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Verification of quantum networks using the GHZ paradox

  • Huan Ye 1 ,
  • Xue Yang , 1, 2, ,
  • Ming-Xing Luo , 1,
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  • 1School of Information Science and Technology, Southwest Jiaotong University, Chengdu 610031, China
  • 2School of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu 610059, China

Authors to whom any correspondence should be addressed.

Received date: 2024-04-03

  Revised date: 2024-07-03

  Accepted date: 2024-07-05

  Online published: 2024-08-16

Copyright

© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

The Greenberger–Horne–Zeilinger (GHZ) paradox shows that it is possible to create a multipartite state involving three or more particles in which the measurement outcomes of the particles are correlated in a way that cannot be explained by classical physics. We extend it to witness quantum networks. We first extend the GHZ paradox to simultaneously verify the GHZ state and Einstein–Podolsky–Rosen states on triangle networks. We then extend the GHZ paradox to witness the entanglement of chain networks consisting of multiple GHZ states. All the present results are robust against the noise.

Cite this article

Huan Ye , Xue Yang , Ming-Xing Luo . Verification of quantum networks using the GHZ paradox[J]. Communications in Theoretical Physics, 2024 , 76(10) : 105102 . DOI: 10.1088/1572-9494/ad5f83

1. Introduction

The concept of entanglement plays a crucial role in modern physics, challenging classical notions and providing a deeper understanding of the interconnected nature of quantum systems. Introduced by Einstein, Podolsky and Rosen (EPR) [1], entanglement showcases strong correlations between particles, regardless of the distance between them. This highlights the non-local and non-classical nature of quantum interactions [24]. Despite initial skepticism, entanglement has been experimentally verified and forms the basis for various applications in quantum information science, such as quantum computing and quantum cryptography [5, 6]. Its existence underscores the limitations of classical physics and emphasizes the need for quantum mechanics to describe the behavior of particles at the microscopic level accurately [7].
One important method used to verify the shared bipartite entanglement involves violating the so-called Bell inequality [812]. The verified nonlocality in physics is a key feature of quantum mechanics, which showcases strong correlations between particles that defy classical explanations. The study of nonlocality has profound implications for our understanding of the interconnected nature of the quantum world and has led to debates and investigations into the fundamental principles of reality. This has been further expended to verify multipartite entanglement [1319]. One example is demonstrated by the Greenberger–Horne–Zeilinger (GHZ) theorem [2022], which presents a new perspective by proving the incompatibility between quantum properties and physical realism in a non-inequality form. This approach, along with other non-inequality studies like the Cabello [23, 24] and Hardy theorems [2527], delves into the nature of quantum nonlocality and highlights the intricate relationship between quantum phenomena and classical theories. By categorizing quantum nonlocality into different strengths, such as quantum steering [2830], Bell nonlocality [31] and quantum entanglement [5, 6], researchers continue to explore the profound implications of quantum mechanics on our understanding of the Universe.
The main goal of this paper is to extend the GHZ paradox to witness quantum chain networks consisting of GHZ states [20] and EPR states [1]. In the second section, we introduce the main idea of the GHZ paradox. In the third section, we first extend the GHZ paradox to simultaneously verify three GHZ states. And then, we verify the triangle network consisting of one GHZ state and three EPR states. In the last section, we extend the GHZ paradox to verify chain networks consisting of multiple GHZ states. All the present results are robust against the noise and can be used to feature the experimental quantum networks.

