Quantum LOSR networks cannot generate graph states with high fidelity

Quantum networks lead to novel notions of locality and correlations and an important problem concerns the question of which quantum states can be experimentally prepared with a given network structure and devices and which not. We prove that all multi-qubit graph states arising from a connected graph cannot originate from any quantum network with bipartite sources, as long as feed-forward and quantum memories are not available. Moreover, the fidelity of a multi-qubit graph state and any network state cannot exceed $9/10$. Similar results can also be established for a large class of multi-qudit graph states.

A quantum state is said to be entangled if it is not a convex combination of product states.To detect the quantum entanglement of two particles, witnesses [14,15] (often related to the fidelity of a target state [11,16,17]) and the PPT criterion [18] are standard methods.In the multiparticle case a state can be entangled, but separable for a given bipartition, then it is called biseparable [13].
If the state cannot be written as a convex combination of any biseparable states, it is said to be genuine multiparticle entangled (GME) [11][12][13].As for the detection of GME states, the fidelity with a highly entangled state, such as the Greenberger-Horne-Zeilinger (GHZ) state is one of the most used in experiments [7,19].For example, an n-partite state with a GHZ fidelity exceeding 1/2 is GME.Including GHZ state as a special example, graph states [20][21][22] play an eminent role in entanglement theory and its applications.
Recently, the concept of genuine multiparticle entanglement has been debated [23,24], and novel notions appropriate for the network scenario have been introduced and studied [25][26][27].In a network, states can be prepared by distributing particles from multiple smaller sources to different parties and applying local channels, see Fig. 1 for an example.In this fundamental scenario the local operations rely on a globally shared classical variable (Local operations and shared randomness, LOSR), e.g., a predefined protocol with shared randomness.The scenario of local operations assisted by classical communication (LOCC) gives more power to create distributed quantum states.But, communication based on outcome of local operations requires considerable time, either in the scenario of distributed quantum computation where local operations takes a while, or in the case that nodes in the network are far away from each other.Consequently, it would also require the usage of quantum memories or feed-forward techniques, which are expensive resources for current quantum technologies.Moreover, device-independent quantum information protocols are frequently related to Bell scenarios, where communication is impossible due to the space-like separation.Overall, an n-partite state is called genuine network multipartite entangled (GNME) if it cannot be created via LOSR in the network approach using (n − 1)-partite sources only.Besides this, other quantum correlations like quantum nonlocality and quantum steering have also been generalized to quantum networks recently [28][29][30][31][32][33][34][35].
For three-qubit states in the triangle network, a witness derived from the fidelity [25,34] and semidefinite programming methods based on the inflation technique [25,36] can be useful.The disadvantage of these approaches is that both of them are hard to generalize to more complicated quantum networks.An analytical method based on symmetry analysis and inflation techniques [37] was proposed recently [27] and can overcome some of the difficulties.Explicitly, it was shown there that all n-qubit graph states with n ≤ 12 are not available in networks with bipartite sources, and it was conjectured that this no-go theorem hold for multi-qubit graph states with an arbitrary number of particles.
In this paper we prove that all multi-qudit graph states with a connected graph, where the multiplicities of the edges are either constant or zero, cannot be prepared in any network with only bipartite sources.In fact, this result holds also for all the states whose fidelity with some of those qudit graph states exceeds a certain value.More specifically, our results exclude the generation of any multi-qubit graph state with a fidelity larger than 9/10 in networks.This proves the conjecture formulated in Ref. [27], it also may provide interesting connections to other no-go theorems on the preparability of graph states in different physical scenarios, such as spin-models with two-body interactions [38,39].
Network entanglement.-Thedefinition of network entanglement is best explained using the example of the triangle scenario, see Fig. 1.Here, one has three bipartite quantum source states ς x for x = a, b, c, and three local channels E (λ) X for X = A, B, C, where λ is the shared random variable with probability p λ .The global state can be prepared in this scenario has the form For a given state ϱ, the question arises whether it can be generated in this manner, and this question was in detail discussed in Refs.[25][26][27].More generally, one can introduce quantum network states as follows: A given hypergraph G(V, E), where V is the set of vertices and E is the set of hyperedges (i.e., sets of vertices) describes a network where each vertex stands for a party and each hyperedge stands for a source which dispatches particles to the parties represented by the vertices in it.This quantum network is a correlated quantum network (CQN), if each party can apply a local channel depending on a shared random variable.Then, in complete analogy to the definition in Eq. ( 1) one can ask whether a state can be prepared in this network or not.More properties of these sets of states (e.g., concerning the convex structure and extremal points) can be found in Refs.[25,27].
