Quantum hyperentanglement and its applications in quantum information processingPublished in Science Bulletin 62(1), 46–68 (2017).

Quantum hyperentanglement and its applications in quantum information processing1

Abstract

Hyperentanglement is a promising resource in quantum information processing with its high capacity character, defined as the entanglement in multiple degrees of freedom (DOFs) of a quantum system, such as polarization, spatial-mode, orbit-angular-momentum, time-bin and frequency DOFs of photons. Recently, hyperentanglement attracts much attention as all the multiple DOFs can be used to carry information in quantum information processing fully. In this review, we present an overview of the progress achieved so far in the field of hyperentanglement in photon systems and some of its important applications in quantum information processing, including hyperentanglement generation, complete hyperentangled-Bell-state analysis, hyperentanglement concentration, and hyperentanglement purification for high-capacity long-distance quantum communication. Also, a scheme for hyper-controlled-not gate is introduced for hyperparallel photonic quantum computation, which can perform two controlled-not gate operations on both the polarization and spatial-mode DOFs and depress the resources consumed and the photonic dissipation.

Keywords: Quantum hyperentanglement, high-capacity quantum communication, concentration and purification, hyperparallel photonic quantum computation, quantum information processing

pacs:
03.67.Dd, 03.67.Hk, 03.65.Ud

I introduction

Quantum information processing (QIP) has attracted considerable interest and attention of scientists in a variety of disciplines with its ability for improving the methods of dealing and transmitting information (1); (2). Entanglement is a distinctive feature of quantum physics (3), and it is very useful in QIP, including both quantum communication and quantum computation. Entangled photon systems are the natural resource for establishing quantum channel in long-distance quantum communication, especially in quantum repeaters (4) for some important tasks of communication, such as quantum key distribution (5); (6); (7), quantum secret sharing (8), and quantum secure direct communication (9); (10); (11); (12); (13). In experiment, the entangled photon systems are usually prepared by the spontaneous parametric down-conversion (SPDC) process in nonlinear crystal (14); (15); (16). In the conventional protocols for quantum information processing, the entanglement in one degree of freedom (DOF) of photon systems is selected in the SPDC process. In fact, there are more than one DOF in a quantum system, such as the polarization, spatial-mode, orbit-angular-momentum, frequency, and time-bin DOFs in a photon system.

Hyperentanglement, the simultaneous entanglement in multiple DOFs of a quantum system, has been studied extensively in recent years. It is a promising candidate for QIP with its high-capacity character. In experiment, hyperentanglement can be generated by the combination of the techniques used for creating entanglement in a single DOF (17). With this method, many different types of hyperentangled states can be prepared (18); (19); (20); (21); (22); (23); (24); (25), such as the polarization-spatial hyperentangled state (18), polarization-spatial-time-energy hyperentangled state (19), and so on. Hyperentanglement is a fascinating resource for quantum communication and quantum computation. On one hand, it can assist us to implement many important tasks in quantum communication with one DOF of photons, such as quantum dense coding with linear optics (26), the complete Bell-state analysis for the quantum states in the polarization DOF (27); (28); (29); (30); (31), the deterministic entanglement purification (31); (32); (33); (34), and the efficient quantum repeater (35). On the other hand, hyperentanglement can be used directly in some important applications in QIP. For example, it can improve the channel capacity of quantum communication and speedup quantum computation largely.

In the applications of hyperentanglement, the complete hyperentangled-Bell-state analysis (HBSA) (36); (37); (44); (38); (39); (41); (40); (42); (45); (43), hyper-teleportation of quantum state with more than one DOF (36), hyperentanglement swapping (37), hyperentanglement concentration (46); (47); (48); (50); (52); (49); (51); (53), hyperentanglement purification (47); (54); (55); (56); (57), and universal entangling quantum gates for hyperparallel photonic quantum computation (59); (58); (60); (61); (62) are very useful and important. HBSA is the prerequisite for high-capacity quantum communication protocols with hyperentanglement and it is used to distinguish the hyperentangled states. Also, in the practical application of hyperentanglement in quantum communication, the hyperentangled photon systems are produced locally, which leads to the decoherence of the hyperentanglement when the photons are distributed over a channel with environment noise or stored in practical quantum devices. Quantum repeater is a necessary technique to overcome the influence on quantum communication from this decoherence (4). In high-capacity quantum repeater with hyperentanglement, hyperentanglement concentration and hyperentanglement purification are two passive ways to recover the entanglement in nonlocal hyperentangled photon systems. They are not only useful but also absolutely necessary in long-distance high-capacity quantum communication with hyperentanglement as the self-error-rejecting qubit transmission scheme (63) do not work in depressing the influence of noise from both a long-distance channel and the storage devices for quantum states. Moreover, quantum repeaters for long-distance quantum communication require the entangled photons with higher fidelity (usually 99%) beyond that from faithful qubit transmission schemes.

