Error-detected generation and complete analysis of hyperentangled Bell states for photons assisted by quantum-dot spins in double-sided optical microcavities

We construct an error-detected block, assisted by the quantum-dot spins in doublesided optical microcavities. With this block, we propose three error-detected schemes for the deterministic generation, the complete analysis, and the complete nondestructive analysis of hyperentangled Bell states in both the polarization and spatial-mode degrees of freedom of twophoton systems. In these schemes, the errors can be detected, which can improve their fidelities largely, far different from other previous schemes assisted by the interaction between the photon and the QD-cavity system. Our scheme for the deterministic generation of hyperentangled twophoton systems can be performed by repeat until success. 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Introduction
Quantum entanglement plays a critical role in quantum information processing [1] and it is a key quantum resource in quantum communication, such as quantum teleportation [2], quantum dense coding [3,4], quantum key distribution [5], quantum secret sharing [6], quantum secure direct communication [7,8], and so on. The complete and deterministic analysis of Bell states is required to achieve some important tasks in quantum communication. In 1999, two schemes of Bell-state analysis (BSA) [9,10] for teleportation with only linear optical elements were proposed. However, it is impossible to deterministically and completely distinguish the four Bell states in polarization with only linear optical elements [11][12][13]. In 2005, Barrett et al. [14] proposed an analyzer for distinguishing all four polarization Bell states using weak nonlinearities. In 2015, Sheng et al. [15]  [29] demonstrated experimentally the generation of a twophoton six-qubit hyperentangled state in three DOFs. A hyperentangled photon pair, which is hyperentangled in both the polarization and spatial-mode DOFs with 16 orthogonal Bell states, can be produced by spontaneous parametric down-conversion source with the β barium borate crystal. However the quantum efficiency of this method is low and the the multiphoton generation probability is high, which will limit the application of hyperentanglement in quantum information processing. Moreover, one cannot distinguish the 16 Bell states completely with only linear optics. In 2010, Sheng et al. [30] proposed the first scheme for the complete hyperentangled BSA (HBSA) of all the two-photon polarization-spatial hyperentangled states with cross-Kerr nonlinearity. In 2016, Li et al.
[31] presented a simplified complete HBSA with cross-Kerr nonlinearity. As the solid state system can provide giant nonlinearity, it is viewed as a promising candidate for quantum information processing and quantum computing. In 2012, Ren et al.
[32] proposed a scheme for complete HBSA assisted by quantum-dot spins in a oneside optical microcavity. In the same year, Wang et al. [33] proposed two interesting schemes for hyperentangled-Bell-state generation (HBSG) and HBSA by quantum-dot spins in doublesided optical microcavities. In 2015, Liu et al. [34] proposed two schemes for HBSG and HBSA assisted by nitrogen-vacancy centers in resonators.
The electron spin in a GaAs-based or InAs-based charged quantum dot (QD) is an attractive candidate for solid-state quantum information processing. The electron-spin coherence time of a charged QD can be maintained for more than 3μs [35, 36] and the electron spin-relaxation time can be longer (∼ ms) [37,38]. Moreover, it is comparatively easy to embed the QDs in the solid-state cavities and it can be easily manipulated fast and initialized [39][40][41]. Based on a singly charged QD inside an optical resonant cavity, many interesting schemes for quantum information processing have been proposed [32, 33, 42-47]. In the ideal condition, the fidelities and the efficiencies of these schemes can be 100%. In realistic condition, their fidelities and the efficiencies would be affected by the parameters of the QD-cavity systems more or less. In 2011, Kastorynao et al. [48] proposed a scheme for the preparation of a maximally entangled state utilizing the decay of the cavity to improve the fidelity, which can herald the error. In 2012, Li et al. [49] proposed a robust-fidelity scheme for atom-photon entangling gates, which is adequate for high-fidelity maximally entangling gates even in the weak-coupling regime based on the scattering of photons off single emitters in one-dimensional waveguides.
