Generalized sequential state discrimination for multiparty QKD and its optical implementation

Sequential state discrimination is a strategy for N separated receivers. As sequential state discrimination can be applied to multiparty quantum key distribution (QKD), it has become one of the relevant research fields in quantum information theory. Up to now, the analysis of sequential state discrimination has been confined to special cases. In this report, we consider a generalization of sequential state discrimination. Here, we do not limit the prior probabilities and the number of quantum states and receivers. We show that the generalized sequential state discrimination can be expressed as an optimization problem. Moreover, we investigate a structure of generalized sequential state discrimination for two quantum states and apply it to multiparty QKD. We demonstrate that when the number of receivers is not too many, generalized sequential state discrimination for two pure states can be suitable for multiparty QKD. In addition, we show that generalized sequential state discrimination for two mixed states can be performed with high optimal success probability. This optimal success probability is even higher than those of quantum reproducing and quantum broadcasting strategy. Thus, generalized sequential state discrimination of mixed states is adequate for performing multiparty QKD. Furthermore, we prove that generalized sequential state discrimination can be implemented experimentally by using linear optics. Finally, we analyze the security of multiparty QKD provided by optimal sequential state discrimination. Our analysis shows that the multiparty QKD guarantees nonzero secret key rate even in low channel efficiency.

In 2013, Bergou et al. 27 proposed the sequential state discrimination strategy. This strategy can consist of many receivers (called Bob 1, Bob 2 … Bob N), who are separated and are not allowed to perform classical communication with each other. In sequential state discrimination, a sender Alice sends one out of two quantum states, with a prior probability to Bob 1. It is assumed that all receivers are aware of the prior probabilities, before they perform sequential state discrimination. Bob 1 performs a measurement to discriminate Alice's quantum state. After the measurement, Bob 1 sends his post-measurement state to Bob 2. Then, Bob 2 performs his measurement to discriminate Bob 1's post-measurement state. This process is sequentially performed. The purpose of sequential state discrimination is to maximize the probability that all receivers successfully discriminate Alice's quantum state. According to Bergou et al. 27 , the optimal (maximum) success probability is non-zero, in general. It implies that Bob + I 1 can obtain information about Alice's quantum state, from Bob I's post-measurement state. This result not only enables us to know the property of a non-projective measurement but also allows its application to multiparty QKD strategy. Bergou et al. 27 and Pang et al. 28 separately investigated the sequential state discrimination of two pure states with equal prior probabilities. Solis-Prosser et al. 29 implemented the sequential state discrimination trategy. In their work, two polarized single photon states with equal prior probabilities, were considered. Moreover, Zhang et al. 30 investigated the sequential state discrimination of two pure states with unequal prior probabilities. Hillery and Mimih 31 considered the sequential state discrimination of N symmetric pure state, with equal prior probabilities.
However, most studies of sequential state discrimination have been focused on special cases. In other words, the generalized structure of sequential state discrimination has not been investigated yet [32][33][34] . Therefore, in this report, we consider a generalization of sequential state discrimination. That is, in constructing sequential state discrimination, we do not limit the prior probabilities and the number of quantum states and receivers. Moreover, we consider the most general case of quantum states, in which every quantum state can be either pure 32 or mixed state 33 . Because unambiguous discrimination of general mixed states is not known yet [35][36][37] , sequential state discrimination for general mixed states is beyond the scope of this paper. However, if mixed states are given in the form of Herzog's work 36 , we can build generalized sequential state discrimination of two mixed states. Also, in terms of success probability, generalized sequential state discrimination provides better result in mixed states than in pure states.
First, we show that generalized sequential state discrimination can be expressed to a mathematical optimization problem. This optimization problem provides an optimal positive-operator-valued-measurement (POVM) condition, as well as an optimal success probability. Exploiting this structure, we explicitly investigate the generalized sequential state discrimination of two quantum states. Naturally, our investigation contains previous works 27,28 . Also, we apply it to multiparty QKD. We show that if the number of receivers is too many, generalized sequential state discrimination of two pure states can be performed, with very small optimal success probability. It means that generalized sequential state discrimination of two pure states can be suitable for multiparty QKD, only when the number of receivers is not too many. Meanwhile, generalized sequential state discrimination of two mixed states can be performed, with high optimal success probability. Especially, its optimal success probability exceeds those of quantum reproducing 27 and quantum broadcasting 38,39 strategy. It implies that generalized sequential state discrimination of two mixed states can be more suitable for multiparty QKD than other strategies.
In addition, we show that linear optics can be used to experimentally implement generalized sequential state discrimination. Here, our models can be implemented by modifying the Banaszek model 40 or the Huttner model 41 . We show that generalized sequential state discrimination of binary coherent states 34 can be implemented optimally. Moreover, we show that generalized sequential state discrimination of two mixed states can be implemented optimally. Further, we consider mixed states, which consists of coherent states. When an information carrier is a coherent state, which is robust in a noisy environment 42 , our model can be suitable for implementing a realistic multiparty QKD.
Finally, we analyze the security of multiparty QKD based on optimal sequential state discrimination. It is known that B92 protocol provides unconditional security 43 . Therefore, one can guess that the QKD based on generalized sequential state discrimination guarantees security. To show this, we evaluate the secret key rate 44 of multiparty QKD based on generalized sequential state discrimination. Our result tells that the multiparty QKD guarantees nonzero secret key rate even in low channel efficiency. In addition, our multiparty QKD is composed of the method based on prepare and measure 24,45,46 and is more robust in noise than the QKD of multipartite entanglement.

