Asymmetrical hyperentanglement concentration for entanglement of polarization and orbital angular momentum

1. Introduction

Quantum entanglement is a unique physical nonlocal phenomenon which is considered to be the most nonclassical manifestations of quantum formalism. The quantum state is entangled, if quantum state with pairs or groups of particles must be described as a whole, which means each particle cannot be described independently and the measurement of one particle will change the states of the rest particles. The entangled state, especially the maximally entangled state, is an important physical resource which has been widely exploited in quantum information process (QIP), such as quantum computation [1], quantum key distribution (QKD) [2, 3], quantum teleportation [4], quantum dense coding [5, 6], quantum security direct communication (QSDC) [7–11] and so on. In addition, the entangled photons and entanglement swapping [12] are exploited in the quantum repeater to improve the distance of quantum communication due to the limitations of current technologies in the quantum communication or quantum network. The entangled state are composed of different kinds of particles in a single degree of freedom (DOF), e.g. polarization, energy, spatial, momentum direction and so on. Different DOFs or different kinds of particles can also form entangled states which are called hyperentanglement [13] and hybridentanglement, respectively. The hyperentangled quantum state has attracted much more attention due to its extensive applications in quantum communication and quantum information processing. For example, it can be used to improve the channel capacity of quantum communication, assist in completing the Bell state analysis [14–16], help implement the deterministic entanglement purification protocols [17–19] (EPP), demonstrate basic physical problems [20] and so on. Several kinds of DOFs assisting QIPs have been reported in theory and experiments, such as polarization-time-energy [16, 21], polarization-momentum [22–23], polarization-spatial modes [17, 21, 24, 25], and polarization-spatial-time-bin entanglement [26–28]. Orbital angular momentum (OAM), especially the entanglement of OAM [29] is a novel kind of quantum resources for the multi-free degree space property which can beat the channel capability limit for linear optic QIP. It allows one to encode a single photon with a higher-dimensional quantum space for the OAM can take any integer value [30–32] which has led to many applications in qutrit quantum communication protocols [33, 34]. In 2008, Barreiro et al. [34] demonstrated the quantum advantage of hyperentanglement resorting to photons entangled in spin and OAM simultaneously. They proposed a dense coding experiment with linear optics which breaks through the channel capacity limit and designed an apparatus for spin-obit Bell-state analysis (BSA). In 2015, D. Bhatti et al. [35] investigated two- and multi-photon entangled states using OAM and spin angular momentum and presented the experimental setup to produce two-photon polarization-OAM entangled states.

In practice, the entangled photon pairs will inevitably interact with environment during their distribution or storage processes. The decoherence and loss of quantum states will influence the efficiency and the security of quantum communication processes, which will limit the communication distance. Several protocols are proposed to enhance the communication efficiency and increase distance, such as quantum repeater [36, 37], quantum error correction code (QECC) [38], entanglement purification [39–44], decoherence-free subspace (DFS) [45–50] and so on. Entanglement concentration is also an effective way to preserve the fidelityof entanglement channels, which is employed to distill maximally entangled states from an ensemble of partially entangled pure states [51]. Many kinds of entanglement concentration protocols have been proposed and discussed based on different quantum states [52–58]. In 2003, A. Vaziri et al. [59] presented the experimental realization of higher dimensional entanglement concentration of OAM entangled photons. The purification of hyperentangled states considering different physical systems and degrees of freedom has attracted more and more attention in recent years. Concentration of hyperentangled states with different physical systems and DOFs has attracted much more attention recently [2–28, 60–62]. In 2012, Chen [63] presented a hyperentanglement concentration protocol (Hyper-ECP) and a Shannon dimensionality measurement protocol for the polarization-OAM partially hyperentangled states with known parameters. Both of the remote parties performed local operations on the two DOFs of a photon. Here, we propose practical asymmetrical Hyper-ECPs of polarization-OAM entanglement with known and unknown parameters in different initial states, respectively.

In this paper we offer two Hyper-ECPs of two-photon entanglement with OAM and polarization DOFs. The first one is to concentrate one maximally hyperentangled Bell-like state from two partially hyperentangled states with unknown parameters. The second concentration protocol has higher success probability and only needs one partially entangled state with known parameters. Both of the protocols can be extended to multi-photon hyperentanglement concentration as in [61], and the efficiencies should be the highest in different cases. Our protocols can be exploited in long distance communication effectively and expediently.

