1. Introduction

Quantum entanglement is a key resource in quantum computation and quantum communication, as it provides the tremendous power for speeding up the computation [1–3] and improves security of communication [4–13]. Nonlocal maximal entanglement provides a useful way for establishing quantum communication channels in quantum communication protocols including quantum teleportation [4], quantum dense coding [5,6], quantum key distribution [7,8], quantum secret sharing [9], quantum secure direction communication [10–13], and so on. In the quantum network established by the quantum entanglement, a basic problem for communication between remote nodes is to hold nearly perfect entanglement. Unfortunately, the quality of entanglement between two nodes decreases exponentially with the length of the communication channel due to the decoherence and photon losses in the entanglement storage and distribution processes, which may limit the communication distance to hundreds of kilometres [14]. This bottleneck can be overcome by using the quantum repeater originally proposed by Briegel et al. in 1998 [15]. The principle of quantum repeater can be described as follows: In order to generate entanglement between two remote nodes, say quantum memories $A$ and $Z$, entanglement is first created independently within short elementary links, i.e. between the quantum memories $A$ and $B$, $C$ and $D$,…, $W$ and $X$, $Y$ and $Z$. After entanglement swapping, entanglement is then generated between quantum memories $A$ and $D$,…, $W$ and $Z$. Through cascading entanglement swapping, entanglement is generated between the terminal quantum memories $A$ and $Z$. In the seminal DLCZ protocol, Duan et al. proposed the "from-the-memory" quantum repeater using atomic ensembles as quantum memories [16]. In a quantum repeater, nested entanglement purification [17,18] is an ingredient component to distill high-fidelity entangled states from low-fidelity entangled ones within short elementary links [15,16]. There are two ways for entanglement purification in the "from-the-memory" quantum repeater. One way is to convert the entangled states stored in the quantum memories into entangled photons first [19] and subsequently purify entangled photons [20]. The other way is to directly purify entangled quantum memories.

Quantum stationary systems are usually acted as quantum memories in the network of quantum communication and quantum computation. Diamond nitrogen vacancy centers, atomic ensembles, single trapped ions, and quantum dots (QDs) are frequently-used quantum stationary systems in quantum repeaters [21–33], and many interesting entanglement purification protocols (EPPs) on these quantum stationary systems have been proposed [34–45]. A charged QD is one of the most promising quantum stationary systems playing the role of quantum memory in the quantum network, due to the long coherence time of an electron spin in a QD. The coherence time of an electron spin in a charged QD can maintain 3$\mu s$ [46,47], and the relaxation time of an electron spin can reach the order of $ms$ [48,49] through the technique of spin echo. Meanwhile, fast preparation of the superposition state of the electron spin [50,51] has been demonstrated, and fast manipulation [52–55] and detection [56] of the electron spin state in a QD have been reported in recent years. Furthermore, in view of realizing a QD embedded in a microcavity (QD-microcavity system) [57,58], the interaction between a QD-microcavity system and a circularly polarized photon has attracted much attention for its applications in quantum information processing [59,60], especially in quantum repeater using QD systems and entanglement purification on electron spins in QDs [31,40].

Since the first EPP proposed by Bennett et al. [17], a number of approaches, such as controlled-NOT (CNOT) gate, Toffoli gate, parity check, and hyperentanglement, have been considered to assist completing entanglement purification [20,61–70]. Parity check is an important method which has been used to reserve the two pairs of entangled systems in the same parity mode and discard the ones in the different parity modes. In this method, it is especially important to correctly judge the parity mode with nondestruction on the entangled state. However, using the interaction between a QD-microcavity system and a circularly polarized photon, the parity check is faithful only in specifically ideal conditions, which is difficult to be accomplished. In the realistic experiment, the interaction between a QD-microcavity system and a circularly polarized photon turns imperfect due to the realistic experiment parameters deviate from the specifically ideal ones, and it will lead to an unfaithful parity check, including wrong judgement on the parity mode and destruction on the entangled system. If an unfaithful parity check makes a wrong judgement on the parity mode, the fidelity of the mixed states cannot reach a desired one after purification. Moreover, if an unfaithful parity check has a destruction on the entangled system, the maximally entangled states in the mixed states would be changed into partially entangled states (the detailed information can be found in Sec. 4).