Different from conventional parallel quantum computation in which the states of quantum systems in one DOF or equivalent are used to encode information, hyperparallel photonic quantum computation performs universal quantum gate operations on two-photon or multi-photon systems by encoding all the quantum states of each photon in multiple DOFs (two or more DOFs) as information carriers (59); (58); (60); (61); (62). With hyperparallel photonic quantum logic gates, the resource consumption can be reduced largely and the photonic dispassion noise can be depressed in quantum circuit (60). Moreover, the multiple-photon hyperentangled state can be prepared and measured with less resource and less steps by using the hyperparallel photonic quantum logic gates, which may speedup the quantum algorithm (59); (58).

In this review, we will overview the development of hyperentanglement and its applications in QIP in recent several years. We will first review the preparation of hyperentanglement, and then introduce the applications of the hyperentanglement in quantum communication, including hyper-teleportation of an unknown quantum state in more than two DOFs and hyperentanglement swapping. We also highlight how to improve the entanglement of nonlocal hyperentangled photon systems with hyperentanglement concentration and hyperentanglement purification. At last, the principle of a polarization-spatial hyper-controlled-not (hyper-CNOT) gate is described for hyperparallel quantum computing.

Ii preparation of hyperentanglement

Hyperentangled states offer significant advantages in QIP due to the presence of quantum correlations in multiple DOFs. In this section, we will introduce the preparation of hyperentangled states of photon systems. In the first part, we overview the preparation of entangled photon pairs with the SPDC process in nonlinear crystals. In the second part, we overview the preparation of hyperentangled photon systems with the combination of the techniques used for creating entanglement in single DOF.

ii.1 Preparation of entanglement in single DOF

Generally speaking, the most extensive method used to generate an entangled state is the SPDC process in a nonlinear crystal. When a pump laser beam shines a nonlinear birefringent crystal, the idler photon and the signal photon are generated probabilistically from the crystal. The maximal probability can be achieved by satisfying two matching conditions. One is the phase-matching:

(1)

and the other is energy-matching:

(2)

Here represents the wave vector and denotes the frequency. Usually, there are two common kinds of phase-matching adopted in experiment, depending on the extraordinary and the ordinary polarizations of the pump photon and the two SPDC photons. The type-I phase-matching is and the type-II phase-matching is .

In the type-I phase-matching, two SPDC photons are both ordinary and have the same polarizations. To generate an entangled state, two crystals with orthogonal optical axes can be used (15). The principle is shown in Fig. 1. To satisfy the phase-matching condition, two correlated photons are emitted over opposite directions of the cone surface. By selecting one pair of the correlated wavevector modes, the polarization entangled states can be prepared. Here and represent the horizontal and vertical polarization states of a photon, respectively. An alternative way to prepare an entangled state with type-I phase-matching is using a single crystal and a double passage of the laser beam after reflection on a mirror (16).

Figure 1: Type-I polarization entanglement sources (15).

In the type-II phase-matching, the two degenerate photons are emitted over two different mutually crossing emission cones (14). The emission directions of the signal and idler photons are symmetrically oriented with respect to the propagation direction of the pump photon. The two entangled photons are generated along the direction of the intersection of the two cones. Since the ordinary and extraordinary photons have orthogonal polarization states, the polarization entangled states are prepared with type-II phase-matching. If the two cones only intersect at one point, it is called the collinear SPDC process and orthogonally polarized photons are indistinguishable at exactly this point. The type-II collinear down-conversion is more commonly used in experiment, as it offers a trivial way to deterministically separate the photon pair by their polarization and to work with each photon separately. For the non-collinear type-II SPDC process which is shown is Fig. 2, the two emission cones have two intersection directions, which can be made indistinguishable with respect to their polarization, and then the entangled state is generated.

Figure 2: Non-collinear type-II phase-matching spontaneous parametric down-conversion process (14).

Actually, by obeying these two matching conditions, entanglement in other DOFs can be prepared, such as frequency, time-bin, and spatial-mode DOFs. The SPDC photon pairs are coherently emitted at different emission times as long as the interaction time of the pump wave with the crystal is shorter than the coherence time of the pump photons. The photons are automatically generated into an energy-entangled state due to the nature of the SPDC process. In a word, the energy-time correlations are presented in all SPDC photon pairs. The spatial-mode entangled state can be generated by selecting more correlated directions. And if the frequency of the idler and signal photons are not the same, they are always entangled to fulfill the energy-matching condition.

Besides spin, photons possess a further angular momentum, the orbital-angular-momentum (OAM), described by the Laguerre-Gaussian mode . Under the collinear phase-matching conditions, the OAM of these photons should satisfy (consider simple situation ). Therefore, when the pump beam is a Gaussian TEM beam, the two generated photons have opposite as

(3)

which is an OAM entangled state. Here denotes the probability of creating a signal photon with OAM and an idler one with .

ii.2 Hyperentanglement in more than one degree of freedom

The techniques used for creating single DOF entanglement can be combined to generate hyperentanglement, which is entangled in more than one DOF in the same time. The first proposal of an energy-momentum-polarization hyperentangled state with a type-II phase-matching was presented by Kwiat (17) in 1997. The schematic diagram is shown in Fig. 3. The photons emitted from conjugate points are all energy-time entangled. And the photons generated from and are automatically in a polarization entangled state since these two cones have opposite polarizations, which are indistinguishable at and . However, these photons have definite momentum. Photons emitted from are entangled in momentum with the definite polarization state. The quantum state with photons generated along the directions are entangled in momentum, energy-time, and polarization simultaneously.