2. Asymmetrical hyperentanglement concentration protocol with unknown parameters

Suppose the nonlocal two photons are partially hyperentangled in polarization and OAM DOFs, initially. The state can be written as [21]

Here$|H〉$ and$|V〉$ represent the horizontal state and vertical state of polarization DOF, respectively.$|l〉$ and$|-l〉$ denote the OAM eigenstates of photon in Laguerre-Gauss mode [64, 65] with l helicity but without radial dependence. The subscripts A and B make a distinction between the photons belonging to Alice and Bob who have a great distance off. α, β, δ and η are normalized constants satisfying$\left|\alpha {|}^2+\right|\beta {|}^2=1$ and$\left|\delta {|}^2+\right|\eta {|}^2=1$. In most of the QIPs, the maximally hyperentangled Bell-like states with$\alpha =\beta =\frac{1}{\sqrt{2}}$ and$\delta =\eta =\frac{1}{\sqrt{2}}$ are used to transmit information as quantum channel. So we wish to concentrate the following state from Eq. (1)

When α, β, δ and η are unknow parameters, the Hyper-ECP can be implemented with the help of another partially hyperentangled state Eq. (1). At the very beginning of Hyper-ECP, the two pairs of two-photon state as Eq. (3) are prepared by the source and shared by two remote users, Alice and Bob, respectively.

The subscripts A₁, A₂, B₁, and B₂ are used to distinguish the four photons. The photons A₁, A₂ (B₁, B₂) are send to Alice (Bob) through different paths as shown in Fig. 1.

 

Fig. 1 The schematic illustration of the Hyper-ECP with unknown parameters. BS: beam splitter; PBS: polarization beam splitter;

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FBS: polarization beam splitter which$|F〉=\frac{1}{\sqrt{2}}(|H〉+|V〉)$ transmitted and$|S〉=\frac{1}{\sqrt{2}}(|H〉-|V〉)$ reflected;$D_1^A,D_2^A,D_1^B,D_2^B$: single photon detectors which distinguish$|F〉$ and$|S〉$; FBH: forked binary holograms, which can divide photon into upper and lower path based on its OAM of+l and$-l$ respectively, and its OAM changed into zero; SLM: spatial light modulator, which produces OAM with values$+l\left(-l\right)$;$R_i^l$: adding a phase factor i to a quantum state with an OAM number of$-l$;${\sigma }_{x\left(z\right)}^p$:${\sigma }_{x\left(z\right)}$ on polarization DOF;${\sigma }_{x\left(z\right)}^l$:${\sigma }_{x\left(z\right)}$ on OAM DOF.

First of all, Alice and Bob make the${\sigma }_x^p$ (a Pauli operator σ_x acting on the polarization DOF) operation on A₂ and B₂, respectively. The quantum state should be written as following,

Then Bob sends photons B₁ and B₂ to a PBS which consolidates them and makes a parity comparison of polarization states by postselection. The photon in horizontal state passes through the PBS and the vertical one is reflected. The even-parity polarization section of underline in Eq. (4) is chosen by the postselection process, and the following discuss bases on this case. It is noted that the photons in$|HH〉$ will exchange the route after PBS as described in Eq. (5).

The superscript1 and2 denote the two routes following PBS. The following process is showed in Fig. 1, and we will give a particular introduction later. On Alice’s side,$A1$ and$A2$ photons first enter$\pm l-$OAM splitters, which consist of forked binary holograms (FBHs) and single-mode fiber [34]. The action of the FBH is a binary plane-wave phase grating, which divides an incoming photon in the state$|l〉(|-l〉)$ into “up” (“down”) path. The state of photon is transformed into Gaussian state with0 OAM. The state of the photons is following,

u, u^′, d, and d^′ are used to distinguish different lines in Fig. 1. The photon with l ($-l$) OAM enters the u (d) route after passing through FBH${ }_1$ or u^′ ($d^{\prime }$) route following FBH${ }_2$, and its OAM is changed into zero. The photons in line u ($u^{\prime }$) pass through a spatial light modulator (SLM) which changes the Gaussian beam into OAM of+l ($-l$) before getting into BS${ }_1$. After the photon in line d or d^′ passes through the beam splitter (BS) BS${ }_2$, it is detected by special single photon detectors (SPDs) D${ }_1^A$ and D${ }_2^A$. The SPDs can distinguish the$|F〉$ and$|S〉$ polarization states of the photons, where$|F〉=\frac{1}{\sqrt{2}}(|H〉+|V〉)$ and$|S〉=\frac{1}{\sqrt{2}}(|H〉-|V〉)$ are two superposition states of horizontally and vertically polarized states. After passing through the devices BS${ }_1$ and BS${ }_2$, the photons in line u and d^′ or d and u^′ will be chosen by postselection process. On the other hand, Bob makes R_i operation on the OAM DOF ($R_i^l$) of photon in line1, which will change the state$|-l〉$ into$i|-l〉$ and keep state$|l〉$ constant. Then the photon passes through a Mach-Zehnder (M-Z) interferometer to perform${\sigma }_x^l$ operation on the photons in the horizontal state, and the vertical state photons do not operate. Photon in line2 passes through the FBH first, which will change it into Gaussian state. Then Bob makes a special polarization measurement as Alice after the photon passing the BS${ }_3$. The non normalized quantum state before the SPDs can be represented as follows,

