Following the development of quantum computing and quantum algorithms,^[1] quantum cryptography has gradually become one of the most important research topics. Quantum secret sharing is one of the more significant branches of quantum cryptography. There are two kinds of quantum secret sharing protocols: sharing classic information and sharing quantum information. Sharing quantum information is also called the quantum information splitting (QIS) protocol. Hillery et al.^[2] first proposed a quantum information splitting protocol in 1999. The QIS is used to share secret quantum information from one sender to several distant agents through quantum teleportation, via shared entangled states and classical communication channel. Every agent can recover the secret quantum information with the help of other agents. Subsequently, the quantum information splitting protocols of an arbitrary single-particle state,^[3,4] two-particle state^[5,6] and some other types of protocols^[7,8] and applications^[9,10] have successfully been developed.

However, all of these protocols have focused on the symmetry scenario; that is, agents in the agreement have the same authority to recover the quantum secret message. Gottesman^[11] first pointed out that a more applicable protocol should include an asymmetrical case where the agent authority to recover a secret state is different. In 2010, Wang et al.^[12] successfully realized a hierarchical quantum information protocol (HQIS) for distributing an arbitrary qubit to three agents via the four-qubit entangled state proposed by Yeo and Chua.^[13] The three agents in that agreement have different authorities on the recovery of the quantum secret message. The agent in the upper grade only needs one of the other two agents to work together, while the agent in the lower grade needs the cooperation with two agents. In 2010, Wang et al.^[14] proposed a scheme with six-photon cluster state, where a boss transmits quantum secret to five distant agents who are divided into two grades. In 2011, Wang et al.^[15] proposed a multiparty HQIS protocol with a multipartite-entanglement channel and classical communication. In 2014, an HQIS of an arbitrary two-qubit state via two four-qubit cluster states was proposed by Xu et al.^[16] Some of these protocols have been implemented by using the maximally entangled state. However, an initially maximally entangled state may easily evolve into a non-maximally entangled or mixed state due to the inevitable environment effect in the actual communication. Therefore, Peng et al.^[17,18] proposed a hierarchical and probabilistic information splitting protocol that is based on the non-maximally entangled state and realized the transmission of arbitrary single-qubit. Subsequently, with an unknown eight-qubit cluster state. Bai et al.^[19] implemented a hierarchical and probabilistic QIS. In their protocol, the sender can distribute an arbitrary single-qubit state to seven distant agents who are divided into two grades.

According to Xu et al.’s protocol,^[16] we propose a hierarchical and probabilistic quantum information splitting protocol via non-maximally entangled states. In our protocol, two four-qubit cluster states are shared by four participants, forming a quantum channel. The sender transmits an arbitrary two-qubit state to three agents with two Bell measurements and broadcasts measurement results via classical channel. Here, we assume that the quantum channel is noiseless and that the classical channel is certified. Bob with high authority can successfully recover the quantum secret message by his cooperation with Charlie or David. Charlie and David have low authorities. Any one of them needs the cooperation with two agents to recover the secret. In addition, the agent needs to introduce two auxiliary particles and performs CNOT operation. Eventually, the positive operator-valued measurement (POVM)^[20] operation is performed to distinguish non-orthogonal states to obtain the secret state. Moreover, due to the symmetry of the four-qubit cluster state, we further propose a hierarchical and probabilistic quantum information splitting protocol where there are m agents with high authority and n agents with low authority. If an agent with high authority recovers the secret state, then he or she only needs the help from all the other agents in the upper grade and one of the agents in the lower grade. However, if an agent with low authority recovers the secret state, then he or she needs the cooperation with all agents.

