Exploring Bell nonlocality of quantum networks with stabilizing and logical operators

In practical quantum networks, a variety of multi-qubit stabilized states emitted from independent sources are distributed among the agents, and the correlations across the entire network can be derived from each agent's local measurements on the shared composite quantum systems. To reveal the Bell non-locality in such cases as a quantum feature, minimal knowledge of the emitted stabilizer state is required. Here, we demonstrate that knowing the stabilizing and logical operators indeed provides a new way of exploring Bell non-locality in quantum networks. For the qubit distribution in quantum networks, the associated nonlinear Bell inequalities are derived. On the other hand, to violate these inequalities, one can design local incompatible observables using minimal knowledge of the emitted states. The tilted nonlinear Bell inequalities tailored for specific non-maximal entangled stabilizer states and a way of achieving the maximal violation are also explored.


I. INTRODUCTION
In the seminal work of John Bell, it was shown that the quantum correlations arising from spatially separated systems can break the limits of classical causal relations [1,2].
In classical physics, the realism and locality of spacelike events constrain the strength of classical correlations bounded by the Bell inequalities. Quantum theory inconsistent with local realism predicts stronger correlations that violate the Bell inequalities. Thanks to quantum information science, two-particle and multi-particle quantum correlations have been extensively investigated. As a distinct feature from classical physics, Bell nonlocality has led to applications in quantum information processing including private random number generation [3], quantum cryptography [4], and reductions in communication complexity [5].
In typical Bell experiments on statistical correlations, a source emits a state comprising two or more particles that are shared between two or more distant observers, who each perform local measurements with a random chosen setting and then obtain the measurement outcomes. To reveal the Bell nonlocality of an entangled state, the local observables should be set deliberately. For example, to violate the Bell inequalities tailored to stabilizer states such as graph states, it is helpful to take the stabilizers as a reference for finding suitable measurement settings [6][7][8]. To emphasize the role of stabilizing operators and logical bit-flip operators, we review the Clauser-Horne-Shimony-Holt (CHSH) inequality and its modification below. Let the two-qubit entangled state (0 < φ < π 2 ) |φ = cos φ 0 + sin φ 1 (1) be the codeword of the [ [2,1,2]] stabilizer-based quantum error-detecting code with the logical states 0 = |00 and 1 = |11 . The stabilizer generator, logical bit-flip, and phase-flip operators are σ z ⊗ σ z , σ x ⊗ σ x , and I ⊗ σ z , respectively (σ x , σ y , and σ z are Pauli matrices and I is the identity operator). In the bipartite Bell test, the CHSH operator where A i and B j are local observables, and the CHSH inequality states that B CHSH c ≤ 2, where the · denotes the expectation value of ·. To violate the CHSH inequality, for the first-qubit, we assign and for the second-qubit, we assign It is easy to verify that φ |B CHSH | φ = 2 cos µ φ |σ z ⊗ σ z | φ + 2 sin µ φ |σ x ⊗ σ x | φ = 2(cos µ + sin µ sin 2φ) = 2 1 + sin 2 2φ cos(µ − µ 0 ) where tan µ 0 = sin 2φ. Some remarks are in order. First, the operators (  [7][8][9][10][11][12]. If the multi-qubits mixed states involve two stabilizing operators, their nonlocality can be verified by the violation of stabilizer-based Bell-type inequalities [13,14]. Second, observables A 0 and A 1 can be regarded as the results of "cutting and mixing" the stabilizing operator and the logical bit-flip operator into the local observables: Cutting means cutting the first qubit observables from σ z ⊗ σ z and σ x ⊗ σ x ; mixing means linearly superposing the two cut observables. In what follows, the local observables on the source side will be constructed in a similar way. In addition, note that the local observables anticommute; i.e., {A 0 , A 1 } = {B 0 , B 1 } = 0. Third, the observable B 0 is the phase-flip operator, which can be exploited in the tilted CHSH operators B β-CHSH = βB 0 + B CHSH with the Bell inequality B β-CHSH c ≤ 2 + β. By setting β = 2/ 1 + 2 tan 2 2φ, the maximal