In this paper, we construct an error-detected block assisted by the QD spins in doublesided optical microcavities. With this block, we propose three schemes for the deterministic HBSG, the complete HBSA, and the complete nondestructive HBSA of hyperentangled Bell states in both the polarization and spatial-mode DOFs of two-photon systems. As the errors can be detected in these schemes, they possess the advantage of having high fidelity, far different from other previous schemes by others, especially in our scheme for the deterministic generation of hyperentangled two-photon systems as it can be performed by repeat until success. Moreover, our schemes work in both the weak coupling regime and the strong coupling regime. These features make our schemes more useful in high-capacity quantum communication with hyperentanglement in the future.

Interaction between a circularly polarized light and a single charged QD in a double-sided microcavity
A singly charged electron In(Ga)As QD or a GaAs interface QD is embedded in an optical resonant double-sided microcavity with two mirrors partially reflective in the top and the bottom, as shown in Fig. 1(a). The optical excitation of a photon and an excess electron injected into the  QD can create an exciton X − consisting of two electrons bound to one heavy hole with negative charges, as shown in Fig. 1(b). If the excess electron in the QD is in the spin state | ↑ , only the photon in the state |R ↑ or |L ↓ can be resonantly absorbed to create the exciton state | ↑↓⇑ ; otherwise, if the excess electron in the QD is in the spin state | ↓ , only the photon in the state |L ↑ or |R ↓ can be resonantly absorbed to the exciton state | ↓↑⇓ . Here | ⇑ and | ⇓ represent the heavy-hole spin state | + 3 2 and | − 3 2 , respectively. The dipole interaction process can be represented by Heisenberg equations for the cavity field operator a and the exciton where ω, ω c , and ω X − are the frequencies of the photon, the cavity, and X − transition, respectively. g is the coupling constant between the X − and the cavity. κ and κ s represent the cavity decay rate and the leaky rate, respectively. γ is the exciton dipole decay rate. H and G are the noise operators related to reservoirs. a in , a in , a r , and a t are the input and output field operators. In the weak excitation approximation, the reflection coefficient r (ω) and the transmission coefficient t(ω) of the QD-cavity system can be described by [42, 43, 51] When the QD is uncoupled to the cavity (cold cavity), that is g = 0, the reflection r 0 (ω) and the transmission t 0 (ω) coefficients can be written as ( If ω = ω c = ω X − , the reflection and transmission coefficients of the coupled cavity and the uncoupled cavity can be simplified as (4) The rules for optical transitions in a realistic QD-cavity system can be described as [43], 3. An error-detected block for the interaction between the circularly-polarized photon and the QD-cavity system The schematic diagram of our error-detected block for the interaction between the circularlypolarized photon and the QD-cavity system is shown in Fig. 2, which is constructed with a 50 : 50 beam splitter (BS), two half-wave plates (H pi ), two mirrors (M i ), a single-photon detector (D), and a QD-cavity system. The QD in the cavity is prepared in the state |ϕ + = 1 . If the photon is in the right-circularly polarized state |R , one injects it into our error-detected block from the path i 1 and lets it pass through BS and H pi (i = 1, 2). After being reflected by the mirror M i (i = 1, 2), the state of the whole system composed of the photon and the QD in the cavity is changed from the state |Φ 0 = |R |i 1 ⊗ 1 After the interaction between the photon and the QD-cavity system, the state becomes Subsequently, the photon is reflected by M i (i = 1, 2) and passes through H pi (i = 1, 2) again, and the state evolves to At last, the photon passes through BS again and the final state can be described as Here D = 1 2 (t + r + t 0 + r 0 ) and T = 1 2 (t + r − t 0 − r 0 ) are the reflection coefficient and the transmission coefficient of the error-detected block, respectively. Similarly, if the QD in the cavity is prepared in the state |ϕ − = 1 √ 2 (| ↑ −| ↓ ), the outcome of the process can be described as One can see that there are two parts of the outcome. If the photon is reflected from the error-detected block with probability of |D| 2 , the polarization of the photon and the state of the QD