Results
Scenario of Sequential State Discrimination. The concept of generalized sequential state discrimination can be understood as a game, consisting of a sender Alice and N receivers such as Bob 1, Bob 2, …, Bob N(see Fig. 1). In this scenario, every party acts as follows: Alice prepares a quantum state { , , } ( 1) on Alice's quantum state, for unambiguous discrimination. Here, M j (1) is a POVM element, corresponding to a measurement outcome j. If Bob 1 obtains a conclusive outcome ( ≠ j 0), he thinks Alice's quantum state as j ρ . If Bob 1 obtains an inconclusive result (j 0 = ), he cannot figure out which quantum state Alice had prepared. If Bob 1 obtains a conclusive result, he sends a post-measurement state to Bob 2. Because in generalized sequential state discrimination every receiver should perform unambiguous discrimination, the post-measurement state of Bob 1 is given as † ( 2) . Then, Bob 2 sends his post-measurement state † σ σ δ ∝ K K i j i j ij (2) ( 2) (1) (2) to Bob 3. This process is sequentially conducted from Bob 3 to Bob N . The average success probability of generalized sequential state discrimination is given as ). The purpose of generalized sequential state discrimination is to maximize the average success probability, as expressed in Eq. (1). In the process, every receiver should obey the following rules: Rule 1. Bob 1, Bob 2, …, Bob N 1 − performs a nonoptimal unambiguous discrimination. However, Bob N performs an optimal unambiguous discrimination.
Rule 2. Classical communication is forbidden between every receiver. If Bob I N {1, 2, , 1} ∈ … − performs an optimal unambiguous discrimination, Bob + I 1 cannot obtain any information from Bob I's post-measurement state. Moreover, if one of the receivers sends his measurement outcome through classical communication, an eavesdropper can steal the measurement outcome without being noticed by any receivers. Thus, it is reasonable that Rules 1 and 2 should be imposed on every receiver. construction of the optimization problem. In this section, we express generalized sequential state discrimination as an optimization problem. To construct the optimization problem, we should involve not only two rules but also POVM conditions for every receiver. First, we should consider a POVM that performs an unambiguous discrimination. . If the set of quantum states S n satisfies the following theorem, there exists a POVM that performs an unambiguous discrimination on S n .   (2) ( 2) 0 , Bob 2 also discriminates i (1) σ which is the post-measurement state of Bob 1, without an error. Then, this process is sequentially performed from Bob 3 to Bob N .
This equality corresponds to the POVM condition, which performs an unambiguous discrimination on a pure state (S 2 ). If every receiver performs an optimal unambiguous discrimination, Eq. (5) becomes a strict equality. Then, the overlap between post-measurement states becomes one, according to Eq. (4). Hence, to obey Rule 1, the POVM of Bob 1, Bob 2, …, Bob N 1 − should not satisfy the equality of Eq. (5). Furthermore, because every submatrix M ab should be positive-semidefinite, Eq. (5) is also involved in the POVM condition, performing an unambiguous discrimination of n pure states.
Generalizing POVM for mixed state discrimination. In this section, we investigate the generalized sequential state discrimination of mixed quantum states. Unfortunately, an explicit form of POVM, that performs an unambiguous discrimination of arbitrary mixed states is unknown. That is because we do not know how to deal with Theorem 1. When mixed states can be expressed in the form given by Herzog's work 36  www.nature.com/scientificreports www.nature.com/scientificreports/ r r r r r r r r r r r r , , , , 0 are linearly independent, every sub-POVM is obtained, using similar process as Theorems 2 and 3. Therefore, we obtain POVM that performs unambiguous discrimination, which completes the proof of Theorem. □ With the help of Theorem 5, we can apply a method that deals with the discrimination of pure states into a mixed-state case. If ≠ i 0, the POVM element M i can be expressed as According to the completeness condition, M 0 is given as Hence, the post-measurement state i σ is expressed as We can obtain the Kraus operator K 0 , corresponding to M 0 , by exploiting Theorem 3. Because every eigenvector of σ i should satisfy s s r r Both Eqs. (9) and (10) imply the following meaning. if either α ij or α kj is nonzero, |〈 | 〉| s s ij kj is larger than r r ij kj |〈 | 〉|. That is, the support of two post-measurement states is more overlapped than that of Alice's mixed states. If an optimal unambiguous discrimination is performed, Eq. (10) becomes a strict equality. Moreover, according www.nature.com/scientificreports www.nature.com/scientificreports/ to Eq. (9), s s ij kj |〈 | 〉| becomes equal to 1. Therefore, for all i, supp( ) s upp( ) cannot be discriminated, without any error. Now, let us consider the case where every ρ i has a different rank. Without loss of generality, we can assume an inequality such as }   I  I  n  I  n  m  I  I  n  I  n  m  I  I  n  I  n Combining Eq. (11) with Eq. (12), we can express generalized sequential state discrimination as following optimization problem 32 : Now, let us investigate the geometric properties of each C I ( ) . The set of POVM that performs an unambiguous discrimination is convex and C I ( ) is also convex: It is important to investigate the relation between C I ( ) and + C I ( 1) , to analyze the generalized sequential state discrimination. In our previous work 32 , we proposed the following conjecture: . Considering the discrimination problem of two pure states, we can confirm that Conjecture 1 holds. Conjecture 1 has the following meaning. When the real vector ( , , ) has at least one nonzero component α I 1 ( ) , Bob I can obtain partial information of Alice's quantum state, by performing a measurement on Bob I 1 − 's post-measurement state, with a nonzero probability. Conjecture 1 implies that options for POVM that Bob I 1 + can choose are limited, as Bob I obtains the information. In the extreme case, if Bob I obtains the maximal information, then Bob + I 1 cannot construct a POVM to unambiguously discriminate Bob I's post-measurement state, which means that Bob I 1 + cannot obtain any information from Bob I's post-measurement state. Hence, we can propose the following conjecture: Conjecture 2. If 32 C ( , , ) is the only element of + C I ( 1) . If Alice prepares a pure state from S 2 , both Conjecture 1 and Conjecture 2 hold 32 . In the case of N 3 = , we can numerically check that both conjectures 1 and 2 are correct.
