Nonlocal correlations in quantum networks distributed with different entangled states

We initiate the study of the nonlocal correlations in generic asymmetric quantum networks in a star configuration. Therein, the diverse unrelated sources can emit either partially or maximally entangled states, while the observers employ varying numbers of measurement settings. We propose nonlinear Bell inequalities tailored to the distributed entangled states. Specifically, we demonstrate that the algebraic maximal violations of the proposed nonlinear Bell inequalities are physically achievable within the quantum region. To achieve this, we construct the segmented Bell operators through the cut-graft-mix method applied to the Bell operators in the standard Bell tests. Furthermore, we devise the fitting Bell operators using the sum-of-square approach.


Introduction
Bell nonlocality, one of the distinguished quantum features from classical physics, discloses the inconsistency between quantum theory and classical causal relations in predicting the outcome correlations of a composite system [1,2].In a standard Bell test, a source, whether classical or quantum, initially emits a state involving two or more particles that are then received by spatially separated observers.Next, these observers each perform local measurements with random measurement settings as the inputs and then obtain the measurement outcomes as the outputs.Local realism imposed the constraints on the correlation strength of spacelike events that is upper-bounded by the Bell inequalities, while quantum theory predicts stronger correlations that violate these Bell inequalities.Thanks to the rise of quantum information science, the extensive study on Bell nonlocality has yielded a variety of proposed Bell inequalities [3], and, on the other hand, brought practical applications in quantum cryptography and quantum communication [4][5][6][7].In particular, it is essential to certify the quantum properties solely on the limited capability of measurement statistics using untrusted devices, [8].For example, to self-test an entangled state, the device-independent approach leverages specific Bell inequalities tailored to this state [4,[9][10][11][12].Nevertheless, the number of Bell inequalities tailored to partially entangled states or entangled subspaces remains limited [13][14][15][16].
Given the long-distance, multi-user quantum networks as the ultimate goal of quantum communication, it is intriguing to explore the nonlocal correlations in quantum networks comprising multiple independent quantum sources and a set of spatially separated observers.Even in classical networks, those sophisticated causal relations lead to stronger constraints than those in the standard Bell tests with a single source.Most Bell-type inequalities for various classes of quantum networks are nonlinear due to the non-convexity of the multipartite correlation space [17][18][19].In the simplest two-source quantum network, the linear and nonlinear bilocal inequalities are extensively studied [20][21][22][23][24][25].Recently, Bell nonlocality in the star-shaped network [28][29][30][31][32][33][34] and other broader classes of quantum networks has also been widely studied [17,30,35,36].In particular, the notion of asymmetric networks is introduced.Therein, different edge observers receiving only a particle have different numbers of measurement settings [25][26][27].
In nearly all literature on bilocality or K-locality quantum networks, the quantum sources emit two-qubit EPR pairs or multi-qubit Greenberger-Horne-Zeilinger (GHZ) states that are maximally entangled [37].The proposed Bell inequalities tailored for bilocality or K-locality quantum networks can reach the quantum upper bound if all sources emit maximally entangled states [20,24,28,31].In this work, we introduce Bell inequalities tailored for quantum networks with sources emitting partially entangled states, thus reaching the maximal violation within the quantum region.
Although partially entangled states can be exploited to device-independent randomness generation at optimal rate [13,14], to our knowledge, Bell nonlocality of quantum network distributed with the non-maximally entangled states was studied only in [38].In this work, we explore Bell nonlocality with variant entangled states distributed in quantum networks.Therein, the quantum sources can emit either partially or maximally entangled states, while the receivers can have different numbers of inputs [25].We aim to propose the Bell inequalities tailored to asymmetric networks, where their algebraic maximum violations can be physically realized.To achieve this, we introduce the 'cut-graft-mix' method to construct the Bell operators from those in the standard Bell tests [38].For example, considering quantum sources emitting the two-qubit partially entangled states, we cut the associated Bell operators of the tilted Clauser-Horne-Shimony-Holt (CHSH) inequality tailored to this entangled state into three segmented Bell operators.We then graft (make tensor products of) the segmented Bell operators according to the particle distribution in the networks.Finally, we obtain the nonlinear Bell inequalities by mixing (adding) the expectation values of the grafted Bell operators [38].In addition, we devise customized Bell inequalities tailored to the Bell states using the sum-of-squares decompositions [15], enabling the use of the associated Bell operators that are divided into segmented Bell operators.