2. GHZ paradox

The GHZ paradox states that for a three-particle GHZ state with a set of observables that commute with each other, measuring these observables will result in outcomes that are fundamentally different between quantum theory and classical local realism. The GHZ paradox [20] and its implications offer a more profound revelation of the essence of nonlocality, showcasing the conflict between the two theories in a more straightforward manner.
Assuming all three-particle spins are in the state 1/2, and the system is in the following GHZ state:
$\begin{eqnarray}| \phi {\rangle }_{{abc}}=\displaystyle \frac{1}{\sqrt{2}}(| 000\rangle +| 111\rangle ).\end{eqnarray}$
By using a Pauli matrix we obtain the common eigenstates of the following four quantities, and their corresponding eigenvalues and eigen-equations are as follows:
$\begin{eqnarray}{\sigma }_{x}^{a}{\sigma }_{y}^{b}{\sigma }_{y}^{c}| \phi \rangle =-| \phi \rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{y}^{a}{\sigma }_{x}^{b}{\sigma }_{y}^{c}| \phi \rangle =-| \phi \rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{y}^{a}{\sigma }_{y}^{b}{\sigma }_{x}^{c}| \phi \rangle =-| \phi \rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{x}^{a}{\sigma }_{x}^{b}{\sigma }_{x}^{c}| \phi \rangle =| \phi \rangle ,\end{eqnarray}$
where σx and σy are Pauli matrices.
In local realism, each operator represents an observable quantity that is an objectively existing value, i.e. ${m}_{x(y,z)}^{a(b,x)}\in \{\pm 1\}$. Therefore, the above equation can be written as:
$\begin{eqnarray}{m}_{x}^{a}{m}_{y}^{b}{m}_{y}^{c}=-1,\end{eqnarray}$
$\begin{eqnarray}{m}_{y}^{a}{m}_{x}^{b}{m}_{y}^{c}=-1,\end{eqnarray}$
$\begin{eqnarray}{m}_{y}^{a}{m}_{y}^{b}{m}_{x}^{c}=-1,\end{eqnarray}$
$\begin{eqnarray}{m}_{x}^{a}{m}_{x}^{b}{m}_{x}^{c}=1.\end{eqnarray}$
Here, the first three equations can be clearly seen to be contradictory when multiplied together with the fourth equation, indicating that quantum theory cannot be explained by local realism. This means that, from the local measurements on the correlated three-particle spins, the GHZ paradox demonstrates the contradiction between quantum theory and classical theory [20].

3. GHZ paradox on triangle network

In this section we extend the GHZ paradox to simultaneously verify the multiple entangled states by using triangle networks. The nonlocality in quantum networks has been recently discussed for verification of the quantum features of entanglement [3237] and can be applied to quantum communication [38, 39] and distributed quantum computation [40]. Most of the proposed results are dependent on the nonlinear Bell inequalities. There are few similar results from the GHZ paradox [41]. We resolve this problem by using the generalized GHZ paradox.

