esa-sceva/satcom-chunk-collection huggingface dataset:
## 4 Results  
### Distribution of Global Ecosystem-Scale Optimum Temperature for Photosynthesis  
The distribution of global \(T_{opt}^{coo}\) based on long-term CSIF was shown in Figure 3a. In general, the average \(T_{opt}^{coo}\) over the vegetated areas was 20.17 \(\pm\) 5.57 \({}^{\circ}\)C and \(T_{opt}^{coo}\) varied from 15\(\sim\)30 \({}^{\circ}\)C in approximately 82.47% of the vegetated area. The global \(T_{opt}^{coo}\) presented a decreasing trend from low latitudes to high latitudes, except that the average \(T_{opt}^{coo}\) in the vegetated area of the Qinghai-Tibet Plateau was 10.58 \(\pm\) 3.55 \({}^{\circ}\)C, showing a significant spatial gradient. In this study, \(T_{opt}^{coo}\) calculated by the in situ GPP was used to verify the global \(T_{opt}^{coo}\) based on satellite SIF, and the relationship between them was shown in Figure 3b. Satellite SIF-derived \(T_{opt}^{coo}\) was comparable to that calculated by in situ GPP (R\({}^{2}\) = 0.63, RMSE = 3.15 \({}^{\circ}\)C), which provided support for estimating global \(T_{opt}^{coo}\) using CSIF data. Among all vegetation types, EBF, CRO, DBF, SAV, and WSA had higher \(T_{opt}^{coo}\), while the mean \(T_{opt}^{coo}\) of ENF, DNF, and GRA were lower (Figure 3c). When we calculated the global GPP, if there were too many invalid data and the temperature response curve was not statistically significant, we used the average optimal temperature of the vegetation type in Figure 3c to replace the optimal temperature of this pixel.  
Figure 3: (**a**) Global ecosystem-scale optimum temperature for photosynthesis (T\({}_{\rm opt}^{\rm coo}\)); (**b**) Relationship between in situ GPP and satellite SIF-derived \(T_{\rm opt}^{\rm coo}\); (**c**) \(T_{\rm opt}^{\rm coo}\) for different vegetation types. The vegetation types include evergreen needleleaf forest (ENF), evergreen broadleaf forest (EBF), deciduous needleleaf forest (DNF), deciduous broadleaf forest (DBF), mixed forest (MF), closed shrubland (CSH), open shrubland (OSH), woody savanna (WSAV), savanna (SAV), grassland (GRA), permanent wetland (WET), and cropland (CRO).
# Fractional Derivative Gradient-Based Optimizers for Neural Networks and Human Activity Recognition
## 3 Results
### Experiment 3
#### 3.3.3 Experiment 3.3  
Finally, the same experiments 3.2 and 3.2 were repeated with 10 folds, and similar results were obtained. An overview of the accuracies of both groups, group 1 (FSGD, FSGDP, FAdagrad, FAdadelta) and group 2 (FRMSProp, Fadam, FAdamP), is shown as boxplots in Figures 18 and 19. The boxplots correspond to each optimizer listed in groups 1 and 2, for \(\
u\in[0.1,1.9]\) with increments of 0.1.  
In Figure 18, it is possible to appreciate the highest accuracy for FAdagrad at \(\
u=1.7\), whereas in Figure 19, the highest accuracy is for FAdam at \(\
u=1.2\).  
Figure 16: Experiment 3.2: FAdam cross-folding accuracies, \(K=4\) folds, \(\
u\in[0.1,1.9]\).  
Figure 17: Experiment 3.2: FAdamP cross-folding accuracies, \(K=4\) folds, \(\
u\in[0.1,1.9]\).  
Once the experiments have been carried out, it is convenient to mention that, similar to the integer case, a geometric interpretation can be given to Equation (24) as a gradient  
Figure 19: Experiment 3.2: Group 2, cross-folding accuracies, \(K=4\) folds, \(\
u\in[0.1,1.9]\).  
