General quantum broadcast and multi-cast communications based on entanglement

Yan Yu; Nan Zhao

doi:10.1364/OE.26.029296

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 2N -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, Bob₁, Bob₂,·⋯, Bob_n. For simplicity, we denote A as Alice, B₁, B₂, ⋯, B_n as N receivers, respectively.

Table 1. Similarities and differences of the information obtained by receivers between two models of the applications.

View Table

To achieve the above tasks of quantum multi-party communications, sender Alice as well as the N receivers(B₁, B₂, ⋯, B_n) take use of the 2N -qubit entangled state that is made of (N − 2) identical EPR pairs |ϕ⁺〉 and the four-qubit cluster state |Φ⁴〉. We assume one of EPR pair used in the channel is in the state

| ϕ^{+} 〉 = \frac{1}{\sqrt{2}} (| 00 〉 + | 11 〉),

then the (N − 2) |ϕ⁺〉, taking the joint state

\begin{matrix} | ϕ 〉 = \prod_{1}^{n - 2} | ϕ^{+} 〉 \\ = \prod_{1}^{n - 2} \frac{1}{\sqrt{2}} (| 00 〉 + | 11 〉), \end{matrix}

with the four-qubit Cluster state

| Φ^{4} 〉 = \frac{1}{2} (| 0000 〉 + | 0101 〉 + | 1010 〉 - | 1111 〉),

together as the quantum channel, that is

\begin{matrix} {| Ψ 〉}_{2 n} = {| ϕ^{+} 〉}_{1, n + 1} \otimes {| ϕ^{+} 〉}_{2, n + 2} \otimes \dots \otimes {| ϕ^{+} 〉}_{n - 2, 2 n - 2} \otimes {| Φ^{4} 〉}_{n - 1, n, 2 n - 1, 2 n} \\ = \frac{1}{\sqrt{2}} {(| 00 〉 + | 11 〉)}_{1, n + 1} \otimes \frac{1}{\sqrt{2}} {(| 00 〉 + | 11 〉)}_{2, n + 2} \otimes \dots \otimes \frac{1}{\sqrt{2}} {(| 00 〉 + | 11 〉)}_{n - 2, 2 n - 2} \\ \otimes \frac{1}{2} {(| 0000 〉 + | 0101 〉 + | 1010 〉 - | 1111 〉)}_{n - 1, n, 2 n - 1, 2 n}, \end{matrix}

where the ith-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 w₂ is

w_{2} = (\begin{matrix} α^{2} & α β & α β & - β^{2} \\ α β & - α^{2} & β^{2} & α β \\ α β & β^{2} & - α^{2} & α β \\ β^{2} & - α β & - α β & - α^{2} \end{matrix}),

then the general basis vector W_n can be constructed as

W_{n} = \otimes_{2}^{n} w_{i} (i = 2, 3, \dots, n),

where

w_{i} (\begin{matrix} α & β \\ β & - α \end{matrix}) (i = 3, 4, \dots, n) .

For instance, one can rewrite the mutually orthogonal basis vector W₄(N = 4) composed of w_i(i = 2, 3, 4), i.e.

\begin{matrix} W_{4} = w_{4} \otimes w_{3} \otimes w_{2} \\ = (\begin{matrix} α & β \\ β & - α \end{matrix}) \otimes (\begin{matrix} α & β \\ β & - α \end{matrix}) \otimes w_{2} \\ = (\begin{matrix} α^{2} w_{2} & α β w_{2} & α β w_{2} & β^{2} w_{2} \\ α β w_{2} & - α^{2} w_{2} & β^{2} w_{2} & - α β w_{2} \\ α β w_{2} & β^{2} w_{2} & - α^{2} w_{2} & - α β w_{2} \\ β^{2} w_{2} & - α β w_{2} & - α β w_{2} & α^{2} w_{2} \end{matrix}), \end{matrix}

which is adopted in the following discussion.

