Device-independent quantum secret sharing with advanced random key generation basis

Quantum communication aims to guarantee the unconditional communication security based on the fundamental principles of quantum mechanics. As a multipartite cryptographic primitive QSS1 ; QSS2 ; QSS3 , quantum secret sharing (QSS) is one of the main quantum communication branches. QSS divides the key of the dealer into multiple parts and distributes each part to a player. Any subset of players cannot reconstruct the distributed key, which can only be reconstructed when all the players cooperate QSS1 .

The first QSS protocol based on the Greenberger-Horne-Zeilinger (GHZ) state was proposed in 1999 QSS1 . Since then, QSS has been widely studied both in theory and experiment QSS2 ; QSS3 ; QSS4 ; QSS5 ; QSS6 ; QSS7 ; QSS8 ; QSS9 ; QSS10 ; MDIQSS1 ; MDIQSS2 ; MDIQSS3 ; QSS11 ; QSS12 ; QSS13 ; QSS14 ; QSS15 ; QSS16 ; QSS17 ; QSS18 ; QSS19 ; QSS20 ; QSS21 . Like other quantum communication branches, such as quantum key distribution (QKD), the practical imperfect experimental devices may cause serious security loopholes cryptography1 ; cryptography2 ; cryptography3 . The device-independent (DI) paradigm provides a promising method to guarantee the security based on the nonlocal correlation of the measurement results nonlocal1 . DI type protocols can defend against all possible attacks from practical experimental devices and provide the highest level of security DIrandom1 ; DIrandom2 ; DIQKD1 ; DIQKD2 ; DIQKD3 . The research of DI-type protocols started in the DI QKD system DIQKD1 ; DIQKD2 ; DIQKD3 . Since 2020, DI theory have been introduced into the quantum secure direct communication field DIQSDC1 ; DIQSDC2 ; DIQSDC3 . In 2022, DI QKD’s experiments have achieved significant breakthroughs DIQKDe1 ; DIQKDe2 ; DIQKDe3 , thus proving the feasibility of the DI theory. The research on DI QSS has been started since 2019. Refs. DIQSS2 ; DIQSS3 proved DI QSS’s correctness and completeness under a causal independence assumption regarding measurement devices. In 2024, the DI QSS protocol based on the violation of Svetlichny inequality was proposed DIQSS1 . This protocol first characterized DI QSS’s performance in a practical noisy communication scenario by numerically simulating its key generation rate, global detection efficiency threshold, noise threshold, and the maximal communication distance between any two users. Similar as DI QKD protocols DIQKD2 ; DIQKD3 , DI QSS protocol requires quite high global detection efficiency and has low noise resistance DIQSS1 . In particular, as DI QSS requires to distribute multi-photon entanglement to multiple users while DI QKD only requires to distribute EPR state to two users, DI QSS is more vulnerable to channel loss and limited detection efficiency. It requires ultra-low-noise settings with the global detection efficiency as high as 96.32% and the noise threshold of 7.148% DIQSS1 .

In the DI QKD field, some active improvement strategies have been put forward to enhance its noise resistance and reduce the high detection efficiency requirement, including noise preprocessing DIQKD9 ; DIQKD10 ; DIQKD11 , postselection DIQKD10 ; DIQKD11 ; DIQKD12 ; DIQKD13 , random key generation basis DIQKD12 ; DIQKD15 , and advantage distillation DIQKD17 . For example, the adoption of the noise preprocessing strategy increase DI QKD’s noise threshold from initial 7.1492% to 8.0848% DIQKD10 ; DIQKD12 . By combining the random postselection and noise preprocessing strategies, DI QKD’s global detection efficiency threshold was further reduced to less than 87.49%. Benefit for the improvement strategies, the first DI QKD experiment in the optical platform is experimentally demonstrated DIQKDe3 . In 2021, the DI QKD protocol with random key generation basis strategy was proposed DIQKD15 , which effectively improved the noise tolerance threshold from the initial 7.15% to 8.24%. However, the estimation of Eve’s conditional entropy bound is quite complicated, which requires about 5000 core hours and thus largely increases the difficulty of parameter optimization. Soon later, Ref. DIQKD12 simplified the DI conditional entropy bound and provided the optimal entropy in a much shorter time (less than one second). It largely reduced the difficulty of parameter optimization and further improved DI QKD’s noise tolerance threshold to 8.36%. The noise preprocessing combined with the random random postselection strategy was also introduced into the DI QSS protocol DIQSS1 , which can reduce its global detection efficiency threshold from 96.32% to 94.30% and increase the noise threshold from 7.148% to 8.072%.