The paper is organized as follows.In section 2 we focus on the two-source networks.We review the tilted CHSH inequality tailored to the two-qubit partially entangled states and the cut-graft-mix method.Then we consider two cases: (i) Two sources emit the same partially entangled states, and (ii) In asymmetric networks, one source emits the partially entangled states, while the other emits the maximally entangled states.We further extend our study to the K-locality star-shaped networks as a generalization of the two-source networks.We also consider two cases: (i) K sources emit the same partially entangled states, and (ii) Some of these K sources each emit the same partially entangled states and the other emit Bell states.Conclusions are made in section 4. Throughout the paper, the superscripts in brackets of the local observables or states denote the particle/qubit index and are omitted where there is no confusion.In addition, ⟨•⟩ LHV and ⟨•⟩ Q denote the expectation values of the (segmented) Bell operator • within the classical and quantum realms, respectively.

Bilocality inequalities for partially entangled states
where the non-negative parameter β is defined as: The tilted CHSH inequality reads It is illustrative to prove (3) 1 ), and ). Regarding the local realism, the absolute value of the expectation of B l reads l a 1 P (a where l = 2, 3, y 2 = 0, y 3 = 1, ⟨A x ⟩ λ = ax=−1,1 a x P(a x |x, λ), B y λ = by=−1,1 b y P(b y |y, λ), and The first inequality is obtained using the inequality ´Ω •dµ(λ) ⩽ ´Ω |•| dµ(λ), and the second inequality is obtained using the inequality B y l λ ⩽ 1.Since In addition, since b Notably, for example, it is easy to verify that the classical bound ⟨tCHSH⟩ OPT LHV can be achieved by setting the deterministic outcomes a 0 = a 1 = b 0 = b 1 = 1 for any λ.
In the quantum region, the local observables of qubit i are assigned as: where 0 < µ < π 2 , and the local observables of the qubit j are assigned In addition, given an arbitrary two-qubit state |ϕ⟩ shared between Alice and Bob, the Tsirelson inequality in the quantum region reads where , and the equality of the inequality in (7) holds with the matching condition [15] tan µ = sin 2θ.
That is, if the parameter β meets the condition (2) and parameter pair (θ, µ) meets the condition (8), the tilted CHSH-Bell operator tCHSH is tailored to the partially entangled state |ψ θ ⟩.As a remark, one can regard the states 0 = |00⟩ and 1 = |11⟩ as the logical states of [2, 1, 2] quantum stabilizer code.The codeword state |ψ θ ⟩ is stabilized by the operator s = σ z ⊗ σ z , and the logical phase-flip and bit-flip operators are which indicate the usefulness of the stabilizing and logical operators in devising Bell operators [38].Finally, it is straightforward to verify the inequality

Cut-graft-mix method and bilocal Bell inequalities
Here we consider the following two-source networks.Source e 1 emits the states of particles 1 and 2 that are received by Alice 1 and Bob, respectively.Source e 2 emits the states of particles 3 and 4 that are received by Bob and Alice 2 , respectively.If the source e 1 (e 2 ) emits |ψ (12) θ ⟩ (|ψ )), and B 13 = B (2) )).In the following, let the matching conditions ( 2) and ( 8) hold for the β, µ, and θ in tCHSH i and |ψ θ ⟩, and hence ⟩ achieves the maximal violation of the Bell inequality (3).We demonstrate how to construct the Bell inequality tailored to the product state |ψ (12) θ ⟩ |ψ (34) θ ⟩ using the cut-graft-mix method.(i) Cut the tCHSH i is into three segmented Bell operators : B i1 , B i2 , and B i3 .(ii) Graft/tensor the segmented Bell operators in a one-to-one way, resulting in the three extended Bell operators 1 .As a result, we have , and Notably, if θ = π 4 , then the states ψ become maximally entangled and the parameter β = 0 according to (2).In this case, we have ⟨B 11 ⟩ = ⟨B 21 ⟩ = I 1 = 0 and the bilocal inequality in (10) is the same as that in [20,35].