Figure 3: Schematic diagram of the type-II phase-matching curves. The upper half (solid lines) are three extraordinary-polarized cones with different wavelengths (17). The ordinary-polarized cones are shown in the lower half (dashed lines) with corresponding wavelength. The numbered points represent the areas of directions to extract hyperentangled states.

In 2005, Yang et al. (18) generated a two-photon state entangled both in polarization and spatial-mode DOFs to realize the all-versus-nothing test of local realism. The setup of generation is shown in Fig. 4. In their experiment, the pump pulse passes through the nonlinear BBO (-barium borate) crystal twice. The first passage of laser prepares a polarization-entangled pairs in the spatial modes and with a small probability. Then the pump beam is reflected by a mirror and goes through the crystal a second time (again). In this time, it probabilistically generates a state in another two path modes and . The generation probabilities of two passages can be adjusted to equal. Therefore, if there is perfect temporal overlap of modes and ( and ), the two possible ways of producing may interfere, which results a spatial mode entangled state . can be achieved by adjusting the distance between the mirror and the crystal. Then, a maximally hyperentangled state in both polarization and spatial mode () is generated. In their experiment, the generation rate of entangled photon pairs achieves per second.

Figure 4: Schematic diagram of the setup to generate hyperentanglement in both polarization and spatial-mode DOFs (18).

In the same year, an experimental demonstration of a photonic hyperentangled system which simultaneously entangled in polarization, spatial-mode, and time-energy was reported (19). In their experiment, the entangled pairs are prepared with the type-I phase-matching, and two BBO crystals with orthogonal optical axes are used, which produce pairs of horizontally and vertically polarized photons, respectively. Since the spatial modes emitted from each crystal are indistinguishable, the photon pairs are polarization entangled. Moreover, photon pairs from a single nonlinear crystal are entangled in OAM. And according to the energy-matching condition, each pair is entangled in energy too. The generated state can be written as

(4)

Here and represent the Laguerre-Gauss modes carrying and 0 OAM, respectively. and denote the relative early and late emission time of photons, respectively. indicates the OAM spatial-mode balance prescribed by the source and selected via the mode-matching conditions. By collecting only OAM state, the state in the spatial subspace is also a Bell state. The total dimension of this hyperentangled system is . To verify the quantum correlations, they tested each DOF against the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality, and the results showed for each DOF the Bell parameter exceeded the classical limit. They also fully characterized the polarization and spatial-mode state subspace by tomography and obtained the maximum fidelity .

In 2009, Vallone et al. (20) also realized a two-photon six-qubit hyperentangled state which is entangled in polarization and two longitudinal momentum DOFs. The system used to generate the state consists of two type-I BBO crystal slabs. The polarization entanglement is created by spatially superposing the two perpendicularly polarized emission cones of each crystal. Since the two nonlinear crystal are cut at different phase matching angles, the photon pairs will be created along the surfaces of two cones, called the “internal” () and “external” () ones. Coherence and indistinguishability between these two emission cones are guaranteed by the coherence length of the pump beam. The double longitudinal momentum entanglement is generated by singling out four pairs of correlated modes with an eight-hole screen, shown in Fig. 5. The hyperentangled state, which is a product of one polarization entanglement and two longitudinal momentum entanglement, can be written as

(5)

Here and refer to the left and right sides of each cone. The three relative phases , , and can be adjusted in experiment.

Figure 5: Schematic diagram of generating hyperentanglement in both polarization and two longitudinal-momentum DOFs (20). Two BBO crystal slabs are used to generate hyperentangled photon pairs (solid line) and (dashed line). and denote the external and internal cones, respectively. and refer to the left and right sides of each cone, respectively. For simplicity, the positive lens which transforms the conical parametric emission of the crystal into a cylindrical one is not shown in the figure.
Figure 6: Model labeling of the hyperentangled state prepared with type-I BBO crystal (21). To get the six-qubit cluster state, HWPs oriented at are placed on the modes of and HWPs oriented at are placed on the modes of .

The hyperentangled states prepared in the previous protocols are the product states of different entangled DOFs. In 2009, Ceccarelli et al. (21) generated a two-photon six-qubit linear cluster state by transforming a two-photon hyperentangled state which is originally entangled in polarization and two linear momentum DOFs. First, they generated the following six-qubit hyperentangled state by SPDC in a Type-I BBO crystal, shown in Fig. 6.