violation can be achieved by |φ [15,16]. That is, the nonmaximally entangled state can maximally violate the tilted Bell inequalities involving the phase-flip operators. One can generalize the above approach to the multi-qubit case. For example, let |ψ = cos ψ 0 +sin ψ 1 be the codeword of the [ [5,1,3]] stabilizer-based quantum error-correcting code (SQECC), and let Alice hold the first qubit and Bob hold the other qubits. In this case, the useful stabilizing operator and logical bit-flip operator are g = σ z ⊗ σ z ⊗ σ x ⊗ I ⊗ σ x and σ ⊗5 x , respectively. The observables of the first and second qubits of the stabilizing operators and the bit-flip operator are the same as those in (2) and (3), respectively. The last three qubits are termed idle qubits since the observable σ x is always measured on them, while the outcome of the fourth qubit is discarded if Bob measures B ′ 1 . In the Bell test, Alice randomly measures one of the observables in (2), whereas Bob randomly measures The CHSH-like inequality can be written as  In addition, each observer can perform a joint measurement on the qubits at hand, which could result in strong correlations across the network. Unlike the typical linear Bell-type inequalities, most Bell-type inequalities for various classes of networks are nonlinear, which implies the nonconvexity of the multipartite correlation space [17][18][19][20]. In the two-source case as the simplest quantum network, bilocal and nonlocal correlations were thoroughly investigated [21,22]. Next the Bell-type inequalities for star-shaped and noncyclic networks were studied [17,18]. Recently, broader classes of quantum networks based on locally causal structures have also been investigated [19,20,23,24]. In particular, computationally efficient algorithms for constructing Bell inequalities have been proposed [20]. These Bell-type inequalities are tailored for networks with quantum sources emitting either two-qubit Bell states or generalized GHZ states. The stabilizing operators and logical operators implicitly play substantial roles in setting up local joint observables. On the other hand, regarding the potential applications of encrypted communication in quantum networks, stabilizer quantum error-correcting codes (SQECCs) can be more useful in quantum network cryptography, such as in quantum secret sharing [25] and secure quantum key distribution protocols [26].
Revealing Bell nonlocality in a network is required for detecting potential eavesdropping attacks. Moreover, a structured quantum state with stabilizing operators and logical operators is more useful in the engineering of quantum networks [27][28][29][30][31].
In this work, we study the Bell nonlocality of quantum networks. Hereafter, we consider a K-locality network N as shown in Fig i = n i . In the classical networks, source i emits n i particles described by hidden state λ i ; in the quantum networks, it emits n i qubits in quantum state ψ (i) , which can be either a stabilizer state or a codeword of a SQECC. In the Bell test, observer In the following, the index pair (i, j) denotes the j-th particle emitted from source i, and (i, The remainder of this paper is organized as follows: In Sec. II, we investigate the classical networks, where general local causal models (GLCMs) are introduced. Then, Bell inequalities associated with classical networks are proposed. We study Bell nonlocality of K-locality quantum networks in Sec. III. First we review the stabilizer states and SQECCs. Then we demonstrate how to violate the proposed Bell inequalities using deliberated local observables. It will be shown that, the local observables can be made up of "cut-graft-mixing" stabilizing operators and logical operators. Notably, there are two alternative nonlocal correlations. One is due to the entanglement of the logical states themselves, where two suitable stabilizing operators are exploited to construct the local observables. The other nonlocality results from the entanglement due to the superposition of logical states, in which a stabilizing operator and a logical operator can be suitably exploited in this case. We illustrate these two kinds of nonlocality in terms of the 5-qubit code, which is the smallest stabilizer code that protects against single-qubit errors. In Sec. IV, we propose Bell inequalities tailored for nonmaximally entangled states distributed in a quantum network. Finally, conclusions are drawn in Sec. V.