To prove (10), note that the i-th classical source emits the system represented by the local random variable as the hidden state λ i in the measure space (Ω i , Σ i , µ i ) with the normalization condition ´Ωi dµ i (λ i ) = 1.In the two-source Bell test scenario, all classical systems are in the hidden state in the measure space (Ω, Σ, µ), where Ω = Ω 1 ⊗ Ω 2 and the measure of Λ 2 is given by µ On receiving the particle from the source s, Alice s randomly chooses the classical bit x s ∈ {0, 1} as the measurement setting and then measures the local observable A where x2 ).Denote the local expectation as and Let where m = 2, 3, Before proceeding further, we introduce the useful result.
Lemma Let x sm be non-negative real numbers and s, m be natural numbers, then where the equality holds if [28,32].
To prove (10), we have Here, the equality of the first inequality holds if ⟨∥B km ∥⟩ LHV ; the equality of the second inequality holds if ⟨∥B 1m ∥⟩ LHV = ⟨∥B 2m ∥⟩ LHV ∀m = 1, 2, 3. Eventually, the classical bound β + 2 is tight and achievable by setting all outcomes fixed and unchanged for any λ 1 and λ 2 .
In the quantum version of the two-source networks, the product state ⟩ is prepared and distributed among these three observers.As a result, where the first inequality comes from ( 14) by setting x sm = ⟨B sm ⟩ Q , M = 3, and K = 2; the second inequality is due to (9).As a result, the equalities of the first and second inequalities hold simultaneously if |ϕ Some remarks are in order.We can use an alternative grafting way to construct the following segmented Bell operators 1 ), and , where 1 , and 1 B (3) 0 and the measurement assignments are the same as those in ( 5) and (6).It is easy to verify the Bell inequality and the Tsirelson inequality where the proofs of ( 17) and ( 18) are very similar to those of ( 10) and ( 16), respectively.The classical bound β + 2 in ( 17) is still achievable by setting all outcomes to be 1.However, the equality in (18) does not hold, since That is, the RHS of ( 18) as the algebraic maximal violation of the bilocal Bell inequality ( 17) is not operationally achievable in the quantum region.
In addition, according to the measurement assignment, two local observables B 0 = B (2) 0 B (3) , which indicates that Bob can measure B 0 and B 1 simultaneously without disturbance.Instead of measuring B i directly, he can perform the Bell-state measurement as the entanglement swapping with the measurement basis [18,39,40].Let the post-selection state of the Bell-state measurement |ϕ Bell ⟩, and Bob's outcome is set to be ⟨ϕ Bell |B i | ϕ Bell ⟩ ∈ {−1, 1}.However, in the Bell test of ( 18), Bob's three observables are Finally, since B ′ 0 = B 0 , Bob can measure B ′ 0 directly or perform the Bell-state measurement using the measurement basis S Bell.

Asymmetric bilocal quantum network
In the bilocal asymmetric quantum network proposed in [25], two quantum sources emit Bell states, but the numbers of measurement settings for Alice 1 and Alice 2 are different.Here, we explore another type of the bilocal asymmetric quantum network as follows.Besides the difference between the numbers of measurement settings, these two quantum sources emit different entangled states.As shown in figure 1, source e 1 emits the Bell-state |ψ (12) π/4 ⟩ and source e 2 emits the partially entangled state |ψ (34) θ ⟩.The goal is to devise a nonlinear Bell inequality tailored to the product state |ψ (12) π/4 ⟩ ⊗ |ψ (34) θ ⟩ distributed in this bilocal scenario.This entails constructing a specific Bell operator E 1 = E 11 + E 12 + E 13 such that |ψ (12) π /4 ⟩ can attains the optimal Bell value, denoted as , and E 1j π/4 = B 2j θ > 0 for any j = 1, 2, 3.
