## Abstract

Broadcast and multi-cast communications are two important applications in quantum information science. Such applications guarantee the implementation of quantum information theory. In this paper, we propose two general schemes for the quantum broadcast and multi-cast communications to ensure that the central party (sender) can broadcast arbitrary single-qubit state to multiple receivers synchronously. Moreover, it is guaranteed that the information among the multiple receivers is different to satisfy the requirement in multi-cast communications. In particular, the proposed schemes indicate the probabilities, of which the multiple receivers obtain the quantum states successfully, could reach 1. We expect this works will shed some light for the prospective research on multi-party quantum communications and quantum cryptography.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum information science breaks limitations of conventional information transfer in communications and computations, since quantum entanglement is employed as the resource of information processing. As an important feature in quantum information science, multi-mode correlation, squeezing, and entanglement works have been studied experimentally and theoretically [1–8]. With the merits of entanglement, many information transmission tasks can be securely completed, such as quantum teleportation (QT) [9–14], quantum key distribution(QKD) [15–23], quantum secret sharing (QSS) [24–30], quantum secure direct communication(QSDC) [31–34], etc. Since the first theoretical scheme for teleporting an arbitrary single-qubit was presented in 1993 [9], teleportation of an arbitrary single-qubit state has been studied theoretically and experimentally using four-partite GHZ state [35], SLOCC equivalent W-class state [36], and Cluster state [37], etc. When compared with such methods, the multi-party QT based on the high dimensional entangled quantum channel is considered as one of the promising approaches for teleporting the arbitrary single-qubit state, and several related works have been extended recently [38–56].

In 2005, Deng et al. [39] introduced a symmetric multi-party controlled teleportation scheme which teleports an arbitrary two-qubit entangled state. In 2007, Xia et al. [40] gave a remote state preparation scheme, where a single-qubit quantum state is transmitted in multi-party collaboration. Dai et al. [41] investigated a remote state preparation scheme, where an entangled two-qubit state is transmitted from one sender to either of two receivers. However, the receivers can not obtain the quantum states to be prepared synchronously. In 2009, Ishizaka and Hiroshima [44] presented a multi-party quantum teleportation scheme, where the quantum states can be jointly prepared by multi-party. In 2010, Nguyen [45] invented a joint remote preparation scheme for teleporting single- and two-qubit states, where the W- and W-type states are employed as the quantum channel. Cao et al. [49] showed a deterministic joint remote preparation scheme in 2012, where an arbitrary qubit can be transferred with the benefit of EPR pairs. Zhou et al. [50] developed a three-party remote state preparation scheme, nonetheless, its procedure is complicated. In 2016, Wang et al. presented the efficient remote preparation via four-qubit cluster-type entangled states with multi-party [54]. A probabilistic controlled remote state preparation scheme was shown in 2018, where an arbitrary two-qubit state via partially entangled states with multi-party was presented [56]. Although over the past few decades the schemes of quantum multi-party communications are constantly presented and improved, the communication requirements about the low complexity and synchronous diverse information transmission are not satisfied.

Combining with the idea of prepare-and-measure schemes [57], we consider two main applications about quantum multi-party communications, including quantum broadcast and multi-cast communications. In this work, we present two transmission schemes, in which arbitrary single-qubit state from central party (sender) could be synchronously delivered to *N* receivers via 2*N* -qubit entangled state. The first scheme is adopted to transmit the same arbitrary single-qubit state synchronously. Besides, the information among the multiple receivers is different to satisfy the requirement in the second one. In particular, we show that the propose schemes are optimal with high efficiency and low complexity.

The paper is arranged as follows. A brief introduction and assumptions are illustrated in Sec. 1 and 2, respectively. Sec. 3 describes the (*N* + 1)-party quantum broadcast scheme while Sec. 4 deals with the quantum multi-cast scheme, where the different arbitrary single-qubit state have to be synchronously delivered to *N* receivers. In Sec.5, we apply the general schemes to two simple examples. Finally, a brief discussion and conclusion are drawn in Sec. 6.