In this paper, we propose an imperfect-interaction-free EPP for QD systems by constructing a faithful parity check in the nearly realistic condition. The faithful parity check has nondestruction on the entangled state and makes correct judgement on the parity mode of an entangled state in the nearly realistic condition. With the faithful parity check, the imperfect-interaction-free EPP can guarantee the desired fidelity of the mixed states and keep the maximally entangled states from being changed into partially entangled ones. Meanwhile, this imperfect-interaction-free EPP relaxes the requirement of experiment techniques as it can be implemented with nonidentical QD-microcavity systems in a nearly realistic condition. Furthermore, this imperfect-interaction-free EPP can be repeated until success, as the faithful parity check is a heralded one, whose failure can be alarmed by the single-photon detector. These distinctive features make this imperfect-interaction-free EPP for a quantum stationary system more useful in a quantum repeater.

2. Faithful parity check for imperfect-interaction-free EPP

Let us consider the interaction between a QD-microcavity system and a circularly polarized photon first [71]. A singly charged QD (e.g. a self-assembled In(Ga)As QD or a GaAs interfacial QD) inside a double-sided microcavity is shown in Fig. 1(a). Both the bottom and the top distributed Bragg reflectors of the double-sided microcavity are partially reflective with the same reflection coefficients. Meanwhile the microcavity supports both two circularly polarized modes [72–75]. The circularly polarized photon $|R^{\uparrow }\rangle$ or $|L^{\downarrow }\rangle$ with $S_{z}=+1$ is coupled to the dipole transition from the state $|\uparrow \rangle$ to the exciton ($X^{-}$) state $|\uparrow \downarrow \Uparrow \rangle$; The circularly polarized photon $|R^{\downarrow }\rangle$ or $|L^{\uparrow }\rangle$ with $S_{z}=-1$ is coupled to the dipole transition from the state $|\downarrow \rangle$ to the exciton ($X^{-}$) state $|\downarrow \uparrow \Downarrow \rangle$, as shown in Fig. 1(b). Here, $|\uparrow \rangle$ and $|\downarrow \rangle$ denote the states of the excess electron spin in a QD with $J_{z}=\pm \frac {1}{2}$, respectively. $|\uparrow \downarrow \Uparrow \rangle$ and $|\downarrow \uparrow \Downarrow \rangle$ denote the states of the exciton consisting of two electrons bound to one heavy hole with $J_{z}=\pm \frac {3}{2}$, respectively.

 

Fig. 1. (a) Schematic diagram for an electron-spin of a QD inside a double-sided optical microcavity. (b) Schematic description of the relevant exciton energy levels and the spin selection rules for dipole transitions of negatively charged excitons. $|R^{\uparrow }\rangle$($|R^{\downarrow }\rangle$) and $|L^{\uparrow }\rangle $($|L^{\downarrow }\rangle $) represent the right-circularly polarized photon and the left-circularly polarized photon propagating along (against) the normal direction of the cavity Z axis (quantization axis), respectively.

Download Full Size | PDF

The dynamic process of the interaction between a QD-microcavity system and a circularly polarized photon can be represented by Heisenberg equations of motion for the cavity field operator $\hat {a}$ and the exciton $X^{-}$ dipole operator $\hat {\sigma }_{-}$ in the interaction picture [76],

 

Fig. 2. (a) Schematic diagram for the circuit unit. BS is a 50:50 beam splitter, which performs a spatial-mode Hadamard operation [$|i\rangle \rightarrow \frac {1}{\sqrt {2}}(|j_{1}\rangle +|j_{2}\rangle ), |i_{1}\rangle \rightarrow \frac {1}{\sqrt {2}}(|j_{1'}\rangle +|j_{2'}\rangle ), |i_{2}\rangle \rightarrow \frac {1}{\sqrt {2}}(|j_{1'}\rangle -|j_{2'}\rangle )$] on a photon. H$_{pi(i')(i=1,2)}$ is a half-wave plate, which performs a polarization Hadamard operation [$|R\rangle \rightarrow \frac {1}{\sqrt {2}}(|R\rangle +|L\rangle )$,$|L\rangle \rightarrow \frac {1}{\sqrt {2}}(|R\rangle -|L\rangle )$] on a photon. M$_{i(i=1,2)}$ is a mirror, and PD is a single-photon detector. (b) Schematic diagram for quantum circuit of the faithful parity check on the electron spins in QDs, say QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ as an example. UBS is an unbalanced beam splitter with the transmission coefficient $\cos \theta =1/\sqrt {1+(T_{1}T_{2})^{2}}$ and the reflection coefficient $\sin \theta =T_{1}T_{2}/\sqrt {1+(T_{1}T_{2})^{2}}$. BS is a 50:50 beam splitter. U$_{\mathrm {A(C)}}$ represents the circuit unit in which the QD$_{\mathrm {A(C)}}$-microcavity system is embedded. X$_{i(i=1,2)}$ is a half-wave plate which performs a polarization bit-flip operation $\sigma _{x}^{p}=|R\rangle \langle L|+|L\rangle \langle R|$ on the photon. PD$_{i(i=a,b)}$ is a single-photon detector. DL is a time-delay device.