We outline the construction of E 1 involving three segment Bell operators E 11 , E 12 , E 13 .For more details, please refer to the appendix.There are three operators z , and x ⊗ σ (2) x that stabilize the Bell state |ψ (12) Using the sum-of-squares decompositions, we construct the semi-definite operators γ = 2 √ 3 3 i =1 α i P † i P i , where the Hermitian operator P i = s i − I and α i > 0. Upon assigning the measurements, γ can be expressed as and three segmented Bell operators E 11 , E 12 , E 13 are listed in (A2), (A3), and (A4), respectively.Then we adjust the three positive parameters α 1 , α 2 , and α 3 to meet the following condition within the quantum realm Applying the cut-graft-mix method to E 1 and tCHSH 2 , we obtain three asymmetric segmented Bell operators: For the proposed bilocal scenario, the loose Bell inequality reads where f(α 1 , α 2 , α 3 ) is defined in (A6).The proof of ( 20) is similar to that of (10) with the term B 1m in (15) being replaced by E 1m for m = 1, 2, 3.However, if these two conditions That is, the RHS of ( 20) as the algebraic classical bound is operationally unattainable.In the quantum region, the expectation value of segmented Bell operators of I A m can be factorized as 14), the Tsirelson inequality reads where the equality holds with the delicate matching condition (19), and hence the RHS of ( 21) as the algebraic quantum bound is achievable in physical realization.

K-locality networks
Here we delve into the K-locality scenario in the star configuration as a straightforward generalization of the bilocality scenario as follows.There are K independent sources e 1 , ..., e K and K + 1 observers Alice 1 , ..., Alice K , and Bob.The source e k emits particles 2k − 1 and 2k sent to Bob and Alice k , respectively.In the measurement processing, Alice k randomly chooses the classical bit x k ∈ {0, 1} as the measurement setting and then measures the observable A (2k) x k on the particle 2k; Bob randomly chooses the classical bits y ∈ {0, 1} and then measures the joint observable B y on the received K particles.In the classical version of the K-locality networks, each source emits two classical particles as information carriers of local hidden variable λ k ; in the quantum version, the quantum source e k emits two entangled qubits in the pure state |ψ The Bell operator associated with the particles from the source e k is denoted by 1 ), and In the K-source Bell test scenario, all classical systems exist in the hidden state x k on the classical particle 2k by a (2k) x k ∈ {−1, 1}, which is determined by λ k ; and denote the outcome of Bob's observable B y is by b y ∈ {−1, 1}, which is determined by Λ K .The joint conditional probability distribution of the measurement outcomes is given by where the a K = (a xK ) and x K = (x 1 , ..., x K ).The local expectations are denoted as and New J. Phys.26 (2024) 033026 where m = 2, 3, i 1 = 1, i 2 = 4, y 2 = 0, and y 3 = 1, and ⟨∥B k1 ∥⟩ LHV .As a result, the Bell inequality in the star-network configuration reads Here the classical bound β + 2 is achievable by setting all the measurement outcomes to be 1 for any hidden state Λ K .
In the quantum region, the product state ⟩ is prepared and distributed in the star networks, letting ⟨B km ⟩ Q .The Tsirelson inequality in the K-locality networks reads where the first and second inequalities are due to ( 14) and ( 9), respectively.As a result, the quantum bound 8 + 2β 2 are achievable by letting the state emitted from e k be |ϕ ⟩, and assigning the measurement σ z → 1  2 cos µ (A . Bob's joint observables are assigned as B 0 = σ ⊗k z and B 1 = σ ⊗k x .In this case, the optimal value 8 + 2β 2 in ( 26) is operationally achievable in the quantum region.