Figure 18: Experiment 3.2: Group 1, cross-folding accuracies, \(K=4\) folds, \(\
u\in[0.1,1.9]\).  
at the region of interest on the error surface. The main benefit of fractional \(v\)-derivative over the integer version is the expansion of the region of search around the test point given by \(v\) and the weights \(w^{l}_{kj}\)[33]. In this sense, the benefit is provided by the factor \(f^{v}_{w}\) of Equation (25), and to explore the effect of non-locality, a 3D plot is shown in Figure 20. It is possible to identify four regions with different behavior:  
1. Region A: For \(v\to 0^{+}\). The factor \(f^{v}_{w}\) follows \(|w^{l}_{kj}|\) and it could produce divergence for \(w^{l}_{kj}\) and consequently for \(f^{v}_{w}\), and for the fractional gradient \(D^{v}_{w^{l}_{kj}}\,E_{i}\).
2. Region B: For \(v=1\). The integer case corresponds to the red line \(f^{v}_{w}=1\). No context information is considered, just the local point.
3. Region C: For \(1<v<2\). The surface \(f^{v}_{w}\) reaches a local maximum \(f^{v^{*}}_{w}=8.75\) at \(v^{*}=1.77\) for \(\epsilon=0.01\).
4. Region D: For \(v\to 2^{-}\). \(f^{v}_{w}\) tends to zero as fast as \(|w^{l}_{kj}|\) increases. It promotes that small weights increase its values and move to the flatter region, where they will stabilize. The Cartesian axes in Figure 20 are as follows:  
* \(x\)-axis. The weights \(w^{l}_{kj}\).
* \(y\)-axis. Fractional value \(v\in(0,2)\).
* \(z\)-axis. The fractional factor \(f^{v}_{w}\) that modifies the integer order gradient.  
In Figure 20 there is a yellow plane at \(f^{v}_{w}=1\) used as reference for the red line of the integer case.  
It is noteworthy that in region C, the experimental accuracies of the fractional optimizers (depending of \(v\)) follow a similar behavior to \(f^{v}_{w}\). For now, it is just an observation that merits exploring the possible relationship between the optimal fractional order \(v^{*}\) and the region of best accuracy for fractional optimizers.
# Prestack Seismic Inversion via Nonconvex L\({}_{1-2}\) Regularization
## 2 Method
### Decorrelation Processing of Inversion Parameters  
According to prior work in this field and statistical analyses of logging curve data, it has been determined that the P impedance, S impedance, and density values that are input into the above forward model are not independent of one another. An additional problem arises because these three parameters have different dimensions. To improve the accuracy of our inversion results, we decorrelate the P impedance, S impedance, and density parameters using singular value decomposition (SVD).  
The relative changes in the parameters to be inverted are expressed in matrix form as  
\[\mathbf{X}=\left[\begin{array}{cccc}e_{p}^{1}&e_{p}^{2}&\cdots&e_{p}^{N}\\ e_{s}^{1}&e_{s}^{2}&\cdots&e_{s}^{N}\\ e_{\rho}^{1}&e_{\rho}^{2}&\cdots&e_{p}^{N}\end{array}\right] \tag{14}\]  
where \(N\) is the number of sampling points and \(e_{p}\), \(e_{s}\), and \(e_{\rho}\) are the natural logarithmic series of the P impedance, S impedance, and density values obtained from the logging curve, respectively. Based on Equation (18), we define the symmetric covariance matrix as:  
\[\mathbf{C_{x}}=\frac{\mathbf{X}\mathbf{X}^{\mathrm{T}}}{N-1}=\left[\begin{array} []{cccc}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{array}\right] \tag{15}\]  
The SVD of this matrix is  
\[\mathbf{C_{x}}=\mathbf{v}\mathbf{u}\mathbf{v}^{-1}=\mathbf{v}\mathbf{u} \mathbf{v}^{\mathrm{T}} \tag{16}\]  
Substituting Equation (15) into Equation (16) and rearranging the variables yields:  
\[\mathbf{u}=\mathbf{v}^{\mathrm{T}}\mathbf{C_{x}}\mathbf{v}=\left[\begin{array} []{cccc}\sigma_{p}^{2}&&&\\ &\sigma_{s}^{2}&&\\ &&\sigma_{p}^{2}\end{array}\right]=\frac{\mathbf{v}^{\mathrm{T}}\mathbf{X} \mathbf{X}^{\mathrm{T}}\mathbf{v}}{N-1} \tag{17}\]where \(\mathbf{v}=\left[\begin{array}{ccc}v_{11}&v_{12}&v_{13}\\ v_{21}&v_{22}&v_{23}\\ v_{31}&v_{32}&v_{33}\end{array}\right]\)  
For a single-layer interface, \(\mathbf{u}=\left[\begin{array}{ccc}\sigma_{p}^{2}&\\ &\sigma_{s}^{2}&\\ &&\sigma_{p}^{2}\end{array}\right]\) can be used as a priori constraints.  