It is similar to broadcast communications, we suppose that the basic mutually orthogonal basis vector v_n_−1,_n of multi-cast communication is

v_{n - 1, n} = (\begin{matrix} α_{n - 1} α_{n} & α_{n - 1} β_{n} & β_{n - 1} α_{n} & - β_{n - 1} β_{n} \\ α_{n - 1} β_{n} & - α_{n - 1} α_{n} & β_{n - 1} β_{n} & β_{n - 1} α_{n} \\ β_{n - 1} α_{n} & β_{n - 1} β_{n} & - α_{n - 1} α_{n} & α_{n - 1} β_{n} \\ β_{n - 1} β_{n} & - β_{n - 1} α_{n} & - α_{n - 1} β_{n} & - α_{n - 1} α_{n} \end{matrix}),

then the general basis vector V_n can be constructed as

V_{n} = \otimes_{1}^{n - 2} v_{i} v_{n - 1, n},

where

v_{i} = (\begin{matrix} α_{i} & β_{i} \\ β_{i} & - α_{i} \end{matrix}) (i = 1, 2, \dots, n - 2) .

Taking N = 4 as an example, according to Eq. (10) V₄ turns to

\begin{matrix} V_{4} = v_{1} \otimes v_{2} \otimes v_{34} \\ = (\begin{matrix} α_{1} & β_{1} \\ β_{1} & - α_{1} \end{matrix}) \otimes (\begin{matrix} α_{2} & β_{2} \\ β_{2} & - α_{2} \end{matrix}) \otimes v_{34} \\ = (\begin{matrix} α_{1} α_{2} v_{34} & α_{1} β_{2} v_{34} & β_{1} α_{2} v_{34} & β_{1} β_{2} v_{34} \\ α_{1} β_{2} v_{34} & - α_{1} α_{2} v_{34} & β_{1} β_{2} v_{34} & - β_{1} α_{2} v_{34} \\ β_{1} α_{2} v_{34} & β_{1} β_{2} v_{34} & - α_{1} α_{2} v_{34} & - α_{1} β_{2} v_{34} \\ β_{1} β_{2} v_{34} & - β_{1} α_{2} v_{34} & - α_{1} β_{2} v_{34} & α_{1} α_{2} v_{34} \end{matrix}), \end{matrix}

which is applied in the Sec. 5.

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,

\begin{array}{l} U_{0} = | 0 〉 〈 0 | + | 1 〉 〈 1 |, U_{1} = | 0 〉 〈 0 | - | 1 〉 〈 1 |, \\ U_{2} = | 0 〉 〈 1 | - | 1 〉 〈 0 |, U_{3} = | 0 〉 〈 1 | + | 1 〉 〈 0 | . \end{array}

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

| ψ 〉 = α | 0 〉 + β | 1 〉,

where 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:

(S₁₁) The broadcast channel shared by (N + 1) participants is the 2N-qubit entangled state |Ψ〉₂_N shown in Eq. (4).

(S₁₂) Alice sends N particles (n + 1, n + 2, ⋯·, 2n) from the state |Ψ〉₂_n to N receivers (B₁ B₂, ⋯, B_n) respectively while keeping the other N particles (1, 2, ⋯, n) to herself.

(S₁₃) 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, ⋯, 2ⁿ), i.e.

M_{1, 2, \dots, n} = (\begin{matrix} | ω_{1} 〉 \\ | ω_{2} 〉 \\ ⋮ \\ | ω_{2^{n}} 〉 \end{matrix}) = W_{n} S^{T},

where S = {|00 ⋯ 00〉, |00 ⋯ 01〉,·⋯, |11 ⋯ 11〉} is the normal orthogonal basis of the dⁿ-dimensional Hilbert space, and S^T is the transposition of S. Accordingly, one can rewrite the Eq. (4) as

\begin{matrix} {| Ψ 〉}_{2 n} = {(\frac{1}{\sqrt{2}})}^{n} [{| ω_{1} 〉}_{12 \dots n} {(α | 0 〉 + β | 1 〉)}_{n + 1} {(α | 0 〉 + β | 1 〉)}_{n + 2} \dots {(α | 0 〉 + β | 1 〉)}_{2 n} \\ + {| ω_{2} 〉}_{12 \dots n} {(α | 0 〉 + β | 1 〉)}_{n + 1} {(α | 0 〉 + β | 1 〉)}_{n + 2} \dots {(β | 0 〉 - α | 1 〉)}_{2 n} \\ + \dots \\ + {| ω_{2^{n}} 〉}_{12 \dots n} {(β | 0 〉 - α | 1 〉)}_{n + 1} {(β | 0 〉 - α | 1 〉)}_{n + 2} \dots {(β | 0 〉 - α | 1 〉)}_{2 n}] . \end{matrix}

(S₁₄) 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, ⋯, 2ⁿ) and broadcasts her measurement outcome via classical communication to B₁, B₂, ⋯, B_n, synchronously.