Although the above improvement can reduce DI QSS’s experimental difficulty, the experimental demonstration of DI QSS is still a big challenge. Meanwhile, the DI QSS protocol DIQSS1 only uses one basis combination to generate the key and discards the measurement results for the other basis combination, which is a waste of precious entanglement resources. For further enhancing DI QSS’s noise resistance, and reducing the global detection efficiency and the entanglement resource consumption, we propose the DI QSS protocol with advanced random key generation basis strategies, which combine the random key generation basis, noise preprocessing, and postselection strategies. We develop the technique in Ref. DIQKD12 to simplify Eve’s conditional entropy bound and numerically simulate its performance in the practical noise environment. The advanced random key generation basis strategy can increase DI QSS’s noise tolerance threshold from 7.147% to 9.231% (29.16% growth) and reduce the global detection efficiency requirement from initial 96.32% to 93.41% ($q\rightarrow 0.5$). In this way, the maximal secure communication distance between any two users can be increased from 0.26 km to 1.43 km (about 5.5 times). Based on the above features, our DI QSS protocol with advanced random key generation basis strategy can effectively facilitate DI QSS’s experimental implementations and application in the future quantum communication field.

The paper is organized as follows. In Sec. II, we briefly describe the tripartite joint key generation process and the detection of genuine tripartite nonlocal correlations. In Sec. III, we present the DI QSS protocol with the random key generation basis. In Sec. IV, we reduce the conditional entropy bound problem of the two-basis variant to qubit system to estimate its key generation rate in the practical noisy channel environment. In Sec. V, we combine the noise preprocessing strategy to further enhance DI QSS’s resistance to channel noise. Finally, we make some discussion and draw a conclusion in Sec. VI.

In the DI QSS, three users’ devices are successively repeated for $n$ ($n\rightarrow\infty$) rounds. Their measurement results have two purposes: (1) some rounds of measurement results are used to generate the joint key bits and estimate the qubit error rate (QBER); (2) some rounds of measurement results are used to estimate the genuine tripartite nonlocal correlation among three users’ results for security checking.

Here, we explain our DI QSS protocol with random key generation basis strategy. Similar to previous DI protocols DIQKD1 ; DIQKD2 ; DIQKD3 ; DIQSDC1 ; DIQSDC2 ; DIQSDC3 ; DIQKD4 ; DIQKD5 ; DIQKD6 , the security of this DI QSS protocol can be guaranteed by two basic assumptions, say, the quantum physics is correct and all users’ physical locations are secure. Meanwhile, the three users must be legitimate and honest. Three users’ devices are repeatedly used for $n$ ($n\to\infty$) rounds. The schematic diagram of DI QSS protocol with random key generation basis strategy is shown in Fig. 1, which includes the following five steps:

Step 1 Entanglement distribution. The central entanglement source prepares $n$ pairs of three-photon polarization GHZ states (1). The source divides the three photons in each GHZ state into $E_{1}$, $E_{2}$, and $E_{3}$ sequences. All the photons in the three sequences are sequentially transmitted to Alice, Bob, and Charlie, respectively.

Step 2 Measurement basis selection. Three users independently and randomly select the measurement bases to measure all photons in the sequences $E_{1}$, $E_{2}$, and $E_{3}$, respectively. The measurement bases $A_{i}$, $B_{j}$, and $C_{k}$ are described in Sec. II.1, where $i,k\in\{1,2\}$ and $j\in\{1,2,3\}$. The tripartite measurement results are denoted as $a_{i}$, $b_{j}$, and $c_{k}$, where $a_{i},b_{j},c_{k}\in\{-1,+1\}$. The detector’s no-click event $\perp$ is randomly noted as $-1$ or $+1$, which is called the postselection operation.

Step 3 Parameter estimation. After all measurements, Alice, Bob, and Charlie sequentially announce the measurement basis for each photon. According to the measurement basis selection, the parameter estimation can be categorized into three cases.