Three-qudit GHZ states and the inflation technique.-Theinflation technique [37] turns out to be an useful tool to study network entanglement [25,27].Unless otherwise stated, we consider networks with bipartite sources only.For a given network as the triangle network depicted in Fig. 1 and also in Fig. 2(a), the copies of sources are sent to different copies of the parties as in Fig. 2(b) and Fig. 2(c).In principle, the source states may also be FIG. 2. The triangle network in (a) and two kinds of its inflation in (b) and (c).For convenience, we use (d) and (e) as short notations of (b) and (c), respectively.Here the broken edge e with only one vertex v means that we ignore or trace out the particles not for the party represented by v from the source state represented by e.
wired differently in different kinds of inflation.For convenience, here we use also edges with only one vertex as in Fig. 2(d) and Fig. 2(e), which means that the particles, which have not appeared in the state, are traced out in the source states.
The key idea of the inflation method is the following: If the three-particle state ρ can be prepared in the network, the six-particle states γ and η can be prepared in the inflated networks as well.The states γ and η share some marginals with ρ and with each other.So, if one can prove that six-particle states with these desired properties do not exist, the state ρ is not reachable in the original network.
To see how the idea of inflation works in practice, we take the three-qudit GHZ state [40] as an example.The three-qudit GHZ state |iii⟩ is a stabilizer state, whose stabilizers include where the unitary operators are , with ⊕ to be the addition modulo d, and For a given network state ρ, we can consider the value ⌈M ⌉ ρ of any stabilizer M in Eq. (2), i.e., the expectation value ⟨Π  M to be the projector into the eigenspace of M with eigenvalue +1.If one of ⌈M ⌉ ρ does not equal to 1, we can conclude that this network state ρ cannot be the state |GHZ⟩⟨GHZ|.
Let us consider two kinds of inflation of the trianglenetwork as in Fig. 2, where the corresponding network states are denoted as γ, η.Roughly speaking, the source between B, C is broken in the γ inflation.Note that inflation η is actually a trivial inflation in triangular network.By comparing Fig. 2 For convenience, here ⌈Z B Z † C ⌉ η stands for ⌈Z B Z † C ⌉ η BC , where η BC is the reduced state of η on parties B and C. We use such shorthand notations throughout the whole manuscript without confusion.By applying Lemma 4 in Appendix A in Supplemental Information(SI), we have Under the assumption that where θ d = 0 when d is even and θ d = π 2d when d is odd.This assumption can be ensured whenever F(|GHZ⟩⟨GHZ|, ρ) ≥ 3/4 according to Lemma 5 in SI.
If ρ is indeed the GHZ state, then Eq. ( 3)-( 7) cannot hold simultaneously, which is a contradiction.Combined with Lemma 5, which implies Hence, either the assumption does not hold, then F(|GHZ⟩⟨GHZ|, ρ) < 3/4; or the assumption holds and so do the inequalities in Eq. ( 8).Since the upper bound in Eq. ( 8) is always larger than 3/4 for any dimension d, the upper bound in Eq. ( 8) holds whatever the assumption holds or not.We remark that we only used GHZ states as an example to introduce our general method, a tighter bound in Appendix D on the fidelity exists [25].
Multi-qudit graph states.-Thethree-qudit GHZ state is a special case of a multi-qudit graph state [22].In the same spirit, we can also derive no-go theorems for network states with multi-qudit graph states as targets.Each graph state is associated with a multigraph G = (V, E), defined by its vertex set V , and the edge set E, where the edge between vertices i, j with multiplicity m ij is denoted as ((i, j), m ij ).Without loss of generality, we will only consider multi-partite graph states, i.e. number of parties (vertices) is at least 3. Due to the periodicity as follows, the multiplicity can be limited to where d is dimension of Hilbert space for a single qudit.Besides, we denote N i the neighborhood of vertex i, and define unitary stabilizers g i = X i ⊗ j∈Ni Z mij .Note that g d i = I.For a given multigraph G = (V, E), the corresponding graph state is the unique common eigenvector of the operators {g i } i∈V with eigenvalue +1 [22,[41][42][43].This eigenvector |G⟩ is called the graph state associated to the graph G. Theorem 1. Multi-qudit graph states with connected graph and multiplicities either being constant m or 0 cannot be prepared in any network with bipartite sources.