Suppose that there are four legitimate participants Alice, Bob, Charlie, and David in the communication. Sender Alice wants to teleport an unknown two-qubit state

Alice then sends particles 2 and 6 to Bob, particles 3 and 7 to Charlie, and particles 4 and 8 to David. Alice keeps particles 1 and 5 in her hand. Consequently, the four participants share two four-particle cluster states, thereby forming the quantum channel required by the protocol. The state of the whole system is written as

Subsequently, Alice should perform two joint measurements on her pair qubits (x,1) and (y,5) by using the Bell basis {Φ^±, Ψ^±}, respectively. The four Bell states are given by

It is known that Alice can obtain one of the 16 kinds of measurement results with equal possibility. The 16 kinds of measurement results and the corresponding joint states for the three agents after the measurement are summarized in Table 1. Alice needs to broadcast the measurement results to all her agents through a classical channel. In this way, the quantum information is spilt into the entangled states shared with the three agents. According to the quantum non-cloning theorem, only one agent’s qubits can collapse into the secret state. As a result, all three agents are able to successfully recover the secret quantum message with the cooperation of other agents. Case 1

Agent Bob with high authority is assigned to recover Alice’s secret state. Without loss of generality, we assume that Alice’s measurement result is |Ψ⁺ Φ⁻⟩_x1y5, the states of the other agents will collapse into the following state:

Table 1.

Table 1.

Table 1. Relationships between Alice’s measurement results (AR) and the corresponding joint states (JS) shared with the other agents. . AR JS |Φ+ Φ+⟩x1y5 |Φ+Φ−⟩x1y5 |Φ+Ψ+⟩x1y5 |Φ+Ψ−⟩x1y5 |Φ−Φ+⟩x1y5 |Φ−Φ−⟩x1y5 |Φ−Ψ+⟩x1y5 |Φ−Ψ−⟩x1y5 |Ψ+Φ+⟩x1y5 |Ψ+Φ−⟩x1y5 |Ψ+Ψ+⟩x1y5 |Ψ+Ψ−⟩x1y5 |Ψ−Φ+⟩x1y5 |Ψ−Φ−⟩x1y5 |Ψ−Ψ+⟩x1y5 |Ψ−Ψ−⟩x1y5

Table 1. Relationships between Alice’s measurement results (AR) and the corresponding joint states (JS) shared with the other agents. .

To recover Alice’s secret state, Charlie and David should make two single-qubit measurements on their particles and transmit the measurement results to Bob via a classical channel. Obviously, if Charlie and David measure their particles by using the basis {|0⟩, |1⟩}, then their results are always correlated with each other. Thus, Bob can deduce the Charlie’s (David’s) results from David’s (Charlie’s) results. This shows that only one of the other two agents is needed to inform Bob of his measurement results. Suppose that Charlie or David’s measurement result is |01⟩, then Bob’s state will collapse into

Table 2.

Table 2.

Table 2. Relationships between Charlie or David’s measurement results (CDR) and Bob’s operations (BO). . CDR BO |00⟩ I ⊗ Z |01⟩ I ⊗ I |10⟩ Z ⊗ Z |11⟩ Z ⊗ I

Table 2. Relationships between Charlie or David’s measurement results (CDR) and Bob’s operations (BO). .

After the above operation, Bob’s state eventually becomes

To demonstrate the realization of the protocol in detail, we still make the above assumptions; that is, after a series of operations, Bob’s state is as shown in Eq. (9). It is obvious that Bob’s state is different from the secret qubits sent by Alice. To successfully obtain Alice’s secret state, Bob needs to take the initial state of two auxiliary qubits m and n as initial state |00⟩_mn and performs two CNOT operations, with qubits 2 and 6 used as the controlled qubits while the auxiliary qubits m and n as the target qubits. Now, Bob’s state becomes

This discussion is based on the assumption that Alice’s measurement result is |Ψ⁺ Φ⁻⟩_x1y5 and Charlie or Bob’s measurement result is |01⟩. Taking all of the cases into account, we can work out the success probability of this scheme to be

Case 2

The agent with low authority is able to recover the secret state. In this protocol, both Charlie and David have low power to recover the secret state. Here, we take Charlie as an example. Equation (7) can now be rewritten as

where Bob and David perform single-particle measurements on their pair qubits (2,6) and (4,8) by using the basis {|+⟩, |−⟩}, respectively. From Eq. (18), it can be seen that Charlie needs the cooperation of two agents to successfully recover the secret qubits. Without loss of generality, suppose that Bob’s measurement result is |−+⟩ and David’s measurement result is |+ +⟩, then Charlie’s quantum state will collapse into Charlie performs two Hardamard transformations on pair qubits particles 3 and 7, then Bob’s state is To recover the secret state, Charlie then performs Z operation on qubits 7 and 3 separately. The state shown in Eq. (21) transforms into

