1. Introduction

Quantum entanglement plays an important role in quantum information science and technology. It is a unique quantum resource for many branches of quantum information processing (QIP), such as quantum key distribution [1,2], quantum dense coding [3,4], quantum teleportation [5], quantum secret sharing [6], quantum secure direct communication [7,8], and quantum computation [9,10]. Of the many different kinds of entangled states, Bell states of two qubit systems, which are the maximally entangled and mutually orthogonal basis states for the Hilbert space, are the most popular and crucial ones. In many quantum cryptography and quantum communication schemes, a key step is the implementation of the joint Bell-state measurement [11]. The number of the bases that can be unambiguously discriminated usually determines the capacity of the quantum communication process. Therefore, much effort has been made to perform the Bell-state analysis (BSA) on various quantum systems, and the ultimate goal is the complete and faithful discrimination of the Bell states.

Photon systems have been identified as ideal candidates for QIP. However, the complete BSA of photon pairs in one degree of freedom (DOF) cannot be achieved using only linear-optical elements. It has been proven that the four two-photon Bell states can only be classified into three groups, and the optimal success probability of state analysis is 50% with only linear optics [12–14]. By the use of a quantum nondemonlition detector (QND), the success probability can be improved to unity [15]. On the other hand, photons possess several DOFs, such as polarization, spatial mode, time bin, frequency, and orbital angular momentum (OAM), and each DOF can be manipulated independently. In particular, photonic hyperentanglement [16,17], which involves photons simultaneously entangled in several DOFs, can be used to realize the complete BSA in one DOF [18–24]. In 1998, Kwiat et al. [18] first investigated the method to completely distinguish the four polarization Bell states based on the hyperentanglement in both the polarization and momentum DOFs. In 2003, Walborn et al. [19] proposed a hyperentanglement-based scheme for the complete BSA in the polarization DOF or the momentum DOF with linear optics. The complete BSA experiments have been demonstrated with polarization-time-bin [21], polarization-momentum [22], and polarization-OAM [23] hyperentanglements. In 2017, Williams et al. [24] reported the demonstration of superdense coding over optical fiber links with a complete BSA enabled by polariztion-time-bin hyperentanglement. In addition to the complete BSA, hyperentanglement has many important applications, such as deterministic entanglement purification [25–29], high-capacity quantum communication [30–33], and efficient quantum computation [34–36]. Hence, the generation [37–39] and manipulation [40–42] of hyperentanglement have been widely studied in the past years.

Hyperentangled-Bell-state analysis (HBSA) of photons in more than one DOF, which is very useful for QIP based on the photonic hyperentanglement, has attracted much attention in recent years [43–46]. Considering two photons simultaneously entangled in two qubitlike DOFs, there are 16 orthogonal hyperentangled Bell states in a large Hibert space. It is shown that the 16 two-photon hyperentangled Bell states can only be classified into seven groups with only linear optics. When QNDs are utilized, the 16 states can be discriminated completely. In 2010, Sheng et al. [47] proposed the first two-step HBSA scheme for the complete discrimination of the 16 hyperentangled Bell states in the polarization and spatial-mode DOFs, using QNDs constructed with cross-Kerr nonlinearities. In this scheme, the spatial-mode entanglement is analyzed in the first step and the two photons are preserved using two different QNDs. The polarization entanglement is analyzed in the second step with the information obtained in the first step and another QND. The principle can be generalized to perform the hyperentangled Greenberger-Horne-Zeilinger-state analysis [48]. In 2016, Li et al. [49] proposed an efficient self-assisted complete polarization-spatial HBSA scheme, using less QNDs constructed with cross-Kerr nonlinearities. This scheme preserves not only the photons but also the spatial-mode entanglement in the first step, and uses the preserved spatial-mode entanglement as an ancillary to analyze the polarization entanglement in the second step. The self-assisted principle can also be used to analysis the polarization-time-bin hyperentanglement [50]. Recently, an interesting scheme was presented for the complete analysis of 16 hyperentangled Bell states in the polarization and first longitudinal momentum DOFs, assisted by the auxiliary hyperentanglement in the frequency and second longitudinal momentum DOFs [51].