However, with a multi-layer interface, we first obtain \(\mathbf{r}^{\prime}\) via decorrelation during the actual inversion process. Next, we convert \(\mathbf{r}^{\prime}\) into \(\mathbf{r}\) :  
\[\mathbf{r}=\tilde{\mathbf{V}}\mathbf{r}^{\prime} \tag{18}\]  
where, \(\tilde{\mathbf{V}}=\left[\begin{array}{ccc}\mathbf{V}_{11}&\mathbf{V}_{12}& \mathbf{V}_{13}\\ \mathbf{V}_{21}&\mathbf{V}_{22}&\mathbf{V}_{23}\\ \mathbf{V}_{31}&\mathbf{V}_{32}&\mathbf{V}_{33}\end{array}\right]\) and \(\mathbf{V}_{ij}=\text{diag}\Big{(}\mathbf{v}_{ij}^{1}\mathbf{v}_{ij}^{2} \cdots\mathbf{v}_{ij}^{N}\Big{)}\), for i,j = 1,2,3.  
Once the decorrelation is completed, we can redefine the inversion objective function shown in Equation (19):  
\[\mathbf{x}=\text{arg}\min\bigg{\{}\frac{1}{2}\|\mathbf{G}^{\prime}\mathbf{x}- \mathbf{b}\|_{2}^{2}+\lambda(\|\mathbf{x}\|_{1}-\|\mathbf{x}\|_{2})\bigg{\}} \tag{19}\]  
where \(\mathbf{G}^{\prime}=\mathbf{G}\tilde{\mathbf{V}}\),\(\mathbf{x}=\mathbf{r}^{\prime}\).
## 4 Hierarchical Energy Balancing Protocol (HEBP) Design
### Optimizing WSN Performance with HEBP
#### 4.3.2 HEBP Performance  
The performance of HEBP compared to that of DSMP, the previously considered protocol for WS powered by an energy harvesting process, is shown in Figure 14.  
* In Figure 14a, we observed for a data size set of 512 bits that each WS transmits up to 23 Mbits of data, unlike the at-most 10 kbits of data as shown in Figure 12b.
* In Figure 14b, the data size is set to 1024 bits, and it was observed that the maximum data size had doubled; each WS transmitted up to more than 45 Mbits of data.
* Figure 14c compares the performances achieved in the case of DSMP and HEBP; this was for a variable data size between 512 and 4096 bits. It was first observed that all the WSs of the network transmit information, unlike the result obtained with the DSMP, where only 5 of the 12 WSs could communicate.
* Figure 14d shows the effective range of HEBP; for this, the number of active WSs for each protocol is compared as a function of the data size. It shows that with HEBP, the 12 WSs deployed on the system transmit data when the data size is less than 1100 bits.  
Figure 13: HEBP design stage.
# User Comment-Guided Cross-Modal Attention for Interpretable Multimodal Fake News Detection
## 2 Review of Fake News Detection Approaches and Multimodal Learning Methods
### Context-Based Fake News Detection  
Context-based approaches identify false information by examining the social environment surrounding news propagation, including user characteristics, propagation patterns, temporal dynamics, and social network structures. These methods recognize that fake news often exhibits distinctive behavioral patterns during propagation through social networks [30; 31].  
Social network analysis research has revealed characteristic propagation patterns of fake news. [PERSON] et al. [32] analyzed large-scale Twitter data and found that false news spreads faster, farther, and deeper than true news. [PERSON] et al. [33] further demonstrated that fake news exhibits higher engagement rates and distinctive user interaction patterns. [PERSON] et al. [34] showed that false news tends to propagate within echo chambers and filter bubbles, forming identifiable network topological features.  
Graph-based methods have shown unique advantages in modeling complex relationships. [PERSON] et al. [35] proposed Hierarchical Social Attention (HSA) networks that utilize hierarchical social network structures and user comments for rumor identification. [PERSON] et al. [36] developed graph convolutional networks capable of simultaneously processing complex relationships among users, content, and propagation patterns. [PERSON] et al. [37] designed recurrent neural networks that model temporal evolution characteristics of rumors by constructing propagation tree structures. Additionally, tensor factorization with sparse and graph regularization has been applied to capture intricate social network structures for fake news detection [38].  