Case 1: Bob chooses the measurement basis $B_{2}$ or $B_{3}$. In this case, all three users announce the measurement results, which are used to estimate the Svetlichny polynomial $S_{ABC}$ and the CHSH polynomial $S$. When $2<S\leq 2\sqrt{2}$ (equivalent to $4<S_{ABC}\leq 4\sqrt{2}$), the corresponding results at three users’ locations have genuine nonlocal correlation. The user can bound the key leakage rate to Eve, thus the photon transmission process is secure and the communication continues. When $S\leq 2$ (equivalent to $S_{ABC}\leq 4$), the measurement results are only classical correlated. The users cannot detect Eve’s attacking. In this way, the users regard that the photon transmission process is not secure, and the communication should be terminated.

Case 2: When the measurement basis combination is $\{A_{1}B_{1}C_{1}\}$ or $\{A_{2}B_{1}C_{2}\}$. Alice randomly announces a part of her measurement results, and Bob (Charlie) announces his corresponding measurement results to estimate the QBER $\delta$. The users preserve the remaining unannounced measurement results as the raw key bits.

Case 3: When the measurement basis combination is $\{A_{1}B_{1}C_{2}\}$ or $\{A_{2}B_{1}C_{1}\}$, the users have to discard their measurement results.

Step 4 Error correction and private amplification. The users repeat the above process until all the users have obtained a sufficient number of secure key bits. Then, the users perform the error correction and private amplification on the raw key bits, eventually forming a series of secure key bits.

Step 5 Secret reconstruction. Charlie announces his key bit $K_{C}$, and Bob combines his key bit $K_{B}$ to reconstruct the key bit $K_{A}$ delivered by Alice based on the encoding rule in Sec. II.1.

In this section, we evaluate the performance of the DI QSS protocol with random key generation basis strategy in the practical noisy environment against collective attacks. We assume that Alice (Charlie) uses measurement basis $A_{1}$ ($C_{1}$) and $A_{2}$ ($C_{2}$) with probabilities $p$ and $\bar{p}=1-p$, respectively.

The users generate the raw key bits when the measurement basis combination is $\{A_{1}B_{1}C_{1}\}$ or $\{A_{2}B_{1}C_{2}\}$. In the asymptotic limit of a large number of rounds, we calculate the asymptotic key generation rate $r_{\infty}$, which is the ratio of the extractable key length to measurement rounds $n$ ($n\to\infty$). Taking into account the “sifting” effect, the asymptotic key generation rate can be written as DIQKD12 ; DIQKD15

$\displaystyle r_{\infty}=p^{2}r_{111}+\bar{p}^{2}r_{212}=\left(p^{2}+\bar{p}^{2}\right)\left(\lambda r_{111}+\bar{\lambda}r_{212}\right),$

(9)

where $\lambda=p^{2}/(p^{2}+\bar{p}^{2})$ represents the matching weight for two key generation basis combinations, and $\bar{\lambda}=1-\lambda$. We adopt the Devetak-Winter bound DWrate ; DWrate2 to estimate the key generation rates corresponding to the measurement basis combination $\{A_{1}B_{1}C_{1}\}$ and $\{A_{2}B_{1}C_{2}\}$. The Devetak-Winter bound is a universal method for calculating the key rate in the quantum cryptography field, which has been widely used in QKD and QSS systems DWQKD1 ; DWQKD2 ; DWQKD3 ; DWQKD4 , and extended to DI QKD and one-side DI QSS systems under the collective attack assumption DIQKD9 ; DIQKD17 ; DWQSS1 ; DWQKD5 ; DWQKD6 . In the asymptotic limit of a large number of rounds, the key generation rates corresponding to the measurement basis combination $\{A_{1}B_{1}C_{1}\}$ and $\{A_{2}B_{1}C_{2}\}$ can be provided by DWrate ; DWrate2 ; DIQSS1

	$\displaystyle r_{111}$	$\displaystyle=$	$\displaystyle H\left(A_{1}|E\right)-H\left(A_{1}|B_{1},C_{1}\right),$
	$\displaystyle r_{212}$	$\displaystyle=$	$\displaystyle H\left(A_{2}|E\right)-H\left(A_{2}|B_{1},C_{2}\right),$		(10)

where $H(|)$ is the conditional von Neumann entropy. $H\left(A_{1}|E\right)$ and $H\left(A_{2}|E\right)$ estimate the uncertainty of Eve’s knowledge about Alice’s key bits when he knows Charlie’s key bits, say, the key secrecy rate to Eve corresponding to the measurement basis combinations $\{A_{1}B_{1}C_{1}\}$ and $\{A_{2}B_{1}C_{2}\}$, respectively. $H\left(A_{1}|B_{1},C_{1}\right)$ and $H\left(A_{2}|B_{1},C_{2}\right)$ quantify the tripartite raw key irrelevance with the measurement basis combinations $\{A_{1}B_{1}C_{1}\}$ and $\{A_{2}B_{1}C_{2}\}$, respectively. Therefore, Eq. (9) is rewritten as