It is worth noting that, in the case β = 0, the Bell inequalities ( 10) and ( 25) are reduced to the bilocal and K-local Bell inequalities [20,24,28,31].In details, given β = 0 and the product state ⟩ distributed in the star-shaped quantum networks, we have It is easy verified that the upper bound can be obtained by setting µ = θ = π 4 [20,24,28,31].On the other hand, if θ ̸ = π 4 , the Bell inequality ( 25) can be violated by choosing the suitable parameter µ such that 2 cos µ + 2 sin µ sin 2θ > 2. In the following, we consider the asymmetric K-locality networks as follows.Without loss of generality, let the quantum source e k ′ emit two-qubit Bell state i, where 1 ⩽ k ′ ⩽ K ′ ; the quantum source e k ′ ′ emit the partially entangled state |ψ ), and ).Using the cut-graft-mixing method, denote the segmented Bell operators by In details, we have and . By replacing the segmented operator B k ′ m in ( 25) with E k ′ m , we derive the K-local inequality in the classical K-locality asymmetrical networks that reads Notably, the equality of the first inequality in (27) does not hold because ⟨∥E k ′ m ∥⟩ LHV is not equal to ⟨∥B km ∥⟩ LHV , and hence the algebraic classical bound in the third line is operationally inaccessible.In quantum networks, the measurement settings of the first 2K ′ qubits are reassigned as follows.Assign the measurement setting of the qubit 2k ′ as σ 3 ), and σ z → , where 1 where the equality of the inequality holds if As a result, Tsirelson inequalities for the asymmetric K-locality quantum networks reads To achieve the RHS of (29) as quantum bound, the quantum source e k ′ and e k ′ ′ each emits the entangled states |ϕ In the end, it is worth noting that K-locality scenario can be expanded to encompass another network topology.Specifically, the qubits 1, 3, .., (2K − 1) are sent to different t observers Bob 1 , Bob 2 ,…, and Bob t instead of a single observer, Bob.If Bob j receives n

Conclusions
In conclusion, we explore the bi-nonlocality and K-nonlocality in asymmetric quantum networks with sources emitting variant entangled states.We derive bi-local and K-local nonlinear Bell inequalities, where the algebraic upper bounds in the RHS of ( 20) and ( 27) are loose, meaning that such upper bounds are not achievable using local hidden variables.On the other hand, our ultimate goal is to devise the nonlinear Bell with their algebraic maximal violations can be achieved experimentally, which indicates that the RHS in the equalities ( 29) and ( 21) are, mathematically speaking, tight and operationally achievable in the quantum region.Our findings shed light on the fundamental aspects of nonlocal correlations in star-networks scenarios and may find applications in various quantum information tasks such as quantum state certification.

⩾ 2 )
with the given input y, he measures the joint observable B y in the asymmetric networks.The Bell inequalities tailored for this network are the same as those tailored for the star-shaped network.
in lines.The measure space of the local hidden variable λ shared between Alice and Bob share is denoted by (Ω, Σ, µ), and µ(λ) is the measure of λ with the normalization condition ´Ω dµ(λ) = 1.According to the local causal relations, Alice and Bob measure the local observables A x and B y with the outcomes denoted by be a x and b y , respectively, where x, y ∈ {0, 1} and a x , b y ∈ {1, −1}.{In the following, the tilted CHSH-Bell operator in (1) is divided into three} segmented Bell operators denoted by add the roots of |⟨I 1 ⟩|, |⟨I 2 ⟩|, and |⟨I 3 ⟩|, which finally leads the bilocal Bell inequality and we have Bell }, where H denotes the 2 × 2 Hadamard matrix.Let the post-selection state of the Bell-state measurement be |ϕ ′ Bell ⟩.The outcome is set as ⟨ϕ and B ′ 2 = σ x ⊗ σ z that are not pairwise compatible.In this case, if Bob is to measure either B ′ 1 or B ′ 2 , he can perform Bell-state measurement using the basis S ′ Bell = {(H ⊗ I) |ϕ⟩ | ∀ |ϕ⟩ ∈ S