## 2. Assumptions

To illustrate our works, we detail the similarities and differences between two models of the applications in Tab.1. In our schemes, we suppose a trusted central party, Alice, transmits the arbitrary single-qubit state to *N* legal users, Bob_{1}, Bob_{2},·⋯, Bob* _{n}*. For simplicity, we denote

*A*as Alice,

*B*

_{1},

*B*

_{2}, ⋯,

*B*as

_{n}*N*receivers, respectively.

To achieve the above tasks of quantum multi-party communications, sender Alice as well as the *N* receivers(*B*_{1}, *B*_{2}, ⋯*, B _{n}*) take use of the 2

*N*-qubit entangled state that is made of (

*N*− 2) identical EPR pairs |

*ϕ*

^{+}〉 and the four-qubit cluster state |Φ

^{4}〉. We assume one of EPR pair used in the channel is in the state

*N*− 2) |

*ϕ*

^{+}〉, taking the joint state

*i*th-qubit is entangled with (

*N*+

*i*)th-qubit(

*i*= 1, 2, ⋯

*, n*).

To fulfill the preparation of the desired state, sender Alice measures her qubits in an appropriate basis. We provide a series of measurement bases comprehensively with the details as follows.

In the case of broadcast communication, we suppose that the basic mutually orthogonal basis vector *w*_{2} is

*W*can be constructed as where

_{n}For instance, one can rewrite the mutually orthogonal basis vector *W*_{4}(*N* = 4) composed of *w _{i}*(

*i*= 2, 3, 4), i.e.

It is similar to broadcast communications, we suppose that the basic mutually orthogonal basis vector *v _{n}*

_{−1,}

*of multi-cast communication is*

_{n}*V*can be constructed as where

_{n}Taking *N* = 4 as an example, according to Eq. (10) *V*_{4} turns to

We then give the basic tools and some assumptions that will be employed in our schemes. The four unitary operations {*U _{i}*}(

*i*= 0, 1, 2, 3) can transfer any one of the four Bell states into another,

In order to ensure transmission of the arbitrary single-qubit state with perfect fidelity and unit probability when all parties participate, our study requires two assumptions to be made: 1. All participants verify that they share a perfect entangled channel; 2. Every participant in the schemes is honest. The security of proposed protocols is ensured under above assumptions that there are no attacks. Next, we proceed to discuss the schemes in detail.

## 3. Quantum broadcast scheme

Firstly, we propose the general scheme of broadcasting the same arbitrary single-qubit state with (*N* + 1)-party. Without loss of generalization, we suppose that the sender Alice broadcasts the same arbitrary single-qubit state |*ψ*〉 to a group of receivers synchronously. Here |*ψ*〉 reads

*α*,

*β*are known to sender while unknown to receivers, and satisfy |

*α|*

^{2}+ |

*β|*

^{2}= 1

We then detail the broadcast scheme of the same arbitrary single-qubit state in the following steps:

(S_{11}) The broadcast channel shared by (*N* + 1) participants is the 2*N*-qubit entangled state |Ψ〉_{2}* _{N}* shown in Eq. (4).

(S_{12}) Alice sends *N* particles (*n* + 1, *n* + 2, ⋯·, 2*n*) from the state |Ψ〉_{2}* _{n}* to

*N*receivers (

*B*

_{1}

*B*

_{2}, ⋯,

*B*) respectively while keeping the other

_{n}*N*particles (1, 2, ⋯,

*n)*to herself.

(S_{13}) Alice carries out a *N* -particle *N* -dimensional projective measurement on Particles (1, 2, ⋯, *n*) in a set of mutually orthogonal basis vectors *M*_{1,2,⋯,}* _{n}* = {|

*ω*〉

_{i}_{1,2,⋯,}

*}(*

_{n}*i*= 1, 2, ⋯, 2

*), i.e.*

^{n}*S*= {|00 ⋯ 00〉, |00 ⋯ 01〉,·⋯, |11 ⋯ 11〉} is the normal orthogonal basis of the

*d*-dimensional Hilbert space, and

^{n}*S*is the transposition of

^{T}*S.*Accordingly, one can rewrite the Eq. (4) as

(S_{14}) Alice performs a projective measurement *M* 1_{,2,⋯,}* _{n}* on particles (1, 2, ⋯,

*n)*under the general basis {|

*ω*〉

_{i}_{1,2,⋯,}

*}(*

_{n}*i*= 1, 2, ⋯, 2

*) and broadcasts her measurement outcome via classical communication to*

^{n}*B*

_{1},

*B*

_{2}, ⋯

*, B*, synchronously.

