Quantum entanglement is a significant physical resource which plays a fundamental role in many meaningful applications of quantum information processing.^[1] For example, quantum entanglement can be used in quantum teleportation,^[2] quantum key distribution,^[3] quantum secure direct communication,^[4,5] and some other important quantum communication protocols.^[6–18] Hence, entanglement is considered to be the information carrier in quantum information and should be created first.

Typically, there are two types of entanglements. The first one is encoded in discrete variables, such as horizontal and vertical polarization states of photons.^[19] The second one is encoded in continuous variables, such as the entangled coherent state (ECS).^[20] In recent years, there is another type of entanglement which has been widely studied. It is the hybrid entanglement.^[21–29] The hybrid entangled state (HES) means that the entanglement is generated between different degrees of freedom of a particle pair. In 2006, van Loock et al. proposed the first hybrid quantum repeater protocol using bright coherent light.^[21] In 2013, Sheng et al. first discussed the entanglement purification for the hybrid entangled state.^[26] Recently, an important work about the generation of a hybrid entangled state was reported.^[27] Interestingly, Lee and Jeong put forward a new type of hybrid entanglement.^[28] Based on the HES, they proposed near-deterministic quantum teleportation and resource-efficient quantum computation. In their protocol, a qubit is encoded in the combination of polarization states and coherent states. The orthogonal basis to define an optical hybrid qubit is {|0_L〉 = |+〉|α〉, |1_L〉 = |−〉|−α〉}, where and | ± α〉 are coherent states with amplitudes ±α. In particular, they showed that near-deterministic quantum teleportation can be performed using linear optics and hybrid qubits.

In quantum communication protocols, the quantum channel must be maximally entangled. Unfortunately, the maximally entangled states usually collapse into nonmaximally entangled states or even mixed states because of the imperfect channel in the practical process of storage and transmission. In order to obtain the high quality and secure quantum communication, we should first repair the degraded quantum entanglement channel. Entanglement concentration is an important method that we can distill the maximally entangled states from the less-entangled states.^[30] In 1996, Bennett et al. proposed the first entanglement concentration protocol (ECP) with collective measurement, which is based on the Schimidt decomposition.^[30] Until now, many ECPs have been proposed.^[30–80] For example, Zhao et al. proposed ECPs based on the linear optical element.^[33] In 2008, Sheng et al. presented a recyclable ECP based on cross-Kerr nonlinearities^[34] and the efficiency of their protocol was improved obviously. In 2012, Sheng and Deng first presented the basic principle of efficient single-photon-assisted ECP for partially entangled photon pairs.^[35,36] Subsequently, with the help of the single qubit, the ECPs for arbitrary W states,^[51–62] cluster states,^[63–69] cat states,^[71] χ type entangled states,^[72] hyper-entangled states,^[74–77] and hybrid entangled states^[78] were also proposed.

In this paper, we will focus on the entanglement of the HES, which has the different form of Ref. [78]. We propose three ECPs for different types of HESs. The first ECP is to concentrate the Bell-type HES. The second and the third ECPs are to concentrate the W-type and the cluster-type HES, respectively. In the first ECP, we exploit an ancillary single coherent state and use the beam splitter (BS) to complete the task. In the second ECP, we require two coherent states, and in the third ECP, we require a single coherent state and a pair of ECS to realize the concentration. All protocols are based on the linear optics, which makes them feasible in future experiments.

This paper is organized as follows. In Section 2, we will describe the ECP for an arbitrary less-entangled Bell-type HES. In Section 3, we will propose the ECP for an arbitrary less-entangled W-type HES. In Section 4, we will discuss the ECP for an arbitrary less-entangled cluster-type HES. In Section 5, we will present a discussion.

As shown in Fig. 1, Alice and Bob share the less-entangled HES of the form

Fig. 1.

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	Fig. 1. Schematic diagram of concentrating an arbitrary less-entangled Bell-type HES. Alice and Bob share the HES, and own the hybrid states denoted as a and b, respectively. Bob prepares the ancillary single coherent state c. BS1 is used to perform a parity check measurement. BS2 is used to obtain the Bell-type maximally HES.

Therefore, the whole system can be written as

In this section, we will extend the above protocol to the case of arbitrary W-type HES of the form

Fig. 2.

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	Fig. 2. Schematic diagram showing the principle of the proposed ECP for arbitrary W-type HES. Alice, Bob, and Charlie share the HES, and own the hybrid qubits a, b, and c, respectively. Alice and Charlie prepare the ancillary single coherent states d and e, respectively.

