With the emergence of classical communication security problems, quantum communication has been studied more extensively. In this paper, a novel probabilistic hierarchical quantum information splitting protocol is designed by using a non-maximally entangled four-qubit cluster state. Firstly, the sender Alice splits and teleports an arbitrary one-qubit secret state invisibly to three remote agents Bob, Charlie, and David. One agent David is in high grade, the other two agents Bob and Charlie are in low grade. Secondly, the receiver in high grade needs the assistance of one agent in low grade, while the receiver in low grade needs the aid of all agents. While introducing an ancillary qubit, the receiver’s state can be inferred from the POVM measurement result of the ancillary qubit. Finally, with the help of other agents, the receiver can recover the secret state probabilistically by performing certain unitary operation on his own qubit. In addition, the security of the protocol under eavesdropping attacks is analyzed. In this proposed protocol, the agents need only single-qubit measurements to achieve probabilistic hierarchical quantum information splitting, which has appealing advantages in actual experiments. Such a probabilistic hierarchical quantum information splitting protocol hierarchical is expected to be more practical in multipartite quantum cryptography.

Quantum information splitting, one of the core contents of quantum information science, means that secret information is split in some way and each sub-secret is managed by different agents. Only legal agents can work together to recover secret information. The concept of quantum information splitting and quantum state sharing [

Quantum information splitting is possible if using a non-maximally entangled quantum source. In this condition, the success rate is probabilistic. Hence, this kind of protocol is referred to as probabilistic quantum information splitting protocol (PQIS). In the references [

Researchers have focused on the diversity of feasible entangled states, which could realize probabilistic HQIS protocol, such as GHZ states [

In this work, a novel probabilistic hierarchical quantum information splitting protocol (probabilistic HQIS) is designed to teleport an arbitrary one-qubit secret state

In the protocol, something needs to be noticed. First, the shared quantum source and secret state are both uncertain and probabilistic. So, the receiver can only recover the secret state probabilistically under the cooperation of other agents. Apart from sender Alice’s Bell measurement, the other agents Bob, Charlie and David only need to perform single-qubit measurement on his own qubit while one of them could recover the secret state. Second, for the sake of the secret state’s recovery, the receiver often needs to entangle one or more auxiliary qubits with the receiver’s qubits in the probabilistic schemes. In the protocol, after introducing one auxiliary qubit

We assume the particle 1, 2, 3 and 4 are entangled

Assume that the sender Alice wants to teleport an arbitrary one-qubit secret state

where the coefficients

where the coefficients

As shown in

Then the whole system can be expressed as

Now, Alice needs to perform a joint measurement on her particles

After Alice informs her measurement result, the states of the other three would collapse into the following four possible outcomes:

If the measurement result of Alice is

It’s important to note that only the authorized agent can recover the secret state

In the protocol, there is only one agent David in high grade. Suppose David is the receiver, we rewrite

Next, Bob or Charlie needs to perform

Suppose Alice and Bob’s measurement results are

At last, David performs a unitary operation

Alice’s result | Bob’s or Charlie’s result | State obtained by David | David’s operation |
---|---|---|---|

However, the state obtained by David is slightly different from the original secret state

Then David needs to perform a controlled-NOT operation

where

Hence, they cannot be differentiated deterministically by usual projection measurement. To distinguish the two states of particle

where

Note that

for all

The probabilities of obtaining the value of

Similarly,

As shown above, David can obtain the state of qubit

Once the state of particle

Up to now, we only consider the success probability when Alice and Bob’s measurement results are

Alice’s result | Bob’s (Charlie’s) result | Operators of POVM | Success probability |
---|---|---|---|

Now, we consider the agent in low grade (Bob or Charlie) to recover the secret state. In this case, the agent in low grade needs the cooperation of all agents. Since Bob and Charlie have the same authority, they have the same recovery process. What follows is the process of Bob’s recovering the secret state

To help Bob recover the original secret state, Charlie and David need to measure their own particle in the

Alice’s result | Charlie’s result | David’s result | State obtained by Bob | Bob’s operation |
---|---|---|---|---|

As shown in

Just as proposed in case 1, Bob needs to introduce an auxiliary qubit

Then, Bob needs to perform a CNOT operation

where

Because the permutation of Bob’s particle 2 and Charlie’s particle 3 cannot change the quantum source, Bob and Charlie are in

Alice’s result | Charlie’s (David’s) result | Operators of POVM | Success probability |
---|---|---|---|

Assume there is an attacker named Eve, he attempts to steal the secret information from the four legitimate agents Alice, Bob, Charlie and David. So, there are two ways of eavesdropping: intercept-measure-resend attack and entanglement attack.

For the former, Eve would intercept and measure the qubits sent by Alice in a random basis. Then Eve resends fake qubits to the other legitimate agents to disturb the secret’s recovery. But Eve would introduce abnormal error rates of the process inevitably. Hence, the intercept-measure-resend attack can be detected, and the teleportation of the secret state should be aborted. If the quantum channel is noiseless, then the error rate is equal to 0. In this scenario, the state of Eve’s system and the original system is a simply separable state, or product state, which means there is neither quantum nor classical correlation between these two systems. Therefore, Eve cannot gain any information from the original quantum state.

For the latter, during the distribution of particle 2, 3, 4, Eve entangles an ancillary qubit

In summary, there are some merits in the paper. First of all, the shared quantum source, a non-maximally entangled four-qubit cluster state, is robust against quantum decoherence. It’s obvious that the generation and preservation of four-qubit non-maximally entangled states is easier than maximally six-qubit states, which has an attractive advantage in the experimental realization. The symmetry of cluster states helps the expansion of HQIS protocol. Then the secret state is arbitrary, which means the strong applicability and generality of the proposed protocol. What’s more, each agent has different grades so that there exists a hierarchy in the protocol. The receiver cannot recover the secret state successfully only if the cooperation of the other agents. Together with the non-maximally quantum source, the receiver could recover the secret state in a certain success probability. In other words, non-maximally quantum source and hierarchy of the protocol help expand the research scope of usual QIS protocols. In addition, only by single-qubit measurements can the receiver recover the secret state, which brings convenience to experimental realization.