_{n}(S_{15}) According to the measurement results, *B*_{1}, *B*_{2}, ⋯, *B _{n}* corresponding perform the unitary operations

*U*on their own particle, respectively, as shown in Fig. 1.

_{i}Next, the general broadcast scheme of the same arbitrary single-qubit state is given, and Fig. 2 shows the schematic diagram for broadcasting the same arbitrary single-qubit states with (*N* + 1) parties.

For *N* receivers, we suppose that the measurement results of particles (1, 2, ⋯*, n*) is $|{\omega}_{{2}^{n}}\u3009$, then the particles (*n* +1), (*n* +2), ⋯, 2*n* are all in the state (*β|*0〉−*α*|1〉). Next, receivers *B*_{1}, *B*_{2}, ⋯, *B _{n}* can reconstruct the state |

*ψ*〉 by performing the unitary operation −

*U*

_{2}, respectively. Other cases are compiled in Fig. 1. As all the quantum states can be reconstructed by multiple receivers under the measurements of {|

*ω*〉

_{i}_{1,2,⋯,}

*}(*

_{n}*i*= 1, 2, ⋯, 2

*), the success probability of the broadcast scheme theoretically can reach 1.*

^{n}In short, by following the steps outlined above, the (*N* + 1) -party quantum broadcast scheme of the same arbitrary single-qubit state can be successfully performed via the 2*N* -qubit entangled channel. In next section, the general (*N* + 1)-party quantum multi-cast scheme will be illustrated.

## 4. Quantum multi-cast scheme

We suppose that there are *N* receivers taking part in the quantum multi-cast process. In such a case, the sender Alice will transmit the following *N* different arbitrary single-qubit states |*ψ*〉* _{i}*(

*i*= 1, 2, ⋯,

*n)*to multiple receivers synchronously,

*α*,

_{i}*β*

_{i}(

*i*= 1, 2, ⋯,

*n*) are known to sender while unknown to receivers, and satisfy |

*α*|

_{i}^{2}+ |

*β*|

_{i}^{2}= 1. It is worth emphasizing that |

*ψ*〉

_{1}≠ |

*ψ*〉

_{2}≠ ⋯ ≠ |

*ψ*〉

*.*

_{n}Then the quantum multi-cast scheme can be described as follows:

(S_{21}) The channel shared by (*N* + 1) participants is the 2*N* -qubit entangled state |Ψ〉_{2}* _{N}* shown in Eq. (4).

(S_{22}) Alice sends *N* particles (*n* + 1, *n* + 2, ⋯, 2*n*) to *N* receivers (*B*_{1}, *B*_{2}, ⋯*, B _{n}*) via entangled channel, respectively, and reserves particles (1, 2, ⋯,

*n*).

(S_{23}) Alice carries out a *N* -particle *N* -dimensional projective measurement on Particles (1, 2, ⋯, *n*) in a set of mutually orthogonal basis vectors ${P}_{1,2,\cdots ,n}=\left\{{|{\mu}_{i}\u3009}_{1,2,\cdots ,n}\right\}(i=1,2,\cdots ,{2}^{n})$,

*S*= {|00 ⋯ 00〉, |00 ⋯ 01〉, ⋯, |11 ⋯ 11〉} is the normal orthogonal basis of the

*d*-dimensional Hilbert space, and

^{n}*S*is the transposition of

^{T}*S*. Since above basis, Eq. (4) becomes

(*S*_{24}) Alice performs a projective measurement *P*_{1,2, ⋯,}* _{n}* on particles (1, 2, ⋯,

*n*) under the general basis {|

*μ*〉

_{i}_{1,2, ⋯,}

*}(i = 1, 2, ⋯, 2*

_{n}*) and broadcasts her measurement outcome via classical communication to*

^{n}*B*

_{1},

*B*

_{2}, ⋯,

*B*synchronously.

_{n}(S_{25}) According to the results, *B*_{1}, *B*_{2}, ⋯, *B _{n}* respectively perform the corresponding unitary operations on their own particle, as shown in Fig. 3.