From Fig. 2, the whole less-entangled HES combined with the two single coherent states can be written as

Then they let the coherent states in spatial modes a₁, d₁, e₁, and c₁ pass through the BSs, respectively. Therefore, the whole system can be rewritten as

In this section, we will discuss the ECP for the cluster-type HES, where the initial less-entangled HES can be written as

As shown in Fig. 3, subsequently, they let the coherent state in spatial modes b₁ and e₁ pass through BS1. Meanwhile, they also let the coherent state in spatial modes c₁ and f₁ pass through BS3. The state |φ〉_ab1c1de1f1 evolves to

In the second step, they first prepare the ancillary single coherent state in spatial mode g as shown in Eq. (21). The state |φ〉_ab3c3d combined with |φ〉_g can be written as

The success probability to obtain the maximally entangled HES from the initial partially entangled state is P₃ = 4|N₃N₄ξγβδ|².

Thus far, we have described our protocols which mainly use BS and QND to complete the task. We described the ECPs for Bell-type HES, W-type HES, and cluster-type HES, respectively. With the post-section principle, we can ultimately obtain a maximally entangled HES with some probability. In Figs. 4–6, we show the success probability of each ECP.

Fig. 4.

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	Fig. 4. The success probability P1 of obtaining a maximally entangled Bell-type HES. P1 is altered with the coefficient γ of the initial partially entangled state, where γ ∈ (0, 1). The α is the amplitude of the coherent state.

Fig. 5.

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	Fig. 5. The success probability P2 of obtaining a maximally entangled W-type HES. P2 is altered with γ and β, which are coefficients of the initial partially entangled state, where γ ∈ (0, 1), β ∈ (0, 1), and γ2 + β2 < 1. The amplitude of the coherent state (α) is set to 2.

Fig. 6.

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	Fig. 6. The success probability P3 of obtaining a maximally entangled cluster-type HES. P3 is altered with β, where we assume that ξ = 1/2, γ = 1/2, and . The amplitude of the coherent state (α) is set to 2.

In Fig. 4, we show the success probability P₁ altered with γ. We let α = 0.5, 1, and 4 respectively. In Fig. 5, we show the success probability P₂ altered with γ and ξ. In this ECP, P₂ can reach the maximum value 1/4 with . In Fig. 6, we show the total success probability P₃ altered with β. We let γ = ξ = 1/2. It shows that P₃ can reach the maximum value 1/8 with β = 1/2. Meanwhile, the other three coefficients are equal to 1/2.

In our ECPs, the BS plays a key role to realize the concentration. We take the first ECP for example. As shown in Fig. 1, we choose the case that the spatial mode c₂ has no photon. It means that we pick up the even parity states |α〉|α〉 and |−α〉|−α〉, which is similar to the previous ECPs.^[34,51] We should point out that a coherent state |α〉 also has some probability to have no photon. Then, it will induce an error. Fortunately, the error probability is |〈0|α〉|². It is about 10⁻⁵ if α = 2, which can be neglected. In our ECPs, the last step is to decrease the amplitude of the coherent state to recover it to the form of the original state. Actually, such an operation is not always necessary. In the practical transmission, the coherent state may suffer from the photon loss, which will decrease the amplitude of the coherent state. We should exploit the amplification technology to increase the coherent state. Interestingly, our ECPs can also increase the amplitude of the coherent state. The increased amplitude of the coherent state will become an advantage of these ECPs. In the second and the third ECPs, the whole protocols are both divided into two steps. Actually, in a practical experiment, both steps should be performed simultaneously because of the post-selection principle.

We have proposed three ECPs for three types of HES, respectively. First, we described the ECP for the Bell-type HES with arbitrary coefficients, and the probability of the ECP is 2|N₀γβ|². Second, we introduced the ECP for the W-type HES. We used two single coherent states to obtain the maximally entangled W-type HES with the probability of 3|N₁N₂γβξ|². Finally, we described the ECP for the cluster-type HES. With the help of a pair of auxiliary particles of ECS and an auxiliary single coherent state, we obtained the maximally entangled cluster-type HES, and this ECP has a success probability 4|N₃N₄ξγβδ|². These ECPs are all based on the linear optics which is feasible in future experiments. We hope that these ECPs based on HESs are useful in many applications of quantum information.