would not change. The reflected photon would be detected by the detector and the click of the detector represents the case in which the corresponding task of the error-detected block runs fail. If the photon is transmitted from the error-detected block with probability of |T | 2 , the polarization of the photon is flipped and the superposition state of the QD is changed from |ϕ + to |ϕ − or from |ϕ − to |ϕ + . We can utilize the transmitted photon and the QD-cavity to accomplish the tasks of deterministic HBSG, complete HBSA, and complete nondestructive HBSA with a high fidelity. |D| and |T | are affected by the g/κ and κ s /κ as shown in Fig. 3. From Fig. 3, one can see that there exists a zero value of reflection coefficient, which results from the destructive interference. For the condition g 2 /κγ 1 and κ s /2κ 1, the final state |Φ f 1 = |L |i 2 |ϕ − and |Φ f 2 = |L |i 2 |ϕ + of the error-detected block is obtained when the QD is prepared in the state |ϕ + and |ϕ − , respectively.

Deterministic photonic hyperentanglement generation
A two-photon four-qubit hyperentangled Bell state can be described as Here the subscripts P and S denote the polarization and the spatial-mode DOFs, respectively. The subscripts A and B represent the two photons in the hyperentangled state, respectively. |R and |L represent the right-circular and the left-circular polarizations of photons, respectively. |a 1 (|b 1 ) and |a 2 (|b 2 ) are the different spatial modes for photon A (B). The four Bell states in the polarization DOF can be expressed as The four Bell states in the spatial-mode DOF can be written as The principle of our scheme for the two-photon polarization-spatial hyperentangled-Bellstate generation (HBSG) constructed with two error-detected blocks and some linear optical elements is shown in Fig. 4. The QDs are prepared in the initial states |ϕ + 1 = |ϕ + 2 and the two photons A and B with the same frequency are prepared in the same initial state The process for HBSG can be described in detail as follows. First, one injects photon A into the quantum circuit from the left input, followed by photon B. The time interval Δt exists between photons A and B, and Δt should be less than the spin coherence time Γ. CPBS 1 will transmit the photon to path a 1 (b 1 ) and reflects the photon to path a 2 (b 2 ) when photon A (B) is in the state |L and |R , respectively. When photon A (B) emerges in path a 2 (b 2 ), it passes through the error-detected block consisting of the QD-cavity 1 system and X 1 . Before the two wave packets split by CPBS 1 interfere at BS 2 , the state of the whole system is changed from Ω 0 = |ϕ A |ϕ B |ϕ + 1 |ϕ + 2 to Ω 1 . Here Ω 1 is written as Then BS 2 , which performs the spatial-mode Hadamard operation |a 1 (b 1 ) → 1 √ 2 (|c 1 (|d 1 ) + |c 2 (d 2 ) , |a 2 (b 2 ) → 1 √ 2 (|c 1 (|d 1 ) − |c 2 (d 2 ) on photons transmitted, will lead photon A (B) to paths c 1 (d 1 ) and c 2 (d 2 ). In path c 2 (d 2 ), photon A (B) passes through CPBS 2 which transmits the photon A (B) in |L and reflects the photon in |R , respectively. Then photon A (B) in |L takes a σ p x operation by X 2 and passes through the error-detected block consisting of QDcavity 2 system, sequently. Photon A (B) in |R passes through the same error-detected block and takes a σ p x operation by X 3 , sequentially. At last, the two wave packets union at CPBS 3 in path a 2 (b 2 ). The state of the whole system becomes From Eq. (15), one can see the relationship between the measurement outcomes of the two QD-cavity systems and the polarization-spatial hyperentangled Bell states of the two-photon system. If QD 1 and QD 2 are in the states |ϕ + 1 and |ϕ + 2 , respectively, the two-photon system is in the hyperentangled Bell state |φ + P |φ + S . If QD 1 and QD 2 are in the states |ϕ + 1 and |ϕ − 2 , respectively, the two-photon system is in the hyperentangled Bell state |φ − P |ψ + S . When QD 1 and QD 2 are in the states |ϕ − 1 and |ϕ + 2 , respectively, the two-photon system is in the hyperentangled Bell state |ψ + P |φ − S . When QD 1 and QD 2 are in the states |ϕ − 1 and |ϕ − 2 , respectively, the two-photon system is in the hyperentangled Bell state |ψ − P |ψ − S . Therefore, one can generate deterministically a polarization-spatial hyperentangled Bell state of the two-photon system by measuring the states of the two QDs. Other polarization-spatial hyperentangled Bell states can be generated with some appropriate single-qubit operations.