$a1$,$a2$ and b denote different routes showed in Fig. 1 in detail.$D_1^A$,$D_2^A$,$D_1^B$ and$D_2^B$ are also used to distinguish the routes followed by different SPDs. The maximally hyperentangled Bell-like state as Eq. (2) can be obtained by postselecting the situations in which the detectors$D_1^A\left(F\right)$,$D_1^B\left(F\right)$click and the photon passes line$a1$, or$D_2^A\left(F\right)$,$D_1^B\left(F\right)$ click and the photon passes line$a2$ with a success probability of$|\alpha \beta \delta \eta {|}^2/4$. In order to achieve the highest probability, the two users can make proper operations according to the measurement results of the SPDs. For example, if the$D_1^A\left(S\right)$ ($D_2^A\left(S\right)$) detector clicks, Alice makes a σ_z operation on polarization DOF (${\sigma }_z^p$) [66] on the photon which passes through line$a1$ ($a2$). Bob makes σ_z operation on the polarization DOF (${\sigma }_z^p$) when the SPD$D_1^B\left(S\right)$ clicks. The operation via measured results of$D_1^A$,$D_2^A$,$D_1^B$, and$D_2^B$ is shown in TABLE I. After using single-photon detectors and postselection, the success probability will be enhanced to the highest$4|\alpha \beta \delta \eta {|}^2$ in the unknown parameters case. The success probability with$\left|\alpha \right|\left(\in \left[0,1\right]\right)$ and$\left|\delta \right|\left(\in \left[0,1\right]\right)$ is shown in Fig. 3(a).

Table 1. The Relationship of Operations and Measurement Results of $D_1^A$, $D_2^A$, $D_1^B$, and $D_2^B$.

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3. Asymmetrical hyperentanglement concentration protocol with known parameters

In some of the practical process, users may known the values of parameters α, β, δ, and η. For example, Alice and Bob obtained parameter information by state estimation. The nonlocal state estimation of hyperentanglement has been well studied [63, 67, 68]. In this case, they may develop more efficient concentration protocols than the unknown parameters case. We will introduce a Hyper-ECP and give a brief discussion of the probability in this section. Each one of Alice and Bob initially have one photon of the hyperentangled state as in Eq. (1). Without loss of generality, we assume$\left|\alpha \right|\le \left|\beta \right|$ and$\left|\delta \right|\le \left|\eta \right|$ in the following discussion. In our asymmetrical protocol, the concentration of polarization DOF and OAM DOF can be implemented by Alice and Bob, respectively. It can also be accomplished by one of them. The concentration devices of the two DOFs is shown in Figs. 2(a) and 2(b), respectively. Here we discuss the first case as an example. The later case has the same result.

In order to distill the maximum polarization entangled state, Alice and Bob allow their photons to pass through the devices in Figs. 2(a) and 2(b), respectively. UBS represents an unbalanced beam splitter with the transmittance$T=\frac{|\alpha {|}^2}{|\beta {|}^2}$ and reflectivity$R=\frac{\left|\beta {|}^2-\right|\alpha {|}^2}{|\beta {|}^2}$. It can be implemented by a adjustable beam splitting device which has been discussed a lot in other work [60, 61, 69]. It can also be replaced with optical attenuator in our protocol. Photon passes by the PBS${ }_2$ when the detector D’ is clicked. Then it is guided to the input port of the devices in Fig. 2(b), which concentrate the entangled state on OAM DOF. The UBS in Fig. 2(b) is an unbalanced beam splitter with the transmittance$T^{\prime }=\frac{|\delta {|}^2}{|\eta {|}^2}$ and reflectivity$R^{\prime }=\frac{\left|\eta {|}^2-\right|\delta {|}^2}{|\eta {|}^2}$. The non normalized quantum state postselected by photon appeared in path 1 is

The UBS can be changed into adjustable intensity modulator (AIM) [70] with attenuation rate R. The theoretical total success probability of our hyper-ECPs with known parameters is$4|\alpha \delta {|}^2$if Alice makes a${\sigma }_z^l$ operation on the photon in path 2, which is higher than the unknown parameters case, and also the maximum value of known parameters case [27, 28, 61]. Therefore, if the number of initial states is large, it is necessary to perform state estimation before concentration, which makes the protocol more efficient and practical. The success probability of this case with$\left|\alpha \right|\left(\in \left[0,1\right]\right)$ and$\left|\delta \right|\left(\in \left[0,1\right]\right)$ is shown in Fig. 3 (b).