Optimization problem for the mixed states case. Now, let us consider a mixed states case. When Alice prepares i ρ , which is expressed as Eq. (6) Here, C j I int, ( ) and ∂ C I ( ) are respectively defined as , are × m m principal submatrices. Then, we can express generalized sequential state discrimination of mixed states, as following optimization problem 33 : hen m 1 = , this optimization problem describes the generalized sequential state discrimination of pure states. Note that the j-th constraint only affects sub- . From this property, this optimization problem can be partitioned into the following sub-optimization problems 33 :  ecause every sub-optimization problem in the case of mixed states is the same as that in the pure state case, if every mixed state is expressed as Eq. (6), we can apply the method for pure states to the generalized sequential state discrimination of mixed states. Unfortunately, Eq. (6) is not the case of the most general mixed state. However, if we use these mixed states as an information carrier, the optimal success probability of the generalized sequential state discrimination can exceed that of the quantum reproducing 27 and the quantum broadcasting strategy 39 . This implies that generalized sequential state discrimination is a more suitable strategy for application to multiparty QKD than the other two strategies. Furthermore, Eq. (6) can be implemented using linear optics. In the next sections, we explain these advantages in detail.
Generalized Sequential State Discrimination of two Quantum States. In this section, we consider the optimization problem proposed in the previous section. We deal not only with the problem of two pure states but also with the problem of two mixed states in a multi-receiver case.
Generalized sequential state discrimination of two pure states. Here, we consider the generalized sequential state discrimination with an arbitrary N . First, let us consider the case of = N 3. The three receivers are denoted as Bob, Charlie, and David, and each POVM of Bob, Charlie, and David corresponds to the two dimensional real vectors ( , ) 1 2 α α , ( , ) 1 2 β β , and ( , ) 1 2 γ γ , respectively. According to Eq. (12), each real vector should satisfy , ( , ) , ( , ) where C X int ( ) and ∂ ∈ C X B C D ( { , , }) X ( ) are respectively defined as 32 www.nature.com/scientificreports www.nature.com/scientificreports/ The set of POVM, labeled as ∈ X {B, C, D}, can be expressed as . Therefore, Eq. (13) becomes 32 To solve this problem, we need to consider the equality constraint of Eq. (15). David's optimal condition of generalized sequential state discrimination can be obtained by finding a tangential point ( , ) 1 2 γ γ between a plane P q q ( ) ( ) s (B,C,D) 1 1 1 1 2 2 2 2 α β γ αβ γ = + and a surface (1 )(1 ) When this tangential point is substituted into Eq. (15), Eq. (15) becomes the following optimization problem: The detailed derivation of Eq. (16) can be found in the Methods section. Because this optimization problem is difficult to solve analytically, one may apply a numerical method to solve it. Therefore, for the numerical method, a penalty function may be used to solve this constrained optimization problem 49 .
To search for the optimal condition of α α β β ( , , , 2 , we need to find the condition where the derivative P / s i (B,C,D) β ∂ ∂ becomes zero. The condition that β β ( , ) 1 2 satisfies the zero derivative is given as β β is analytically expressed as Because α α = 1 2 and β β = 1 2 , 1 2 γ γ = also holds (see the Methods section). This condition is equal to that obtained by Bergou et al. 27 In this case, the objective function of Eq. (16) is expressed as We can easily check that Eq. (19) is maximized when s 1 holds. In that case, an optimal success probability can be analytically described as . This success probability is equal to the result of Bergou et al. 27 However, this success probability is not optimal in general. That is because the equality condition P ( / , / ) 0 = does not guarantee optimum. Furthermore, we cannot confirm that the optimal condition satisfies the additional constraints 1 2 α α = and β β = 1 2 . Therefore, we should check whether the maximum of Eq. (19) is really equal to that of Eq. (16). We plot the maxima of both Eqs. (16) and (19) in Fig. 2. In the Fig. 2 www.nature.com/scientificreports www.nature.com/scientificreports/ line shows the maximum of Eq. (19). The red circles (blue dots) shows the maximum of Eq. (16) with (without) the additional constraint α α = 1 2 and β β = 1 2 . If the overlap 1 2 ψ ψ |〈 | 〉| becomes smaller, the maxima of both Eqs. (16) and (19) become to coincide with. When the overlap is small, the optimal strategy of every receiver is to discriminate two pure states of Alice. Because the prior probabilities of two pure states are identical, the optimal measurement of every receiver does not show any bias to a specific pure state. Therefore, the condition of α α β β γ γ = = = , , 2 should be included in the optimality conditions. If the overlap ψ ψ |〈 | 〉| 1 2 becomes larger, the maximum of Eq. (16) becomes larger than that of Eq. (19). In this case, , α α β β = = , and 1 2 γ γ = are not optimal conditions anymore for the generalized sequential state discrimination. Especially, we observe that at least one of 1 γ and γ 2 becomes zero. This means that it is an optimal strategy when Bob, Charlie, and David discriminate one of Alice's two pure states.