(6)

Here and correspond to the up and down sides of the emission cones, respectively. By encoding the qubits 1 and 4 with DOF, qubits 2 and 5 with DOF, and qubits 3 and 6 with DOF, the desired two-photon six-qubit linear cluster state is described as

The transformation from the hyperentangled state to the cluster state is carried out by applying half-wave plates (HWPs) oriented at on the internal modes and HWPs oriented at on the left modes. The fidelity of the generated state is measured and is obtained, which is 7 % better than the best previous result for six-qubit graph state with six particles. The characterization and application of this state (22) were also investigated in 2010.

Later, the hyperentangled state has been extended to ten-qubit Schrödinger cat state in experiment, which carries the genuine multi-qubit entanglement (23). Although the previous schemes demonstrated the hyperentangled states with really high dimensions, they were only focused on two-photon states. In 2010, Gao et al. (23) generated the genuine multipartite hyperentanglement. The generation in their demonstration is composed of two steps. In the first step, a five-photon polarization entangled cat state is prepared by post-selection. Two pairs of entangled photons are produced by SPDC in the state and a single photon is prepared in state. The principle of the first step is shown in Fig. 7. Two polarizing beam splitters (PBSs), which transmit the horizontal state and reflect the vertical state , are used to post-select the five-photon cat state . In detail, the situation that each numbered spatial mode has one and only one photon kept, which corresponds to the desired state. Then, each photon is guided to a PBS, and the ten-qubit hyperentangled state is produced:

(8)

Here and signify the two spatial modes of each photon.

Figure 7: Experiment setup for generating ten-qubit five-photon hyperentangled state (23). (a) Generation of five-photon polarization entangled state with postselection. (b) Creation of polarization-spatial mode entanglement with PBS. and are two different spatial modes.

So far, the generation of entangled state is implemented via the nonlinear optical process of the SPDC in different types of nonlinear crystals. Recently, some works have been focused on waveguides due to their high efficiency and on-chip integratability. In 2014, a hyperentangled photon source in semiconductor waveguides was proposed and demonstrated, which offers an alternative path to realize an electrically pumped hyperentangled photon source (24). They utilized phase-matching in Bragg reflection waveguides to produce hyperentangled pairs through two type-II SPDC processes. The ideal hyperentangled state in mode and polarization DOFs is

(9)

Here and denote the total internal reflection (TIR) mode and Bragg mode, respectively. The fully entangled fraction of the generated state is calculated, whose maximum value can achieve 0.99.

Figure 8: Experiment principle for generating hyperentangled states between atomic ions (25). (a) Internal energy levels of ion. The and are ground states and is the excited state. and are the polarized photons emitted due to the energy decay. (b) The setup of generating spin entanglement between two atomic ions confined in separated Paul traps.

Usually, the entanglement can only be generated locally. Since the photon is one of the most ideal candidates for quantum communication, most of the previous hyperentanglement generation schemes are based on photons. Actually, other physical entity can also be used as the carrier of hyperentanglement. For example, Hu et al. (25) proposed a scheme of generating four-qubit hyperentangled state between a pair of distant non-interacting atomic ions which are confined in Paul traps. The state is entangled in both spin and motion DOFs. The atomic ions with a configuration move along one direction in the Paul trap. The principle for generating hyperentangled states between atomic ions is shown in Fig. 8. First, the ions in each trap are excited to the excited state in spin DOF. Then they decay along two possible channels and accompanied by the emission of a (with the spin ) or (with the spin ) polarized photon, respectively. Therefore, the system consisting of the spin state of the ion and the polarization of the emitting photon is in a maximally entangled state:

(10)

Then the two emitting photons from two traps are guided to a PBS. By post-selecting the case that each atom emits a single photon, the two ions are entangled in spin DOF as

(11)

Here and denote the left and the right traps, respectively, as shown in Fig. 8b. The motion DOF of both ions is initially in the ground state . Then the entanglement is transferred to the motion DOF with a sequence of laser pulses.

(12)

Finally, by repeating the first step, the following hyperentangled state can be produced,

This proposal is experimentally feasible, although it has not been demonstrated in labs.

Iii High-capacity quantum communication with hyperentanglement

iii.1 Status of Bell-state analysis for photonic quantum systems

Bell-state analysis (BSA), which is used to distinguish the four orthogonal Bell states of a two-particle quantum system in one DOF, is the prerequisite for quantum communication protocols with entanglement and it is one of the important parts in quantum repeaters. In 1999, two linear optical BSA protocols were proposed by Vaidman’s (64) and Lütkenhau’s (65) groups, respectively, where the success probability is 50%. When hyperentanglement is used to assist the analysis of Bell states, one can completely distinguish all the four Bell states of a two-photon system in one DOF. For example, in 1998, Kwiat and Weinfurter (27) proposed two complete BSA protocols by using the hyperentanglement, which can distinguish the four orthogonal Bell states in polarization DOF with the success probability 100%. In 2003, Walborn et al. (28) presented two complete BSA protocols for photon pairs entangled in one DOF with hyperentanglement, resorting to linear optical elements. Schuck et al. (29) and Barbieri et al. (30) demonstrated the complete BSA protocols in experiment by assisting hyperentanglement.