 

Fig. 2 The schematic illustration of the Hyper-ECP for an initial state with known parameters. (a) This part is used to concentrate the Bell state of polarization DOF. UBS₁ is an adjustable beam splitting with transmittance T, and can be replaced by an adjustable intensity modulator. (b) This part is used to concentrate the maximal entangled state of OAM. The UBS₂ has Tʹ transmittance here.

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Fig. 3 (a) and (b) depict the success probability of our hyper-ECP for each two-photon four-qubit system with unknown and known parameters, respectively. The dash line in (b) corresponds to the highest probability 12.5% in unknown parameters case.

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4. Discuss and summary

Hyperentanglement concentration is an effective solution for improving the entanglement of nonlocal photon systems. We have discussed the hyperentanglement concentration protocols in the case of known parameters and unknown parameters, respectively. The two-photon four-qubit hyperentangled state is in the spin-orbit modes. The success probability of the unknown parameters and known parameters initial states are$4|\alpha \beta \delta \eta {|}^2$ and$4|\alpha \delta {|}^2$ ($\left|\alpha \right|<\left|\beta \right|$ and$\left|\delta \right|<\left|\eta \right|$), respectively. Both of their probability reach the theoretical maximum value by using the improved measurement device and postselection. The concentration with known parameters initial states has a higher success probability and it only requires one copyof the less-entangled state. Therefore, if the number of states to be concentrated is large, the state estimation is necessary.

The success probability discussed above is based on the ideal situation, which means the efficiency of the elements such as PBS, BS are perfect. We also assume the efficiency of the SPDs is100%. In practice, they do not work ideally. The imperfect elements and detectors will decrease the probability of the hyper-ECPs. In this case, the two parties can still obtain the maximally hyperentangled states by postselection and the photon detector. Single Photon detector can be added to the$a2$ path. If no photons are detected, the maximum hyperentangled state can be concentrated from the post-selection of the$a1$ path. The generalized Shannon dimensionality [63] can be introduced to evaluate the performance.

In long-distance high-capacity quantum communication with multi-parties, the hyperentangled GHZ state of multi-photon system is an important resource. Both methods can be extended to concentrate the N-photon partially hyperentangled GHZ state [27, 28, 61]:

Here the subscripts$1,2,\cdots ,N$ denote the N photons which are held by N remote parties, respectively. For the first case, the parameters of the initial sates are unknown. All of the users share two identical copy of the partially entangled states$|\varphi {〉}_{1,2,\cdots ,N}$ and$|\varphi {〉}_{1^{\prime },2^{\prime },\cdots ,N^{\prime }}$. Two of the users named Alice and Bob do the same work as Alice and Bob in our first protocol. Each one of the other parties first performs$R_i^l\left({\sigma }_x^p\right)$ operation on his or her first (second) photon. Then the second photon passes through the same device as the device in path 2 of Fig. 1. In order to obtain the maximum entangled state, each user operates on the first photon based on the measurement result of the second photon. Operations$\{I$,${\sigma }_z^p$,${\sigma }_z^p{\sigma }_z^l$ and${\sigma }_z^l\}$ correspond to measurements$\{D_1^i\left(F\right)$,$D_1^i\left(S\right)$,$D_2^i\left(F\right)$ and$D_2^i\left(S\right)\}$ ($i=1,2,\cdots ,N-2$ denotes the ith user other than Alice and Bob), respectively. It is easy to see the success probability to obtain the maximally hyperentangled GHZ state as in Eq. (9) is the same as that of the two-photon Hyper-ECP.

For the second case, only one copy of partially hyperentangled state is need to concentrate the maximally hyperentangled state with known parameters. Two parties concentrate the polarization and OAM DOFs as showed in Figs. 2(a) and 2(b), respectively, and the other$N-2$ parties do nothing. Both of the operations can also be performed by one party. The success probability in this case has the same success probability as two-parties protocol. Moreover, other maximally hyperentangled Bellstates or GHZ states can also be obtained by performing appropriate single-photon operations. It is easy to get the conclusion that both of our protocols can be extended to the multi-photon hyperentangled system without increasing the experimental difficulty and reducing the success probability.

In summary, we have proposed two hyper-ECPs for two-photon four-qubit states entangled in two DOFs in two kinds of conditions. Both of the protocols can be used to concentrate the hyperentanglement of N photons in GHZ states. Only linear-optical elements, such as BS, PBS, SLM, and FBH, are employed to accomplish the concentration work, which makes our protocol more convenient to be implemented in current technologies. Different from the protocol in [63], the concentration of polarization DOF and OAM DOF are accomplished independently by Alice and Bob respectively in our work. This will greatly reduce the amount of per user’s operations and the number of devices that each photon passes through during the quantum information process. The total efficiency in practice will be increased. All these characteristics show that our protocol is practical and useful to improve the entanglement with nonlocal multi-photon polarization and OAM DOFs in the long-distance high-capacity quantum information processing.