The singly charged semiconductor quantum dot (QD) is a promising solid-state physical system for QIP. The QD-confined electron spin can be used to store and process quantum information due to the long electron-spin coherence time (typically around several microseconds), which is limited by the spin-relaxation time. And the spin initialization, manipulation, and measurement have been well investigated in the past decades. Current micro- and nano-fabrication techniques provide very efficient ways to incorporate the QDs into optical resonant cavities, which is of great importance for the realization of coherent manipulations of quantum information and the development of practical quantum devices [52–55]. For example, based on the optical giant circular birefringence (GCB) induced by a single-electron spin in a QD inside a single-sided or double-sided optical microcavity due to the effect of cavity quantum electrodynamics (QED), one can realize different types of photon-spin quantum entangling gate (quantum interface) [56–58], which can be widely used in designing scalable and all-optical QIP schemes [59–63], including photonic complete BSA [64–66] and HBSA [67–71] schemes. In 2012, Ren et al. [67] proposed a complete HBSA scheme using QNDs constructed with single-sided QD-cavity systems, based on the ideal GCB in the reflection geometry. In the same year, Wang et al. [68] presented a different complete HBSA scheme with double-sided QD-cavity systems based on the ideal GCB in the reflection/transmission geometry. In 2016, Wang et al. [69] proposed an interesting complete HBSA scheme and a complete nondestructive HBSA scheme using error-detected blocks constructed with double-sided QD-cavity systems, which can work effectively even when the ideal GCB cannot be realized. In 2018, Zeng presented a self-assisted complete HBSA scheme using single-sided QD-cavity systems to construct QNDs [70].

In this paper, we present a complete and faithful HBSA scheme for two-photon quantum systems hyperentangled in both the polarization and spatial-mode DOFs, using a failure-heralded and fidelity-robust quantum swap gate for the polarization states of two hyper-encoded photons (P-SWAP gate). Compared with the previously proposed complete HBSA schemes, our scheme can also distinguish the 16 two-photon polarization-spatial hyperentangled Bell states completely, but has several distinct features: (i) Our scheme uses the P-SWAP gate rather than the parity-check QNDs, which significantly simplifies the state analysis process. There is no need to pause the state analysis process midway to read out the parity information and confirm the spatial modes of the two photons. After all the operations and measurements have been successfully performed on the photons, we can identify the initial hyperentangled Bell state according the final single-photon measurement outcomes. So the scheme can be implemented in one shot. (ii) The P-SWAP gate is constructed with only one double-sided QD-cavity system and some linear-optical elements, which greatly reduces the number of the required QD-cavity system and saves the quantum resource. (iii) The failure of the P-SWAP gate operation is heralded by the clicks of single-photon detectors in the quantum circuit and the gate fidelity is robust when the operation succeeds, since the reflection and transmission coefficients of the double-sided QD-cavity system only appear in the global coefficient of the final output state of the gate. This makes our scheme works faithfully (without the error introduced by the infidelity of the P-SWAP operation), even when the QD-cavity parameters cannot satisfy the conditions for the ideal GCB. Therefore, our scheme may be more feasible and useful, which can be faithfully used in various applications, such as quantum hyperdense coding, teleportation via hyperentangled channels, hyerentanglement swapping, and others.

2. Interaction between an incident photon and a singly charged QD in a double-sided optical microcavity

As shown in Fig. 1(a), we consider a singly charged semiconductor QD, for instance, a self-assembled In(Ga)As QD or a GaAs interface QD, placed at the antinode of a double-sided optical micropillar cavity, where the two GaAs/Al(Ga)As distributed Bragg reflectors (DBRs) and the transverse index guiding provide the three-dimensional confinement of light. The two DBRs of the microcavity are made partially reflective (double sided) and symmetric, and the cross section is made circular.

 

Fig. 1. (a) A singly charged QD in a double-sided optical micropillar cavity. The two distributed Bragg reflectors (DBRs) and the transverse index guiding provide the three-dimensional confinement of light. The two DBRs of the microcavity are made partially reflective (double sided) and symmetric, and the cross section is made circular. (b) The relevant energy levels and the optical spin selection rule for $X^-$ transitions due to the Pauli exclusion principle and the conservation of total spin angular momentum (see text).