Download Full Size | PDF

Suppose an electron spin in a QD embedded in a double-sided microcavity is initialized in the state $|\varphi \rangle _{s}=\alpha |\uparrow \rangle +\beta |\downarrow \rangle$ with $|\alpha |^{2}+|\beta |^{2}=1$. A photon in the right-circularly polarized state $|R\rangle$ is injected into the circuit unit from the input path $i$. After the photon passes through the circuit unit, the state of the system consisting of the photon and the electron spin in the QD is changed from the state $|\Phi \rangle _{0}=|R\rangle (\alpha |\uparrow \rangle +\beta |\downarrow \rangle )$ to the state

To faithfully check the parity mode of the entangled state of two electron spins in two QDs, such as QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$, one injects a right-circularly polarized photon from the left-input port $in$ into the quantum circuit of the faithful parity check, as shown in Fig. 2(b). In the process of the parity check, the photon will interact with the first circuit unit U$_{\mathrm {A}}$ consisting of QD$_{\mathrm {A}}$-microcavity system and the second circuit unit U$_{\mathrm {C}}$ consisting of QD$_{\mathrm {C}}$-microcavity system, where the two circuit units can be nonidentical and the construction of each one is the same as the one shown in Fig. 2(a). If the initial state of the electron spins in QDs is $|\phi ^{\pm }\rangle =\frac {1}{\sqrt {2}}(|\uparrow \uparrow \rangle \pm |\downarrow \downarrow \rangle )$, after the photon passes through the unbalanced beam splitter UBS, the quantum state of the whole system consisting of the photon and the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ is changed from $|\Psi \rangle _{0}$ to $|\Psi \rangle _{1}$. Here,

In the process of parity check, the single-photon detector PD$_{1}$ clicks with a probability of $\frac {D_{1}^{2}}{1+(T_{1}T_{2})^{2}}$, and the single-photon detector PD$_{2}$ clicks with a probability of $\frac {T_{1}^{2}D_{2}^{2}}{1+(T_{1}T_{2})^{2}}$. The click of PD$_{1}$ or PD$_{2}$ marks the failure of this round of parity check, and the parity mode of entangled state of the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ can be checked through injecting another photon into the quantum circuit directly (or after an operation $\sigma _{z}$ on the electron spin in QD$_{\mathrm {C}}$ in the second circuit unit). That is, the failure of the parity check can be alarmed by the click of PD$_{1}$ or PD$_{2}$, and the parity check can be repeated until success.

With the faithful parity check as shown in Fig. 2(b), the even-parity states $|\phi ^{\pm }\rangle$ can be distinguished from the odd-parity states $|\psi ^{\pm }\rangle$ by the click of single-photon detector PD$_a$ or PD$_b$. From the final states in the nearly realistic condition, as shown in Eq. (11) and Eq. (12), one can see that the maximally entangled states $|\phi ^{\pm }\rangle$ and $|\psi ^{\pm }\rangle$ are not changed (such as being destructed into partially entangled states). Meanwhile, one can see that in the nearly realistic condition, there is no wrong judgement being made, as the click of the single-photon detector PD$_b$ (PD$_a$) does’t happen when the pair of entangled electron spins in QDs is in the even-parity states $|\phi ^{\pm }\rangle$ (odd-parity states $|\psi ^{\pm }\rangle$). The only difference between the final states $|\Psi \rangle _{f1}$ $(|\Psi \rangle _{f2})$ in the nearly realistic condition and the ideal final states $|\phi ^{\pm }\rangle |R\rangle _{a}$ $(|\psi ^{\pm }\rangle |R\rangle _{b})$ in the specifically ideal condition is an integrate coefficient $\sqrt {\frac {2(T_{1}T_{2})^{2}}{1+(T_{1}T_{2})^{2}}}$, which only affects the efficiency of the parity check.

