Cloning scheme for multipartite entangled pure states via photonic quantum walk

WANG Guocui; LIN Zhi; LI Xikun; YANG Qing; YANG Ming

doi:10.3969/j.issn.1007-5461.2025.02.006

0　Introduction

Information is a physical entity^[1]. Classical information can be precisely measured, perfectly cloned, broadcasted and erased. However, as dictated by the no-cloning theorem, there is no way to perfectly duplicate a system that is initially in an unknown quantum state^[2], and it is one of the most fundamental differences between classical and quantum information. Although the no-cloning theorem for pure states has been extended to the cases where general mixed states cannot be broadcast^[3], imperfect cloning remains permissible, leading to the development of quantum cloning machines (QCMs) across diverse systems^[4], such as nuclear magnetic resonance^[5], single photons^[6]. Various QCMs have been successively proposed, encompassing transformations of QCMs with optimal fidelity^[7-9] as well as perfect probabilistic cloning protocols^{[10, 11]}. These advancements have catalyzed applications of quantum cloning in quantum information processing^[12], quantum cryptography^[13] and the investigation of quantum-to-classical transition^[14]. Therefore, the study of quantum cloning is of interest for reasons of both fundamental and practical applications.

Quantum entanglement^[15], a cornerstone of quantum mechanics, plays a pivotal role in cloning protocols. Early efforts by Bužek and Hillery^[16] introduced universal cloning machine for Bell states, while subsequent works explored whether (and how well) entanglement could be cloned and entanglement cloning in higher dimensions^[17-20]. Notably, the group of Pan^[21] experimentally demonstrated two-photon entangled state cloning using a Pauli cloning machine, achieving fidelity above 50% for maximally entangled inputs. However, their method requires auxiliary maximally entangled states and discards photons during post-selection, leading to resource inefficiency. Furthermore, extending this approach to multipartite systems results in diminished fidelity. Addressing these limitations, this work introduces novel cloning schemes for unknown multipartite entangled states that achieve higher fidelity with reduced resource consumption.

1　Photonic quantum walk

Quantum walk (QW) is an extension of the classical random walk in the quantum mechanics and the concept of quantum random walk was firstly proposed by Aharonov et al.^[22] in 1993. It is useful for a wide range of applications including search algorithms^[23], quantum simulation^[24] entanglement swapping^[25] and preparation of entanglement state^[26]. Currently, the implementation of QW has been proposed or realized in different physical systems, including ion traps^[27], optical lattices^[28] and single photons^[29], and so on. Based on the fact that the propagation speed of quantum random walk is far beyond the classical characteristics, the application of quantum random walk in the field of quantum cloning, namely the cloning of entangled states, is proposed. This may provide more options for quantum cloning.

At present, the coin QW model is widely used in photonic system. The evolution of a photonic QW consists of two parts: coin flipping and conditional position shift. The whole evolution of coin flipping and conditional position shift can be expressed as $U = S (C \otimes I)$ , where $C$ is the coin flipping operator, and $C \in U (2)$ , $S$ is the conditional position shift operator, and $S$ can be expressed as

S = \sum_{x} (|x + 1〉 〈x| \otimes |H〉 〈H| + |x - 1〉 〈x| \otimes |V〉 〈V|) . (1)

In the above equation, x represents the spatial coordinate, $|H〉$ and $|V〉$ denote the horizontal and vertical polarization states of the photon, respectively. The expression of the conditional position shifts operator Sreveals that a QW can induce the coupling between the spatial degree of freedom (DOF) and the polarization DOF of a photon, so it is possible to effectively manipulate the polarization states of photons by using the spatial DOF as the auxiliary DOF, and thus there is no collapse of the polarization states in the measurement process, which will preserve the quantum states in polarization DOF as much as possible. Leveraging this unique advantage of QW, this paper innovatively presents a QW-based scheme for cloning unknown multiphoton polarization entangled states, where the fidelity of cloned states is relatively high, and the waste of quantum resource is reduced greatly. Moreover, the processes of cloning are simple with only linear optical devices, which greatly reduces the difficulty and complexity of experimental implementation.