Like Case 1, Charlie needs to introduce two auxiliary particles for CNOT operation in order to successfully obtain the secret state. Charlie also needs to perform a POVM operation with different coefficients. Finally, we can obtain the probability of success in this mode to be

The relationship between the coefficient and the probability is shown with Fig. 1. No matter which kind of four-particle cluster prepared by Alice is set to be a quantum channel, we ultimately hope that the successful probability of splitting quantum information is maximum; that is, from Eqs. (23) and (17) the maximum can be obtained. In Fig. 1, the three curves correspond to different values of a and c; that is, the three curves represent different four-particle cluster states. It can be seen from the figure that no matter what kind the four-particle cluster is, the success probability of the scheme is inversely proportional to the value of ω. As mentioned previously, ω satisfies the inequality ω ≥ 16max {(bd)²,(ac)²,(bc)²,(ad)²}. Thus, the scheme is an optimal strategy if ω = 16max {(bd)²,(ac)²,(bc)²,(ad)²}. In other words, the POVM is determined after the four-particle cluster has been identified. In particular, the probability is a maximum value of 1 when a = c = 1/2 and ω = 16max {(bd)²,(ac)²,(bc)²,(ad)²}. At this time, the four-particle cluster is in the maximally entangled state.

Fig. 1.

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	Fig. 1. (color online) Relationships between the value of ω and the total success probability P, which satisfies P = (16a2c2b2d2)/ω × 16 in the three cases, with coefficients satisfying a2 + b2 = 1/2 and c2 + d2 = 1 / 2.

Due to the symmetry of the four-qubit cluster state, our protocol can be extended to multiparty agents. To illustrate this point more clearly, we first consider the case where there are two agents called Bob_i (i = 1,2) with high authority and three agents called Charlie_j (j = 1, 2, 3) with low authority. The sender Alice prepares two two-qubit entangled states in Eq. (2) and eight single qubits whose initial states are all in the state |0⟩. Alice then performs some CNOT operations to entangle these single qubits into cluster states. These two entangled states then become

Subsequently, Alice sends the particles x_{i + 1} and y_{i + 1} to Bob_i (i = 1, 2), and the particles x_{j + 3} and y_{j + 3} to Charlie_j (j = 1, 2, 3). Alice keeps the particles x₁ and y₁ in her hand. Consequently, the four participants share two six-qubit clusters, thus forming a quantum channel required by the protocol. The state of the whole system is written as

To distribute |δ⟩_xy in Eq. (1) to five agents, the sender Alice should perform two joint measurements on her pair qubits (x,x₁) and (y,y₁) by using the Bell basis {Φ^±, Ψ^±}, respectively. It is clear that Alice can obtain 16 kinds of measurement results. Like the protocol in the three agents, Alice should also broadcast the result to five agents. Case 1

The agent with high authority is assigned to recover Alice’s secret state. Here, we take Bob₁ as an example. Without loss of generality, we assume that Alice’s measurement result is |Ψ⁺Ψ⁺⟩_xx1yy1, the states of the other agents will collapse into the state

To recover Alice’s secret state, the agents with low authority should make two single-qubit measurements on their particles by using the basis {|0⟩,|1⟩}. Their measurement results are always correlated with each other. This shows that only one of these agents is needed to inform Bob₁ of his measurement results. Bob₂ with high authority should measure his particles by using the basis {|+⟩, |−⟩}. Suppose that the measurement result of the agent with low authority is |00⟩ and Bob₂’s is |+ −⟩, then the state of Bob₁ will be

Case 2

An agent with low authority is to recover the secret state. Here, we take Charlie₁ as an example. Equation (27) can now be rewritten as

Bob_i (i = 1, 2) and Charlie_j (j = 2, 3) perform single-particle measurements on their pair particles by using the basis {| + ⟩, | −⟩}, respectively. From Eq. (30), it can be seen that Charlie₁ needs the cooperation with all the other agents to successfully recover the secret qubits. Without loss of generality, suppose that the measurement result of the agents with high authority is |+ − + +⟩ and the measurement result of the agents with low authority is |+ − + −⟩, then Charlie₁’s quantum state collapses into Charlie₁ performs Hardamard transformations and Pauli operations on his particles according to other agents’ measurement results. Using the same method, Charlie₁ introduces two auxiliary qubits and performs a POVM operation with different coefficients. Finally, we can obtain the probability of success in this mode to be