If Bob, Charlie, and David only discriminate one of Alice's pure state, the generalized sequential state discrimination can be expressed as the following optimization problem: From the equality P / 0  ). The small graph shows the optimal success probability in the region of < < . s 0 005. The solid black line shows the optimal success probability when three receivers discriminate every two pure state of Alice's 27 . The black dashed line as Eq. (21) shows the optimal success probability when three receivers discriminate only one of two pure states of Alice's (Eq. (21) is a generalization of the result of Pang et al. 28 ). The red circles and the blue dots display the optimal success probability from Eq. (16). More specifically, the red circles (the blue dots) shows the optimal success probability when the constraint conditions of 1 s 0 017559, the red circles and the blue dots coincide with the solid black line, which shows that our result agrees with that of Bergou et al. 27 . Further, the blue dots are larger than the solid black line, but are smaller than the black dashed line. Therefore, if > . s 0 017559, the optimal condition of generalized sequential state discrimination does not include the constraint conditions of 1 2 α α = and β β = www.nature.com/scientificreports www.nature.com/scientificreports/ Next, we consider the case of N 4 = . In this case, let us denote the four receivers as Bob, Charlie, David, and Eliot. Each POVM of the four receivers corresponds to the two dimensional real vectors α α ( , ) 1 2 , β β ( , ) 1 2 , ( , ) 1 2 γ γ , and ( , ) 1 2 δ δ . According to Eq. (12), each real vector should satisfy Here, C int (E) and ∂C (E) are respectively defined as   Moreover, the POVM condition for Eliot is expressed as . Hence, we can obtain the following optimization problem: Now, let us consider the equality constraint of Eq. (23). The optimal condition of the generalized sequential state discrimination for Eliot is given as a tangential point δ δ ( , ) 1 2 between a plane α β γ δ and a surface (1 )(1 ) If this tangential point is substituted into the objective function of Eq. (23), the following optimization problem can be obtained: The detailed derivation of Eq. (24) is provided in the Methods section. When the constraints , ,  27 If the overlap is small, this equality constraint is included in the optimal condition of sequential state discrimination. Because the prior probabilities of two pure states are identical, the optimal measurement of every receiver does not show any bias to a specific pure state. If we do not add these constraints, the optimal success probability becomes larger than that provided by Bergou et al. 27 . In this case, we observe that at least one of 1 δ and δ 2 becomes zero. This implies that it is optimal only when four receivers discriminate only one out of two pure states of Alice.
If Bob, Charlie, David, and Eliot only discriminate one out of two pure states, the generalized sequential state discrimination is described as the following optimization problem: The success probability satisfies the following inequalities: (1 ) . When q q , it is optimal for the four receivers to discriminate every two pure state of Alice.
, it is optimal for the four receivers to discriminate only one out of two pure states.
From the previous results, we can consider the generalized sequential state discrimination for arbitrary N , in an inductive manner. We can construct an optimization problem of the generalized sequential state discrimination for any N as follows:  ). The small graph shows the optimal success probability in the region of < < . s 0 00035. The solid black line shows the optimal success probability when four receivers discriminate every two pure state of Alice's 27 . The black dashed line as Eq. (26) shows the optimal success probability when four receivers discriminate only one of two pure states of Alice's (Eq. (26) is a generalization of the result of Pang et al. 28 ). The red circles and the blue dots display the optimal success probability from Eq. (24). More specifically, the red circles (the blue dots) shows the optimal success probability when the constraint conditions of α α = , the red circles and the blue dots coincide with the solid black line, which shows that our result agrees with that of Bergou et al. 27 . Further, the blue dots are larger than the solid black line, but are smaller than the black dashed line. Therefore, if > .
s 0 001282, the optimal condition of generalized sequential state discrimination does not include the constraint conditions of α α = www.nature.com/scientificreports www.nature.com/scientificreports/ If prior probabilities are all equal, the optimal success probability is obtained as . Equation (28) satisfies the result of Bergou 27 and Pang 28 . If ψ ψ |〈 | 〉| < S N ( ) 1 2 , the optimal condition contains I ( ) ∀ . This property was argued by Bergou et al. 27 When ∈ … N {2, 3, , 7}, the value of S N ( ) is numerically given as When N becomes larger, S N ( ) rapidly converges to zero. This implies that when the overlap between two pure states is not small enough, discriminating two pure states when many receivers participate is not a good strategy. Therefore, sequential state discrimination can be applicable for multiparty QKD in the case of a suitable number of receivers. That is because, in multiparty QKD, all receivers are required to discriminate every quantum state 24,27 .
Generalized sequential state discrimination of two mixed states. Here, let us consider the generalized sequential state discrimination of two mixed states. For convenience, we will consider rank-2 mixed states such as¯¯ρ Equation (29) has the same form as that of Eq. (6). Like in Eq. (6), every POVM corresponds to a real vector ¯( , , , ) We can describe the generalized sequential state discrimination of two mixed states as the following optimization problem 33 Then, Eq. (30) can be divided into the following two optimization problems:    , it is optimal that every receiver discriminates Alice's two pure state. However, If |〈 | 〉| < r r S N ( ) 1 2 or r r S N ( ) 1 2 |〈 | 〉| <¯, discriminating only one out of Alice's two pure states is optimal.