2　1-to-2 cloning scheme for bipartite entangled states

The schematic diagram of the scheme for cloning bipartite entangled states is shown in Fig. 1, where the two qubits 1 and 2 in the input entangled state are input together with two independent blank qubits 3 and 4 into the QCM, and two approximate cloned states of the input state will be generated on the output qubits $1^{'}$ , $3^{'}$ and $2^{'}$ , $4^{'}$ (gray dash-dot lines), or $1^{'}$ , $4^{'}$ and $2^{'}$ , $3^{'}$ (red dashed lines), respectively.

Figure 1.Schematic diagram of the QCM, which can generate two approximate clones of an unknown input bipartite quantum state on the two input qubits and two blank qubits, with the clones indicated by red dashed lines as Clone $1$ and Clone $2$ , or by gray dash-dot lines as Clone $1^{'}$ and Clone $2^{'}$

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Suppose that Charles wants to make two copies of an unknown entangled state for Alice in the local place and for Bob at a distant location, respectively, so the unknown entangled state needs to be cloned. Charles prepares a pair of polarization entangled photons (1, 2) in the state ${|ψ〉}_{12} = α |H H〉 + β |V V〉$ and the photon 2 is distributed to Bob through the multicore fiber^[30]. Meanwhile, Alice and Bob introduce the photon 3 and the photon 4 in the horizontal polarization state $|H〉$ in their own locations, respectively.Here $α$ and $β$ are complex coefficients satisfying the normalization condition ${|α|}^{2} + {|β|}^{2} = 1$ , without loss of generality, it is assumed that both $α$ and $β$ are real numbers, i.e. $α, β \in ℝ$ . Next the photons (1, 2, 3, 4) will be input into the QCM, and Alice, Bob will each obtain an approximate clone of the input quantum entangled state after a series of cloning operations. The cloning process depicted in Fig. 2 is subsequently elaborated upon.

Figure 2.The optical circuit for cloning bipartite entangled states. $H_{i} (i = 1, 2)$ denotes HWPs, with $H_{1} = \frac{1}{\sqrt[]{2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}]$ and $H_{2} = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ corresponding to their specific functions; BDi (i = 1, 2)and M are beam displacers and mirror, respectively; PR denotes phase retarder

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The QCM in Fig. 2 can be regarded as a two-step QW, and the polarization operations of the two steps are

\begin{matrix} \begin{matrix} C_{1,1} = C_{1,2} = C_{1,3} = C_{1,4} = \frac{1}{\sqrt[]{2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}], (2) \end{matrix} \end{matrix}

\begin{matrix} C_{2,1} = C_{2,2} = C_{2,3} = C_{2,4} = σ_{x} = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] . (3) \end{matrix}

Where $C_{i, j}$ describes the polarization operators for the $i$ -th step when the photon is in the line j. Following the polarization operation by a conditional position shift S [Eq.(1)] according to the outcome of the polarization operators for each step. S can be achieved by birefringent calcite beam displacers (BDs) and the polarization operators in Eqs. (2-3) can be realized by setting the half wave plates (HWPs) in different directions. The operation of an HWP is described by the matrix

\begin{matrix} H_{θ} = [\begin{matrix} c o s 2 θ & s i n 2 θ \\ s i n 2 θ & - c o s 2 θ \end{matrix}], (4) \end{matrix}

where $θ \in (0, π / 2)$ , represents the angle between the optical axis of HWP and the horizontal polarization.

In addition to the polarization operations, a key operation must be introduced in the QCM scheme, i.e. the trajectory swapping operation for photons, to swap the trajectory information of the two photons without changing their polarization states. The function of this trajectory swapping operation can be described as follows

a_{φ_{1}, x_{n}}^{†} |0〉 \to a_{φ_{1}, y_{m}}^{†} |0〉; a_{φ_{2}, x_{m}}^{†} |0〉 \to a_{φ_{2}, y_{n}}^{†} |0〉, (5)

where $|0〉$ is a vacuum state, $φ$ in $a_{φ_{1}, x_{n}}^{†}$ labels the polarization state, $n$ , $m$ and $x$ , $y$ , label the lines and the current position of the line, respectively. This operation will directly lead to the coupling between two photons, so the state of the four-photon quantum system will evolve to the proper state by the appropriate trajectory swapping operation.

