Introduction

Entanglement plays an important role in quantum information processing (QIP)1. For example, entangled photons can act as the information carriers in quantum communication, such as quantum key distribution2,3 and quantum dense coding4,5. Also the entangled photons can serve as the quantum channel in quantum teleportation6, quantum secret sharing7,30,31,32, complete deterministic entanglement purification33,34, or assisting complete Bell-state analysis26,27,35. The complete distinguishing of the hyperentangled Bell states is required in most fundamental quantum communication processes which exploits the nonlocal correlation of hyperentanglement entanglement, such as establishing hyperentangled channel for superdense coding29 and the multi-DOF quantum teleportation for single photons32. In 2007, Wei et al.36 proved that with the help of linear optics, one can only distinguish 7 states out of the group of 16 orthogonal hyperentangled Bell states in two DOFs and the upper bound of the maximal number of mutually distinguishable n-qubit Bell-like states is , which is true for n = 1 and n = 2. In 2011, Lynn et al.37 provided a more general proof of this bound for Bell-state distinguishability. And if nonlinear optics is introduced, these 16 orthogonal Bell states can be distinguished completely38,39,40. In 2010, Sheng et al.38 proposed a scheme to distinguish the 16 hyperentangled Bell states completely with the help of cross-Kerr nonlinearity and discussed the application of this scheme in quantum communication. When a combined system composed of a single photon and a coherent probe beam passing through a cross-Kerr medium, a phase shift θ is picked up on the coherent probe beam. With the action of the cross-Kerr nonlinearity, one can distinguish the even-parity states from the odd-parity states in spatial-mode DOF of a two-photon system without destroying the two-photon system in the other DOF. Although the cross-Kerr nonlinearity in the optical single photon regime has been widely presumed, it remains quite controversial for the lack of experimental supporting with current techniques41,42.

The artificial atom and optical cavity coupled system is an essential platform for the realization of QIP. And the description of the system using cavity quantum electrodynamics (QED) plays an important role for information exchange between static and flying qubits in quantum communication networks and it has been demonstrated that, even in the bad-cavity regime, a measurable nonlinear phase shift between single photons can be achieved in a cavity QED system43. This nonlinearity can be realized by a variety of physical systems, such as a leaky resonator interacting with an atom or a quantum dot44,45,46. In 2010, Bonato et al.46 proposed the first proposal that uses interface between the photon and the spin of an electron confined in a quantum dot embedded in a microcavity operating for Bell-state analysis in the weak coupling regime. Also, a further incentive to study HBSA based on cavity coupling system lies in the recent advances of such systems39,40. In 2012, Ren et al.39 presented complete HBSA with the nonlinear optics based on a quantum dot(QD)-one-sided cavity system. In a one-sided cavity, due to the spin selection rule, the right circularly polarized light and the left circularly polarized light pick up two different phase shifts after being reflected from the QD-cavity system and then, after two photons reflected by a cavity, the parity state of this photon pair in polarization DOF can be determined by measuring the state of the excess electron of the auxiliary QD without destroying the two-photon quantum system. However, in this work, there are four auxiliary QD-cavity coupled units which lead to in average 4 times two-qubit interactions between the photons and QDs and the auxiliary QDs are all required to be prepared in a certain superposition spin state of the excess electron of the QDs and should be measured to read the parity information of the photon-pairs. In the same year, by using the double-sided QD-cavity system, Wang et al.40 presented a scheme for complete analysis of the hyperentangled Bell in both polarization and spatial-mode DOFs which requires only two auxiliary QD-cavity coupled units with 4 two-qubit interactions between the photons and QDs. In 2015, Liu et al.47 presented a scheme for the generation and analysis of hyperentanglement assisted by two nitrogen-vacancy (NV) centers in diamonds coupled with microtoroidal resonators. In these schemes, the nonlinearity between the photons and the auxiliary particles is used to construct the two-qubit controlled-phase operations which plays a critical role in HBSA protocols.

In this paper, we show that the complete differentiation of 16 hyperentangled Bell states in both polarization and spatial-mode DOFs for two-photon system can be efficiently achieved based on a two-qubit SWAP gate by using a three-level Λ-typle atom-cavity coupled unit interacting with single photons in reflection geometry. By replacing the usual two-qubit controlled-phase operations using the two-qubit SWAP gates, the interaction between the photons and the auxiliary particle is reduced to three times and there is only one auxiliary particle required in our scheme. The initialization requirement of the auxiliary particle is reduced since it works as a temporary quantum memory and it is not required to be measured. Moreover, because the state of the auxiliary particle remains unchanged after the HBSA operation, within the decoherence time, the auxiliary particle can be repeatedly used in next HBSA operations. Compared with the previous HBSA schemes, the required experimental resource and the engineering complexity of the HBSA operation in our scheme is greatly simplified. And it is proved that the present scheme can both work in the weak- and strong-coupling regimes with current technologies. Finally, we discuss the feasibility of our scheme.

Results

The model of single-sided cavity and three-level Λ-type system

Here we consider the case that an atom is trapped in a single-sided optical cavity and the atom is assumed to be a three-level Λ-type system as shown in Fig. 1. The degenerate ground states of the atom, i.e., and , are considered to be the qubit states and the excited level to be the ancillary state. The optically allowed transitions can only be excited by the single V-polarized (H-polarized) photon under the selection rules. The Hamiltonian describes the interaction between the atom and the electric cavity field which is given by , here we set . represents the light-matter interaction strength, and are the corresponding creation and annihilation operators for the k-polarization cavity field, respectively. The Hamiltonian describes the interaction between the cavity field and the input-output fiber mode which is given by . Here and are the annihilation and creation operators for the k-polarized photon in the fiber mode and denotes the cavity-photon dam** rate through the output mirror.