(S₁₅) According to the measurement results, B₁, B₂, ⋯, B_n corresponding perform the unitary operations U_i on their own particle, respectively, as shown in Fig. 1.

Fig. 1 The measurement results and unitary operations U_i of each receiver in (N+1)-party quantum broadcast scheme of the same arbitrary single-particle state. Here, α|0〉 + β|1〉 and β|0〉 − α|1〉 are represented by + and −, respectively.

Download Full Size | PDF

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.

Fig. 2 The schematic principle for the quantum broadcast scheme of the same arbitrary single-qubit state. A 2N -qubit entangled state is employed as the quantum channel. M_1,2,⋯,_n stands for a projective measurement on particles (1, 2, ⋯, n), which is performed by Alice. U stands for Hadamard operation which is performed by N receivers on their own particle. A qubit is represented by a dot.

Download Full Size | PDF

For N receivers, we suppose that the measurement results of particles (1, 2, ⋯, n) is $| ω_{2^{n}} 〉$ , then the particles (n +1), (n +2), ⋯, 2n are all in the state (β|0〉−α|1〉). Next, receivers B₁, B₂, ⋯, B_n can reconstruct the state |ψ〉 by performing the unitary operation −U₂, 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, ⋯, 2ⁿ), the success probability of the broadcast scheme theoretically can reach 1.

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 2N -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,

{\begin{matrix} {| ψ 〉}_{1} = & α_{1} | 0 〉 + β_{1} | 1 〉 \\ {| ψ 〉}_{2} = & α_{2} | 0 〉 + β_{2} | 1 〉 \\ ⋮ \\ {| ψ 〉}_{n} = & α_{n} | 0 〉 + β_{n} | 1 〉, \end{matrix}

where 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:

(S₂₁) The channel shared by (N + 1) participants is the 2N -qubit entangled state |Ψ〉₂_N shown in Eq. (4).

(S₂₂) Alice sends N particles (n + 1, n + 2, ⋯, 2n) to N receivers (B₁, B₂, ⋯, B_n) via entangled channel, respectively, and reserves particles (1, 2, ⋯, n).

(S₂₃) Alice carries out a N -particle N -dimensional projective measurement on Particles (1, 2, ⋯, n) in a set of mutually orthogonal basis vectors $P_{1, 2, \dots, n} = {{| μ_{i} 〉}_{1, 2, \dots, n}} (i = 1, 2, \dots, 2^{n})$ ,

P_{1, 2, \dots, n} = (\begin{matrix} | μ_{1} 〉 \\ | μ_{2} 〉 \\ ⋮ \\ | μ_{2^{n}} 〉 \end{matrix}) = V_{n} S^{T},

where S = {|00 ⋯ 00〉, |00 ⋯ 01〉, ⋯, |11 ⋯ 11〉} is the normal orthogonal basis of the dⁿ-dimensional Hilbert space, and S^T is the transposition of S. Since above basis, Eq. (4) becomes

\begin{matrix} {| Ψ 〉}_{2 n} = {(\frac{1}{\sqrt{2}})}^{n} [{| μ_{1} 〉}_{12 \dots n} {(α_{1} | 0 〉 + β_{1} | 1 〉)}_{n + 1} {(α_{2} | 0 〉 + β_{2} | 1 〉)}_{n + 2} \dots {(α_{n} | 0 〉 + β_{n} | 1 〉)}_{2 n} \\ + {| μ_{2} 〉}_{12 \dots n} {(α_{1} | 0 〉 + β_{1} | 1 〉)}_{n + 1} {(α_{2} | 0 〉 + β_{2} | 1 〉)}_{n + 2} \dots {(β_{n} | 0 〉 - α_{n} | 1 〉)}_{2 n} \\ + \dots \\ + {| μ_{2^{n}} 〉}_{12 \dots n} {(β_{1} | 0 〉 - α_{1} | 1 〉)}_{n + 1} {(β_{2} | 0 〉 - α_{2} | 1 〉)}_{n + 2} \dots {(β_{n} | 0 〉 - α_{n} | 1 〉)}_{2 n}] . \end{matrix}

(S₂₄) Alice performs a projective measurement P_{1,2, ⋯,}_n on particles (1, 2, ⋯, n) under the general basis {|μ_i〉_{1,2, ⋯,}_n}(i = 1, 2, ⋯, 2ⁿ) and broadcasts her measurement outcome via classical communication to B₁, B₂, ⋯, B_n synchronously.