	$\displaystyle r_{\infty}$	$\displaystyle=$	$\displaystyle\left(p^{2}+\bar{p}^{2}\right)\Big{[}\left(\lambda H\left(A_{1}|E\right)+\bar{\lambda}H\left(A_{2}|E\right)\right)$		(11)
		$\displaystyle-$	$\displaystyle\left(\lambda H\left(A_{1}|B_{1},C_{1}\right)+\bar{\lambda}H\left(A_{2}|B_{1},C_{2}\right)\right)\Big{]}.$

For further increasing the noise tolerance threshold of our DI QSS protocol, we adopt the noise preprocessing strategy in our DI QSS protocol. Noise preprocessing is an important active improvement strategy, which can effectively increase the noise tolerance threshold of DI QKD DIQKD9 ; DIQKD12 ; DIQKD13 and DI QSS protocols DIQSS1 . Here, we combine the noise preprocessing with the random key generation basis strategies, which is called the advanced random key generation basis strategy.

The DI QSS protocol only requires minor changes as follows. When the measurement basis combination is $\{A_{1}B_{1}C_{1}\}$ or $\{A_{2}B_{1}C_{2}\}$, Alice flips her measurement result ($+1\rightarrow-1$, $-1\rightarrow+1$) with the probability of $q$. This operation essentially introduces some artificial noise to Alice’s measurement results, which can further decrease the correlation among three users’ measurement results, and increase Eve’s uncertainty about the key bits. As a result, the net effect on the key generation rate can be positive. After all the measurements, Alice will announce the flip probability $q$ in the error correction process. Then, three users apply a hash function to the raw key bits to get the final secure key bits.

With the advanced random key generation basis strategy, the total noise qubit error rate (NQBER) $\delta_{q}$ consists of two components. First, the initial measurement results do not suffer from bit-flip errors but are flipped by Alice with the probability of $q$. Second, the initial measurement results suffer from bit-flip error and are not flipped by Alice with the probability of $\bar{q}=1-q$. Therefore, $\delta_{q}$ is given by

$\displaystyle\delta_{q}=q(1-\delta)+(1-q)\delta=q+(1-2q)\delta.$

(26)

In this way, the tripartite raw key irrelevance becomes

$\displaystyle H(A_{1}|B_{1},C_{1})_{q}=H(A_{2}|B_{1},C_{2})_{q}=h(\delta_{q}).$

(27)

According to the derivation of the DI QSS protocol with noise preprocessing strategy DIQSS1 , after the noise preprocessing, the lower bound of the total key secrecy to Eve is

	$\displaystyle H\left(A^{q}|E\right)=\lambda H\left(A_{1}^{q}|E\right)+\bar{\lambda}H\left(A_{2}^{q}|E\right)$
	$\displaystyle\geq g\left(\sqrt{\lambda\left\langle\bar{A_{1}}\otimes B\right\rangle^{2}+\bar{\lambda}\left\langle\bar{A_{2}}\otimes B^{\prime}\right\rangle^{2}},q\right),$		(28)

where

$\displaystyle g(x,q)=1+\phi\left(\sqrt{\left(1-2q\right)^{2}+4q\left(1-q\right)x^{2}}\right)-\phi\left(x\right).$

The right part of Eq. (V) has the same bound of that in Eq. (IV.1) DIQKD12 (see Appendix A).

Finally, by substituting Eq. (24), Eq. (26)-Eq. (V) into Eq. (11), we can obtain a lower bound of the key generation rate $r_{\infty}^{q}$ of DI QSS with the advanced random key generation basis strategy as

	$\displaystyle r_{\infty}^{q}$	$\displaystyle\geq$	$\displaystyle\left(p^{2}+\bar{p}^{2}\right)\Big{[}g\left(\tilde{E}_{\lambda}\left(2\sqrt{2}\left(1-2\delta\right)\right),q\right)$		(30)
		$\displaystyle-$	$\displaystyle h\left(q+\left(1-2q\right)\delta\right)\Big{]}.$

When $q=0$, Eq. (30) degenerates into Eq. (25).