Proof.Here we explain the main idea, the detailed proof is provided in Appendix B in SI.The structure of the proof is in the same spirit as the one for |GHZ⟩ state.
The key point is to keep necessary marginal relations in different kinds of inflations and finally derive a contradiction.
For a connected graph G with no less than three vertices, there are always three vertices A, B, C such that (A, B), (A, C) are two edges.If (B, C) is not an edge, we call (A, B, C) an angle.Otherwise, we call (A, B, C) a triangle.
To carry out the proof, we have to carefully group the neighborhoods of vertices A, B, C and choose proper stabilizers of the graph state correspondingly.Here we consider the case where (A, B, C) is a triangle as an example.Then we can partition all the vertices into 4 groups as in Fig. 3, where T ABC is the common neighborhood of A, B, C, J AB is the common neighborhood of A, B but not C, E A is the neighborhood which is not shared by B, C and so on.
By choosing C (the t-th power of g C ), we have the marginal relations where S 4 ′ is related to S 4 by changing party C ′ in the support to C. These marginal relations can be verified by comparing the supports of each operator in different kinds of inflation.
However, S 3 , S 4 ′ do not commute in the inflation η.More precisely, we have that where 0 ≤ θ t,d ≤ π/6 by choosing t properly.
Similarly as the analysis for the GHZ state, the relation that S 3 = S 1 S 2 and conditions in Eqs.(9 -11) lead to This is in fact a universal bound for arbitrary configuration of equal-multiplicity multi-qudit graph states.
Note that for qubit graph states, the multiplicity is either 1 or 0, this leads to the following theorem.
Theorem 2. Any multi-qubit graph state with connected graph cannot be prepared in a network with only bipartite sources, with 9/10 as an upper bound of fidelity between graph state and network state.This follows from the fact that θ t,2 = 0.
In order to formulate a more general statement, note that the key ingredients in the proof of Theorem 1, were that all parties can be grouped in a special way which fits to the algebraic relations S 3 = S 1 S 2 for commuting S 1 , S 2 , moreover, it was needed that S 3 , S 4 ′ have no common eigenvectors with eigenvalue +1.This leads to a a more general theorem for the states with a set of stabilizers.
Theorem 3.For a given pure state σ with commuting (unitary or projection) stabilizers {S 1 , S 2 , S 3 = S 1 S 2 , S 4 }, it cannot be prepared in bipartite network if 1. all the parties can be grouped into {G i } 4 i=1 such that S i has no support in G i , see also Fig. 4(a Proof.Here we provide the main steps of the proof without diving into details.The first condition implies that, those four operators do not have common support for all of them.Hence, the set of parties can be divided into the four parts as illustrated in Fig. 4(a).By comparing the supports of the operators in different kinds of inflation as in Fig. 4, we have the marginal relations Through those marginal relations, we can relate ⌈S 3 ⌉ η and ⌈S 4 ′ ⌉ η to f = F(σ, ρ).Finally, the second condition leads to the result that f < 1.In fact, ) FIG. 4. General inflation scheme in bipartite network similar to Fig. 3.In (a), Gi's are group of parties in a partition, the green, red, blue and dashed circles stand for the supports for operators S1, S2, S3, S4, respectively.In (b), (c) and (d), we replace the label of each group by the indices of the Si's which has this group as support.The green shadow represents a multipartite source relating to all the groups in it.The sources which have not been changed in the whole proof are omitted in the figures.where λ ′ < 2 is the maximal singular value of {S 3 , S † 4 ′ }.
Note that all the bipartite sources in the network among parties in supp(S 1 ) have not been touched during the inflation procedure, so the proof still holds even if there is a multipartite source just affecting this set of parties.By exhaustive search and applying Theorem 3 to multi-qudit graph states, we can figure out the situations where all n-partite qudit graph states in dimension d cannot arise from a network with bipartite sources, see Theorem 10, which is a consequence of Theorem 3, and discussions therein in Appendix C in the SI.The result is summarized in Table I.Moreover, any prime-dimensional graph state satisfying special structures as in Theorem 10 cannot be prepared in a network with bipartite sources.
Conclusion and Discussions.-Herewe have developed a toolbox to compare multi-qudit graph states and states which are generated in a quantum network without memory and feed forward.By combining those tools related to symmetry and the inflation technique, we proved that all multi-qudit graph states, where the non-zero multiplicities are a constant, cannot be prepared in the quantum network with just bipartite sources.The result can also be generalized to a larger class of multi-qudit graph states and quantum networks with multipartite sources, in the case that the generalization does not affect the necessary marginal relations during the inflation procedure.The more general case with multi-partite sources is an interesting topic for future research.Furthermore, we provided a fidelity estimation of the multi-qudit graph states and network states based on a simple analysis.