3.1. Three GHZ states

Consider a triangle quantum network consisting of three GHZ states, as shown in figure 1. Here, each party of a, b or c share three particles of GHZ states. This can then be regarded as a high-dimensional entanglement, where the local dimension is 8 for each party. Previous methods have mainly been used to show the nonlocality of bipartite [6, 42]. Our goal here is to verify the nonlocality of these quantum networks by extending the GHZ paradox. Although the present network will make use of the source independently, it is beyond the previous network model [43].
Figure 1. The GHZ paradox with three nodes. It consists of three GHZ states shared by three parties.
Let all the GHZ states be $| \phi {\rangle }_{{abc}}=\tfrac{1}{\sqrt{2}}(| 000\rangle +| 111\rangle )$. The total state of the triangle network is given by
$\begin{eqnarray}| {\rm{\Phi }}\rangle =| {\phi }^{(1)}\rangle | {\phi }^{(2)}\rangle | {\phi }^{(3)}\rangle .\end{eqnarray}$
Inspired by the GHZ paradox [20], i.e. from equations (2)–(5) we can obtain that
$\begin{eqnarray}\begin{array}{l}{\otimes }_{i=1}^{3}{\sigma }_{{x}_{a}}^{(i)}{\sigma }_{{y}_{b}}^{(i)}{\sigma }_{{y}_{c}}^{(i)}| {\rm{\Phi }}\rangle \\ \quad =\displaystyle \frac{1}{\sqrt{2}}({\sigma }_{{x}_{a}}^{(1)}| 0\rangle {\sigma }_{{y}_{b}}^{(1)}| 0\rangle {\sigma }_{{y}_{c}}^{(1)}| 0\rangle +{\sigma }_{{x}_{a}}^{(1)}| 1\rangle {\sigma }_{{y}_{b}}^{(1)}| 1\rangle {\sigma }_{{y}_{c}}^{(1)}| 1\rangle )\\ \quad \otimes \displaystyle \frac{1}{\sqrt{2}}({\sigma }_{{x}_{a}}^{(2)}| 0\rangle {\sigma }_{{y}_{b}}^{(2)}| 0\rangle {\sigma }_{{y}_{c}}^{(2)}| 0\rangle +{\sigma }_{{x}_{a}}^{(2)}| 1\rangle {\sigma }_{{y}_{b}}^{(2)}| 1\rangle {\sigma }_{{y}_{c}}^{(2)}| 1\rangle )\\ \quad \otimes \displaystyle \frac{1}{\sqrt{2}}({\sigma }_{{x}_{a}}^{(3)}| 0\rangle {\sigma }_{{y}_{b}}^{(3)}| 0\rangle {\sigma }_{{y}_{c}}^{(3)}| 0\rangle +{\sigma }_{{x}_{a}}^{(3)}| 1\rangle {\sigma }_{{y}_{b}}^{(3)}| 1\rangle {\sigma }_{{y}_{c}}^{(3)}| 1\rangle )\\ \quad =-| {\rm{\Phi }}\rangle ,\end{array}\end{eqnarray}$
where ${\sigma }_{{u}_{v}}$ denotes the Pauli matrix σu (u ∈ {x, y, z}) being performed on the particle v ∈ {a, b, c}. This implies that the total state ∣Φ⟩ in equation (10) is an eigenstate of the operator ${\otimes }_{i=1}^{3}{\sigma }_{{x}_{a}}^{(i)}{\sigma }_{{y}_{b}}^{(i)}{\sigma }_{{y}_{c}}^{(i)}$ with the eigenvalue −1.
Similarly, we obtain the following eigenvalue equations for the total state as:
$\begin{eqnarray}{\sigma }_{{x}_{a}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{i}_{2}}^{(3)}{\sigma }_{{y}_{b}}^{(1)}{\sigma }_{{j}_{1}}^{(2)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{y}_{c}}^{(1)}{\sigma }_{{k}_{1}}^{(2)}{\sigma }_{{k}_{2}}^{(3)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{i}_{2}}^{(3)}{\sigma }_{{x}_{b}}^{(1)}{\sigma }_{{j}_{1}}^{(2)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{y}_{c}}^{(1)}{\sigma }_{{k}_{1}}^{(2)}{\sigma }_{{k}_{2}}^{(3)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{i}_{2}}^{(3)}{\sigma }_{{y}_{b}}^{(1)}{\sigma }_{{j}_{1}}^{(2)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{x}_{c}}^{(1)}{\sigma }_{{k}_{1}}^{(2)}{\sigma }_{{k}_{2}}^{(3)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{x}_{a}}^{(1)}{\sigma }_{{x}_{a}}^{(2)}{\sigma }_{{x}_{a}}^{(3)}{\sigma }_{{x}_{b}}^{(1)}{\sigma }_{{x}_{b}}^{(2)}{\sigma }_{{x}_{b}}^{(3)}{\sigma }_{{x}_{c}}^{(1)}{\sigma }_{{x}_{c}}^{(2)}{\sigma }_{{x}_{c}}^{(3)}| {\rm{\Phi }}\rangle =| {\rm{\Phi }}\rangle ,\end{eqnarray}$
where (i1, j1, k1), (i2, j2, k2) ∈ {(xa, yb, yc), (ya, xb, yc), (ya, yb, xc)}.
In the realm of local realism, physical quantities are objectively present. This means in the classical domain, each operator representing an observable quantity is a physical actual element with an objectively existing value. From equation (9) we obtain the classical equivalent form as