Comparison with other discrimination strategies. There are other strategies for multi-party QKD, besides sequential state discrimination. The first one is the quantum reproducing strategy 27 . This strategy is performed as in the following procedure (see Fig. 4 is the probability that Bob I obtains the measurement outcome i. If we assume an equal prior probability, when mixed states have the same eigenvalues, the optimal success probability of the quantum reproducing strategy is derived as (see Method) 21  |〈 | 〉|, respectively. The second case is the quantum broadcasting strategy 27,32 . This strategy is performed as in the following process (see Fig. 5): Bob 1 puts Alice's unknown quantum states into a quantum broadcasting machine, which trans-

P r s rs
σ … , with a probability of less than 1 39 . Here, these states satisfy [ ] broad i is the maximal success probability to succeed in broadcasting when ρ i is given. The optimal success probability of the quantum broadcasting strategy is larger than that of the quantum reproducing strategy. In Fig. 5, quantum broadcasting strategies for three receivers (Bob, Charlie, and David) and four receivers (Bob, Charlie, David, and Eliot) are described. In these cases, the optimal success probabilities of the quantum broadcast strategy are given by , the optimal success probability of quantum broadcasting can be derived, similar to that of quantum reproducing. If Alice prepares one out of two pure states, the optimal success probability of the generalized sequential state discrimination is less than that of quantum reproducing or quantum broadcasting 27 . However, if Alice prepares one out of two mixed states, the optimal success probability of the generalized sequential state discrimination can be larger than that of both the quantum reproducing and the quantum broadcasting strategy. Namely, the generalized sequential state discrimination of mixed states has more potential for application to multiparty QKD than those of the quantum reproducing and quantum broadcasting strategies. It should be noted that sequential state discrimination can be a good candidate for application to multiparty QKD when mixed states are used, in contrast to the result of the pure states obtained by Bergou et al. 27 . = − = . , = . s 0 7, and ¯= . s 0 001. In the case of = N 3(Bob,Charlie, and David), the optimal success probability for each strategy is numerically obtained as = .
. Here, P seq , P rep , and P rep denote the success probability of generalized sequential state discrimination, quantum reproducing, and quantum broadcasting, respectively. In the case of N 4 = (Bob, Charlie, David, and Eliot), optimal success probability for each strategy is numerically obtained as = .
r 0 6, = . r 0 7, = . s 0 7, and = . s 0 001 , are used, the optimal success probability of generalized sequential state discrimination performs better than quantum reproducing and quantum broadcasting. = − = . , s 0 5 = . , and s 5 10 8 = × − . We plotted the optimal success probability of generalized sequential state discrimination and quantum reproducing strategy in Fig. 6. As can be seen in the figure, the optimal success probability of generalized sequential state discrimination exceeds those of the quantum reproducing and quantum broadcasting strategies. Here, we describe a specific way to perform the quantum broadcasting strategy, as shown in Fig. 6, when seven receivers participate. When the given quantum states are non-orthogonal, because the success probability of quantum broadcasting cannot achieve to one, generalized sequential state discrimination also outperforms quantum broadcasting strategy. In the extreme case, if N 1 = , the optimal success probability in Fig. 6 becomes equal to the optimal unambiguous discrimination 21 .
In conclusion, the generalized sequential state discrimination of two mixed states can outperform the other two strategies. It can be implemented using linear optics. In the next section, we will describe its implementation in detail. optical implementation. Here, we explain the method to implement the generalized sequential state discrimination of two coherent states, using linear optics.
Implementation of the POVM for unambiguous discrimination. Suppose that Alice prepares ψ | 〉 ∈ S i n , with a prior probability q i . When an ancilla state | 〉 B is prepared in Bob, his measurement that performs an unambiguous discrimination can be constructed as is an orthonormal basis. Moreover, α i (ᾱ i ) is a conditional probability, that Bob obtains conclusive result i(inconclusive result), when Alice prepares ψ i . It is well known that Eq. (31) is equivalent to Eq. (2), in unambiguous discrimination.Moreover, Eq. (31) shows a way to implement an unambiguous discrimination in real-world settings.
If S 2 consists of two polarized single photon states ψ

a global unitary operator U AB
(B) can be implemented using linear optics. Solis-Prosser et al. 29 used this method to perform a sequential state discrimination of two polarized single-photon states, with equal prior probability. In their model, an ancilla state in Eq. (31) corresponds to a single-photon path. If their model is applied to generalized sequential state discrimination, Bob I should prepare 3 I paths. Therefore, if the number of receivers is large, generalized sequential state discrimination should require an exponentially large number of single photon paths.
However, the sequential state discrimination of coherent states does not require many paths 34 . One can perform the sequential state discrimination by using the modified Banaszek or the Huttner model. In the next subsection. we will explain the modified Banaszek model) 40,41 . Both models can perform an unambiguous discrimination of two nonorthogonal coherent states, with general prior probabilities. Furthermore, both models can achieve an IDP limit [8][9][10][11] . By simply adding beamsplitters in the Banaszek or the Huttner model, one can perform a generalized measurement, that produces post-measurement states 34 .
Implementation of the sequential state discrimination of two pure states. In this subsection, we propose a method to implement the generalized sequential state discrimination of two coherent states. In our previous work 34 , the sequential state discrimination of two coherent states was discussed in the case of two receivers. Figure 6. Plot of the optimal success probability of generalized sequential state discrimination (solid black line), quantum reproducing strategy (black dashed line) and quantum broadcasting strategy (dash-dot black line). We also describe a specific way to perform quantum broadcasting strategy.
