QD the time scale of the excitons can reach hundreds of picoseconds [45][46][47] . The effect of the spin dephasing is mainly due to the hole-spin dephasing, while the hole spin coherence time is at least three orders of magnitude longer than the cavity photon lifetime 48 , so the spin dephasing can be safely neglected. Obviously, in the whole fusion process, the fused large W state is from the items of the initial state with the electron (photon) 1 and 2 in the states | 〉 | 〉 ). While the item with the electron (photon) 1 and 2 in 1 1 would become two smaller W states after measuring the photon 1 (electron 1), which can be recycled using the same fusion mechanism. Therefore the success probability P s and the recyclable probability P r are written as initially sent into the cavity along the z axis, after interacting with QD-cavity system, the joint state of the photon and electron becomes Obviously, the projection measurement of the electron spin can be completed by detecting the reflection and transmission of the photon. The electron spin is projected into the state ↑ for photon's reflection; the electron spin is projected into the state ↓ for photon's transmission.