A. General local causal models
The general local causal models (GLCMs) in classical networks can be described as follows: The i-th source is associated with a local random variable as the hidden state λ i in the measure space (Ω i , Σ i , µ i ) with the normalization condition Ω i dµ i (λ i ) = 1. All systems in the Bell test scenario are considered in the hidden state Λ = (λ i , · · · , λ N ) in the measure space (Ω, Σ, µ), where Ω = Ω 1 ⊗ · · · ⊗ Ω N and the measure of Λ is given by where a = (a 1 , a 2 , · · · , a K ) and x = (x 1 , x 2 , · · · , x K ); hence, we have the correlation On the other hand, in the K-locality condition [20], S (s) can access the hidden state and Eq. (6) can be rewritten as Denote the local expectation as By K-locality with the GLCM, we have and On the receiving side, let n (m) j > 0 if j ∈ {j 1 , · · · , j km } and n (m) j = 0 otherwise. In this case, R (j) receives the hidden states Λ In the measurement phase, R (m) performs the measurement B

B. Bell inequalities
The correlation strength in the proposed K-locality network is evaluated by two quantities: and In the classical scenario, we have and where the inequalities are from (15). Before proceeding further, two useful lemmas are introduced as follows: Lemma 1 (Mahler inequality) Let α k and β k be nonnegative real numbers, and let p ∈ N; The proof can be found in [17].
We obtain the following nonlinear Bell inequality: where the first inequality is from Lemma 1, and the fourth line is a consequence of (14).

III. BELL NONLOCALITY OF A QUANTUM NETWORK
A. Review of stabilizer states and stabilizer-based quantum error-correcting code Let the state ψ (i) emitted from the quantum source i be an n i -qubit stabilizer state.
By definition, an n i -qubit stabilizer state is one that is stabilized by a stabilizer which is a nontrivial subgroup of the Pauli group. In particular, if ψ (i) as a codeword of SQECC, denote the last (k i -th) logical qubit as 0 i and 1 i and the corresponding logical bit-and phase-flip operators as X (i) and Z (i) . Without loss of generality, we have where ϕ 0 i |ϕ 1 i ∈ R, and ϕ 0 i |ϕ 0 i = ϕ 1 i |ϕ 1 i = 1. In addition, In what follows, we exploit g (i) and g ′ (i) as useful stabilizing operators, and X (i) and Z (i) as useful logical operators. Denote the Pauli set of qubit (i, j) as P (i, j) = {X (i, j) , Y (i, j) , Z (i, j) , and since g (i) , h (i) , and h ′ (i) are n i -fold tensor products of the Pauli operators, we have where s (i, j) , t (i, j) , and t ′ (i, j) ∈ P (i, j) ∀i, j. Before proceeding further, some notations are introduced as follows: Denote the qubit index sets as and |D i | + |H i | = n i . The qubits belonging to the sets D 1 , · · · , D N play substantial roles in the proposed Bell inequalities of the quantum networks. Note that qubit (i, j) is called idle if (i, j) ∈ H i and (i, j) → R (k) for some k, and the nonidentity operator of s (i, j) or t (i, j) is denoted by o (i, j) , which will be repeatedly measured on qubit (i, j) in the Bell test. Let Here, we focus on quantum networks fulfilling the following conditions: Regarding the qubits held by S (k) , and regarding the qubits held by R (k) , Combining the constraints (27), (28), and (29), the value K + M must be even. According to the qubit distribution, it is very flexible to choose suitable g (i) , h (i) , and h ′ (i) to implement local observables.