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The optical properties of singly charged QDs are dominated by the spin-dependent optical transitions of the negatively charged exciton $X^-$ which consists of two electrons bound to one hole [72]. The relevant energy levels and the optical spin selection rule for $X^-$ transitions are shown in Fig. 1(b). Here, $\left | \uparrow \right \rangle$ and $\left | \downarrow \right \rangle$ represent the single-electron spin states with ${J_z} = \pm 1/2$, $\left | \Uparrow \right \rangle$ and $\left | \Downarrow \right \rangle$ represent the heavy-hole spin states with ${J_z} = \pm 3/2$, and the two electron spins are antiparallel in the $X^-$ states $\left | { \uparrow \downarrow \Uparrow } \right \rangle$ and $\left | { \downarrow \uparrow \Downarrow } \right \rangle$. The spin-quantization axis $z$ is along the QD growth direction, i.e., the normal direction of the cavity and the propagation direction of the input (or output) light. When we consider an incident single photon interacts with the charged QD, the photon spin ${s_z}$ can also be defined with respect to the $z$ axis. Due to the Pauli exclusion principle and the conservation of total spin angular momentum, there are only two optically allowed dipole transitions between the electron states and the $X^-$ states: the photon in the state $\left | {{L^ \downarrow }} \right \rangle$ or $\left | {{R^ \uparrow }} \right \rangle$ (${s_z} = + 1$) can only couple to the transition $\left | \uparrow \right \rangle \leftrightarrow \left | { \uparrow \downarrow \Uparrow } \right \rangle$, and the photon in the state $\left | {{R^ \downarrow }} \right \rangle$ or $\left | {{L^ \uparrow }} \right \rangle$ (${s_z} = - 1$) can only couple to the transition $\left | \downarrow \right \rangle \leftrightarrow \left | { \downarrow \uparrow \Downarrow } \right \rangle$ [see Fig. 1(b)]. Here, $\left | R \right \rangle$ and $\left | L \right \rangle$ represent the right- and left-circularly polarized states of the photon, respectively. The superscript arrows indicate the photon propagation directions along the $z$ axis.

The reflection and transmission coefficients of the double-sided QD-cavity system can be obtained by solving the Heisenberg equations of motion for the cavity-field operator $\hat a$ and the $X^-$ dipole operator ${\hat \sigma _ - }$ in the interaction picture, and the input-output relations [58]

3. An error-detected circuit unit constructed with a double-sided QD-cavity system and some linear-optical elements

We can construct an error-detected circuit unit with a double-sided QD-cavity system and some linear-optical elements for fidelity-robust QIP. The schematic diagram of the error-detected circuit unit is shown in Fig. 2. Here, C is an optical circulator and D is a single-photon detector. BS represents a 50:50 beam splitter, which performs the Hadamard operation [$\left | {{i_1}} \right \rangle \leftrightarrow \left ( {\left | {{j_1}} \right \rangle + \left | {{j_2}} \right \rangle } \right )/\sqrt 2$ and $\left | {{i_2}} \right \rangle \leftrightarrow \left ( {\left | {{j_1}} \right \rangle - \left | {{j_2}} \right \rangle } \right )/\sqrt 2$] on the spatial-mode DOF of single photons. H (H$_{j1}$ or H$_{j2}$) represents a half-wave plate oriented at ${22.5^ \circ }$, which performs the Hadamard operation [$\left | R \right \rangle \leftrightarrow \left ( {\left | R \right \rangle + \left | L \right \rangle } \right )/\sqrt 2$ and $\left | L \right \rangle \leftrightarrow \left ( {\left | R \right \rangle - \left | L \right \rangle } \right )/\sqrt 2$] on the polarization DOF of single photons. M (M$_{j1}$ or M$_{j2}$) is a mirror. Suppose we always inject the right-circularly polarized photon $\left | R \right \rangle$ into the circuit unit from the path $i_1$ (see Fig. 2). If the electron spin in the QD is initially in a superposition state $\left | + \right \rangle = \left ( {\left | \uparrow \right \rangle + \left | \downarrow \right \rangle } \right )/\sqrt 2$, after the photon passes through the block composed of BS, H$_{j1}$, H$_{j2}$, M$_{j1}$, M$_{j2}$, and QD-cavity system, the total state for the photon and the electron spin is changed from $\left | {{\Phi _0}} \right \rangle = \left | R \right \rangle \otimes \left | + \right \rangle$ to

 

Fig. 2. Schematic diagram of the error-detected circuit unit constructed with a double-sided QD-cavity system and some linear-optical elements. Here, C is an optical circulator and D is a single-photon detector. BS represents a 50:50 beam splitter, which performs the spatial-mode Hadamard operation [$\left | {{i_1}} \right \rangle \leftrightarrow \left ( {\left | {{j_1}} \right \rangle + \left | {{j_2}} \right \rangle } \right )/\sqrt 2$ and $\left | {{i_2}} \right \rangle \leftrightarrow \left ( {\left | {{j_1}} \right \rangle - \left | {{j_2}} \right \rangle } \right )/\sqrt 2$]. H (H$_{j1}$ or H$_{j2}$) represents a half-wave plate oriented at ${22.5^ \circ }$, which performs the polarization Hadamard operation [$\left | R \right \rangle \leftrightarrow \left ( {\left | R \right \rangle + \left | L \right \rangle } \right )/\sqrt 2$ and $\left | L \right \rangle \leftrightarrow \left ( {\left | R \right \rangle - \left | L \right \rangle } \right )/\sqrt 2$]. M (M$_{j1}$ or M$_{j2}$) is a mirror.