(S₂₅) According to the results, B₁, B₂, ⋯, B_n respectively perform the corresponding unitary operations on their own particle, as shown in Fig. 3.

Fig. 3 The measurement results and unitary operations U_i of each receiver in (N+1)-party quantum multi-cast scheme of different arbitrary single-particle state. Here, α_i|0〉 + β_i|1〉 and β_i|0〉 − α_i|1〉 are represented by +_i and −_i, respectively, (i = 1, 2, ⋯, n).

Download Full Size | PDF

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.

Fig. 4 The schematic principle for the quantum multi-cast scheme of N different arbitrary single-qubit state. A 2N -qubit entangled state is employed as the quantum channel. P_{1,2, ⋯,}_n stands for a projective measurement on particles (1, 2, ⋯, n), which is performed by Alice. U stands for Hadamard operation which is performed by N receivers on their own particle. A qubit is represented by a dot.

Download Full Size | PDF

For instance, we suppose that the measurement results of particles (1, 2, ⋯, n) is |μ₁〉, then the particles (n + 1), (n + 2), ⋯, 2n are in the states (α_i|0〉 + β_i|1〉)(i = 1, 2, ⋯, n). Next, B₁, B₂, ⋯, B_n can reconstruct the states |ψ〉_i by performing the unitary operation U₀. 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, ⋯, 2ⁿ), the success probability of the multi-cast scheme theoretically can reach 1.

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(B₁, B₂, B₃, B₄). 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 2N -qubit entangled state that is made of (N − 2)|ϕ⁺〉 and the four-qubit cluster state |Φ⁴〉.

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

\begin{matrix} {| Ψ 〉}_{8} = {| ϕ^{+} 〉}_{1, 5} \otimes {| ϕ^{+} 〉}_{2, 6} \otimes {| Φ_{4} 〉}_{3, 4, 7, 8} \\ = \frac{1}{4} {(| 00 〉 + | 11 〉)}_{1, 5} \otimes {(| 00 〉 + | 11 〉)}_{2, 6} \\ \otimes {(| 0000 〉 + | 0101 〉 + | 1010 〉 - | 1111 〉)}_{3, 4, 7, 8} . \end{matrix}

Here, particles (1234) in the state |Ψ〉₈ belong to Alice, while particle 5,6,7,8 belong to B₁, B₂, B₃, B₄, 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 B₁, B₂, B₃, B₄ synchronously. |ψ〉 is known completely to Alice but unknown to four receivers,

| ψ 〉 = α | 0 〉 + β | 1 〉,

where α, β satisfy the relation |α|² + |β|² = 1.

The measurement basis chosen by Alice is a set of mutually orthogonal basis vectors M₁₂₃₄ = {|ω_i〉₁₂₃₄}(i = 1, 2, _ _ _, 16), according to Eq. (8),

M_{1234} = (\begin{matrix} | ω_{1} 〉 \\ | ω_{2} 〉 \\ ⋮ \\ | ω_{16} 〉 \end{matrix}) = W_{4} S^{T},

The measurement basis chosen of the d ⁴-dimensional Hilbert space, and S^T is the transposition of S. Accordingly, Eq. (20) turns to

\begin{matrix} {| Ψ 〉}_{8} = \frac{1}{4} [{| ω_{1} 〉}_{1234} {(α | 0 〉 + β | 1 〉)}_{5} {(α | 0 〉 + β | 1 〉)}_{6} {(α | 0 〉 + β | 1 〉)}_{7} {(α | 0 〉 + β | 1 〉)}_{8} \\ + {| ω_{2} 〉}_{1234} {(α | 0 〉 + β | 1 〉)}_{5} {(α | 0 〉 + β | 1 〉)}_{6} {(α | 0 〉 - β | 1 〉)}_{7} {(β | 0 〉 - α | 1 〉)}_{8} \\ + \dots \\ + {| ω_{16} 〉}_{1234} {(β | 0 〉 - α | 1 〉)}_{5} {(β | 0 〉 - α | 1 〉)}_{6} {(β | 0 〉 - α | 1 〉)}_{7} {(β | 0 〉 - α | 1 〉)}_{8}] . \end{matrix}

Then Alice performs a projective measurement M₁₂₃₄ on particles (1234) under the basis {|ω_i〉₁₂₃₄}(i = 1, 2, ⋯, 16) and broadcasts her measurement outcome via classical communication to B₁, B₂, B₃, B₄ synchronously.