As depicted in Fig. 2, the objective is to clone the entangled state ${|ψ〉}_{12}$ . The positions 0 of lines 1, 2, 3 and 4 are input ports, the input state ${|ψ〉}_{12}$ is the state to be cloned and the two horizontally polarized photons (3, 4) are input into the QCM as the carriers of blank state. The state of the four-photon system can be expressed as follows

{|ψ^{(0)}〉}_{1234} = (α a_{H, 0_{1}}^{†} a_{H, 0_{2}}^{†} + β a_{V, 0_{1}}^{†} a_{V, 0_{2}}^{†}) \otimes a_{H, 0_{3}}^{†} a_{H, 0_{4}}^{†} |0〉 .

(6)

After traversing the HWPs, the four photons are directed into their respective first BDs, and the state of the four photons after BDs is in the following state

\begin{matrix} {|ψ^{(1)}〉}_{1234} = \frac{1}{4} [α (a_{H, 1_{1}}^{†} + a_{V, - 1_{1}}^{†}) (a_{H, 1_{2}}^{†} + a_{V, - 1_{2}}^{†}) + β (a_{H, 1_{1}}^{†} - a_{V, - 1_{1}}^{†}) (a_{H, 1_{2}}^{†} - a_{V, - 1_{2}}^{†})] \otimes \end{matrix} (7)

(a_{H, 1_{3}}^{†} + a_{V, - 1_{3}}^{†}) (a_{H, 1_{4}}^{†} + a_{V, - 1_{4}}^{†}) |0〉

After the first step of QW, as shown in Fig. 2, the site-dependent ( $x = - 1$ ) paths of lines 1 and 3 (2 and 4) will be swapped, which is the key operation of this coupling mechanism. To ensure that the spatial wave functions of the two photons fully overlap when they terminate on the same lines after trajectory swapping, phase retarders (PRs) are introduced for the purpose of phase compensation. Here, the full overlap of two photon wave functions will lead to the coupling of the photons and thus be indistinguishable, leading to the need to consider quantum statistical effects. At the end of the trajectory swapping, the state of the system evolves into

\begin{matrix} \begin{array}{l} {|ψ^{(1)}〉}^{'}_{1234} = \frac{1}{4} [α (a_{H, 1_{1}}^{†} + a_{V, - 1_{3}}^{†}) (a_{H, 1_{2}}^{†} + a_{V, - 1_{4}}^{†}) + β (a_{H, 1_{1}}^{†} - a_{V, - 1_{3}}^{†}) (a_{H, 1_{2}}^{†} - a_{V, - 1_{4}}^{†})] \otimes \\ (a_{H, 1_{3}}^{†} + a_{V, - 1_{1}}^{†}) (a_{H, 1_{4}}^{†} + a_{V, - 1_{2}}^{†}) |0〉 . \end{array} \end{matrix} (8)

The photons after the trajectory swapping enter the second HWPs and BDs successively, performing the second step of QW [Eq.(3)]. Hence, at the end of this step, the state is

\begin{matrix} \begin{array}{l} {|ψ^{(2)}〉}_{1234} = \frac{1}{4} [α (a_{V, 0_{1}}^{†} + a_{H, 0_{3}}^{†}) (a_{V, 0_{2}}^{†} + a_{H, 0_{4}}^{†}) + β (a_{V, 0_{1}}^{†} - a_{H, 0_{3}}^{†}) (a_{V, 0_{2}}^{†} - a_{H, 0_{4}}^{†})] \otimes \\ (a_{V, 0_{3}}^{†} + a_{H, 0_{1}}^{†}) (a_{V, 0_{4}}^{†} + a_{H, 0_{2}}^{†}) |0〉 = \\ \frac{1}{4} [(α + β) (a_{V, 0_{1}}^{†} a_{V, 0_{2}}^{†} a_{V, 0_{3}}^{†} a_{V, 0_{4}}^{†} + a_{H, 0_{1}}^{†} a_{H, 0_{2}}^{†} a_{H, 0_{3}}^{†} a_{H, 0_{4}}^{†}) + \\ (α - β) (a_{V, 0_{1}}^{†} a_{H, 0_{2}}^{†} a_{V, 0_{3}}^{†} a_{H, 0_{4}}^{†} + a_{H, 0_{1}}^{†} a_{V, 0_{2}}^{†} a_{H, 0_{3}}^{†} a_{V, 0_{4}}^{†})] + {|ψ〉}_{o t h e r} = {|ψ〉}^{'}_{1^{'} 2^{'} 3^{'} 4^{'}} + {|ψ〉}_{o t h e r} . \end{array} \end{matrix} (9)