In Fig. 4, the noise tolerance thresholds of the DI QSS protocol with one key generation basis ($p=100\%$, dashed lines) and two random key generation basis ($p=50\%$, solid lines) both increase with the growth of the flipping probability $q$. When using the same level (same color) of artificial noise (same value of $q$), the noise tolerance thresholds of the DI QSS protocol with the advanced random key generation basis strategy ($p=50\%$, solid lines) are all higher than that only uses one key generation basis ($p=100\%$, dashed lines). In detail, the advanced random key generation basis strategy ($q=40\%$ and $p=50\%$) can improve the noise tolerance of the DI QSS protocol from 7.148% (blue dashed line) to 9.231% (red solid line) (29.16% growth).

We define the global detection efficiency $\eta=\eta_{t}\eta_{d}\eta_{c}$, where $\eta_{d}$ is the detection efficiency of the photon detector and $\eta_{c}$ is the coupling efficiency of the photon to the fiber. The photon transmission efficiency in noisy quantum channel is $\eta_{t}=10^{-\frac{\alpha L}{10}}$, where $L$ denotes the photon transmission distance, $\alpha=0.2$ dB/km for the standard optical fiber. In theory, the growth of noise tolerance threshold would effectively reduce the global detection efficiency threshold. Then, we calculate the global detection efficiency threshold of the DI QSS protocol with the fidelity $F=1$. From Fig. 5, the adoption of the advanced random key generation basis strategy ($q=40\%$) can reduce the global detection efficiency threshold from initial 96.32% to 93.42%, which is further lower than 94.30% corresponding to the adoption of the advanced postselection strategy (combing noise preprocessing with $q=40\%$ and the postselection strategies) DIQSS1 . As a result, the advanced random key generation basis strategy can effectively reduce the requirement on experimental equipment’s performance and facilitate DI QSS’s experimental realization.

Fig. 6 simulates the key generation rate of the DI QSS protocol in five scenarios altered with the photon transmission distance $L$. The five scenarios include the DI QSS without any active improve strategy, with noise preprocessing strategy($q=20\%$), with postselection strategy, with advanced post selection strategy (combining noise preprocessing and post selection strategies), and advanced random key generation basis strategy. Recently, the superconducting nanowire single-photon detector (SNSPD) with a detection efficiency of 98% at 1550 nm band was reported SNSD . Therefore, we assume $\eta_{d}=98\%$ and $\eta_{c}=99\%$ in our simulations. The DI QSS protocol with the advanced random key generation basis strategy ($q=20\%$) has the longest photon transmission distance of 0.799 km. This means that the maximal secure communication distance between any two users achieves 1.384 km. If we increase the value of $q$ to 40%, the maximal secure communication distance between any two users can achieve 1.43 km, which is about 5.5 times of the initial value (0.26 km) DIQSS1 . Meanwhile, DI QSS with the advanced random key generation basis strategy ($q=20\%$) has the highest key generation rate when photons are transmitted over long distances. For example, in the case of $L=0.3$ km, its key generation rate is about 3.5 times of that with the advanced postselection strategy ($q=20\%$). Based on the comparison results, the DI QSS protocol with advanced random key generation basis strategy has big advantages in practical applications.

In the paper, we propose the DI QSS protocol with the advanced random key generation basis, which can effectively improve DI QSS’s noise tolerance and maximal secure photon transmission distance. In this work, we analyze DI QSS’s conditional entropy bound by simplifying it to the qubit bound, and numerically and analytically evaluate its key generation rate. The advanced random key generation basis strategy can effectively increase Eve’s uncertainty about the key generation basis and the measurement results, which eventually improves the protocol’s resistance to noise.

Here, we consider the tripartite DI QSS protocol. Actually, we can use the $m$-party Svetlichny polynomial with the form of CHSH polynomials to extend it to the general multipartite DI QSS protocol with advanced random key generation basis strategy nonlocal4 . The $m$-party Svetlichny inequality has the form of

$\displaystyle S_{m}=\langle S_{m-1}M_{2}\rangle+\langle S_{m-1}^{\prime}M_{1}\rangle\leq 2^{m-1}.$

(31)

We can use the violation of $m$-party Svetlichny inequality to guarantee the nonlocal correlation among the particles in $m$ users and realize the general $m$-party DI QSS with advanced random key generation basis strategy.