Finally, according to Alice’s measurement results, B₁, B₂, B₃, B₄ 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.

Fig. 5 The measurement results and unitary operations U_i of each receiver in five-party quantum broadcast scheme of the same arbitrary single-particle state. Here, α|0〉 + β|1〉 and β|0〉 − α|1〉 are represented by + and −, respectively.

Download Full Size | PDF

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,

{\begin{cases} {| ψ 〉}_{1} = α_{1} | 0 〉 + β_{1} | 1 〉 \\ {| ψ 〉}_{2} = α_{2} | 0 〉 + β_{2} | 1 〉 \\ {| ψ 〉}_{3} = α_{3} | 0 〉 + β_{3} | 1 〉 \\ {| ψ 〉}_{4} = α_{4} | 0 〉 + β_{4} | 1 〉, \end{cases}

where α_i, β_i satisfy the following relation |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 P₁₂₃₄ = {|μ_i〉₁₂₃₄}(i = 1, 2, ⋯, 16),

P_{1234} = (\begin{matrix} | μ_{1} 〉 \\ | μ_{2} 〉 \\ ⋮ \\ | μ_{16} 〉 \end{matrix}) = V_{4} S^{T},

where S is the normal orthogonal basis of the d ⁴-dimensional Hilbert space, and S^T is the transposition of S. Thus, Eq. (20) can be rewrite as

\begin{matrix} {| Ψ 〉}_{8} = \frac{1}{4} [{| μ_{1} 〉}_{1234} {(α_{1} | 0 〉 + β_{1} | 1 〉)}_{5} {(α_{2} | 0 〉 + β_{2} | 1 〉)}_{6} {(α_{3} | 0 〉 + β_{3} | 1 〉)}_{7} {(α_{4} | 0 〉 + β_{4} | 1 〉)}_{8} \\ + {| μ_{2} 〉}_{1234} {(α_{1} | 0 〉 + β_{1} | 1 〉)}_{5} {(α_{2} | 0 〉 + β_{2} | 1 〉)}_{6} {(α_{3} | 0 〉 + β_{3} | 1 〉)}_{7} {(β_{4} | 0 〉 - α_{4} | 1 〉)}_{8} \\ + \dots \\ + {| μ_{16} 〉}_{1234} {(β_{1} | 0 〉 - α_{1} | 1 〉)}_{5} {(β_{2} | 0 〉 - α_{2} | 1 〉)}_{6} {(β_{3} | 0 〉 - α_{3} | 1 〉)}_{7} {(β_{4} | 0 〉 - α_{4} | 1 〉)}_{8}] . \end{matrix}

Then Alice performs a projective measurement P₁₂₃₄ on particles (1234) under the general basis {|µ_i〉₁₂₃₄}(i = 1, 2, ⋯, 16) and broadcasts her measurement outcome via classical communication to B₁, B₂, B₃, B₄ synchronously.

Finally, according to Alice’s measurement results, B₁, B₂, B₃, B₄ 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.

Fig. 6 The measurement results and unitary operations U_i of each receiver in five-party quantum multi-cast scheme of different arbitrary single-particle state. Here, α_i|0〉 + β_i|1〉 and β_i|0〉 − α_i|1〉 are represented by +_i and −_i, respectively, (i = 1, 2, 3, 4).

Download Full Size | PDF

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.

Fig. 7 The quantitative changing diagram in both multi-party broadcast and multi-cast scenarios under the specific cases.

Download Full Size | PDF

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.

Fig. 8 The brief description of two assumptions under the ideal and practical circumstances.

Download Full Size | PDF

In summary, we present two general schemes for (N + 1)-party broadcast and multi-cast communications via the 2N -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).