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From Eqs. (5) and (6), one can observe that there are two terms in each final state, in which the first term is usually defined as the error term and the second term is our desired state. If the photon is transmitted through the block (which corresponds to the second term with the probability amplitude $T$), the polarization of the photon would be flipped and the spin superposition state of the electron would be changed from $\left | + \right \rangle$ to $\left | - \right \rangle$ or from $\left | - \right \rangle$ to $\left | + \right \rangle$. Then we can utilize the transmitted photon and the QD-cavity system to accomplish the task of fidelity-robust QIP. Conversely, if the photon is reflected by the block (which corresponds to the first term with the probability amplitude $D$), the electron spin state would not change and the reflected photon would be detected by the single-photon detector D. Thus the click of the detector indicates the case in which the error is detected and the corresponding task of the circuit unit runs fail. Such an error-detected circuit unit has been used to implement quantum logic gates [73–77], entanglement generation and analysis [69,74], and entanglement purification and concentration [78–80] very recently.

4. Implementation of a failure-heralded and fidelity-robust quantum swap gate for the polarization states of two hyper-encoded photons

Suppose two photons $A$ and $B$ are hyper encoded in the polarization and spatial-mode DOFs as

The quantum circuit for the implementation of a failure-heralded and fidelity-robust P-SWAP gate with the error-detected circuit unit is shown in Fig. 3. Here, CPBS represents a circularly polarizing beam splitter in the basis $\left \{ {\left | R \right \rangle ,\left | L \right \rangle } \right \}$, which transmits the right-circularly polarized photon component and reflects the left-circularly polarized photon component. X represents a half wave plate oriented at ${45^ \circ }$, which performs the polarization bit-flip operation ${\sigma _x} = \left | R \right \rangle \left \langle L \right | + \left | L \right \rangle \left \langle R \right |$. UBS represents an unbalanced beam splitter with the transmission coefficient $T$. S is an optical switch. The electron spin in the QD is initially prepared in the state $\left | + \right \rangle$ and the function of the error-detected circuit unit is described by Eqs. (5) and (6). To implement the P-SWAP gate operation for the two photons $A$ and $B$, we input the photons $A$ and $B$ into the quantum circuit through the spatial modes $l_1$ and $l_2$ ($l=a,b$) in sequence (see Fig. 3). In both the spatial modes $l_1$ and $l_2$, after each photon passes through the CPBS, the left-circularly polarized photon component will pass through the X, and then be injected into the error-detected circuit unit with the reflection coefficient $D$ and transmission coefficient $T$. The right-circularly polarized photon component will pass through the UBS. If there is no click of the single-photon detectors, the two components of each photon are combined again at the other CPBS. After that, a polarization Hadamard operation is performed on each photon through the H and a spin Hadamard operation [$\left | \uparrow \right \rangle \leftrightarrow \left ( {\left | \uparrow \right \rangle + \left | \downarrow \right \rangle } \right )/\sqrt 2$ and $\left | \downarrow \right \rangle \leftrightarrow \left ( {\left | \uparrow \right \rangle - \left | \downarrow \right \rangle } \right )/\sqrt 2$] is performed on the electron spin in the QD. After the above operations, the total state for the two photons and the electron is changed from $\left | {{\Omega _0}} \right \rangle = {\left | \varphi \right \rangle _A} \otimes {\left | \varphi \right \rangle _B} \otimes \left | + \right \rangle$ to (without normalization)

 

Fig. 3. Quantum circuit for the implementation of a failure-heralded and fidelity-robust P-SWAP gate with the error-detected circuit unit. Here, CPBS represents a circularly polarizing beam splitter in the basis $\left \{ {\left | R \right \rangle ,\left | L \right \rangle } \right \}$, which transmits the right-circularly polarized photon component and reflects the left-circularly polarized photon component. X represents a half wave plate oriented at ${45^ \circ }$, which performs the polarization bit-flip operation ${\sigma _x} = \left | R \right \rangle \left \langle L \right | + \left | L \right \rangle \left \langle R \right |$. UBS represents an unbalanced beam splitter with the transmission coefficient $T$. S is an optical switch. Other devices are the same as that in Fig. 2.