In high-capacity long-distance quantum communication, HBSA is also required to attach some important goals, especially in high-capacity quantum repeaters, teleportation of an unknown quantum state in two or more DOFs, and hyperentanglement swapping. In 2007, Wei et al. (44) proposed a HBSA protocol with linear optical elements, which can only distinguish 7 hyperentangled Bell states from 16 hyperentangled Bell states. In order to completely distinguish the 16 hyperentangled Bell states, nonlinear optical elements are required.

The complete HBSA originates from the work by Sheng et al. (36) in 2010. They proposed the first scheme for the complete HBSA of the two-photon polarization-spatial hyperentangled states with cross-Kerr nonlinearity and designed the pioneering model for teleporting an unknown quantum state in more than one DOF. In 2012, Ren et al. (37) introduced another interesting scheme for the complete HBSA for photon systems by using the giant nonlinear optics in quantum-dot-cavity systems and presented the hyperentanglement swapping with photonic polarization-spatial hyperentanglement. In 2012, Wang et al. (38) presented an important scheme for the complete HBSA for photon systems by the giant circular birefringence induced by double-sided quantum-dot-cavity systems. In 2015, Liu and Zhang (39) proposed two important schemes for hyperentangled-Bell-state generation and HBSA assisted by nitrogen-vacancy (NV) centers in resonators. Li and Ghose (41) presented a very simple scheme for the self-assisted complete maximally hyperentangled state analysis via the cross-Kerr nonlinearity and another interesting HBSA scheme (42) for polarization and time-bin hyperentanglement. Up to now, there are several important schemes for the analysis of hyperentangled states (36); (37); (44); (38); (39); (41); (40); (42); (45); (43), including the probabilistic one based on linear optical elements (44) and the one for hyperentangled Greenberger-Horne-Zeilinger (GHZ) states (45). In 2015, Wang et al. (66) demonstrated in experiment the quantum teleportation of an unknown quantum state of a single photon in multiple DOFs by implementing the HBSA of two-photon systems probabilistically with linear optical elements and ancillary entanglement sources. In 2016, Liu et al. (40) gave the original scheme for the complete nondestructive analysis of two-photon six-qubit hyperentangled Bell states assisted by cross-Kerr nonlinearity.

Here, we will introduce two high-capacity quantum communication protocols, including teleportation of an unknown quantum state of a single photon in two DOFs with hyperentanglement (36) and hyperentanglement swapping (36); (37). First, we introduce the complete HBSA for the polarization and spatial-mode DOFs of photon systems (36), which is an important technique in high-capacity long-distance quantum communication. In the second part, we introduce the quantum teleportation protocol based on polarization-spatial hyperentanglement (36). At last, a hyperentanglement swapping protocol (36); (37) is introduced for quantum repeater and quantum communication.

iii.2 Hyperentangled Bell-state analysis

The polarization-spatial hyperentangled Bell state is defined as the two-photon system entangled in both the polarization and spatial-mode DOFs, such as . Here the superscripts and represent the two photons, and the subscripts and represent the polarization and spatial-mode DOFs, respectively. and represent the two spatial modes of photon (). The polarization Bell states and the spatial-mode Bell states are defined as

(14)

where and are the Bell states in the odd-parity mode, and and are the Bell states in the even-parity mode. The 16 orthogonal hyperentangled Bell states can be distinguished completely by using polarization parity-check quantum nondemolition detectors (QNDs) and spatial-mode parity-check QNDs, assisted by cross-Kerr nonlinearity.

Figure 9: Schematic diagram of the HBSA for the spatial-mode Bell states, resorting to the cross-Kerr nonlinearities (36). (a) The spatial-mode parity-check QND for distinguishing the states from . (b) The second step for the analysis of spatial-mode Bell states, which is used to distinguish the state () with the relative phase shift from the state () with the relative phase shift . BS represents a 50:50 beam splitter, which is used to perform a Hadamard operation on the spatial-mode DOF of a photon.

The Hamiltonian of cross-Kerr nonlinearity is described as , where represents the coupling strength of the nonlinear material. () and () are the creation (annihilation) operators. With this cross-Kerr interaction, the system composed of a single photon and a coherent state can be evolved as

(15)

where and represent the Fock states that contain and photon, respectively. represents a coherent state. represents a phase shift with the interaction time . With this cross-Kerr nonlinearity, the HBSA protocol for the 16 polarization-spatial hyperentangled Bell states can be implemented with two steps, including the spatial-mode Bell-state analysis and the polarization Bell-state analysis, shown in Figs. 9 and  10, respectively.