References

1. M. Huber, F. Mintert, A. Gabriel, and B. C. Hiesmayr, “Detection of high-dimensional genuine multipartite entanglement of mixed states,” Phys. Rev. Lett 104, 210501 (2010). [CrossRef] [PubMed]

2. O. A. Castro-Alvaredo and B. Doyon, “Entanglement entropy of highly degenerate states and fractal dimensions,” Phys. Rev. Lett 108, 120401 (2012). [CrossRef] [PubMed]

3. R. Prevedel, G. Cronenberg, M. S. Tame, M. Paternostro, P. Walther, M. S. Kim, and A. Zeilinger, “Experimental realization of dicke states of up to six qubits for multiparty quantum networking,” Phys. Rev. Lett 103, 020503 (2009). [CrossRef] [PubMed]

4. G. Gour, “Evolution and symmetry of multipartite entanglement,” Phys. Rev. Lett 105, 190504 (2010). [CrossRef]

5. H. X. Chen, X. Zhang, D. Y. Zhu, C. Yang, T. Jiang, H. B. Zheng, and Y. P. Zhang, “Dressed four-wave mixing second-order talbot effect,” Phys. Rev. A 90, 043846 (2014). [CrossRef]

6. C. B. Li, Z. H. Jiang, Y. Q. Zhang, Z. Y. Zhang, F. Wen, H. X. Chen, Y. P. Zhang, and M. Xiao, “Controlled correlation and squeezing” in pr3+:y2sio5 to yield correlated light beams,” Phys. Rev. Appl . 7, 014023 (2017). [CrossRef]

7. G. Abdisa, I. Ahmed, X. X. Wang, Z. C. Liu, H. X. Wang, and Y. P. Zhang, “Controllable hybrid shape of correlation and squeezing,” Phys. Rev. A 94, 023849 (2016). [CrossRef]

8. D. Zhang, C. B. Li, Z. Y. Zhang, Y. Q. Zhang, Y. P. Zhang, and M. Xiao, “Enhanced intensity-difference squeezing via energy-level modulations in hot atomic media,” Phys. Rev. A 96, 043847 (2017). [CrossRef]

9. C. H. Bennett, G. Brassard, C. Crpeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett 70, 1895–1899 (1993). [CrossRef] [PubMed]

10. Q. Ai, “Toward quantum teleporting living objects,” Sci. Bull 61, 110–111 (2016). [CrossRef]

11. T. C. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Sci. Bull 61, 163–171 (2016). [CrossRef]

12. T. Gao, F. L. Yan, and Y. C. Li, “Optimal controlled teleportation,” Europhys. Lett . 84, 50001 (2008). [CrossRef]

13. P. Zhou, X. H. Li, F. G. Deng, and H. Y. Zhou, “Multiparty-controlled teleportation of an arbitrary m-qudit state with a pure entangled quantum channel,” J. Phys. A: Math. Theor 40, 13121 (2007). [CrossRef]

14. D. Zhang, X. W. Zha, W. Li, and Y. Yu, “Bidirectional and asymmetric quantum controlled teleportation via maximally eight-qubit entangled state,” Quantum Inf. Process . 14, 3835 (2015). [CrossRef]

15. A. K. Ekert, “Quantum cryptography based on bells theorem,” Phys. Rev. Lett 67, 661 (1991). [CrossRef] [PubMed]

16. C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bells theorem,” Phys. Rev. Lett 68, 557 (1992). [CrossRef] [PubMed]

17. F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A 68, 042315 (2003). [CrossRef]

18. F. G. Deng and G. L. Long, “Bidirectional quantum key distribution protocol with practical faint laser pulses,” Phys. Rev. A 70, 012311 (2004). [CrossRef]

19. W. Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett 91, 057901 (2003). [CrossRef] [PubMed]

20. X. H. Li, F. G. Deng, and H. Y. Zhou, “Efficient quantum key distribution over a collective noise channel,” Phys. Rev. A 78, 022321 (2008). [CrossRef]

21. H. K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett 108, 130503 (2012). [CrossRef] [PubMed]

22. Z. L. Bai, X. Y. Wang, S. S. Yang, and Y. M. Li, “High-efficiency gaussian key reconciliation in continuous variable quantum key distribution,” Sci. China. Physics, Mech. Astron. 59(1), 614201 (2016). [CrossRef]