We assume that the atom is initially in state and the incoming pulse is in the k-polarization state at the beginning. By considering the spontaneous emission of the exited state with the decay rate γ, the general time dependent wave function of the system can be described as48,49

In the state , denotes the atomic state, vac describes the vacuum state in the fiber mode and 0 or 1 means that the number of the photons in the k-polarization state. It is known that , and can be obtained by solving the Heisenberg equations of motion. In the rotating frame, the input-output relation are given by50, or or mixed (for example, , the auxiliary atom is just used as a temporary quantum memory and after the hyper-SWAP operation(Step1–Step3), the atomic state is unchanged. During the whole process of our scheme, the coherence of the atomic state should be maintained, but the initial state of the atom can be prepared arbitrarily, pure or mixed of the ground states. This is an important difference between our scheme and other hyperentangled Bell-state analysis scheme in which the auxiliary particle must be prepared in the pure state and finally should be measured.

In summary, we proposed an efficient HBSA scheme for photonic system by replacing the usual two-qubit controlled-phase operations using the two-qubit SWAP gates. The interaction times between the photons and the auxiliary particle is reduced to three and only one auxiliary particle is required in our scheme. The requirement of the auxiliary particle is reduced since it works as temporary quantum memories and need not to be actively controlled or measured. Moreover, as the state of the auxiliary particle remains unchanged after the HBSA operation, the auxiliary particle can be repeatedly used in the next HBSA operations within the coherence time. Therefore, the engineering complexity of the HBSA operation is greatly simplified compared with the previous HBSA scheme. Exploiting the existing experimental data, our calculation shows that this protocol are insensitive to both cavity decay and atomic spontaneous emission, so it can work in the case of a larger cavity decay rate, i.e., the cavity with a relatively lower-Q factor. All these advantages make this scheme more feasible in practical applications of long-distance quantum communication and scalable quantum computing.

Methods

Average fidelities and efficiencies of the gates

In this part, we give a brief discussion about the experimental implementation of our scheme. The level configuration under our consideration in Fig. 1 can be found in 87Rb; for example, the level with F = 1 (e.g., of Rb) acts as the ground state and the excited state could be . And could be trapped at the center of an optical cavity52. By combining with the long trap** time of the atom in the cavity (typically, the atom trap** times are tens of seconds53), the atom can be considered as a good carrier of stationary qubits. We can calculate the fidelity and the efficiency in the case that the initial atomic state is and the initial hyperentangled photon-pair (marked with A and B) is in the state . For simplicity, the input photons are assumed in resonance with the cavity. The fidelity of HBSA operation on the photon is F which could be described as

If the initial atomic state is and the initial state of the hyperentangled photon-pair AB is , after gate which performs the SWAP gate between the polarization and spatial-mode DOFs of the photon B, the state of the photon-pair AB becomes and then after the ideal hyper-SWAP gate, the state of the hybird system becomes . However, according to Eq.(5) in the unideal resonance case and after the hyper-SWAP gating opertion, the state of the system composed of photon-pair AB and the Λ-type atom becomes

and so the fidelity of the hyper-SWAP gating opertion is , here

The efficiency is given by . The calculated results of the fidelity and the efficiency is shown in Figs 3 and 4. In Fig. 3, when , the atomic spontaneous emission γ and the cavity decay rate show a slight influence on the fidelity F. Even, it is of large decay rate, i.e., called bad cavity, the fidelity of the entangler gate is still above 0.98 where . Using the values of the cavity-QED parameters 52,54, the gate fidelity is . However, in the weak coupling regime, the efficiency reduces rapidly. From Fig. 4, the efficiency is below 0.8 where . Figure 5 shows that the variance of the coupling strength has a stronger influence on the fidelity. The attained fidelity is found to approach the ideal value when . However, the fidelity is larger than 0.99 with . The present scheme can also be realized in other physical systems such as semiconductor quantum dots55 and superconducting system56 for the similar relevant levels. In general solid-state cavity-coupled system, the effective interaction between cavity-coupled qubits is described by the XY model or the Heisenberg exchange interaction. When a CPHASE gate or a CNOT gate is constructed using such interactions, generally, at least twice two-qubit interactions have to be invoked with complicated pulse sequences, but for a SWAP gate or iSWAP gate, only once two-qubit interaction is required57. Therefore, the development of SWAP-gate-based quantum algorithms would pave the way for an easier integration of solid-state qubits into a quantum communication network.

Figure 5
figure 5

The fidelity (in amplitude) functions of the coupling strengths gh/gv at γ/gv = 0.1 and κ/gv = 2.

Full size image

The challenge that two separate input light with different spatial-mode simultaneously interact with an auxiliary particle in cavity can be overcomed by using the optical switch at the single-photon level. One can switch multiple spatial modes into one light path with different time-bins and so the single-sided cavity only interacts with a single spatial mode of light at different time intervals. Recently, the experimental studies of such system have attracted much attention and we notice that the ultrafast all-optical switching by single photons58 has also been experimentally realized in QDCcavity system. Moreover, in ref. 59, the all-optical transistor which uses one photon to control the resonator transmission is also realized.

Additional Information

How to cite this article: Wang, T.-J. and Wang, C. Complete hyperentangled-Bell-state analysis for photonic qubits assisted by a three-level Λ-type system. Sci. Rep. 6, 19497; doi: 10.1038/srep19497 (2016).