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Subsequently, by using the optical switches, we let the photons pass through the quantum circuit for the second time. After the same operations as that in the first time, if there is no click of the single-photon detectors, the total state becomes

In implementing the P-SWAP gate, the clicks of the single-photon detectors in the quantum circuit herald the failure of the gate operation. However, once the operation succeeds, the gate fidelity is, in principle, in unity, since the reflection and transmission coefficients of the double-sided QD-cavity system only appear in the global coefficient of the final state. That is, the gate fidelity is robust to the QD-cavity parameters. Without considering the photon loss due to the absorption and scattering of the linear-optical elements, and the inefficiency of the single-photon detectors, the success probability of the P-SWAP gate can be directly calculated to be $\eta = {\left | T \right |^8}$. We will discuss the success probability by considering currently achievable QD-cavity parameters in Sec. 6.

5. Complete two-photon polarization-spatial hyperentangled-Bell-state analysis using the P-SWAP gate

A two-photon hyperentangled Bell state in both the polarization and spatial-mode DOFs can be written as

The schematic diagram of our complete and faithful HBSA scheme for polarization-spatial hyperentangled two-photon systems is shown in Fig. 4. Here, we use the failure-heralded and fidelity-robust P-SWAP gate with only one QD-cavity system and some linear-optical elements to achieve the complete HBSA. Suppose a two-photon system $AB$ is in one of the 16 hyperentangled Bell states as described by Eq. (11), the process for the complete HBSA can be described as follows.

 

Fig. 4. Schematic diagram of our complete and faithful HBSA scheme for hyperentangled two-photon systems in the polarization and spatial-mode DOFs.

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First, we let the photon $B$ pass through the block composed of X$_1$, CPBS$_1$, and X$_2$, which can exchange the polarization and spatial-mode states of the photon $B$. Then the initial two-photon hyperentangled Bell state is transformed to a state with the polarization DOF of the photon $A$ ($B$) entangled to the spatial-mode DOF of the photon $B$ ($A$). Second, we let the two photons $A$ and $B$ pass through the failure-heralded and fidelity-robust P-SWAP gate in sequence, to exchange the polarization states of the two photons. After the successful P-SWAP operation, the two-photon state is converted into a product of two single-photon hybrid entangled states [81,82]. For example, if the initial state is ${\left | {{\phi ^ + }} \right \rangle ^p}\left | {{\phi ^ + }} \right \rangle _{AB}^s$, after the above operations, the state transformation is

Table 1. The relationship between the final measurement outcomes of the two photons in the spatial-mode and polarization DOFs and the initial hyperentangled Bell states of the two-photon system. The products of two single-photon hybrid entangled states after the successful P-SWAP operations are also contained in the table, where $\left | {{\phi ^ \pm }} \right \rangle _{A(B)}^{p,s} = {(\left | {R{a_1}({b_1})} \right \rangle \pm \left | {L{a_2}({b_2})} \right \rangle )_{A(B)}}/\sqrt 2$ and $\left | {{\psi ^ \pm }} \right \rangle _{A(B)}^{p,s} = {(\left | {R{a_2}({b_2})} \right \rangle \pm \left | {L{a_1}({b_1})} \right \rangle )_{A(B)}}/\sqrt 2$ represent the single-photon hybrid entangled states.

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Our scheme uses the P-SWAP gate rather than the parity-check QNDs. There is no need to pause the state analysis process midway to read out the parity information and confirm the spatial modes of the two photons, which is apparently different from some of the previous complete HBSA schemes. After all the operations and measurements have been successfully performed on the photons, we can confirm the initial hyperentangled Bell state according the final measurement outcomes. So the scheme can be implemented in one shot.