3. Imperfect-interaction-free EPP

The four Bell states of a pair of entangled electron spins in two QDs are $|\phi ^{\pm }\rangle =\frac {1}{\sqrt {2}}|\uparrow \uparrow \rangle \pm |\downarrow \downarrow \rangle$ and $|\psi ^{\pm }\rangle =\frac {1}{\sqrt {2}}|\uparrow \downarrow \rangle \pm |\downarrow \uparrow \rangle$. Affected by the environmental noise, bit-flip errors or phase-flip errors may happen on the entangled systems, which change the maximally entangled state into the mixed state. The phase-flip error can be transformed into the bit-flip error through local Hadamard operations. Here, we discuss imperfect-interaction-free entanglement purification on the QD system in a mixed state with a bit-flip error. To purify the entangled QD system, say QD$_{\mathrm {A}}$ and QD$_{\mathrm {B}}$, another entangled QD system, say QD$_{\mathrm {C}}$ and QD$_{\mathrm {D}}$, is needed. The two pairs of entangled electron spins in the QDs are in the same mixed state with the bit-flip error, which can be expressed as

The schematic diagram for our imperfect-interaction-free EPP for QD system is shown in Fig. 3. The first step of our imperfect-interaction-free EPP is to distinguish the states $|\phi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$ and $|\psi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$ from the states $|\phi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$ and $|\psi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$. This step can be completed through faithful parity check on the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ by Alice and on the electron spins in QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$ by Bob in the nearly realistic condition (the schematic diagram for the faithful parity check is shown in Fig. 2(b)). After Alice performs the faithful parity check on the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ and Bob performs the same operation on the electron spins in QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$, the pure states of the identical two pairs of electron spins in the QDs are classified into two cases based on the results of the parity check.

 

Fig. 3. Schematic diagram for imperfect-interaction-free EPP for QD system. Faithful parity check is the first step of the EPP, which is performed on the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ (QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$) by Alice (Bob). M represents measurement performed on the electron spins in QD$_{\mathrm {C}}$ and QD$_{\mathrm {D}}$ in the orthogonal basis $\{|+\rangle ,|-\rangle \}$, which is the second step of the EPP.

Download Full Size | PDF

In case (1), the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ and the electron spins in QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$ are in different parity-mode states, which corresponds to the states $|\phi ^{+}\rangle _{AB}|\psi ^{+}\rangle _{CD}$ and $|\psi ^{+}\rangle _{AB}|\phi ^{+}\rangle _{CD}$. In this case, Alice and Bob cannot identify which pair of entangled electron spins in QDs has the bit-flip error. Therefore, they discard these two pairs of electron spins in QDs.

In case (2), the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ and the electron spins in QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$ are in the same parity-mode state, which corresponds to the states $|\phi ^{+}\rangle _{AB}|\phi ^{+}\rangle _{CD}$ and $|\psi ^{+}\rangle _{AB}|\psi ^{+}\rangle _{CD}$. In detail, when the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ and the electron spins in QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$ are both in even-parity states, i.e. the single-photon detector PD$_a$ at Alice’s side and the single-photon detector PD$_a$ at Bob’s side would click respectively in the parity check process, the four electron spins in the four QDs are projected into a mixed state composed of states $|\Omega \rangle _{1}$ and $|\Omega \rangle _{2}$. Here,

Thus, the first round of our imperfect-interaction-free EPP is completed, and the state of the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {B}}$ is transformed into