$\begin{eqnarray}{m}_{{x}_{a}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{i}_{2}}^{(3)}{m}_{{y}_{b}}^{(1)}{m}_{{j}_{1}}^{(2)}{m}_{{j}_{2}}^{(3)}{m}_{{y}_{c}}^{(1)}{m}_{{k}_{1}}^{(2)}{m}_{{k}_{2}}^{(3)}=-1,\end{eqnarray}$
$\begin{eqnarray}{m}_{{y}_{a}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{i}_{2}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{j}_{1}}^{(2)}{m}_{{j}_{2}}^{(3)}{m}_{{y}_{c}}^{(1)}{m}_{{k}_{1}}^{(2)}{m}_{{k}_{2}}^{(3)}=-1,\end{eqnarray}$
$\begin{eqnarray}{m}_{{y}_{a}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{i}_{2}}^{(3)}{m}_{{y}_{b}}^{(1)}{m}_{{j}_{1}}^{(2)}{m}_{{j}_{2}}^{(3)}{m}_{{x}_{c}}^{(1)}{m}_{{k}_{1}}^{(2)}{m}_{{k}_{2}}^{(3)}=-1,\end{eqnarray}$
$\begin{eqnarray}{m}_{{x}_{a}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{c}}^{(2)}{m}_{{x}_{c}}^{(3)}=1,\end{eqnarray}$
where ${m}_{{u}_{v}}^{(i)}$ denotes the classical values ±1 of the measurement being performed on the particle v of the GHZ state with respect to the quantum Pauli matrix σu. Now, multiplying all the equations implies that
$\begin{eqnarray*}\begin{array}{l}{\left[{m}_{{y}_{a}}^{(1)}{m}_{{y}_{a}}^{(2)}{m}_{{y}_{a}}^{(3)}{m}_{{y}_{b}}^{(1)}{m}_{{y}_{b}}^{(2)}{m}_{{y}_{b}}^{(3)}{m}_{{y}_{c}}^{(1)}{m}_{{y}_{c}}^{(2)}{m}_{{y}_{c}}^{(3)}\right]}^{18}\\ \quad \times {\left[{m}_{{x}_{a}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{c}}^{(2)}{m}_{{x}_{c}}^{(3)}\right]}^{10}=-1.\end{array}\end{eqnarray*}$
This leads to a contradiction as the left side is positive and the right side is negative. This means that the fact derived from local realism cannot be simultaneously satisfied. This highlights the contradictory system between local realism and quantum theory.
Considering experimental verification based on this contradiction, we extend the inequalities to show the nonlocality of noise states. In particular, define the following operator
$\begin{eqnarray*}\begin{array}{rcl}{\rm{O}} & = & -{x}_{1}^{{\prime} }{x}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{x}_{1}^{{\prime} }{x}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{x}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{x}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{x}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl} & & -{x}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{x}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{y}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{y}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{x}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{x}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl} & & -{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{rcl} & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{2}^{{\prime} }{y}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{2}^{{\prime\prime} }{y}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime }\\ & & +{x}_{1}^{{\prime} }{x}_{2}^{{\prime} }{x}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{2}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{\prime\prime\prime }{x}_{3}^{\prime\prime\prime },\end{array}\end{eqnarray}$
where xi, zj, yk are measurement operators on the i, j, kth particles, respectively. It can be easily checked that the total state ∣Φ⟩ is an eigenstate of the operator. This means O∣Φ⟩ = 28∣Φ⟩.
In the realm of local realism, these elements can only take values of +1 or −1, i.e. ∣O∣ ≤ 26. This fact can be used to witness the noise states. In fact, suppose we have a Werner state [44] ${\rho }_{w}=w| \phi \rangle \langle \phi | +\tfrac{1-w}{8}{\rm{I}}$, where w ∈ [0, 1], ∣φ⟩ denotes the GHZ state and I denotes the identity operator. It follows that
$\begin{eqnarray}\mathrm{Tr}({\rm{O}}{\rho }_{w}{\rho }_{w}{\rho }_{w})=28{w}^{3}\geqslant 26,\end{eqnarray}$
if $w\gt \sqrt[3]{26/28}\approx 0.9756$. This means the present extended GHZ paradox can witness the noisy GHZ state with noise visibility w = 0.9756. The present result can be easily extended for a triangle network consisting of 2n − 1 GHZ states with any integer n ≥ 1.