Our scheme can also be directly used for analyzing the polarization-entangled Bell states, as the polarization-entangled Bell states can be seen as the specific hyperentangled Bell states with certain spatial modes. For example, suppose a two-photon system $AB$ is in one of the four polarization-entangled Bell states as described by Eq. (12), we can input the photons $A$ and $B$ into the quantum circuit shown in Fig. 4 through the input ports $a_1$ and $b_1$, which means the initial two-photon state has the decided spatial modes. After the same process as that for the complete HBSA and before the photons are measured, the two-photon state evolves as

6. Discussion and summary

Up to now, we have fully presented our complete and faithful two-photon polarization-spatial HBSA scheme. The key element for our scheme is the failure-heralded and fidelity-robust P-SWAP gate. As mentioned above, the gate fidelity is robust to the QD-cavity parameters when the operation succeeds, as the reflection and transmission coefficients only appear in the global coefficient of the output state. This makes our scheme works faithfully (without the error introduced by the infidelity of the P-SWAP operation), which can be used in various applications, especially the safe and high-capacity quantum communication processes [78]. The efficiency of our scheme is mainly limited by the success probability of the P-SWAP gate, which has been calculated to be

 

Fig. 5. Calculated success probability of the P-SWAP gate as a function of $g/\kappa$ and $\kappa _s/\kappa$. Here, $\omega = {\omega _c} = {\omega _{{X^ - }}}$ is assumed and $\gamma = 0.1\kappa$ is taken by considering the practical QD-cavity parameters.

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Although the fidelity of the P-SWAP gate is robust to the QD-cavity parameters, there may exist some other imperfections that should be considered. As noted by Hu et al. [56,57], the perfect spin-selection rule only holds for an ideal QD that is symmetric in both the QD shape and the strain field distribution such that there is no spin-level mixing or splitting at zero magnetic field. A realistic QD which is generally asymmetric could be made symmetric via thermal annealing [85], tuning the QD size [86], or applying an electric/magnetic field [87]. On the other hand, the heavy-light hole mixing in a realistic QD can also affect the spin-selection rule [57,58]. For self-assembled In(Ga)As QDs, the hold mixing in the valence band is in the order of a few percent [88,89], which would reduce the gate fidelity by a few percent. The hold mixing could be reduced by engineering the shape and size of QDs or by utilizing different types of QDs.

The electron-spin coherence time $T_2$ in the QD is an important parameter in implementing our HBSA scheme. Considering the spin decoherence, the high-fidelity P-SWAP gate operation can only be achieved when the operational time is much shorter than the spin-coherence time. It has been shown that In(Ga)As or GaAs charged QDs have the long electron-spin coherence time (${T_2} \sim \mu$s) [90,91], which is limited by the spin-relaxation time (${T_1} \sim$ ms) [92,93]. Recent experiments also show that spin echo or dynamical decoupling techniques can be used to preserve the spin coherence. Based on the ultrafast optical spin echo technique, $T_2 = 1$ $\mu$s has been demonstrated in a In(Ga)As QD [94]. On the other hand, the exciton dephasing can also reduce the gate fidelity by amount of $\left [ {1 - \exp \left ( { - \tau /\Gamma } \right )} \right ]$, where $\tau$ is the cavity photon lifetime and $\Gamma$ is the exciton coherence time [58]. In self-assembled In(Ga)As QDs, as the optical coherence time of excitons can be ten times longer than the cavity lifetime [95], the optical dephasing of the $X^-$ can only slightly reduce the fidelity by less then $10\%$. The spin dephasing of the $X^-$ can be safely neglected in our considerations, as the spin dephasing is mainly related to the hole spin, whose coherence time can be three orders of magnitude longer than the cavity photon lifetime [96].

The electron-spin initialization, manipulation, and measurement required in our scheme can be efficiently achieved. For example, the spin superposition state can be prepared by optical pumping and/or optical cooling followed by spin-flip Raman transitions [97], or single-spin rotations using nanosecond electron spin resonance (ESR) microwave pulses [91] or picosecond/femtosecond optical pulses [98]. The photon-spin entangling gate based on the ideal GCB can also be used to initialize and measure the spin. The spin manipulation and measurement time scales could be much shorter than the spin coherence time.

In summary, we have presented a complete and faithful two-photon polarization-spatial HBSA scheme, using a failure-heralded and fidelity-robust P-SWAP gate constructed with a double-sided QD-cavity system. The HBSA is implemented with three steps: first, the polarization and spatial-mode states of the photon $B$ are exchanged, by which the hyperentangled Bell state is transformed to the state with the polarization DOF of the photon $A$ ($B$) entangled to the spatial-mode DOF of the photon $B$ ($A$); second, the polarization state of the photon $A$ is exchanged with the polarization state of the photon $B$, which transforms the two-photon state into the product of two single-photon hybrid entangled states; at last, the single-photon hybrid entangled states are measured. Compared with previous schemes, our scheme not only simplifies the state analysis process and saves the quantum resource, but also guarantees the robust fidelity and relaxes the experimental requirement. Therefore, our scheme may be more feasible and useful for practical applications based on the photonic hyperentanglement.