The setup of the spatial-mode Bell-state analysis is shown in Fig. 9, which is constructed with the spatial-mode parity-check QNDs. After the two photons and pass through the quantum circuit shown in Fig. 9a in sequence, the state of the quantum system composed of the two-photon system in the spatial-mode DOF and the coherent state evolves to

(16)

Then the coherent beam is detected by an X-quadrature measurement, and the states and cannot be distinguished. Hence the Bell states can be distinguished from the Bell states with the homodyne-heterodyne measurements. If the coherent state has a phase shift (), the spatial-mode state of the two-photon system is one of the states . If the coherent state has no phase shift, the spatial-mode state of the two-photon system is one of the states .

Subsequently, the two photons and are put into the BS shown in Fig. 9b in sequence, and the spatial-mode state of the two-photon system is transformed into

(17)

After the two photons and the coherent beam pass through the cross-Kerr medium in Fig. 9b, the four spatial-mode Bell states can be distinguished by the X-quadrature measurement on the coherent beam. If the coherent state has a phase shift , , , or , the spatial mode of the two-photon system is , , , or , respectively. The initial spatial-mode state (or ) corresponds to the spatial mode or , and the initial spatial-mode state (or ) corresponds to the spatial mode or .

Figure 10: Schematic diagram of the HBSA for the polarization Bell states, resorting to the cross-Kerr nonlinearities (36). PBS represents a polarizing beam splitter, which transmits the photon in the horizontal polarization and reflects the photon in the vertical polarization , respectively. H represents a half-wave plate which is used to perform a Hadamard operation on the polarization DOF of a photon. or () represents a single-photon detector.

The setup of the polarization Bell-state analysis is shown in Fig. 10, which is constructed with the polarization parity-check QND. After the two photons and pass through the polarization parity-check QND shown in Fig. 10 in sequence, the state of the system composed of the two-photon system in the polarization DOF and the coherent state evolves to

(18)

After the X-quadrature measurement is performed on the coherent beam, the Bell states can be distinguished from the Bell states . If the coherent state has a phase shift (), the polarization state of the two-photon system is one of the states . If the coherent state has no phase shift, the polarization state of the two-photon system is one of the states .

Subsequently, the two photons and are put into the H shown in Fig. 10, and the polarization state of the two-photon system is transformed into

(19)

Then the four polarization Bell states can be distinguished by the result of four single-photon detectors. If the detectors (or ) click, the initial polarization state is (or ). If the detectors (or ) click, the initial polarization state is (or ). In this way, one can completely distinguish the 16 hyperentangled Bell states by using the spatial-mode parity-check QNDs, polarization parity-check QND, and single-photon detectors.

iii.3 Teleportation with a hyperentangled channel

The quantum teleportation protocol is used to transfer the unknown information of a quantum state between between the two remote users (67). With hyperentanglement, two-qubit unknown information can be transferred by teleporting a photon (36).

The principle of quantum teleportation protocol with hyperentanglement is shown in Fig. 11. The photon is in the state , and the photon pair is in a hyperentangled Bell state , where the photons and are obtained by the two remote users Alice and Bob, respectively. Alice can transfer the two-qubit information of photon to Bob by performing HBSA on the two photons and .

Figure 11: Schematic diagram of the quantum teleportation protocol with hyperentanglement (36).

The state of the three-photon system can be rewritten as

(20)

After Alice performs HBSA on the two photons and , photon will be projected to a single-photon quantum state in two DOFs. If the outcome of HBSA for the photon pair is , , , or , the state of photon is projected to , , , or , respectively. If the polarization (spatial-mode) state of photon pair is (), Bob should perform a polarization (spatial-mode) phase-flip operation () on photon after Alice publishes the result of HBSA. If the polarization (spatial-mode) state of photon pair is (), Bob should perform a polarization (spatial-mode) bit-flip operation () on photon . If the polarization (spatial-mode) state of photon pair is (), Bob should perform a unitary operation () on photon . Then, Bob can obtain the unknown single-photon state . Here, , , , , , and .

iii.4 Hyperentanglement swapping

Entanglement swapping is used to obtain the entanglement between two particles that have no interaction initially, and it has been widely applied in quantum repeaters and quantum communication protocols. The principle of hyperentanglement swapping is shown in Fig. 12. The photon pairs and are initially in the hyperentangled Bell states and , respectively. Here,

(21)

The photons and belong to Alice. The photons and belong to Bob and Charlie, respectively. That is, Alice shares a hyperentangled photon pair with Bob, and she also shares a hyperentangled photon pair with Charlie. The task of hyperentanglement swapping is to obtain the hyperentangled Bell state , which can be implemented by performing HBSA on photon pair .

Figure 12: Schematic diagram of the hyperentanglement swapping protocol (36); (37).

The state of the four-photon system can be rewritten as

(22)

After Alice performs HBSA on the photon pair , the correlation between the two photons can be created. If the outcome of HBSA for the photon pair is , , , or , the state of the photon pair is projected to , , , or , respectively. If the polarization (spatial-mode) state of the photon pair is (), Bob should perform a unitary operation () on photon after Alice publishes the result of HBSA. If the polarization (spatial-mode) state of the photon pair is (), Bob should perform a unitary operation () on photon . If the polarization (spatial-mode) state of the photon pair is (), Bob should perform a unitary operation () on photon . Now, Bob and Charlie can share a hyperentangled photon pair in the state .