More effort should be contributed to the fidelity analysis in the future, such as introducing more types of inflation in Ref. [27], and generalizing the techniques in Ref. [25] for the states other than GHZ states.Another interesting project for further study is to consider other families of states, like Dicke states or multi-particle singlet states, which are not described by a stabilizer formalism.Finally, from the fact that graph states cannot be prepared in the simple model of a network considered here, the question arises, which additional resources (such as classical communication) facilitate the generation of such states.Characterizing these resources will help to implement quantum communication in networks in the real world.
Note added: While finishing this manuscript, we became aware of a related work by O. Makuta et al. [44].Albeit those two works originate from the same conjecture in Ref. [27], the techniques and results are different from few perspectives.Especially, we have only made use of two kinds of inflation and Theorem 2 here holds for all dimensions but with limited multiplicities.The resulting fidelity between the graphs states and network states has also different estimations.The application of Theorem 3, like in combination with Theorem 10, can cover more situations.
For discussions about fidelity analysis, it is convenient to develop the following mathematical tools.Lemma 4. For three unitary operators S 1 , S 2 , S 3 = S 1 S 2 which commute with each other, we have where ⌈M ⌉ ρ is the expectation value ⟨Π M ⟩ ρ , and Π M is the projector into the eigenspace of M with eigenvalue 1.
Proof.⌈S⌉ ρ ≥ F(σ, ρ) holds since Π S ⪰ σ and ρ ⪰ 0. For an arbitrary vector, we can decompose it as |u⟩ + |v⟩ where ⟨u|v⟩ = 0, Π S |u⟩ = 0. Note that This implies where the last inequality is from the fact that S + S † is hermitian and all its eigenvalues are no less than −2.Thus, On the other hand, the fact that ⟨Re Lemma 6.For a given set of operators {S i }, where S is a matrix with (i, j)-th element ⟨{S i , S † j }⟩, {•, •} is used as notation for the anticommutator, and λ max (S) is the maximal eigenvalue of S.
Proof.For a given normalized complex vector {c i }, denote M = i c i S i .We have Note that S is hermitian.
Consequently, the inequality Since {c i } can be an arbitrary normalized vector, we have In the case that F(σ, ρ) ≥ 1/2, by combining Lemma 5 and Lemma 6, we have Lemma 7.For a given set of unitary operators {S i } n i=1 satisfying S i S j = −e iθij S j S i where θ ij ∈ (−π, π], we have where ρ is an arbitrary mixed state, θ = max i̸ =j |θ ij |.
Proof.In the case that {S i } is a set of unitaries, and S i S j = −e iθij S j S i , we have Consequently, According to Frobenius theorem, Together with Eq. ( 30), this leads to We remark that, in the case that S i S j = −S j S i , i.e., θ ij = 0, Eq. ( 28) reduces to the result proposed in Ref. [45], which have been used in the comparison of qubit graph states and network states in Ref. [27].However note that these inequalities are not tight.If we choose an eigenstate of S 1 and suppose which does not saturate the upper bounds unless all θ ij = 0.

Appendix B. Proof of Theorem 1 and Theorem 3
Here we provide the rest of proof of Theorem 1 and the analysis of the fidelity.To proceed, we introduce some necessary notations.Definition 8. Given a network N = (V, E) and a subset T of vertices, the reduced network is given by N | T = (T, E| T ) where E| T = {e ∩ T |e ∈ E} and e ∩ T is the intersection of T and e by treating e as a subset of vertices.
We have one remark.For two edges e and e ′ in different kinds of inflation, we say e ∩ T = e ′ ∩ T if and only if e ∩ T and e ′ ∩ T have same elements, besides, e and e ′ are copies from the same edge in the original network.Proof.We first prove the theorem for the case where (A, B, C) is a triangle, then prove for the angle case.