4. Discussion

The interaction between a QD-microcavity system and a circularly polarized photon is a useful platform in quantum information processing. When the realistic experiment parameters deviate from the specifically ideal ones, the perfect interaction between an artificial atom inside a microcavity (such as QD-microcavity system) and a circularly polarized photon turns into the imperfect one, which results in the wrong judgement on the parity mode and the destruction on the measured entangled state of a parity check in the realistic condition. Let’s take the parity check in the entanglement purification protocol using the perfect interaction in the specifically ideal condition [80] as an example to illustrate the bad effect on the parity check in the realistic condition. In the specifically ideal condition, after the parity check, the states $\frac {1}{\sqrt {2}}(|00\rangle \pm |11\rangle )|\varphi ^{+}\rangle$ and $\frac {1}{\sqrt {2}}(|01\rangle \pm |10\rangle )|\varphi ^{+}\rangle$ of the hybrid system consisting of the measured system and the auxiliary system would be respectively changed into $\frac {1}{\sqrt {2}}(|00\rangle \pm |11\rangle )|\varphi ^{+}\rangle$ and $\frac {1}{\sqrt {2}}(|01\rangle \pm |10\rangle )|\varphi ^{-}\rangle$, which means the final state of the auxiliary system ($|\varphi ^{+}\rangle$ or $|\varphi ^{-}\rangle$) would reflect the parity mode of the measured system (even or odd), and meanwhile the states $\frac {1}{\sqrt {2}}(|00\rangle \pm |11\rangle )$ and $\frac {1}{\sqrt {2}}(|01\rangle \pm |10\rangle )$ of the measured system would not be changed. However, in the realistic condition, the states $\frac {1}{\sqrt {2}}(|00\rangle \pm |11\rangle )|\varphi ^{+}\rangle$ would be changed into $\frac {1}{\sqrt {2}}(|00\rangle \pm \xi |11\rangle )|\varphi ^{+}\rangle \pm \zeta |11\rangle |\varphi ^{-}\rangle$ after the parity check. Obviously, the state of the auxiliary system remains unchanged for correct judgement, and the maximally entangled states $\frac {1}{\sqrt {2}}(|00\rangle \pm |11\rangle )$ of the measured system are destructed into partially entangled states $\frac {1}{\sqrt {2}}(|00\rangle \pm \xi |11\rangle )$. While the state of the auxiliary system is changed for wrong judgement, and the state of the auxiliary system is projected into $\pm \zeta |11\rangle$. For the state $\frac {1}{\sqrt {2}}(|01\rangle \pm |10\rangle )|\varphi ^{+}\rangle$, after the parity check in the realistic condition, it would be changed into $\frac {\epsilon }{\sqrt {2}}(|01\rangle \pm |10\rangle )|\varphi ^{+}\rangle +\frac {\varepsilon }{\sqrt {2}}(|01\rangle \pm |10\rangle )|\varphi ^{-}\rangle$, which means there would be a wrong judgement on the parity mode when the state of the measured system remains unchanged with the possibility of $|\epsilon |^{2}$. Here, $\xi =\frac {r_{0}^{2}+r^{2}}{2}$, $\zeta =\frac {r_{0}^{2}-r^{2}}{2}$, $\epsilon =-\frac {r_{0}+r}{2}$, and $\varepsilon =\frac {r-r_{0}}{2}$ are the functions of the reflection coefficients ($r_{0}$ and $r$) of the circularly polarized photon after interacting with an artificial atom inside a microcavity and they vary with the realistic experiment parameters such as the coupling strength, decay rate, and so on. Therefore, the imperfect interaction due to the realistic experiment parameters would lead a parity check to an unfaithful one.

The unfaithful parity check with the wrong judgement would make the fidelity of the final state lower than the desired one in the entanglement purification process. For example, the parity check makes correct judgements on the parity mode of the states $|\phi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$, $|\psi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$, and $|\phi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$, but it makes a wrong judgement on the parity mode of the state $|\psi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$. That is, Alice and Bob may misdeem that the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ and electron spins in QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$ are in the same parity-mode states when they are in the state $|\psi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$. Then they will reserve the electron spins in QD$_{\mathrm {A}}$, QD$_{\mathrm {B}}$, QD$_{\mathrm {C}}$, and QD$_{\mathrm {D}}$ when they are in the states $|\phi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$, $|\psi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$, and $|\psi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$, but only discard the electron spins in QDs when they are in the state $|\phi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$. Thus, after the second step, the first round of the entanglement purification is accomplished and the fidelity of the final state is $F''=\frac {F^{2}+0.5F(1-F)}{F^{2}+(1-F)^{2}+F(1-F)}$, which is lower than the desired fidelity $F'$ in the condition of $1/2<F<1$. We have to admit that there exists a case, in which the parity check makes correct judgements on the parity mode of the states $|\phi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$, $|\phi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$, and $|\psi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$, but it makes a wrong judgement on the parity mode of the state $|\psi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$. That is, the electron spins in QD$_{\mathrm {A}}$, QD$_{\mathrm {B}}$, QD$_{\mathrm {C}}$, and QD$_{\mathrm {D}}$ are only reserved when they are in the state $|\phi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$. Thus, after the second step, the first round of the entanglement purification is accomplished and the fidelity of the final state is $1$. However, this case happens occasionally. More often, the wrong judgement on the parity mode of the state would lead to a lower fidelity of the final state than the desired one. Therefore, making a correct judgements on the parity modes of pure entangled states in the mixed state is of great importance.