All terms of Eq.(9) are divided into two parts: the contribution part ${|ψ〉}^{'}_{1^{'} 2^{'} 3^{'} 4^{'}}$ where there is exactly one photon in each output mode and the non-contribution part ${|ψ〉}_{o t h e r}$ where two photons are in one of the output modes. Then Alice and Bob perform the process of post-selection with the photons and keep only the contribution part ${|ψ〉}^{'}_{1^{'} 2^{'} 3^{'} 4^{'}}$ . At this point, all photons are at the position 0 of the lines. Omitting the factorized spatial DOF, the output state of the system becomes

{|ψ〉}_{1^{'} 2^{'} 3^{'} 4^{'}} = \frac{1}{4} [(α + β) |H H H H〉 + (α - β) |H V H V〉 + (α - β) |V H V H〉 + (α + β) |V V V V〉] . (10)

After normalization of [Eq.(9)], two output cloned states are obtained

ρ = ρ_{1^{'} 3^{'}} = ρ_{2^{'} 4^{'}} = \frac{1}{2} [|H H〉 〈H H| + |V V〉 〈V V| + (α^{2} - β^{2}) (|V V〉 〈H H| + |H H〉 〈V V|)] . (11)

It can be seen from Eq. (10) that after QW and trajectory swapping operations, two identical cloned states $ρ_{1^{'} 3^{'}}$ and $ρ_{2^{'} 4^{'}}$ (symmetric cloning) are obtained at the output ports, i.e., the case of Clone $1^{'}$ and Clone $2^{'}$ (gray dash-dot lines) are obtained in Fig. 1. If the trajectories of the four lines of the photons at position $x = 1$ are swapped, the case of Clone 1 and Clone 2 (red dashed lines) can be obtained at the output ports. Specifically, the site-dependent ( $x = - 1$ ) paths of lines 1, 2 and lines 3, 4 are swapped, and the site-dependent ( $x = 1$ ) paths of the lines 1, 3 and lines 2, 4 are swapped. It is noteworthy that the cloned states $ρ_{1^{'} 3^{'}}$ , $ρ_{2^{'} 4^{'}}$ and $ρ_{1^{'} 4^{'}}$ , $ρ_{2^{'} 3^{'}}$ are exactly identical.

Therefore, the fidelities between the input states and the output states are

F (α) = F_{1^{'} 3^{'}} (α) = F_{2^{'} 4^{'}} (α) = \frac{1}{2} + α \sqrt[]{1 - α^{2}} (2 α^{2} - 1) . (12)

When $1 / \sqrt[]{2}$ ≤ $α$ ＜ $1$ ，the fidelity of the two cloned states $F$ ≥ $0.5$ , and the maximum fidelity $F_{m a x} = 0.75$ when $α = \sqrt[]{0.85}$ . The entanglement of the cloned states $ρ$ can be quantified by analyzing the concurrence^[31], with the specific definition of concurrence (C) outlined as follows

\begin{matrix} \begin{matrix} C (ρ) = C (ρ_{1^{'} 3^{'}}) = C (ρ_{2^{'} 4^{'}}) \end{matrix} = m a x \{λ_{1} - λ_{2} - λ_{3} - λ_{4}, 0\}, (13) \end{matrix}

where $λ_{i} (i = 1, 2, 3, 4)$ are four eigenvalues of the density matrix $X = \sqrt[]{ρ \tilde{ρ}}$ in decreasing order, $\tilde{ρ} = (σ_{y} \otimes σ_{y}) ρ^{*} (σ_{y} \otimes σ_{y})$ and $ρ^{*}$ represents the complex conjugation of state $ρ$ ; $λ_{1} = α^{2}$ , $λ_{2} = |α^{2} - 1|$ , $λ_{3} = λ_{4} = 0$ . If $λ_{1}$ ＞ $λ_{2}$ , $α$ ＞ $1 / \sqrt[]{2}$ and thus the concurrence of the cloned states is

C (ρ) = 2 α^{2} - 1 . (14)

Therefore, compared with the scheme in Ref.[21], The fidelity of this scheme is enhanced when the two cloned states display symmetry. Furthermore, to get the two clones, no quantum system will be traced out from the QCM, which reduces the waste of quantum resources. Eventually, Alice at the local place and Bob at a distant location obtain the cloned states $ρ_{1^{'} 3^{'}}$ and $ρ_{2^{'} 4^{'}}$ , respectively.