The process of the protocol is similar to that of the protocol with five agents. In this multiparty protocol, let agents Bob₁, Bob₂, . . ., and Bob_m be in high grade, and the agents Charlie₁, Charlie₂, . . ., and Charlie_n be in low grade. Alice keeps the particles x₁ and y₁ in her hand. The whole system is

Alice wants to teleport the state |δ⟩_xy shown in Eq. (1). The sender Alice performs two Bell measurements on pair particles (x,x₁) and (y,y₁), respectively. If an agent with high authority recovers the secret state, then all of the other agents with high authority need to make two single-qubit measurements on their particles by using the basis {|+⟩, |−⟩}, and only one of the agents with low authority needs to make two single-qubit measurements on his or her particles by using the basis {|0⟩, |1⟩}. However, if an agent with low authority recovers the secret state, then all of the other agents need to make two single-qubit measurements on their particles by using the basis {|+⟩, |−⟩}. Whether an agent with high authority or an agent with low authority recovers the secret state or not, the probability of success is

Because of the no-cloning theorem and entanglement monogamy, the only way for an outside eavesdropper^[21–25] to obtain the secret state is to adopt the intercept-resend attack. In particular, because which one of the agents will possess the secret state is previously undefined in the scheme, it is supposed that the eavesdropper intercepts all of the particles sent to the agents and resends fake particles to the agents. To keep the quantum correlation among the agents, the eavesdropper prepares two cluster states in Eq. (33), such as the sender in the protocol among the multiparty agents.

However, the quantum entanglement between the agents is destroyed. The agents can easily detect such an attack by performing local measurements on their particles. For example, they all perform the measurement by using the basis {|0⟩, |1⟩}. Under the attack by the eavesdropper, there is no correlation between the measurement results of the sender and the agents. However, in a no-eavesdropping case, where the measurement results of the sender and the agent with high authority are always correlated with each other, and the measurement results of the sender and the agents with lower authority are always anti-correlated with each other. Thus, the eavesdropping attack can always be detected in this way and a subset of entanglement channels will be sacrificed. As a matter of fact, most of entanglement-based quantum communication schemas need the users to utilize quantum correlations and sacrifice a subset of entanglement channels to check their security against an eavesdroppers’ interceptions.

We can exploit the definition, and the total efficiency for quantum state sharing can be defined as

In our protocol, we need two non-maximally entangled states which can be prepared by some methods.^[26,27] According to Ziman et al.’s research,^[28] the POVM operation can also be realized. Thus, our protocol is feasible. Moreover, the operations we perform during communication are all the basic operations which are less complex.

The proposed protocol uses a non-maximally entangled state to implement a hierarchical and probabilistic quantum information splitting protocol, whereas Xu’s protocol is implemented by using the maximally entangled state. Although the protocol of the maximally entangled state can recover the secret information successfully and deterministically, the maximally entangled state is easily influenced by the environment and becomes a non-maximally entangled state or a mixed state in practice. Compared with Peng et al.’s hierarchical and probabilistic quantum information splitting protocol, our proposed protocol achieves the sharing of an arbitrary two-qubit state. Moreover, our protocol can be extended to multiparty participants.

In general, we use a non-maximally entangled state to implement a hierarchical quantum information splitting protocol that shares arbitrary two qubits among three agents. In addition, different agents have different authorities for restoring the secret state. Bob with high authority only needs the cooperation of Charlie or David, while Charlie or David with low authority need the cooperation of the other two agents. In addition, the agent recovers the secret state by introducing two auxiliary particles through using the POVM operations. Due to the symmetry of the cluster state, we further extend the protocol into a multiparty protocol. When the agent with high authority is designed to recover the secret state, he will need the help of all the agents with high authority and one of the agents with low authority. However, the agent with low authority needs to cooperate with all of the agents to recover the secret state.