23. W. Huang, “Improved multiparty quantum key agreement in travelling mode,” Sci. China. Physics, Mech. Astron. 59(12), 120311 (2016). [CrossRef]

24. M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829 (1999). [CrossRef]

25. A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A 59, 162 (1999). [CrossRef]

26. L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004). [CrossRef]

27. F. G. Deng, “Improving the security of multiparty quantum secret sharing against trojan horse attack,” Phys. Rev. A 72, 044302 (2005). [CrossRef]

28. F. G. Deng, H. Y. Zhou, and G. L. Long, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337(4), 329–334 (2005). [CrossRef]

29. F. G. Deng, “An efficient quantum secret sharing scheme with einstein-podolsky-rosen pairs,” Phys. Lett. A 340(1), 43–50 (2005). [CrossRef]

30. F. G. Deng, “Circular quantum secret sharing,” J. Phys. A 39(45), 14089–14099 (2006). [CrossRef]

31. F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the einstein-podolsky-rosen pair block,” Phys. Rev. A 68, 042317 (2003). [CrossRef]

32. F. G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A 69, 052319 (2004). [CrossRef]

33. C. Wang, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005). [CrossRef]

34. C. Wang, “Multi-step quantum secure direct communication using multi-particle green-horne-zeilinger state,” Opt. Commun 253(1), 15–20 (2005). [CrossRef]

35. A. K. Pati, “Assisted cloning and orthogonal complementing of an unknown state,” Phys. Rev. A 61, 022308 (2000). [CrossRef]

36. P. Agrawal and A. Pati, “Perfect teleportation and superdense coding withwstates,” Phys. Rev. A 74, 062320 (2006). [CrossRef]

37. H. J. Briegel and R. Raussendorf, “Persistent entanglement in arrays of interacting particles,” Phys. Rev. Lett 86, 910–913 (2001). [CrossRef] [PubMed]

38. J. M. Liu and Y. Z. Wang, “Remote preparation of a two-particle entangled state,” Phys. Lett. A 316(3–4), 159–167 (2003). [CrossRef]

39. F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72(2), 656–665 (2005).

40. Y. Xia, J. Song, and H. S. Song, “Multiparty remote state preparation,” J. Phys. B At. Mol. Opt. Phys 40(18), 3719 (2007). [CrossRef]

41. H. Y. Dai, P. X. Chen, M. Zhang, and C. Z. Li, “Remote preparation of an entangled two-qubit state with three parties,” Chin. Phys. B 17(1), 27–33 (2008). [CrossRef]

42. D. Wang, Y. M. Liu, and Z. J. Zhang, “Remote preparation of a class of three-qubit states,” Opt. Commun 281(4), 871–875 (2008). [CrossRef]

43. K. Hou, J. Wang, Y. L. Lu, and S. H. Shi, “Joint remote preparation of a multipartite ghz-class state,” Int. J. Theor. Phys 48(7), 2005–2015 (2009). [CrossRef]

44. S. Ishizaka and T. Hiroshima, “Quantum teleportation scheme by selecting one of multiple output ports,” Phys. Rev. A 79, 042306 (2009). [CrossRef]

45. B. A. Nguyen, “Joint remote state preparation via w and w-type states,” Opt. Commun 283(20), 4113–4117 (2010). [CrossRef]

46. S. Y. Ma, X. B. Chen, and M. X. Luo, “Remote preparation of a four-particle entangled cluster-type state,” Opt. Commun 284(16–17), 4088–4093 (2011). [CrossRef]

47. J. F. Song and Z. Y. Wang, “Controlled remote preparation of a two-qubit state via positive operator-valued measure and two three-qubit entanglements,” Int. J. Theor. Phys 50(8), 2410–2425 (2011). [CrossRef]

48. L. R. Long, P. Zhou, Z. Li, and C. L. Yin, “Multiparty joint remote preparation of an arbitrary ghz-class state via positive operator-valued measurement,” Int. J. Theor. Phys 51(8), 2438–2446 (2012). [CrossRef]

49. T. B. Cao, V. D. Nung, and B. A. Nguyen, “Deterministic joint remote preparation of an arbitrary qubit via einstein-podolsky-rosen pairs,” Int. J. Theor. Phys 51(7), 2272–2281 (2012). [CrossRef]