Next, the general quantum multi-cast scheme of different arbitrary single-qubit state is given, and Fig. 4 shows the schematic diagram for broadcasting the *N* different arbitrary single-qubit state with (*N* + 1) parties.

For instance, we suppose that the measurement results of particles (1, 2, ⋯, *n*) is |*μ*_{1}〉, then the particles (*n* + 1), (*n* + 2), ⋯, 2*n* are in the states (*α _{i}*|0〉 +

*β*|1〉)(

_{i}*i*= 1, 2, ⋯,

*n*). Next,

*B*

_{1},

*B*

_{2}, ⋯,

*B*can reconstruct the states |

_{n}*ψ*〉

*by performing the unitary operation*

_{i}*U*

_{0}. Other cases are shown as below in Fig. 3. As all the quantum states can be reconstructed by multiple receivers under the measurements of {|

*μ*〉

_{i}_{1,2, ⋯,}

*}(*

_{n}*i*= 1, 2, ⋯, 2

*), the success probability of the multi-cast scheme theoretically can reach 1.*

^{n}By following the steps outline above, the (*N* + 1)-party quantum multi-cast scheme of different arbitrary single-qubit state can be successfully performed. In next section, we give two examples to test the generality of our schemes by specifying *N.*

## 5. Examples and applications of the general schemes

In this section, we test our general schemes by presenting two examples where *N* = 4, which are the cases of the five-party quantum broadcast and multi-cast schemes with one sender(Alice) and four receivers(*B*_{1}, *B*_{2}, *B*_{3}, *B*_{4}). Especially, we have extended the results of three-party quantum broadcast scheme (*N* = 2) via four-qubit cluster state in [58], and take a more general example *N* = 4(*N >* 2) as the description due to the construction of 2*N* -qubit entangled state that is made of (*N* − 2)|*ϕ*^{+}〉 and the four-qubit cluster state |Φ^{4}〉.

Following our schemes in Sec. 2, the eight-qubit entangled state |Ψ〉_{8} is constructed as the channel,

Here, particles (1234) in the state |Ψ〉_{8} belong to Alice, while particle 5,6,7,8 belong to *B*_{1}, *B*_{2}, *B*_{3}, *B*_{4}, respectively, for both example schemes.

#### 5.1. Five-party broadcast scheme

In this section, we give the five-party broadcast scheme. We suppose that Alice wants to transmit the following arbitrary single-qubit state |*ψ*〉 to *B*_{1}, *B*_{2}, *B*_{3}, *B*_{4} synchronously. |*ψ*〉 is known completely to Alice but unknown to four receivers,

*α*,

*β*satisfy the relation |

*α|*

^{2}

*+ |β|*= 1.

^{2}The measurement basis chosen by Alice is a set of mutually orthogonal basis vectors *M*_{1234} = {|*ω _{i}*〉

_{1234}}(

*i*= 1, 2, _ _ _, 16), according to Eq. (8),

The measurement basis chosen of the *d* ^{4}-dimensional Hilbert space, and *S ^{T}* is the transposition of

*S.*Accordingly, Eq. (20) turns to

Then Alice performs a projective measurement *M*_{1234} on particles (1234) under the basis {|*ω _{i}*〉

_{1234}}(

*i*= 1, 2, ⋯, 16) and broadcasts her measurement outcome via classical communication to

*B*

_{1},

*B*

_{2},

*B*

_{3},

*B*

_{4}synchronously.

Finally, according to Alice’s measurement results, *B*_{1}, *B*_{2}, *B*_{3}, *B*_{4} reconstruct the state by performing corresponding unitary operations on their own particle. The measurement results and unitary operations of four receivers are shown in the Fig. 5. Hence, a five-party broadcast scheme via eight-qubit entangled state is presented.

#### 5.2. Five-party multi-cast scheme

In this section, we extend the five-party multi-cast scheme, which is a case of teleporting four different arbitrary single-qubit states,

*α*,

_{i}*β*satisfy the following relation |

_{i}*α*|

_{i}^{2}+ |

*β*|

_{i}^{2}= 1(

*i*= 1, 2, 3, 4). Especially, |

*ψ*〉

_{1}≠ |

*ψ*〉

_{2}≠ |

*ψ*〉

_{3}≠ |

*ψ*〉

_{4}.