For convenience, we denote T ABC the common neighborhood of A, B, C in the graph G, J AB the common neighborhood of A, B but not C, E A the neighborhood which is not shared by B, C and so on.Those sets of parties can be grouped as in Fig. 5.The structure of the proof is in the same spirit as the one for |GHZ⟩ state.The key point is to keep necessary marginal relations in different kinds of inflation and derive contradiction.Let us choose the three operators Similarly, supp The fact that supp Note that supp(g where S 4 ′ is related to S 4 by changing party C ′ in the support to C. If we assume the graph state |G⟩ can be generated as the network state ρ, Eqs.(34-37) lead to ⟨S 4 ′ ⟩ η = ⟨S 3 ⟩ η = 1, which conflicts with the fact that where sin θ t,d = | cos(tmπ/d)|, and we can choose appropriate t such that 0 ≤ θ t,d ≤ π/6, see below.In fact, similar to the analysis for the GHZ state, conditions in Eqs.(34)(35)(36)(37)(38) lead to that This is a universal bound for arbitrary configuration of equal-multiplicity multi-qudit graph states.Now we prove for the angle case, with same spirit as the triangle case.It suffices to explain how the two conditions in Theorem 3 can be satisfied.
Without loss of generality, we assume (A, B, C) is the angle in the graph for the graph state, that is, there are only two edges (A, B) and (A, C).Now we choose S 1 = g † C , S 2 = g B and S 4 = g t A , then S 3 = g B g † C .The supports of the operators are By grouping the parties into four groups as illustrated in Fig. 6, that is, One can verify that supp(S i ) ∩ G i = ∅, for i = 1, 2, 3, 4. Hence, the first condition in Theorem 3 is satisfied.Since the stabilizers S i 's are in product form, to verify the second condition in Theorem 3 is equivalent to verify that S 3 | G2 and S 4 ′ | G2 has no common eigenvector with eigenvalue 1.
As we can see, this implies that S 3 | G2 and S 4 ′ | G2 cannot share a common eigenstate because where t is chosen such that ω tm ̸ = 1 rather closest to −1.With the inflation process as shown in Fig. 6, we also conclude that the graph state with an angle cannot originate from bipartite network.
We continue to estimate the fidelity F(σ, ρ) between the graph state σ and the network state ρ in both cases with either angle or triangle.In the case that S 3 S 4 ′ = ω tm S 4 ′ S 3 with ω = exp 2πi/d, which implies that the maximal absolute value of the eigenvalues of {S FIG. 7. General inflation scheme in bipartite network similar to Fig. 3.In (a), Gi's are group of parties in a partition, the green, red, blue and dashed circles stand for the supports for operators S1, S2, S3, S4, respectively.In (b), (c) and (d), we replace the label of each group by the indices of the Si's which has this group as support.The green shadow represents a multipartite source relating to all the groups in it.The sources which have not been changed in the whole proof are omitted in the figures.
It is direct to see that Here we also present proof of Theorem 3.
Theorem 3.For a given pure state σ with commuting (unitary or projection) stabilizers {S 1 , S 2 , S 3 = S 1 S 2 , S 4 }, it cannot be prepared in bipartite network if 1. all the parties can be grouped into {G i } 4 i=1 such that S i has no support in G i , see also Fig. 4(a); 2. S 1 , S 2 commutes, and S 3 , S 4 ′ have no common eigenvectors with eigenvalue 1, where S 4 ′ has the support G 2 , G 3 and a copy of G 1 , and S 4 ′ acts there same as S 4 on supp (S 4 ).
Proof.Here we provide the main steps of the proof without diving into details.The first condition implies that, those four operators do not have common support for all of them.Hence, the set of parties can be divided into the four parts as illustrated in Fig. 7(a).By comparing the supports of operators S 1 , S 2 in the original network in Fig. 7(b) and in the inflation in Fig. 7(c), we have the marginal relations The fact that S 3 = S 1 S 2 implies From the comparison of support of operator S 3 in the inflation related to γ and the inflation related to η as shown in Fig. 7(c) and Fig. 7(d), we know that Similarly, by comparing the original network and the one related to η, it holds that Since S 3 , S 4 ′ have no common eigenvectors with eigenvalue 1, the values ⌈S 3 ⌉ η and ⌈S 4 ′ ⌉ η cannot be 1 at the same time, which contradicts with the assumption that S 1 , S 2 , S 4 are stabilizers of ρ.
By grouping the parties into four groups similar to Fig. 3 but with J AB = J CA = T ABC = ∅, that is, one can verify that supp(S i ) ∩ G i = ∅, for i = 1, 2, 3, 4. Hence, the first condition in Theorem 3 is satisfied.Similar to Theorem 1, it suffices to verify that S 3 | G2 and S 4 ′ | G2 has no common eigenvector with eigenvalue 1.
, where m = m AB m CA /h (mod d).By assumption, 1 ≤ m ≤ d − 1.By choosing t suitably as discussed in Appendix B, the second condition can be satisfied.We also have the same upper bound of fidelity between the graph state and the network state, i.e., 0.95495.