50. N. R. Zhou, H. L. Cheng, X. Y. Tao, and L. H. Gong, “Three-party remote state preparation schemes based on entanglement,” Quantum Inf. Process. 13, 513–526 (2014). [CrossRef]

51. H. Q. Liang, J. M. Liu, and S. S. Feng, “Effects of noises on joint remote state preparation via a ghz-class channel,” Quantum Inf. Process. 14(10), 3857–3877 (2015). [CrossRef]

52. Z. Sun, J. Yu, and P. Wang, “Efficient multi-party quantum key agreement by cluster states,” Quantum Inf. Process. 15(1), 373–384 (2016). [CrossRef]

53. Z. Sun, C. Zhang, and P. Wang, “Multi-party quantum key agreement by an entangled six-qubit state,” Int. J. Theor. Phys 55(3), 1920–1929 (2016). [CrossRef]

54. D. Wang, R. D. Hoehn, and L. Ye, “Efficient remote preparation of four-qubit cluster-type entangled states with multi-party over partially entangled channels,” Int. J. Theor. Phys 55(7), 3454–3466 (2016). [CrossRef]

55. B. Cai, G. Guo, and S. Lin, “Multi-party quantum key agreement with teleportation,” Mod. Phys. Lett. B 31(10), 1750102 (2017). [CrossRef]

56. J. Wei, L. Shi, and Z. Xu, “Probabilistic controlled remote state preparation of an arbitrary two-qubit state via partially entangled states with multi parties,” Int. J. Quantum Inf . 16, 1850001 (2018). [CrossRef]

57. Y. K. Wang, I. W. Primaatmaja, E. Lavie, A. Varvitsiotis, and C. C. W. Lim, “Characterising the correlations of prepare-and-measure quantum networks,” arXiv:1803.04796v2.

58. Y. Yu, X. W. Zha, and W. Li, “Quantum broadcast scheme and multi-output quantum teleportation via four-qubit cluster state,” Quantum Inf. Process . 16(2), 41 (2017). [CrossRef]

59. Y. J. Zhou and Y. H. Tao, “Probabilistic broadcast-based multiparty remote state preparation scheme via four-qubit cluster state,” Int. J. Theor. Phys. 57(5), 1–5 (2017).

60. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390(6660), 575–579 (1997). [CrossRef]

61. D. Boschi, S. Branca, F. D. Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett 80, 1121 (1998). [CrossRef]

62. M. Riebe, H. Haffner, and C. F. Roos, “Deterministic quantum teleportation with atoms,” Nature 429, 734–737 (2004). [CrossRef] [PubMed]

63. M. D. Barrett, J. Chiaverini, and T. Schaetz, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737–739 (2004). [CrossRef] [PubMed]

64. J. W. Pan, C. Simon, and C. Brukner, “Entanglement purification for quantum communication,” Nature 410, 1067–1070 (2001). [CrossRef] [PubMed]

65. Z. Zhao, Y. A. Chen, and A. N. Zhang, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54–57 (2001). [CrossRef]

66. C. Y. Lu, X. Q. Zhou, and O. Guhne, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007). [CrossRef]

67. X. C. Yao, T. X. Wang, and P. Xu, “Observation of eight-photon entanglement,” Nat. Photonics 6, 225–228 (2012). [CrossRef]

General quantum broadcast and multi-cast communications based on entanglement

Abstract

1. Introduction

2. Assumptions

3. Quantum broadcast scheme

4. Quantum multi-cast scheme

5. Examples and applications of the general schemes

5.1. Five-party broadcast scheme

5.2. Five-party multi-cast scheme

6. Discussion and conclusion

Funding

References

Cited By

Figures (8)

Tables (1)

Equations (26)

Optics Express

	Quantum	Quantum
	broadcast scheme	multi-cast scheme
Similarities	The arbitrary single-qubit state which have to be synchronously delivered to N receivers.
Differences	\|ψ〉 = α\|0〉 + β\|1〉	$\begin{matrix} {\| ψ 〉}_{1} = α_{1} \| 0 〉 + β_{1} \| 1 〉 \\ {\| ψ 〉}_{2} = α_{2} \| 0 〉 + β_{2} \| 1 〉 \\ ⋮ \\ {\| ψ 〉}_{n} = α_{n} \| 0 〉 + β_{n} \| 1 〉 \end{matrix}$