(1 ) Bob I's post-measurement state can be expressed as β , where f is a real function: f x y z xy x z ( , , ) (1 ) = + − . According to Rule 1, Bob N should perform an optimal unambiguous discrimination on Bob − N 1's post-measurement state. Therefore, Bob N's measurement is same as in the Banaszek model (see Fig. 7). In our optical model, the success probability of the generalized sequential state discrimination is obtained as¯¯¯¯{ The optimal success probability of Eq. (32) is shown in Figs. 8 and 9. In Fig. 8, we assume that the prior probabilities are equal. In Fig. 9, we assume that q 0 4 1 = . and q 0 6 2 = . . The red circles (blue dots) shows the optimal success probability, with (without) the additional con-  . Therefore, we conclude that our model can optimally perform a generalized sequential state discrimination of two coherent states. Because the Huttner model provides the same measurement probability distribution as that of the Banaszek model 34 , generalized sequential state discrimination can be performed optimally, using the modified Huttner model. Unlike the Banaszek model, the Huttner model uses a horizontally polarized coherent state β | 〉 i , mixed with a vertically polarized coherent state γ | 〉, as an information carrier. Therefore, if an eavesdropper attempts to steal information encoded in coherent light, the eavesdropper will inevitably change at least one of two polarized coherent states. Hence, eavesdropping ruins the unambiguous discrimination and produces an error on the receiver's measurement. Therefore, all receivers can notice the fact that an eavesdropper exists by checking whether an error occurs. We compare the optimal success probabilities of Figs. 8 and 9. One can see that in Fig. 10, the solid red line which denotes the optimal success probability of sequential state discrimination for three receivers is larger than the solid black line which denotes the optimal success probability of sequential state discrimination for four receivers. The reason for the difference between two success probabilities is due to the strategy of David. In the case of N 3 = , David becomes the last receiver and chooses optimal unambiguous discrimination. However, in the case of = N 4, David is not the last receiver and should choose nonoptimal unambiguous discrimination.
Implementing the sequential state discrimination of two mixed states. In this subsection, we propose a way to implement the generalized sequential state discrimination of two mixed states. In our model, the mixed states are produced as in the following process (See Fig. 11): First, Alice prepares one out of two coherent states β β | 〉 | 〉 , i ī , with prior probabilities | 〉 | 〉 r r , i i . Second, Alice polarizes the coherent state i β | 〉β | 〉 ( ) i in the horizontal (vertical) direction. After performing two steps, Alice obtains a mixed state as 50 are orthogonal to each other, Eq. (33) is equal to Eq. (6). If Alice wants to build rank-m mixed states, she would perform the following process: First, Alice prepares a coherent state β β β with a prior probability r ij . Second, she passes β ij to the j-th photon path (D j ). Then, she obtains the following mixed states: can be perfectly discriminated by using a polarized beam splitter. We can use the Banaszek model to discriminate the nonorthogonal coherent states and in Fig. 10, we propose an optical model to discriminate the mixed states, expressed as Eq. (33). The Huttner model can also be used to perform the measurement for the generalized sequential state discrimination of two mixed states, in a similar way to that in Fig. 10.

Security analysis of multiparty QKD based on sequential state discrimination -part i: eve's single trial for eavesdropping
In this section, we analyze the security of multiparty QKD, which optimal sequential state discrimination provides. Even though our analysis is confined to the case of four receivers (Bob, Charlie, David, Eliot), it can be consistently extended to the case of arbitrary number of receivers (See Fig. 12). In information theory, the www.nature.com/scientificreports www.nature.com/scientificreports/ classical bit of Alice can be expressed by ∈ i {0, 1} and we use | 〉 | 〉 | 〉 { 0 , 1 , ? } as a computational basis. Here, ? | 〉 is a computational basis that corresponds to the "failure" of Eve.
In QKD, Alice should minimize the prior information of the classical bit. Otherwise, Eve can obtain the prior information of the classical bit without being caught by the sender and the receiver. Alice prepares quantum states ψ | 〉 0 and 1 ψ | 〉 corresponding to classical bit 0 and 1, with identical prior probability. In the view of Alice and Bob, it is the situation where Alice and Bob share the entangled state 0 1 0 1 ψ ψ | 〉 ⊗ | 〉 + | 〉 ⊗ | 〉 43 . Suppose that Eve tries to eavesdrop between Alice and Bob (Later, we will consider the case where the strategy of Eve is a collective attack 43   . This result shows that the optimal success probability of N 3 = is larger than that of N 4 = . It can be understood as follows: For instance, when receivers are three and four, the third receiver called David should use a strategy depending on the existence of an extra receiver. In the case of = N 3, David is the last receiver and should choose optimal unambiguous discrimination. However, in the case of = N 4, David is not the last receiver and should use nonoptimal unambiguous discrimination. Therefore, the optimal success probability of N 3 = is larger than that of N 4 = . Scientific RepoRtS | (2020) 10:8247 | https://doi.org/10.1038/s41598-020-63719-9 www.nature.com/scientificreports www.nature.com/scientificreports/ where η ∈ [0, 1] AB is the efficiency of the quantum channel. As the efficiency is more close to 1, Alice and Bob can be less affected by Eve. Then, the bipartite state σ AB can be given as The purification of the bipartite state AB σ can be found as The joint probability of the case where Alice prepares ψ | 〉 i , Bob obtains j as a result of measurement, and Eve gets bit k can be given as follows: When the prior probability is identical, the optimal measurement of Bob corresponds to the case of α α = = − s 1 N 0 1 1/ . Therefore, one can obtain the following probabilities (The detailed derivation can be found in Method): Figure 11. Schematic of the optical model for performing the generalized sequential state discrimination of two mixed states. Here, BS is a beam splitter, PBS is a polarized beam splitter, D is a beam combiner, PS is a phase shifter, and on-off is an on-off detector.  is considered. Because an inconclusive result of Bob and failure of Eve cannot provide any information, Bob (Eve) can discard the inconclusive result (failure). The post-processing transforms four probabilities of Eq. (38) as follows: . It should be noted that the four probabilities obtained from post-processing are dependent only on the probability of the conclusive result. From these four probabilities, one can evaluate the secret key rate between Alice and Bob in the following way 44 : Here, I X Y ( : )(H X Y ( , )) is Shannon's mutual information (joint Shannon entropy) between X and Y . And because of   Here, BC η is a channel efficiency between Bob and Charlie. In the case where Eve eavesdrops between Charlie and David, the marginal probabilities of Alice, David, and Eve are given as  www.nature.com/scientificreports www.nature.com/scientificreports/ where BC η is a channel efficiency between Charlie and David. And in the case where Eve eavesdrops between David and Eliot, the marginal probabilities of Alice, Eliot, and Eve are obtained as (Here, the index Ē denotes Eliot)  Figure 13 shows K AX E : . Here, η is the efficiency of channel where Eve involves. In Fig. 13, we consider the case of = .
s 0 00128. Also, solid line, dashed line, dash-dot line, and dotted line denote the cases of = X B C D , , and Ē respectively.