Meanwhile, the unfaithful parity check with destruction on the entangled states would also lead the maximally entangled state to the partially entangled one in entanglement purification process. Let us take case (2) of Sec. 3 (i.e. the electron spins in QD$_{\mathrm {A}}$ and QD$_{\mathrm {C}}$ and the electron spins in QD$_{\mathrm {B}}$ and QD$_{\mathrm {D}}$ are both in the even-parity states) as an example to analyze the importance of the nondestruction of a parity check. When the parity check has a destruction on the entangled states in the realistic condition, after the parity check, one of the pure state $|\Omega \rangle _{1}$ in the mixed state of the four electron spins in QD$_A$, QD$_B$, QD$_C$, and QD$_D$ would be changed into $|\Omega \rangle '_{1}=(\mu |\uparrow \uparrow \uparrow \uparrow \rangle +\nu |\downarrow \downarrow \downarrow \downarrow \rangle )_{ABCD}$, where $\mu \neq \nu$. Here, $\mu$ and $\nu$ are quantities varying with the experiment parameters of the QD-microcavity system and the circularly polarized photon in the realistic condition. Thus, Alice and Bob obtain the states

Fortunately, our imperfect-interaction-free EPP can avoid being affected by the imperfect interaction in the nearly realistic condition, as the faithful parity check can make correct judgement on the parity mode of the measured entangled state without destruction. The faithful parity check can correctly identify the parity modes of the pure entangled states in the mixed state of the electron spins in QDs, i.e. correctly distinguish the states $|\phi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$ and $|\psi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$ from the states $|\phi ^{+}\rangle _{AB}\otimes |\psi ^{+}\rangle _{CD}$ and $|\psi ^{+}\rangle _{AB}\otimes |\phi ^{+}\rangle _{CD}$. Meanwhile, the faithful parity check has nondestruction on the measured entangled states in entanglement purification. Therefore, with our imperfect-interaction-free EPP, the desired fidelity of mixed state can be achieved even in the nearly realistic condition, which can be further improved by iterating the purification process, and maximally entangled states $|\phi ^{\pm }\rangle _{AB}$ and $|\psi ^{\pm }\rangle _{AB}$ would not be changed into partially entangled states, which means the entanglement concentration is not required after purification.

Our imperfect-interaction-free EPP relaxes the strict requirement for two absolutely identical QD-microcavity systems and the high requirement for the QD-microcavity system. Embedding a QD in a microcavity can strengthen the interaction between the circularly polarized photon and the electron spin in a QD, while manufacturing two QD-microcavity systems with absolutely identical experimental parameters is a strict requirement in experiment. In our faithful parity check, the transmission coefficients of the two circuit units $T_{1}$ and $T_{2}$, varying with the experiment parameters of two QD-microcavity systems, can be different, which means that the faithful parity check could be constructed with nonidentical QD-microcavity systems without affecting the features of nondestruction and correct judgement on the parity mode in the nearly realistic condition. Therefore, the experimental realization of our imperfect-interaction-free EPP doesn’t need two absolutely identical QD-microcavity systems. Meanwhile, from Eq. (11) and Eq. (12), one can see that the experiment parameters of the QD-microcavity systems would not affect the features of nondestruction and correct judgement of the faithful parity check in the nearly realistic condition, but only affect the integrate coefficient of the final state after parity check, which determines the efficiency of the parity check. From Eq. (11) and Eq. (12), the efficiency of one round of parity check can be obtained as $\eta =\frac {2(T_{1}T_{2})^{2}}{1+(T_{1}T_{2})^{2}}$. For simplicity but with no lack of rigor, we plot the efficiency $\eta$ as a function of $g/(\kappa +\kappa _{s})$ and $\kappa _{s}/\kappa$ with $\omega _{X^{-}}=\omega _{C}=\omega$ and $\gamma /\kappa =0.1$ for both two QD-microcavity systems in the two circuit units, as shown in Fig. 4. When $g/(\kappa +\kappa _{s})>1$, the efficiency $\eta >90.27\%$ and $\eta >74.05\%$ can be obtained with the cases $\kappa _{s}/\kappa =0$ and $\kappa _{s}/\kappa =0.2$ respectively. When $g/(\kappa +\kappa _{s})$ is improved to $2$, the efficiency $\eta =93.92\%$ and $\eta =76.70\%$ can be obtained with the cases $\kappa _{s}/\kappa =0$ and $\kappa _{s}/\kappa =0.2$ respectively. One can see that, utilizing the imperfect interaction in the nearly realistic condition, a faithful parity check is constructed with nearly unity fidelity and the efficiency of it is still acceptable compared with the previous works, the reason of which is that we only transfer the imperfect interaction into a heralded failure by the single photon detectors in the circuit units but not decrease the effective interaction which can be used to assist in completing the parity check.