Recent research has focused more on deep integration of user comments and social signals. [PERSON] et al. [39] proposed the dEFEND framework that uses user comments as external knowledge to enhance detection performance. [PERSON] et al. [40] developed DG-Explain, introducing a dual-generative framework that combines content analysis with comment-based explanations, providing both detection capabilities and interpretable insights. However, existing methods typically treat comments as auxiliary information rather than primary modalities, representing an important research gap.
## 2 Methodology  
### BSS Model  
The SOBI method is a relatively robust BSS algorithm that can be used to decompose mixtures of signals into a set of components or putative recovered sources [29]. To better understand the improved SOBI method, the principle of BSS is introduced first. Figure 1 shows an instantaneous linear BSS system model. \(S(t)=\left[s_{1}(t)s_{2}(t)\ldots s_{n}(t)\right]^{T}\) represents \(n\) statistically independent unknown source signals. \(X(t)=\left[x_{1}(t)x_{2}(t)\ldots x_{m}(t)\right]^{T}\) denotes \(m\) observation signals. Then, let \(X(t)=A*S(t)\), where \(A\) denotes a mixing matrix of \(m*n\) dimensions [29; 30]. The essence of BSS is to find a separation matrix \(B\), so that \(Y(t)=BX(t)\) is an optimal estimate of the source signal, which is obtained from the original observation signal \(X(t)\) through the separation matrix \(B\).  
In this study, the prior information of the source signals (time-series displacements without noise) and the mixing matrix were unknown--only the observed signals (time-series displacement obtained  
Figure 1: Instantaneous linear blind source separation (BSS) system model.  
using GBSAR) were processed to obtain the different source signals. However, in the absence of any underlying assumptions and more information, multiple solutions will be produced by BSS. Therefore, it is necessary to make some underlying assumptions and conditions to constrain the source signal and the mixing matrix, and then solve the problem of the multiple solutions. The underlying assumptions and conditions are described as follows [26; 30]:  
1. Mixing matrix, \(A\), is a column full-rank matrix.
2. The source signals are the zero-mean random signals that are not correlated in the time-space domain and correlated in the time domain.
3. The source signal and noise are independent.
4. The number of source signals is less than or equal to the number of the original signals.  
Moreover, when both the mixing matrix, \(A\), and the source signal, \(S(t)\), are unknown, each column of \(A\) and the corresponding row of \(S(t)\) are multiplied by a factor that is reciprocal to each other. In addition, \(X(t)\) is unchanged, which is shown as follows:  
\[X(t)=A\ast S(t)=\sum_{i=1}^{M}a_{i}s_{i}(t)=\sum_{i=1}^{M}\left(a_{i}/b_{i} \right)\left(b_{i}s_{i}(t)\right), \tag{1}\]  
where \(b_{i}\
eq 0\) is an arbitrary constant. In the case where \(A\) is unknown, as shown in Equation (1), it cannot be determined whether the source signal is \(s_{i}(t)\) or \(b_{i}s_{i}(t)\)--that is, the amplitude of the source signal cannot be determined. Therefore, even if the above underlying assumptions and constraints are applied to the obtained time-series displacement, there are still two uncertainties in BSS--that is, the amplitude and component sequence of the separated source signals are uncertain [29; 31; 32]. However, the separated signal is consistent with the waveform of the source signal. In this study, the uncertainty of the component sequence does not affect the result, and the uncertainty of the amplitude of the separated signals are restored using the improved SOBI algorithm.
## 2 Preliminaries
### BPL Decoding Algorithm for Polar Codes
#### 2.1.2 BPL Decoding Algorithm for Polar Codes  
A polar code of length \(N\) has \((log2N)!\) factor graphs, which is referred to as the \"overcomplete representation\" of polar codes [8]. The study in [20] shows that although non-original factor graphs do not outperform the original factor graph in performance, using different factor graphs for decoding can yield gains in the error-correction performance of BP decoding. Figure 2 presents two of the factor graphs for a code length of \(N=8\)[12].  