By grouping the parties into four groups similar to Fig. 6 but with T ABC = ∅ , that is, One can verify that supp(S i ) ∩ G i = ∅, for i = 1, 2, 3, 4. Hence, the first condition in Theorem 3 is satisfied.Similar to Theorem 1, it suffices to verify that S 3 | G2 and S 4 ′ | G2 has no common eigenvector with eigenvalue 1.The fact that In practice, the conditions in Theorem 10 are not applicable in some cases, for example, the condition T ABC = ∅ does not hold for the fully connected graph states.However, one can still try to find out other kinds of stabilizers which fulfill the two conditions in Theorem 3. Another option is to convert the graph state to another one, which satisfies the conditions in Theorem 10, by applying local complementation operation [46].In the case of qudit graph states, this operation on a given vertex a of the graph is where N a is the neighborhood of vertex a.For the qubit graph state, any non-zero m ij can only be 1.Consequently, the local complementation operation on a given vertex a is m ij → 1 − m ij , ∀i, j ∈ N a .
It is easy to see that the angle (A, B, C) can be mapped to the triangle by local complementation for qubit graph state.This is not true generally for the qudit graph states.Now we can determine whether a given graph state with connected graph can be prepared in a bipartite quantum network based on Theorem 10.For a given graph, we can run over all possible local complementation operations and determine if there exists any equivalent graph such that one of the two conditions is satisfied.If so, then this graph (and the equivalent graphs) cannot be prepared in any bipartite network.Running over d = 3, n = 3, 4, 5, 6; d = 4, n = 3, 4, 5; and d = 5, 7, n = 3, 4 cases, where d is local dimension and n is number of parties, we found all of these graph states cannot be prepared in bipartite network.And for prime dimension and n = 3, the conditions always hold.On the one hand, conditions on support is automatically true for n = 3.On the other, since d is prime, (m XY m Y Z /h) (mod d) ̸ = 0 and (m AB m CA /h ′ ) (mod d) ̸ = 0 also always hold.Thus, those graph states cannot be prepared in any bipartite network.We then summarize those results in Table I in the main text.
We remark that, even up to local complementation, the two conditions in Theorem 3 cannot be satisfied for the 6-dimensional tripartite qudit angle graph states, where m AB = 3, m CA = 2 and m BC = 0.

FIG. 1 .
FIG. 1. Sketch of a triangle quantum network, where three bipartite sources ςa, ς b and ςc are distributed to three parties A, B, C, e.g., the source states ςa are sent to B and C. Each party can apply local operations E (λ) X (X = A, B, C) on the received particles, which are affected by global shared randomness λ.

FIG. 3 .
FIG.3.Fully connected network with bipartite sources and two kinds of its inflation, where the corresponding states are denoted as ρ, γ and η.The sources which have not been changed in the whole proof are omitted in the figure.

FIG. 5 .
FIG.5.Fully connected network with bipartite sources and two kinds of its inflation, where the corresponding states are denoted as ρ, γ and η.The sources which have not been changed in the whole proof are omitted in the figure.

Lemma 9 .Theorem 1 .
Consider two given different kinds of inflation N γ , N η of a network, then the reduced states Tr Tc (γ) = Tr Tc (η) if the reduced networks obey N γ | T = N η | T , where T c is the complement set of T .Qudit graph states with connected graph and multiplicities either the constant m or 0 cannot be prepared in any network with bipartite sources.

1 FIG. 6 .
FIG.6.Three types of inflation and support of related regions in the angle case.

S 3 S 4 ′
= ω −ctm CA S 4 ′ S 3 = ω −t mS 4 ′ S 3 , where m = m AB m CA /h ′ (mod d).By assumption, 1 ≤ m ≤ d − 1.By choosing e suitably as discussed in Appendix B, the second condition can be satisfied.Further, we result in the same upper bound of fidelity between the graph state and the network state, i.e., 0.95495.
); 2. S 1 , S 2 commutes, and S 3 , S 4 ′ have no common eigenvectors with eigenvalue 1, where S 4 ′ has the support G 2 , G 3 and a copy of G 1 , and S 4 ′ acts there same as S 4 on supp (S 4 ).

TABLE I
. For a given dimension d one can consider an npartite graph state and ask whether Theorem 10 can be used to prove the impossibility of its preparation, see discussions in Appendix C.This table identifies situations where all npartite qudit graph states in dimension d cannot be prepared by the network.