3　1-to-2 cloning scheme for tripartite entangled states

When the protocol in Ref.[21] is employed for tripartite entangled states, the fidelities of the resultant two cloned states do not exceed 0.5, and it is still under the premise of consuming three additional bipartite maximally entangled states. To address the limitations, a 1-to-2 cloning scheme for tripartite entangled states based on the photonic QW is proposed, as depicted in Fig. 3.

Figure 3.The optical circuit for cloning tripartite entangled states

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Suppose that Charles wants to make two copies of an unknown tripartite entangled state for Alice in the local place and Bob at a distant location, respectively. It begins with Charles preparing a tripartite polarization entangled state of photons (1, 2, 3) in the form of ${|ψ〉}_{123} = α |H H H〉 + β |V V V〉$ , when $α$ and $β$ are complex coefficients satisfying the normalization condition ${|α|}^{2} + {|β|}^{2} = 1$ . Without loss of generality, it is assumed that both $α$ and $β$ are still real numbers, i.e. $α, β \in ℝ$ . Then the photon 2 is distributed to Bob through the multicore fiber^[30], and Alice and Bob introduce the photon 5 and the photons (4, 6) in the horizontal polarization state $|H〉$ in their own locations, respectively. Now Alice and Bob need to clone the tripartite entangled state ${|ψ〉}_{123}$ .

As shown in Fig. 3, the position 0 of each line is regarded as the input port, and the three photons 1, 2 and 3 in the input entangled state ${|ψ〉}_{123}$ are input together with three independent blank (in horizontal polarization state) photons 4, 5 and 6 into the QCM, and two approximate cloned states of the input state will be generated on the output photons $1^{'}$ , $3^{'}$ , $5^{'}$ and $2^{'}$ , $4^{'}$ , $6^{'}$ , respectively. At this point, the initial state of the entire composite system can be represented as

{|ψ^{(0)}〉}_{123456} = (α a_{H, 0_{1}}^{†} a_{H, 0_{2}}^{†} a_{H, 0_{3}}^{†} + β a_{V, 0_{1}}^{†} a_{V, 0_{2}}^{†} a_{V, 0_{3}}^{†}) \otimes a_{H, 0_{4}}^{†} a_{H, 0_{5}}^{†} a_{H, 0_{6}}^{†} |0〉 . (15)

After passing through the HWPs H₁, the six photons will enter the corresponding first BDs BD₁. Before the trajectory swapping operation, the photon at position $x = 1$ in line 1 passe through an HWP H₁, and then enters BD₂. Appropriate trajectory swapping operation is vital for obtaining the target cloned entangled states. The next step is to perform the trajectory swapping operation: the site-dependent ( $x = - 1$ ) paths of lines 3, 4 are swapped with those of lines 5, 2, respectively and lines 5, 2; the site-dependent ( $x = 1$ ) paths of line 2 and line 6 are swapped and the site-dependent ( $x = 2$ ) path of line 1 is swapped with the site-dependent ( $x = 1$ ) path of line 5. Now the state of the system becomes

\begin{array}{l} \begin{matrix} {|ψ^{(2)}〉}^{'}_{123456} = \frac{1}{8 \sqrt[]{2}} [α (a_{H, 1_{5}}^{†} + a_{V, 0_{1}}^{†} + \sqrt[]{2} a_{V, - 2_{1}}^{†}) (a_{H, 1_{6}}^{†} + a_{V, - 1_{4}}^{†}) (a_{H, 1_{3}}^{†} + a_{V, - 1_{5}}^{†}) + \\ β (a_{H, 1_{5}}^{†} + a_{V, 0_{1}}^{†} - \sqrt[]{2} a_{V, - 2_{1}}^{†}) (a_{H, 1_{6}}^{†} - a_{V, - 1_{4}}^{†}) (a_{H, 1_{3}}^{†} - a_{V, - 1_{5}}^{†})] \otimes \end{matrix} \\ (a_{H, 1_{4}}^{†} + a_{V, - 1_{2}}^{†}) (a_{H, 2_{1}}^{†} + a_{V, - 1_{3}}^{†}) (a_{H, 1_{2}}^{†} + a_{V, - 1_{6}}^{†}) |0〉 . \end{array}