Based on the aforementioned concept of \"overcomplete representation\", the study in [7] proposes the BPL decoding algorithm, which performs close to Successive Cancellation List (SCL) decoding in performance. Since using different factor graphs for decoding results in different decoding performances, simultaneously using multiple factor graphs for decoding and complementing the decoding results can achieve better decoding performance. This is the main reason for the excellent performance of the BPL decoding algorithm for polar codes. Unlike SCL decoding, BPL decoding employs \(N\) completely independent BP decoders operating in parallel, each using a different factor graph for decoding.  
It is evident that the key to the excellent performance of the BPL decoding algorithm lies in formulating a reasonable path selection rule and selecting the optimal decoding result from \(L\) decoding outcomes. Currently, the Euclidean distance calculation formula, defined by Equation (3), is primarily used:  
\[\hat{x}_{BPL}=\operatorname*{arg\,\,}_{x_{i}\in\{1,\ldots,L\}}\min\lVert y- \hat{x}_{i}\rVert. \tag{3}\]  
The BPL algorithm flow diagram of the polar code using traditional path selection criteria is shown in Figure 3. In traditional path selection criteria, each of the \(L\) sub-BP decoders performs an \(\hat{x}=\hat{u}G\) check after each individual iteration. If a decoder passes the check, its current output is stored until all BP decoders stop or the maximum iteration count \(I_{max}\) is reached. Once all BP decoders stop, the Euclidean distance decision maker at the end of the decoders activates. If there are paths that pass the check, only the path with the smallest Euclidean distance among those that pass the check is selected for output.  
Figure 1: Minimal computational unit for BP decoding.  
Figure 2: Minimal two-permutation factor graphs for polar codes of length 8.  
However, due to the stringent conditions of \(\hat{x}=\hat{u}G\), when the maximum iteration count \(I_{max}\) is small, all BP decoders may fail to pass the check, resulting in decoding failure. This paper aims to propose a new path selection criterion that can improve the error-correction performance of BP decoding for polar codes even when the maximum iteration count \(I_{max}\) is small.
## 10 Discussion  
The development of the EPATH method provides a solution for hydraulically integrated robotic component assembly. Choosing suitable parameters is depicted in the flowchart of Figure 16. It starts with the robotic part's design and determining the required volume of carbon particle percentage. Once the parts are designed, the required assembly forces are calculated (\(F_{t}\)). Then, the force is calculated per screw as \(\frac{F_{t}}{N}\), with \(N\) representing the number of screws in the assembly zone. The charts are then used to determine the values \(d,e,l\). Figure 17 represents the \(PTF\) per 1 mm of engagement length. The charts are designed to compensate for the maximum porosity existing in the composite. Depending upon the mechanical design of the robotic part, a length \(l\) is first chosen. Consequently, a pair \((d,e)\) can be selected from the graphs. As an example, suppose that total force \(F_{t}=4500\) N is required at the assembly zone and that \(N=6\) screws are used. A choice of engagement length of 5 mm means that the intermediate's length is \(l=5\) mm. Therefore, the required PTF is calculated as follows:  
\[PTF=\frac{F_{t}}{Nl}=\frac{4500}{6\times 5}=150\,\mathrm{N/mm} \tag{34}\]  
Returning to Figure 17, a line is drawn at 150 N/mm. It intersects the curves at five points: \((d;e)=(16;3,55)\), \((d;e)=(14;3,8)\), \((d;e)=(12;4,15)\), \((d;e)=(10;4,85)\), and \((d;e)=(8;6,7)\). Depending upon the design geometric limitation, one of the combinations can be chosen, or the calculation can be repeated with an engagement length (\(l_{1}>l=5\) mm) to further optimize the \((d,e)\) parameters.  
Figure 16: EPATH geometric value choice procedure, A and B are two parts assembled with the EPATH solution.
# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 3 Results
### \(1\to 3\) Optimal Asymmetric Quantum Universal Clones Multicast Protocol
#### 3.2.2 Proof of \(1\to 3\) Quantum Multicast Protocol
\begin{split}&\sum_{j\
eq r,s}^{d-1}\kappa_{j}|jjj\rangle_{DEF} \otimes(\alpha_{1}^{\prime}|irs\rangle+\beta_{1}^{\prime}|rjs\rangle+\gamma_{ 1}^{\prime}|rsj\rangle+\alpha_{1}^{\prime}|sjr\rangle+\gamma_{1}^{\prime}|srj \rangle)_{M_{1}M_{2}M_{3}}\otimes|0\rangle_{N_{1}}\\ &\qquad+\kappa_{r}|rrr\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|rrs\rangle+\beta_{1}^{\prime\prime}|srr\rangle+\gamma_{1}^{\prime \prime}|rsr\rangle)_{M_{1}M_{2}M_{3}}\otimes|1\rangle_{N_{1}}\\ &\qquad+\kappa_{s}|sss\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|ssr\rangle+\beta_{1}^{\prime\prime}|rss\rangle+\gamma_{1}^{\prime \prime}|ssr\rangle)_{M_{1}M_{2}M_{3}}\otimes|1\rangle_{N_{1}}\end{split}  \tag{72}\]The unnormalized state after Step 9 can be written as  
\[\sum_{j\
eq r,s}^{d-1}\kappa_{j}|jj\rangle_{DEF}\otimes(\alpha_{1}^{ \prime}|irs\rangle+\beta_{1}^{\prime}|rjs\rangle+\gamma_{1}^{\prime}|rsj\rangle+ \alpha_{1}^{\prime}|jsr\rangle+\beta_{1}^{\prime}|sjr\rangle+\gamma_{1}^{ \prime}|srj\rangle)_{M_{1}M_{2}M_{3}} \tag{73}\] \[\qquad+\kappa_{r}|rrr\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|rrs\rangle+\beta_{1}^{\prime\prime}|srr\rangle+\gamma_{1}^{\prime \prime}|rsr\rangle)_{M_{1}M_{2}M_{3}}\] \[\qquad+\kappa_{s}|sss\rangle_{DEF}\otimes(\alpha_{1}^{\prime \prime}|ssr\rangle+\beta_{1}^{\prime\prime}|rss\rangle+\gamma_{1}^{\prime \prime}|ssr\rangle)_{M_{1}M_{2}M_{3}}\]  
The unnormalized state after Step 10 can be written as  
\[\omega^{(P_{1}+P_{2}+P_{3})(r+s)}\Big{\{}\sum_{j\
eq r,s}^{d-1} \kappa_{j}(\alpha_{1}^{\prime}|jrs\rangle+\beta_{1}^{\prime}|rjs\rangle+ \gamma_{1}^{\prime}|rsj\rangle+\alpha_{1}^{\prime}|jsr\rangle+\beta_{1}^{ \prime}|sjr\rangle+\gamma_{1}^{\prime}|srj\rangle)_{M_{1}M_{2}M_{3}} \tag{74}\] \[\qquad+\kappa_{r}(\alpha_{1}^{\prime\prime}|rrs\rangle+\beta_{1} ^{\prime\prime}|srr\rangle+\gamma_{1}^{\prime\prime}|rsr\rangle)_{M_{1}M_{2}M_{ 3}}\] \[\qquad+\kappa_{s}(\alpha_{1}^{\prime\prime}|ssr\rangle+\beta_{1} ^{\prime\prime}|rss\rangle+\gamma_{1}^{\prime\prime}|ssr\rangle)_{M_{1}M_{2}M_ {3}}\Big{\}}\]  
We can easily see that the above state is equivalent to \(\left|\Psi_{2}^{(r,s)}\right\rangle\) as defined by Equation (63) except for a global phase. Hence, the proof is complete for \(r\
eq s\). Next, we give the continuation of the proof for \(r=s\) (Protocol 5). Equation (64) guarantees that the unnormalized state on system \(A\) after Step 3 can be written as  
\[\sum_{j=0}^{d-1}\kappa_{j}^{\prime}|j\rangle, \tag{75}\]  
where \(\{\kappa_{j}\}_{j=0}^{d-1}\) is defined as  
\[\kappa_{j}^{\prime}=\sqrt{\frac{2}{d(d+1)}}\delta_{j}\sqrt{\alpha^{2}+\beta^{ 2}+\gamma^{2}}\quad(j\
eq r),\qquad\kappa_{r}^{\prime}=\sqrt{\frac{2}{d(d+1)} }\delta_{r}(\alpha+\beta+\gamma). \tag{76}\]  
In Step 4, the protocol of [PERSON] et al. successfully works, and the unnormalized state at the target nodes can be written as \(\sum_{j=0}^{d-1}\kappa_{j}|j\rangle_{D}|j\rangle_{E}|j\rangle_{F}\).