Complete polarization-spatial hyperentangled Bell states analysis
The schematic diagram of the complete polarization-spatial HBSA for hyperentangled twophoton systems is shown in Fig. 5. Here we use two error-detected blocks consisting of two QDcavity systems and some linear optical elements to achieve the complete polarization-spatial HBSA. The two QDs are prepared in the initial states |ϕ + 1 = |ϕ + 2 and the hyperentangled two-photon system is in one of the 16 hyperentangled Bell states. The process for our HBSA can be described as follows.
One lets photon A pass through the quantum circuit from the left input port, followed by photon B. And the two photons will be detected sequentially at the output port. The interval time Δt existing between the two photons is less than the spin coherence time Γ. According to the results of the interaction between the circularly polarized photon and the error-detected block, after the two photons A and B pass through quantum circuit and before they are detected by the detectors (D Ri and D Li ), the whole system composed of the two photons and the two QDs evolves as At last, the photons A and B are independently measured in both the polarization and the spatial-mode DOFs with single-photon detectors, and the two QDs are measured in the basis {|ϕ + , |ϕ − }. The relationship between the measurement outcomes and the initial hyperentangled states of the two-photon system AB is shown in Table 1.
From Table 1, one can obtain the complete and deterministic analysis on hyperentangled Bell states of a two-photon system AB. The measurement outcomes of QD 1 and QD 2 reveal the phase information in the polarization and the spatial-mode DOFs, respectively. In detail, when QD 1 (QD 2 ) is in the spin state |ϕ + 1 (|ϕ + 2 ), the two-photon system is in the states |φ + P(S) or |ψ + P(S) in the polarization (spatial-mode) DOF. Otherwise, when QD 1 (QD 2 ) is in the spin state |ϕ − 1 (|ϕ − 2 ), the two-photon system is in the states |φ − P(S) or |ψ − P(S) in the polarization (spatial-mode) DOF. The measurement outcomes of the states of the two-photon system in the polarization DOF reveal the parity information in the polarization DOF. In detail, if the measurement outcome of the state is RR or LL, the two-photon system is in the state |φ ± P in the polarization DOF; otherwise, it is in the state |ψ ± P . The measurement outcomes of the states of the two-photon system in the spatial-mode DOF reveal the parity information in the spatial-mode DOF. In detail, if the measurement outcome of the state is a 1 b 1 or a 2 b 2 , the two-photon system is in the state |φ ± S ; otherwise, it is in the state |ψ ± S . That is, the schematic diagram shown in Fig. 5 can be used for the complete and deterministic HBSA. Table 1. The relationship between the measurement outcomes of the states of the twophoton systems in the polarization and spatial-mode DOFs and the two QDs and the initial hyperentangled states of the two-photon system |Ψ PS . QD 1 and QD 2 Polarization Spatial-mode

Complete nondestructive polarization-spatial hyperentangled Bell-state analysis
The protocol for the complete polarization-spatial HBSA proposed in the former section is destructive. After the analysis of two-photon polarization-spatial hyperentangled Bell states, the two photons are detected at last and they cannot be used for other quantum information processing tasks. In this section, we propose a complete nondestructive polarization-spatial HBSA. The schematic diagram of the complete nondestructive polarization-spatial HBSA is shown in Fig. 6. The four QDs are all prepared in the states |ϕ + 1 = |ϕ + 2 = |ϕ + 3 = |ϕ + 4 and the hyperentangled two-photon system is in one of the 16 hyperentangled Bell states. One lets photon A pass through the quantum circuit from the left input port, followed by photon B. The interval time Δt, which is less than the spin coherence time Γ, exists between the two photons. After the two photons A and B passes through quantum circuit, the evolution of the system composed of two photons and four QDs is shown in Table 2.