Iv Hyperentanglement concentration

iv.1 Development of entanglement concentration

In the practical quantum communication with entanglement, the entangled photon systems are produced locally, which leads to their decoherence when the photons are transmitted over a quantum channel with environment noise or stored in practical quantum devices. Quantum repeater is a necessary technique for long-distance quantum communication and it is used to overcome the influence from this decoherence (4). In fact, the optimal way to overcome the influence on photon systems from channel noise in quantum communication is the self-error-rejecting qubit transmission (63) with linear optics as it is an active way to decrease the influence from channel noise and it is very efficient and simple to be implemented in experiment with current feasible techniques. However, this scheme (63) can only depress most of the influence from the channel noise in the process of photon distribution, as the same as the other active methods for overcoming the influence from noise (68); (69); (70). It does not work in depressing the influence of noise from both a long-distance channel and the storage process for quantum states. Moreover, quantum repeaters for long-distance quantum communication require the entangled photons with higher fidelity (usually 99%) beyond that from faithful qubit transmission schemes (about 90% 96% for a polarization quantum state of photons over an optical-fiber channel with several kilometers). That is, entanglement concentration and entanglement purification are not only useful but also absolutely necessary in long-distance quantum communication.

Entanglement concentration is used to distill some nonlocal entangled systems in a maximally entangled state from a set of nonlocal entangled systems in a partially entangled pure state (71). Before 2013, entanglement concentration is focused on the nonlocal quantum states in one DOF, such as the polarization states of photons, the two-level quantum states of atom systems, or the spins of electron systems. The first entanglement concentration protocol (ECP) was proposed by Bennett et al. (71) in 1996, which is based on the Schmidt projection (71). Also, it is just a mathematic method for entanglement concentration. In 2001, two ECPs were proposed (72); (73) with PBSs for two ideal entangled photon sources. In 2008, Sheng et al. (74) proposed a repeatable ECP to concentrate both bipartite and multipartite quantum systems, and it has an advantage of far higher efficiency and yield than those in Bennett’s ECP (71) and the PBS-based ECPs (72); (73), by iteration of the concentration process two or three times. In fact, depending on whether the parameters of the nonlocal less-entangled states are unknown (71); (72); (73); (74); (75) or known (76); (77); (78), the existing ECPs can be classed into two groups. When the parameters are known, one nonlocal photon system is enough for concentrating the nonlocal entanglement efficiently (76); (77); (78) with far higher yield than those with unknown parameters (71); (72); (73); (74). In 1999, Bose et al. (76) designed the first ECP for nonlocal entangled photon pairs in the less-entangled pure state with known parameters, resorting to the entanglement swapping of a nonlocal entangled photon pair and a local entangled photon pair. In 2000, Shi et al. (77) proposed another ECP based on entanglement swapping and a collective unitary operation on two qubits. In 2012, Sheng et al. (78) presented two ECPs for photon systems in the less-entangled states with known parameters, according to which an ancillary single photon state can be prepared to assist the concentration. In 2012, Deng (79) presented the optimal nonlocal multipartite ECP based on projection measurement. Also, some schemes for concentrating W states have been proposed (80); (81); (82). Moreover, two groups (83); (84) demonstrated in experiment the entanglement concentration of two-photon systems with linear optical elements. A review about entanglement concentration of photon systems in one DOF with cross-Kerr nonlinearity was presented in Ref. (85).

The investigation on hyperentanglement concentration began from 2013. In this year, Ren et al. (46) proposed the parameter-splitting method to extract the maximally entangled photons in both the polarization and spatial-mode DOFs when the coefficients of the initial partially hyperentangled states are known. This fascinating (novel) method is very efficient and simple in terms of concentrating partially entangled states as it can be achieved with the maximal success probability by performing the protocol only once, resorting to linear-optical elements only, not nonlinearity, no matter what the form of the known nonlocal entangled state is, what the number of the DOFs is, and what the number of particles in the quantum system is. They (46) also gave the first hyperentanglement concentration protocol (hyper-ECP) for the unknown polarization-spatial less-hyperentangled states with linear-optical elements only and another hyper-ECP (47) for nonlocal polarization-spatial less-hyperentangled states with unknown parameters assisted by diamond nitrogen vacancy (NV) centers inside photonic crystal cavities. Subsequently, Ren and Long (48) proposed a general hyper-ECP for photon systems assisted by quantum-dot spins inside optical microcavities and another high-efficiency hyper-ECP (49) with the quantum-state-joining method. In the same time, Li and Ghose (50) presented a hyper-ECP resorting to linear optics. In 2015, they also brought forward an efficient hyper-ECP for the multipartite hyperentangled state via the cross-Kerr nonlinearity (51) and another hyper-ECP for time-bin and polarization hyperentangled photons (52). In 2016, Cao et al. (53) presented a hyper-ECP for entangled photons by using photonic module system.