B. Violation of the Bell inequalities in a quantum network
To implement the local observables on the source side, we assign and Since A (k) x k is a unitary Hermitian with an eigenvalue of either 1 or −1. On the receiving side, the local observables B (l) y l for observer R (k) are where B = 0 according to (29). In the measuring phase, S (k) (R (k) ) randomly measures either the observable A To evaluate the correlation strength, let Ψ = N i ⊗ ψ (i) , and we have where · = Ψ |·| Ψ and hence N i=1 g (i) = 1. In addition, A useful lemma is introduced as follows: where θ = 1 K K j=1 θ j . The proof can be found in [20].
where the first inequality in (36) follows from Lemma 2 and the second inequality follows from the Cauchy-Schwarz inequality. To reach the maximal violation, by setting θ 1 = . . . = θ K = θ, and tan θ = C, we obtain Some discussion is in order. The state ψ (i) can contribute to the nonlocality of the quantum network through different facets of its nonlocality. Loosely speaking, there are ways of choosing suitable g (i) and g ′ (i) and k i (2 2(n i −k i ) ) ways of choosing suitable g (i) and X (i) , which reflect the flexibility of testing Bell nonlocality in the quantum networks.
The maximum value of C = 1 (θ = π 4 ), and hence max(|I K,M | As shown in (30) and (31), one can benefit from the stabilizing operators in designing local observables to achieve the maximal violations of Bell inequalities in K-locality quantum networks. In detail, The Bell nonlocality here is due to the superposition of 0 i and 1 i ; if h (i) = g ′ (i) , c i is certain to be 1. The Bell nonlocality in a network is due to that of either the stabilizer state or the logical states of the stabilizer code [[n i , k i , d i ]]. Note that such nonlocality involving the stabilizing operators g (i) and g ′ (i) can be also obtained even by using specific states stabilized by the same stabilizing operators. For example, given source i emitting a four-qubit state with stabilizing operators g (i) = σ ⊗4 z and g ′ (i) = σ ⊗4 x , which both can also stabilize the 4-qubit Eventually, Bell nonlocality of at most |D i |-qubit entanglement in ψ (i) is involved.
As an illustration, let K = N = 2 and M = 1, and let sources 1 and 2 each emit the codeword states of [ [5,1,3]] SQECC with four stabilizer generators (the subscript qubit index (i, j) is shortened as j) The well-known logical bit-flip and phase-flip operators are X ′ = 5 i=1 X i and Z ′ = 5 i=1 Z i . Observers S (1) and S (2) hold qubits (1, 1) and (2, 1), respectively, while observer R (k) holds the other 8 qubits, as shown in Fig. (2). Note that, in the following examples, the bilocal inequality is which is exactly the bilocal inequality for binary inputs and outputs in [21].
Consequently, although the two 5-qubit two logical states 0 and 1 and the codeword states are genuinely entangled [32,33], one can replace the genuinely entangled state with a state involving two-qubit entanglement, either |ϕ or |ϕ ′ , to reach the same correlation strength. It is not known whether the Bell nonlocality of genuine entanglement can be revealed in a K-locality quantum network. Specifically, it is not known whether the genuine entanglement of ψ (i) can be deduced from violations of variant Bell inequalities of a quantum network involving different stabilizing operators.

STATES IN A QUANTUM NETWORK
If c i < 1 for at least one i, Ψ cannot achieve the maximal violation of the nonlinear Bell inequality (21). To explore the Bell inequalities maximally violated by Ψ , recall that in the two-qubit case (N = K = M = 1, n 1 = 2), the tilted CHSH operator B β-CHSH = βB 0 + B CHSH is exploited using the logical phase-flip operator B 0 with appropriate parameter β. Without loss of generality, let i ∈ C for 1 ≤ i ≤ |C| and i ∈ c for |C| + 1 ≤ i ≤ |C| + |c| = N.

Although it is unlikely that
that fulfills the following conditions: (i) if (i ′ , j ′ ) → S (k) , then t ′ (i ′ , j ′ ) = I; and (ii) if (i ′ , j ′ ) → R (k) , then either qubit is not idle. The logical phase-flip operator can be revised as where The outcome of B   (40) is the unit matrix. In this case, we propose the tilted Bell inequalities tailored for where P N, . However, it is very difficult to exploit sum-ofsquares decomposition to find the tailored Bell operators in quantum networks with an extremely large Hilbert space [16,34]. Our strategy is to simplify G β N,K as tilted CHSH inequalities. In detail, according to Lemma 2, we have where equality holds when θ 1 = . . . = θ N = θ. Let the parameter β satisfying and As a result, the state Ψ C ⊗ Ψ c can maximally violate the tailored Bell inequality G βmax N,K = c G N,K (β max , φ, θ max ) ≤ β max + 1 by setting θ 1 = . . . = θ N = θ max . Note that G βmax N,K coincides with the tilted CHSH inequality in the case that |c| K = 1 [16].