From Table 2, one can obtain the complete and deterministic analysis on hyperentangled Bell states of a two-photon system AB without detecting it. The measurement outcomes of QD 1 and QD 3 show the parity information in the polarization DOF and the spatial-mode DOF, Table 2. The relationship between the measurement outcomes of the states of the four QDs and the initial hyperentangled states of the two-photon system |Ψ PS .
respectively. In detail, when QD 1 (QD 3 ) is in the state |ϕ + 1 (|ϕ + 3 ), the two-photon system is in the state |φ ± P(S) in the polarization (spatial-mode) DOF. Otherwise, when QD 1 (QD 3 ) is in the state |ϕ − 1 (|ϕ − 3 ), the two-photon system is in the state |ψ ± P(S) in the polarization (spatial-mode) DOF. The measurement outcomes of QD 2 and QD 4 show the phase information in the polarization DOF and the spatial-mode DOF, respectively. In detail, when QD 2 (QD 4 ) is in the state |ϕ + 2 (|ϕ + 4 ), the two-photon system is in the state |φ + P(S) or |ψ + P(S) in the polarization (spatial-mode) DOF. Otherwise, when QD 2 (QD 4 ) is in the state |ϕ − 2 (|ϕ − 4 ), the two-photon system is in the state |φ − P(S) or |ψ − P(S) in the polarization (spatial-mode) DOF. That is, the schematic diagram shown in Fig. 6 can be used for the complete nondestructive polarization-spatial HBSA in a deterministic way.

Discussion and summary
Assisted by the optical transitions in a QD-cavity system, we construct an error-detected block. With the error-detected block, our schemes for the deterministic HBSG, complete HBSA, and complete nondestructive HBSA of the polarization-spatial hyperentangled two-photon system are proposed. In an ideal condition, |Φ f 1 = |L |i 2 |ϕ − or |Φ f 2 = |L |i 2 |ϕ + can be obtained after the right-polarized photon |R interacting with the error-detected block and the fidelities and the efficiencies of our schemes can be 100%. However, in a realistic condition, the outcomes of the interaction between the right-polarized photon |R and the error-detected block, which are described as Eqs. (9) and (10), are affected by the coupling constant between X − and the cavity g, the cavity decay rate κ, the leaky rate κ s , and the exciton dipole decay rate γ, which would affect the fidelities and the efficiencies as well.