One can see that in Fig. 13, the secret key rate is the lowest in the case of X E = (dotted line).This implies that the best performance of Eve can be obtained between David and Eliot. However, it should be emphasized that the effect depending on the position of eavesdropping is not big.

Security analysis of multiparty QKD based on sequential state discrimination -part ii: eve's multi-trial for eavesdropping
Here, we consider the case where Eve uses quantum memories. By using quantum memories of Eve, she can perform eavesdropping between sender and receivers. Suppose that Alice, Bob, and Charlie are involved in sequential state discrimination as a sender and two receivers. In this case, for eavesdropping, Eve uses two quantum memories: one quantum memory is used between Alice and Bob, and another quantum memory is used between Bob and Charlie (It should be noted that even though we consider the sequential state discrimination comprised of a sender and two receivers, our argument can be extended to the sequential state discrimination comprised of a sender and multi-receivers). Now, if Eve use a quantum memory E B for evesdropping between Alice and Bob (see Fig. 14), system of A B , , and E B can be described as When Bob discards an inconclusive result, ABE B |Γ〉 becomes the following mixed state: Here, → K : Here, Bob's POVM consists of α α α | 〉〈 | i i i . When Eve uses a quantum memory E C for eavesdropping between Bob and Charlie, the eavesdropping of Eve can be expressed as a quantum channel C B C ( ) Λ → . That is, the eavesdropping of Eve transforms ACE B σ as follows: The purification of id ( ) is given as www.nature.com/scientificreports www.nature.com/scientificreports/ Here, ? 0 | 〉 and | 〉 ? 1 are computational basis corresponding to Eve's failure. It should be noted that ? 0 | 〉 and ? 1 | 〉 are orthogonal to each other. The computational basis is orthogonal to | 〉 | 〉 | 〉 00 , 01 , 10 , and 11 | 〉. Unlike system E B , system E C is composed of two subsystems. It is because when Eve eavesdrops between Bob and Charlie, Eve also can eavesdrop between Alice and Bob. When E C1 and E C2 are the subsystems of E C , |Γ〉 ACE E B C can be described in the following way: Here, φ | 〉 + is a maximally entangled state φ | 〉 = | 〉 + | 〉 + ( 00 11 )/ 2. In other word, subsystem E C1 is used for eavesdropping of Alice's quantum state and subsystem E C2 is used for eavesdropping of Bob's post-measurement state.
When Bob and Charlie perform optimal sequential state discrimination, the prior probability is The detailed calculation can be found in Method). And, the marginal probability between Alice and Charlie is given as follows: Here, i ( ) a X is defined in the following way: (The notation of the label can be found in Fig. 15). It should be noted that ? 0 | 〉 and | 〉 ? 1 cannot be expressed by a linear combination of | 〉 | 〉 | 〉 00 , 01 , 10 , and 11 | 〉. The marginal probability of Charlie and Eve is given by π π | 〉〈 | = of Eve. Figure 14. The case where Eve eavesdrops between Alice and Bob and between Bob and Charlie. In (a1), Eve eavesdrops between Alice and Bob and between Bob and Charlie. First, Eve prepares two systems of E B and E C . Then, Eve interacts system E B with system B (The interaction is expressed by the global unitary operator U BE B ). Second, Eve interacts system E C with system C (This interaction is described by the global unitary operator U CE C ). Atfer these interactions, Eve measures system E B and E C globally (We denote this measurement as joint measurement). Here, the structure of joint measurement is determined by a unitary transformation If the unitary transformation V is fixed as the identity, (a1) and (a2) are equivalent. In (a2), Eve measures system E B and system E C locally (We denote this measurement as an individual measurement). The description of (a1) and (a2) is equivalent to the case of (b), where Alice, Charlie, and Eve share ACE E e secret key rate between Alice and Charlie, which is given by is displayed in Fig. 16. For convenience, it is assumed that each channel efficiency is equal to each other AB BC η η η = =¯. In Fig. 16, the solid black line corresponds to the case where Eve measures her subsystem, by the eighteen basis (This is the case where unitary transformation 1 18 is an identity). And, green points correspond to the cases of random unitary transformation. Because Alice and Bob cannot know Eve's system, treating unitary transformation as a random one can be justified. Arbitrary joint measurement. When Eve selects a measurement out of 100000 measurements, the ratio to obtaining nonzero secret key rate increases as η increases. When 0 75 η = .
, the ratio to obtaining nonzero secret key rate becomes . 36 323%. When ¯0 95 η = . , the ratio to obtaining nonzero secret key rate becomes . 99 978%. Specially, when ¯0 975 η = . , nonzero secret key rate can be obtained regardless of a measurement. Figure 15. The label of p in terms of basis of E B and E C . As an example, = p 1 corresponds to the case where the basis of E B is "?" and the basis of E C is "? 0 ", which is denoted as | = 〉 = | 〉 ⊗ | 〉 p 1 ? ?