3.2. GHZ and EPR states

In this section we extend the GHZ paradox to verify the GHZ state and EPR states simultaneously. Consider a triangle network consisting of one GHZ state and three EPR states, as shown in figure 2. Each party shares one GHZ state and two EPR states, i.e. three particles. The total network can be regarded as an 8 × 8 × 8-dimensional entangled state. Our goal is to verify the entanglement of this network.
Figure 2. The GHZ paradox with three nodes. It consists of one GHZ state and three EPR states shared by three parties.
Let the GHZ state be $| \phi \rangle =\tfrac{1}{\sqrt{2}}(| 000\rangle +| 111\rangle )$, and the EPR state be $| \varphi \rangle =\tfrac{1}{\sqrt{2}}(| 01\rangle -| 10\rangle )$. The total state is given by
$\begin{eqnarray}| {\rm{\Phi }}\rangle =| {\phi }^{(1)}{\rangle }_{{abc}}| {\varphi }^{(2)}{\rangle }_{{ac}}| {\varphi }^{(3)}{\rangle }_{{ab}}| {\varphi }^{(4)}{\rangle }_{{bc}}.\end{eqnarray}$
According to quantum mechanics, we have the following properties: σzσzφ⟩ = − ∣φ⟩, σxσxφ⟩ = ∣φ⟩ for the Pauli matrices σz and σx. Combining with equations (2)–(5), we obtain the following eigenvalue equations as:
$\begin{eqnarray}{\sigma }_{{x}_{a}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{j}_{1}}^{(3)}{\sigma }_{{y}_{b}}^{(1)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{k}_{1}}^{(4)}{\sigma }_{{y}_{c}}^{(1)}{\sigma }_{{i}_{2}}^{(2)}{\sigma }_{{k}_{2}}^{(4)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{j}_{1}}^{(3)}{\sigma }_{{x}_{b}}^{(1)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{k}_{1}}^{(4)}{\sigma }_{{y}_{c}}^{(1)}{\sigma }_{{i}_{2}}^{(2)}{\sigma }_{{k}_{2}}^{(4)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{j}_{1}}^{(3)}{\sigma }_{{y}_{b}}^{(1)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{k}_{1}}^{(4)}{\sigma }_{{x}_{c}}^{(1)}{\sigma }_{{i}_{2}}^{(2)}{\sigma }_{{k}_{2}}^{(4)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{x}_{a}}^{(1)}{\sigma }_{{x}_{a}}^{(2)}{\sigma }_{{x}_{a}}^{(3)}{\sigma }_{{x}_{b}}^{(1)}{\sigma }_{{x}_{b}}^{(3)}{\sigma }_{{x}_{a}}^{(4)}{\sigma }_{{x}_{c}}^{(1)}{\sigma }_{{x}_{b}}^{(2)}{\sigma }_{{x}_{b}}^{(4)}| {\rm{\Phi }}\rangle =| {\rm{\Phi }}\rangle ,\end{eqnarray}$
where ((i1, i2), (j1, j2), (k1, k2)) ∈ {((za, zb), (za, zb), (xa, xb)), ((za, zb), (xa, xb), (za, zb)), ((xa, xb), (za, zb), (za, zb))}.
In the realm of local realism, each operator represents an observable quantity. This implies from equation (9) that