In our DI QSS protocol, Alice, Bob, and Charlie all independently select the measurement basis to measure their photons, which results in the final key generation rate decreasing due to the “sifting” effect (only two of the twelve basis combinations are used to generate keys, the measurement results corresponding to the basis combinations $\{A_{1}B_{1}C_{2}\}$ and $\{A_{2}B_{1}C_{1}\}$ should be discarded). In particularly, with the growth of the user or measurement settings number, the “sifting” effect would be more serious. Two possible methods may be applied to avoid this degradation DIQKD11 . The first method is to use the quantum memory. In detail, three users receive and store all quantum states in quantum memories. Then, they publicly announce their basis choices and measure their own photon after agreeing on the basis choices (avoiding the case of choosing $\{A_{1}B_{1}C_{2}\}$ and $\{A_{2}B_{1}C_{1}\})$. The second method is the most hopeful one, where Alice, Bob, and Charlie need to share a long enough pre-shared key in advance to coordinate their measurement basis selection during the key generation rounds.

Finally, we discuss the experimental realization of our DI QSS protocol. The DI QSS protocol requires to distribute high-quality GHZ state through quantum channels. The generation of three-photon GHZ states with high fidelity has been extensively studied GHZstate1 ; GHZstate2 ; GHZstate3 ; GHZstate4 . One common GHZ state generation method depends on the post selection GHZstate1 ; GHZstate3 , in which the generated GHZ state is destroyed by the photon detection and cannot be used. In 2014, Hamel et al. GHZstate2 eliminated the limitation of the outcome post selection by cascading two SPDC sources for directly generating three-photons polarization GHZ states with the fidelity of 86%. Recently, by using the phase-stable source, the fidelity of the target GHZ state was increased to over 96% GHZstate4 . Only the phase-flip error may occur in the practical GHZ state generation GHZstate2 ; GHZstate4 , which will cause the quantum bit error in the key generation process. In this way, the imperfect GHZ state generated in Ref. GHZstate4 would cause the QBER of 4%. This QBER is below the noise threshold 9.231%, so that the current GHZ state generation technology in Ref. GHZstate4 satisfies the requirement of our DI QSS protocol with advanced random key generation basis strategy. Combining the practical high-efficient SNSPD, our DI QSS protocol with advanced random key generation basis strategy has potential to demonstrate with current experimental technology. On the other hand, the photon transmission loss and decoherence caused by the practical noisy channels severely degrade the entanglement and reduce DI QSS’s secret key generation rate. Since DI QSS has to transmit entanglement photon pairs to multiple users, any photon loss that occurs during transmission can break the entanglement, so it has relatively low resistance to photon loss. Quantum repeater provides a promising way to construct high-quality long-distance entanglement channel repeater1 ; repeater2 . Combining quantum repeater with the DI QSS protocol may further increase the secure communication distance and realize the long-distance DI QSS.

In conclusion, DI QSS can provide the highest security for QSS under practical imperfect experimental conditions. However, DI QSS protocol has quite low noise tolerance threshold and high global detection efficiency requirement, which largely limit its experimental demonstration and application. To reduce DI QSS’s experimental requirements and improve its performance in practical noisy experiment, we propose a new DI QSS protocol with advanced random key generation basis strategy. The advanced random key generation basis strategy combine the random key generation basis, noise preprocessing, and postselection strategies, which can effectively increase the total key secrecy rate to Eve, and thus effectively increase DI QSS’s noise tolerance threshold from 7.147% to 9.231% (29.16% growth) and reduce the global detection efficiency requirement from initial 96.32% to 93.41% ($q\rightarrow 50\%$). In this way, the maximal secure communication distance between any two users can be increased from 0.26 km to 1.43 km (about 5.5 times). Meanwhile, our DI QSS can reuse some discarded items in original DI QSS protocol DIQSS1 and reduce the entanglement resource consumption. DI QSS requires to distribute high-quality GHZ state. The requirement for high-quality GHZ state can be satisfied with current GHZ source. Our DI QSS protocol can be extended to the general multipartite condition and has potential to demonstrate with current experimental technology, which can effectively promote DI QSS’s experimental demonstration and applications in the future.

This work was supported by the National Natural Science Foundation of China under Grants No. 12175106 and No. 92365110, the Natural Science Foundation of Jiangsu Province of China under Grants No. BK20240611 and No. BK20240612.