C . And p 2 = corresponds to the case where the basis of E B is "?" and the basis of E C is "? 1 ", which is denoted as , Alice and Charlie can obtain nonzero secret key rate (The method of simulation can be found in Method).
Arbitrary joint measurement. When Eve selects a measurement out of 100000 measurements, the ratio to obtaining nonzero secret key rate increases as η increases. When η = .
0 75 , the ratio to obtaining nonzero secret key rate becomes . 36 193%. When η = . 0 95, the ratio to obtaining nonzero secret key rate becomes 99 854% . . Specially, when 0 98 η = . , nonzero secret key rate can be obtained regardless of a measurement. Arbitrary joint measurement. When Eve selects a measurement out of 100000 measurements, the ratio to obtaining nonzero secret key rate increases as η increases. When η = .

Discussion
In this report, we presented a generalization of sequential state discrimination. In our work, we did not limit the prior probabilities and the number of quantum states and receivers. We could express the generalized sequential state discrimination as a mathematical optimization problem. Because this optimization cannot be solved analytically, a numerical method was applied to the construction of the optimal POVM. Our optimization problems include all the results of the previous work 27 as special cases. Moreover, we applied the generalized sequential state discrimination to multiparty QKD. If Alice prepares one out of two pure states, the generalized sequential state discrimination can be used to perform multiparty QKD when there are a few receivers. It should be noted that if Alice prepares one out of two mixed states, the optimal success probability of generalized sequential state discrimination can exceed that of the quantum reproducing and quantum broadcasting strategies. Therefore, the generalized sequential state discrimination of mixed states has more potential for application to multiparty QKD than the other strategies. Finally, we analyze the security of multiparty QKD provided by optimal sequential state discrimination. Our analysis shows that the multiparty QKD guarantees nonzero secret key rate even in low channel efficiency. Even if we considered discriminating two quantum states, we could extend our argument for generalized sequential state discrimination to more than two quantum states. However, an unambiguous discrimination of more than three quantum states has not been known yet. Therefore, one needs to find a way to discriminate more than three quantum states, without any error.
If can be performed. Using this idea, one may devise sequential state discrimination of general mixed states.

Methods
Derivation of the optimization problem. In this section, we derive the optimization problem of generalized sequential state discrimination. First, a tangential point γ γ ( , ) 1 2 between a plane P s (B,C,D) and a surface γ (C) 2 satisfies the following equality: Combining both above equality and (1 )(1 ) β β should satisfy the inequality constraints in Eq. (16). Because a detailed derivation is too lengthy, we omit the derivation. If we substitute ( , ) 1 2 γ γ into the optimization problem, we can obtain Eq. (16).
www.nature.com/scientificreports www.nature.com/scientificreports/ , we obtain Eq. (20). Although we deals only with the = N 3 case, we can use this method for any N .
optimal success probability of the quantum reproducing strategy. We derive the optimal success probability of the quantum reproducing strategy. To make it simple, we consider the N 2 = case. Then, the success probability is expressed as Figure 16. The secret key rate between Alice and Charlie when Eve eavesdrops between Alice and Bob and between Bob and Charlie. The solid black line denotes the case where the unitary transformation of Eve is the identity. The green dots display the secret key rate when the unitary transformation of Eve is arbitrary. The table shows the ratio of nonzero secret key rate out of 100000 random cases.
This optimization problem is partitioned into the following two sub-optimization problem: The optimal solution of the two problems is given as s 1 Derivation of secret key rate in multiparty QKD -part i: eve's single trial of eavesdropping. Here, we explain the method to obtain the secret key rate of generalized sequential discrimination. Even though the identical prior probability is used in Result, we consider general prior probabilities given by q 0 and q 1 . Then, the entangled state between Alice and Bob is expressed by Here, When q q 0 1 = , the optimal condition is given by α α = = − s 1  Derivation of secret key rate in multiparty QKD -part ii: eve's multi-trial of eavesdropping. Here, we show how to evaluate the secret key rate of the case where Eve performs eavesdropping between Alice and Bob and between Bob and Charlie. Here, prior probabilities q 0 and q 1 are considered to be arbitrary values. Suppose that Eve performs eavesdropping between Alice and Bob, using a quantum memory E B . Then, Alice, Bob, and Eve share the following quantum state: Let us assume that the Kraus operator of Bob is given as ( ) . Then, Kraus operator K K , . Therefore, the argument of this case is identical to the security analysis of Part I.
After Bob performs a post-processing, |Γ〉 ABE B becomes the following tripartite state: I  I  K  I  I  K  I  I  K  I   I  K  I I  K  I  I  K  I I  To evaluate the secret key rate between Alice and Charlie, one must obtain the marginal probabilities P P P P , , , A C CA CE . First, P i ( ) A can be evaluated as follows: Figure 17. The graph of η f ( ). In these graphs, the solid black line, the solid blue line, and the solid red line denote the cases of = .
. s 0 05, 0 10, and 0.15, respectively. These graphs show that η f ( ) is a monotonically increasing function of η. Because η is a noise strength of channel, these graphs tell that noise of channel can make a bad influence on the secret key rate.    In Eqs. (44) and (45), ij δ is Kronecker delta. And, i λ , µ i , and i ν are defined as follows: Simulation method to search for critical channel efficiency (η crit ). When the unitary transformation V and overlap s are determined, the secret key rate is expressed as follows: Here, η = − f IA C I C E ( ) ( : ) ( : ) is a function of the single variable η. In Fig. 17, when V is an identity and s {0 05, 0 10, 0 15} ∈ . .
. , η f ( ) is a monotonically increasing function. Therefore, in the region of , because of f ( ) 0 η > , the secret key rate becomes nonzero. In this case, η crit can be obtained by a bisection method 49 .