According to Eq. (12), the measurement basis chosen by Alice is a set of mutually orthogonal basis vectors *P*_{1234} = {|*μ _{i}*〉

_{1234}}(

*i*= 1, 2, ⋯, 16),

*S*is the normal orthogonal basis of the

*d*

^{4}-dimensional Hilbert space, and

*S*is the transposition of

^{T}*S.*Thus, Eq. (20) can be rewrite as

Then Alice performs a projective measurement *P*_{1234} on particles (1234) under the general basis {|*µ _{i}*〉

_{1234}}(

*i*= 1, 2, ⋯, 16) and broadcasts her measurement outcome via classical communication to

*B*

_{1},

*B*

_{2},

*B*

_{3},

*B*

_{4}synchronously.

Finally, according to Alice’s measurement results, *B*_{1}, *B*_{2}, *B*_{3}, *B*_{4} reconstruct the state by performing corresponding unitary operations on their own particle. The measurement results and unitary operations of four receivers are shown in the Fig. 6. Thus, a five-party multi-cast scheme of four different arbitrary single-qubit state via eight-qubit entangled state is extended.

In the above examples, four receivers can fully reconstruct the arbitrary single-qubit states with our general schemes, and it is clear that our proposed schemes can be specified into any other cases for *N* ≥ 2 (*N* ∈ *Z*_{+}). Note that for *N* = 2, it has been extended in [58]. Therefore, we conclude that our theoretical quantum broadcast and multi-cast schemes are highly general and can be deterministically performed.

## 6. Discussion and conclusion

There is much interest in multi-party quantum communications, where quantum teleportation employing high dimensional entangled quantum channel is one of the promising tools. Quantum broadcast and multi-cast communications are the essential applications of multi-party communications. In our schemes, it is found that the number of measurement bases and qubits of channel is closely related to the number of participants. That is, the number of measurement basis and qubits of channel increases with the number of participants increasing. Especially, the number of measurement basis increases exponentially. We extend the specific cases in the Fig. 7.

Indeed, the optimal success probability can be achieved under two ideal assumptions in our schemes. However, due to the unavoidable interaction between the quantum channel and its ambient environment, it is challenging to generate and maintain the perfect entanglement. It means that it is difficult to realize the schemes in practice based on the current technologies. Taking this into consideration, we dedicate ourselves to the investigation of broadcast and multi-cast communications via partially entangled quantum channels in our future works. Actually, related works have been reported [59]. For the eavesdropper or dishonest receivers, we introduce auxiliary particles and supervisor Charlie to enable Charlie to supervise the receivers and detect the eavesdropping. Our related works of above-mentioned issues are under way. A brief description is shown in Fig. 8.

In summary, we present two general schemes for (*N* + 1)-party broadcast and multi-cast communications via the 2*N* -qubit entangled state, i.e., a sender broadcast the arbitrary single- qubit state to *N* distant receivers synchronously. In the quantum broadcast scheme, receivers can reconstruct the same arbitrary single-qubit state by performing suitable unitary operation. It is guaranteed that the information among the multiple receivers are different to satisfy the requirement in multi-cast communications. In particular, the proposed schemes indicate the probabilities, of which the multiple receivers obtain the quantum states successfully, could reach 1. Moreover, the processes of two proposed schemes are simple, only one projective measurement is needed. As the whole quantum source is used to carry the useful quantum information, the efficiency for qubits approached the maximal value, the multi-party broadcast and multi-cast communications are the optimal choices.

Up to now, the progress of the single-qubit unitary operation in experiment in various quantum systems has been reported [60–63], and four-, five-, six-, eight-photon entangled states are successfully prepared [64–67]. However, the multi-qubit entangled states and measurements in our schemes have not been reported in experiment. Actually, it is difficult to realize our schemes in practice based on the current technologies. With the continuous development of quantum information theory and technology, the technology of equipment and materials will be improved, and the production cost can be greatly reduced. It is means that the proposed quantum broadcast and multi-cast theoretical schemes can be realized in the near future.

## Funding

The Foundational Research Funds for the Central Universities under Grants JB180111; The National Natural Science Foundation of China (No.61301171); Foundation of Science and Technology on Communication Networks Laboratory (KX172600031); Shaanxi Key Research and Development Program (Grant No.2017GY-080).

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