In this section, we overview the hyper-ECPs for high-capacity long-distance quantum communication (46), resorting to the parameter-splitting method (46) and the Schmidt projection method (71), respectively. With the parameter-splitting method (46), the hyper-ECP can be implemented with the maximal success probability (46), resorting to linear optical elements only. With the Schmidt projection method, the success probability of the hyper-ECP is relatively low with linear optical elements (46); (50); (52), and it can be improved by iterative application of the hyper-ECP process with nonlinear optical elements (48); (47).

iv.2 Hyper-ECP with parameter-splitting method

The parameter-splitting method is introduced to concentrate nonlocal partially entangled states with their parameters accurately known to the remote users (46). With this method, only one remote user has to perform local operations with linear optical elements, and the success probability of the ECP can achieve the maximal value. The ECP for polarization (spatial-mode) DOF of photon system is introduced in detail in Ref. (46). Here, we introduce the hyper-ECP for polarization-spatial hyperentangled Bell state by using the parameter-splitting method (46). That is, Alice and Bob obtain a subset of nonlocal two-photon systems in a maximally hyperentangled Bell state by splitting the parameters of the partially hyperentangled Bell states with linear optical elements only.

The partially hyperentangled Bell state is described as

where the subscripts and represent two photons obtained by the two remote users, Alice and Bob, respectively. , , , and are four real parameters that are known to the two remote users, and they satisfy the relation .

The setup of the hyper-ECP (46) for the partially hyperentangled Bell state is shown in Fig. 13b. It is implemented by performing some local unitary operations on both the spatial-mode and polarization DOFs of photon . No operation is performed on photon . To describe the principle of the hyper-ECP explicitly and simply, the four parameters are chosen as and . In other cases, the hyper-ECP can be implemented as the same as this one with or without a little modification.

Figure 13: (a) Schematic diagram of an unbalanced BS (UBS) (46). represents a wave plate which can cause a phase shift between the two spatial modes. (b) Schematic diagram of the polarization-spatial hyper-ECP with the parameter-splitting method (46). UBS represents an unbalanced beam splitter with the reflection coefficient . represents a wave plate which is used to rotate the horizontal polarization with an angle . DL denotes a time-delay device which is used to make the two wavepackets of the two spatial modes arrive at PBS (or PBS) in the same time. () represents a single-photon detector.

First, Alice splits the parameter of the spatial-mode state by performing a unitary operation on spatial mode , resorting to an unbalanced beam splitter (i.e., UBS) with reflection coefficient (shown in Fig. 13a). The state of the photon pair is changed from to . Here

(24)

If photon is not detected in the spatial mode , the spatial-mode state of the photon pair is transformed into a maximally entangled Bell state.

Subsequently, Alice splits the parameter of the polarization state by performing the same polarization unitary operations on the spatial modes and as shown in Fig. 13b. After two spatial modes and pass through PBSs (i.e., PBS and PBS) and , the state of the photon pair is transformed from to . Here

(25)

where represents the vertical polarization of photon after an operation . The wave plate is used to perform a rotate operation on the horizontal polarization .

Finally, Alice lets two spatial modes and pass through PBS, PBS, DL, PBS and PBS, and the state of the photon pair is transformed from to . Here

If photon is not detected in one of the spatial modes and , the polarization state of the photon pair is transformed into a maximally entangled Bell state. That is, the maximally hyperentangled Bell state is obtained. Here

(27)

If photon is detected in one of the spatial modes , , and , the polarization DOF or the spatial-mode DOF of the photon pair will project to a product state, which means the hyper-ECP fails. According to the detection in the spatial modes of photon , Alice can read out whether the hyper-ECP succeeds or not in theory. As the efficiency of a single-photon detector is lower than 100 %, the mistaken of a successful event caused by the detection inefficiency can be eliminated by postselection.

The success probability of this hyper-ECP is , which achieves the maximal success probability for obtaining a maximally hyperentangled Bell state from a partially hyperentangled Bell state. Moreover, this parameter-splitting method is suitable for all the entanglement concentration of photon systems in nonlocal partially entangled pure states with known parameters, including those based on one DOF and those based on multiple DOFs.

iv.3 Hyper-ECP with Schmidt projection method

Here, we mainly introduce two hyper-ECPs for polarization-spatial hyperentangled Bell states with unknown parameters (46). The first one is implemented with linear optical elements (46), which is much easier to achieve in experiment. The second one is implemented with nonlinear optical elements (47), which can improve the success probability by iterative application of the hyper-ECP.

Hyper-ECP with linear optical elements

In the Schmidt projection method, two identical photon pairs and are required, which are in the partially hyperentangled Bell states and , respectively. Here

Here the subscripts and represent two photon pairs shared by the two remote users. Alice has the two photons and , and Bob has the two photons and . , , , and are four unknown real parameters, and they satisfy the relation