(16)

Next, the photon at the positions $x = 2$ and $x = - 2$ of line 1 pass through H₂, and then the photon in line 1 enter BD₃, the state of the system subsequently evolves into

\begin{matrix} \begin{array}{l} {|ψ^{(3)}〉}_{123456} = \frac{1}{8 \sqrt[]{2}} [α (a_{H, 1_{5}}^{†} + a_{V, - 1_{1}}^{†} + \sqrt[]{2} a_{H, - 1_{1}}^{†}) (a_{H, 1_{6}}^{†} + a_{V, - 1_{4}}^{†}) (a_{H, 1_{3}}^{†} + a_{V, - 1_{5}}^{†}) + \\ β (a_{H, 1_{5}}^{†} + a_{V, - 1_{1}}^{†} - \sqrt[]{2} a_{H, - 1_{1}}^{†}) (a_{H, 1_{6}}^{†} - a_{V, - 1_{4}}^{†}) (a_{H, 1_{3}}^{†} - a_{V, - 1_{5}}^{†})] \otimes \\ (a_{H, 1_{4}}^{†} + a_{V, - 1_{2}}^{†}) (a_{V, 1_{1}}^{†} + a_{V, - 1_{3}}^{†}) (a_{H, 1_{2}}^{†} + a_{V, - 1_{6}}^{†}) |0〉 . \end{array} \end{matrix} (17)

Then, the photons at position $x = - 1$ of line 1 is directed to pass H₂ and then enters BD₄, and the photon at position $x = 1$ of line 1 directly enters BD₄. At the same time, the photons in other lines also enter H₂ and then pass BD₄. Then the state of the whole system will evolve into

\begin{matrix} {|ψ^{(4)}〉}_{123456} = \frac{1}{8 \sqrt[]{2}} [α (a_{V, 0_{5}}^{†} + a_{H, 0_{1}}^{†} + \sqrt[]{2} a_{V, - 2_{1}}^{†}) (a_{V, 0_{6}}^{†} + a_{H, 0_{4}}^{†}) (a_{V, 0_{3}}^{†} + a_{H, 0_{5}}^{†}) + \end{matrix}

β (a_{V, 0_{5}}^{†} + a_{H, 0_{1}}^{†} - \sqrt[]{2} a_{V, - 2_{1}}^{†}) (a_{V, 0_{6}}^{†} - a_{H, 0_{4}}^{†}) (a_{V, 0_{3}}^{†} - a_{H, 0_{5}}^{†})] \otimes

(a_{V, 0_{4}}^{†} + a_{H, 0_{2}}^{†}) (a_{V, 0_{1}}^{†} + a_{H, 0_{3}}^{†}) (a_{V, 0_{2}}^{†} + a_{H, 0_{6}}^{†}) |0〉

=

(18)

\frac{1}{8 \sqrt[]{2}} [(α + β) (a_{V, 0_{1}}^{†} a_{V, 0_{2}}^{†} a_{V, 0_{3}}^{†} a_{V, 0_{4}}^{†} a_{V, 0_{5}}^{†} a_{V, 0_{6}}^{†} + a_{H, 0_{1}}^{†} a_{H, 0_{2}}^{†} a_{H, 0_{3}}^{†} a_{H, 0_{4}}^{†} a_{H, 0_{5}}^{†} a_{H, 0_{6}}^{†}) +

(α - β) (a_{V, 0_{1}}^{†} a_{H, 0_{2}}^{†} a_{V, 0_{3}}^{†} a_{H, 0_{4}}^{†} a_{V, 0_{5}}^{†} a_{H, 0_{6}}^{†} + a_{H, 0_{1}}^{†} a_{V, 0_{2}}^{†} a_{H, 0_{3}}^{†} a_{V, 0_{4}}^{†} a_{H, 0_{5}}^{†} a_{V, 0_{6}}^{†})] + {|ψ^{'}〉}_{o t h e r}