The fidelity of the process for deterministic HBSG, complete HBSA, or complete nondestructive HBSA is defined as F = | ψ r |ψ i | 2 . Here |ψ r and |ψ i are the final states in the realistic condition and ideal condition, respectively. The efficiency is defined as the ratio of the number of the output photons to the input photons. The fidelities of our scheme for deterministic HBSG are described as Here F 1 , F 2 , F 3 and F 4 are the fidelities of generating the hyperentangled Bell states |φ + P |φ + S , |φ − P |ψ + S , |ψ + P |φ − S , and |ψ − P |ψ − S , respectively. The fidelity of generating the hyperentangled Bell state |ψ − P |ψ − S is unity. The efficiencies of generating these four hyperentangled states with our scheme of deterministic HBSG are described as The fidelities and the efficiencies of our deterministic HBSG scheme vary with the parameter g/κ in the conditions of γ = 0.1κ and κ s = 0.2κ are shown in Figs. 7(a) and (b), respectively. From Figs. 7(a) and (b), one can see that when g/κ = 0.5, which is a low-Q-factor of a cavity, the fidelities are F 1 = 85.45% and F 2 = F 3 = 92.44% and the efficiencies are η 1 = 42.79%, η 2 = η 3 = 36.32%, and η 4 = 30.83% in the conditions γ = 0.1κ and κ s = 0.2κ. For a strong coupling between the QDs and the cavity, g/κ > 1, the fidelities are F 1 > 95.78% and F 2 = F 3 > 97.86%, and the efficiencies are η 1 > 60.17%, η 2 = η 3 > 57.60%, and η 4 > 55.13% in the conditions γ = 0.1κ and κ s = 0.2κ. The fidelity and the efficiency of our complete HBSA scheme for the hyperentangled Bell state |φ + P |φ + S are described as They vary with the parameter g/κ, shown as the blue solid lines in Figs. 8(a) and (b), respectively. The fidelity and the efficiency of our complete nondestructive HBSA scheme for the hyperentangled Bell state |φ + P |φ + S are given by  The red dashed lines in Figs. 8(a) and (b) describe F HBSG2 and η HBSG2 varying with the parameter g/κ, respectively. When the parameter g/κ is larger than 1, the fidelities of our complete HBSA scheme and complete nondestructive HBSA scheme will be higher than F HBS A1 = 95.78% and F HBS A2 = 91.89%, respectively, for the hyperentangled Bell state |φ + P |φ + S in the conditions γ = 0.1κ and κ s = 0.2κ. And the efficiencies will be higher than η HBS A1 = 60.17% and η HBS A2 = 36.13%.
Besides the parameters g, κ, κ s and γ, the fidelities would also be affected by the exciton dephasing, including the optical dephasing and the spin dephasing of X − . Exciton dephasing reduces the fidelity by the amount of [1 − exp(−τ/Γ)], where τ and Γ are the cavity photon lifetime and the trion coherence time, respectively. The optical dephasing reduces the fidelity less than 10%, because the time scale of the excitons can reach hundreds of picoseconds [52-54], while the cavity photon lifetime is in the tens of picoseconds range for a self-assembled In(Ga)As-based QD with a cavity Q factor of 10 4 −10 5 in the strong coupling regime. The effect of the spin dephasing can be neglected because the spin decoherence time is several orders of magnitude longer than the cavity photon lifetime [55][56][57].
In addition, the fidelity can be affected by the imperfect optical selection induced by heavylight hole mixing [58]. However, this imperfect mixing can be reduced by engineering the shape and the size of QDs or by choosing the types of QDs.
As the three protocols for deterministic HBSG, complete HBSA, and complete nondestructive HBSA are constructed by our error-detected block, the errors are detectable and the fidelities are improved. The low efficiencies can be remedied by repeat until success. Once our protocols succeeds, which means the detectors of the error-detected block don't click, the high fidelities can be obtained. This feature seems more important in the protocol for deterministic HBSG, which means high-quality hyperentangled Bell states can be generated.
In our schemes, the spin superposition state |ϕ + is prepared initially. It can be prepared from the spin eigenstates by using nanosecond electron-spin-resonance pulses or picosecond optical pulses. The preparation time for the spin superposition state can be significantly shorter than Γ because of the ultrafast optical coherent control of electron spins in semiconductor quantum wells and in semiconductor QDs. The measurement of the spin superposition state in the basis {|ϕ + , |ϕ − } is required in each scheme. By applying a Hadamard operation on the electron spin, the spin superposition states |ϕ + and |ϕ − can be transformed into the spin states | ↑ and | ↓ , which can be detected by measuring the helicity of the transmitted or reflected photon.