 

Fig. 4. The efficiency of the faithful parity check vs $g/(\kappa +\kappa _{s})$ and $\kappa _{s}/\kappa$ with $\omega _{X^{-}}=\omega _{C}=\omega$ and $\gamma /\kappa =0.1$.

Download Full Size | PDF

The interaction between the QD-microcavity system and the circularly polarized photon is utilized to construct the faithful parity check in our imperfect-interaction-free EPP. The experiment parameters of the QD-microcavity system and the single photon would only influence the interaction between the QD-microcavity system and the circularly polarized photon, but not influence the features of nondestruction and correct judgement of our faithful parity check in the nearly realistic condition. In a more generally realistic condition, the interaction between the QD-microcavity system and the circularly polarized photon could be decreased by a factor of $[1-exp(-\tau /\Gamma )]$ due to the dephasing of the electron spin in the QD. However, this influence on the features of nondestruction and correct judgement of our faithful parity check can be neglected as an electron spin coherence time $\Gamma \simeq 2.6\mu s$ [81] has been experimentally achieved and the cavity photon time $\tau =4.5ns$ [82] is available. Imperfect optical selection induced by heavy hole mixing would also affect the features of nondestruction and correct judgement of our faithful parity check a little. Fortunately, this imperfect mixing can be suppressed by engineering the shape and the size of QDs or by choosing the appropriate types of QDs [83]. Overall, even taking the realistic condition into account, the faithful parity check can also remain the features of nondestruction and correct judgement.

Our imperfect-interaction-free EPP works in a repeat-until-success pattern. In our EPP, an auxiliary circularly polarized photon is used to assist in accomplishing our faithful parity check. According to the click of the single-photon detector PD$_{a}$ or PD$_{b}$ at the out-put port of the circuit of the faithful parity check, the parity mode of the entangled state can be known. Meanwhile, due to the experiment parameters of the QD-microcavity system and the circularly polarized photon in the realistic condition would deviate from the ones in the specifically ideal condition, there is a possibility that the parity check fails. Fortunately, the failure of the parity check can be alarmed by the click of the single-photon detector in the first circuit unit or the one in the second circuit unit. In this case, the parity mode of the entangled QD systems can be checked through injecting another circularly polarized photon. Therefore, the faithful parity check is a heralded one and the imperfect-interaction-free EPP can be repeated until success.

5. Summary

In conclusion, we propose an imperfect-interaction-free EPP for QD system based on faithful parity check, which can make a correct judgement on the parity mode and have no destruction on the entangled state in the nearly realistic condition. Our imperfect-interaction-free EPP can guarantee the fidelity of the mixed state to a desired one and keep the maximally entangled states in the mixed states from being destructed into partially entangled ones even in the nearly realistic condition. Meanwhile, our imperfect-interaction-free EPP relaxes the strict requirement for two absolutely identical QD-microcavity systems and the high requirement for the QD-microcavity system. In addition, our imperfect-interaction-free EPP can be repeated until success with the heralded function of faithful parity check, where the failure can be alarmed by the single-photon detector. These distinctive features make this imperfect-interaction-free EPP for quantum stationary systems have more practical applications in solid quantum repeaters for a large-scale quantum network.