=

{|ψ〉}^{'}_{1^{'} 2^{'} 3^{'} 4^{'} 5^{'} 6^{'}} + {|ψ^{'}〉}_{o t h e r}

All the terms of Eq. (18) are divided into two parts: the contribution part ${|ψ〉}^{'}_{1^{'} 2^{'} 3^{'} 4^{'} 5^{'} 6^{'}}$ where there is exactly one photon in positions $x = 0$ of lines 1, 2, 3, 4, 5 and 6, and the non-contribution part ${|ψ^{'}〉}_{o t h e r}$ of the other terms. Finally, according to the position distribution of the photons, Alice and Bob post-select the contribution part ${|ψ〉}^{'}_{1^{'} 2^{'} 3^{'} 4^{'} 5^{'} 6^{'}}$ . Notably, the photons output at position $x = - 2$ in line 1 will be discarded. After the process of post-selection, the spatial DOF is factorized out, the output state of the system becomes

\begin{matrix} {|ψ〉}_{1^{'} 2^{'} 3^{'} 4^{'} 5^{'} 6^{'}} = \frac{1}{8 \sqrt[]{2}} [(α + β) |H H H H H H〉 + (α - β) |V H V H V H〉 + \\ (α - β) |H V H V H V〉 + (α + β) |V V V V V V〉] . \end{matrix} (19)

After normalization of [Eq.(17)], two output cloned states are obtained

\begin{array}{l} \begin{matrix} ρ = ρ_{1^{'} 3^{'} 5^{'}} = ρ_{2^{'} 4^{'} 6^{'}} = \frac{1}{2} [|H H H〉 〈H H H| + |V V V〉 〈V V V| + \end{matrix} \\ (α^{2} - β^{2}) (|V V V〉 〈H H H| + |H H H〉 〈V V V|)] . (20) \end{array}

Here they still get two identical cloned states whose fidelity function is identical to Eq. (11). In this scenario, to verify whether these two cloned states are GHZ-type entangled states, employing an entanglement witness^[32] seems to be an appropriate method. It is defined as $W = 3 / 4 I - |G H Z〉〈G H Z|$ , which is a Hermitian operator with negative expectation value $T r [W ρ^{'}] < 0$ for an entangled state $ρ^{'}$ . In this case, the expectation value of $W$ can be expressed as

〈W〉 = T r (W ρ^{'}) = \frac{3}{4} - α^{2} . (21)

When $\sqrt[]{3} / 2 < α$ , the expectation value $〈W〉 < 0$ , that is to say, the both clones $ρ_{1^{'} 3^{'} 5^{'}}$ and $ρ_{2^{'} 4^{'} 6^{'}}$ are in the GHZ-type entangled state like the initial entangled state ${|ψ〉}_{123}$ . Consequently, Alice obtains a cloned state $ρ_{1^{'} 3^{'} 5^{'}}$ of ${|ψ〉}_{123}$ while Bob at a distant location gets a cloned $ρ_{2^{'} 4^{'} 6^{'}}$ .

The proposed schemes align with demonstrated technologies: polarization-entangled states generated via spontaneous parametric down-conversion^[33], heralded single-photon sources^[34], and high-dimensional entanglement distribution through fiber^[30]. The spatial DOF of photons, with the advantages of being easily integrable^[35] and offering high measurement accuracy^[36], combined with phase stability techniques^[37], further ensures the feasibility of the experiment^[38,39]. Finally, it should be pointed out that in order to obtain the event-ready cloned states of the initial entangled states, there must be an ideal "non-demolition" measurement of photons to complete the process of post-selection. Although the detection of photonic states will annihilate the photons within the current technology，by integrating the output ports of quantum cloning machines with application setups and performing the measurement at the end of the applications, this problem can be feasibly avoided by implementing these two processes simultaneously.

4　Conclusion

We present resource-efficient cloning schemes for bipartite and tripartite entangled states using photonic QW. Key advantages of our schemes are three-fold. The fidelities of the cloned states are higher than the previous physical schemes for cloning entanglement. No quantum systems will be traced out during the cloning process, which will reduce the waste of quantum resource. The key operations of the cloning setup include trajectory swapping operation and post-selection processes, which are straightforward and feasible with current technology, ensuring compatibility with current linear optics.