$\begin{eqnarray}\begin{array}{l}{m}_{{x}_{a}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{j}_{1}}^{(3)}{m}_{{y}_{b}}^{(1)}{m}_{{j}_{2}}^{(3)}{m}_{{k}_{1}}^{(4)}{m}_{{y}_{c}}^{(1)}{m}_{{i}_{2}}^{(2)}{m}_{{k}_{2}}^{(4)}=-1,\\ {m}_{{y}_{a}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{j}_{1}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{j}_{2}}^{(3)}{m}_{{k}_{1}}^{(4)}{m}_{{y}_{c}}^{(1)}{m}_{{i}_{2}}^{(2)}{m}_{{k}_{2}}^{(4)}=-1,\\ {m}_{{y}_{a}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{j}_{1}}^{(3)}{m}_{{y}_{b}}^{(1)}{m}_{{j}_{2}}^{(3)}{m}_{{k}_{1}}^{(4)}{m}_{{x}_{c}}^{(1)}{m}_{{i}_{2}}^{(2)}{m}_{{k}_{2}}^{(4)}=-1,\\ {m}_{{x}_{a}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{a}}^{(4)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{b}}^{(4)}=1,\end{array}\end{eqnarray}$
where $(({i}_{1},{i}_{2}),({j}_{1},{j}_{2}),({k}_{1},{k}_{2}))\in \{(({z}_{a},{z}_{b}),({z}_{a},{z}_{b}),({x}_{a},{x}_{b})),(({z}_{a},{z}_{b}),({x}_{a},{x}_{b}),({z}_{a},{z}_{b})),(({x}_{a},{x}_{b})$, $\left.\left.\left({z}_{a},{z}_{b}\right),\left({z}_{a},{z}_{b}\right)\right)\right\}$.
Multiplying the first three equalities of equation (27) implies that
$\begin{eqnarray}\begin{array}{l}{\left[{m}_{{y}_{a}}^{(1)}{m}_{{y}_{b}}^{(1)}{m}_{{y}_{c}}^{(1)}{m}_{{z}_{a}}^{(2)}{m}_{{z}_{b}}^{(2)}{m}_{{z}_{a}}^{(3)}{m}_{{z}_{b}}^{(3)}{m}_{{z}_{a}}^{(4)}{m}_{{z}_{b}}^{(4)}\right]}^{6}\\ \quad \times {\left[{m}_{{x}_{a}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{a}}^{(4)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{b}}^{(4)}\right]}^{4}=-1.\end{array}\end{eqnarray}$
This implies that
$\begin{eqnarray}{m}_{{x}_{a}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{a}}^{(4)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{b}}^{(4)}=1,\end{eqnarray}$
which contradicts the last equation of equation (27). This means the quantum features cannot be derived from local realism. This highlights the contradiction between local realism and quantum theory.
Considering experimental verification with noisy states, we extend the present inequalities to a total inequality. Define an operator as
$\begin{eqnarray}\begin{array}{rcl}{\rm{O}} & = & -{x}_{1}^{{\prime} }{z}_{2}^{{\prime} }{z}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{z}_{3}^{{\prime\prime} }{x}_{4}^{\prime} {y}_{1}^{\prime\prime\prime }{z}_{2}^{{\prime\prime} }{x}_{4}^{\prime\prime} \\ & & -{x}_{1}^{{\prime} }{z}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{z}_{4}^{\prime} {y}_{1}^{\prime\prime\prime }{z}_{2}^{{\prime\prime} }{z}_{4}^{\prime\prime} \\ & & -{x}_{1}^{{\prime} }{x}_{2}^{{\prime} }{z}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{z}_{3}^{{\prime\prime} }{z}_{4}^{\prime} {y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime\prime} }{z}_{4}^{\prime\prime} \\ & & -{y}_{1}^{{\prime} }{z}_{2}^{{\prime} }{z}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{z}_{3}^{{\prime\prime} }{x}_{4}^{\prime} {y}_{1}^{\prime\prime\prime }{z}_{2}^{{\prime\prime} }{x}_{4}^{\prime\prime} \\ & & -{y}_{1}^{{\prime} }{z}_{2}^{{\prime} }{x}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{z}_{4}^{\prime} {y}_{1}^{\prime\prime\prime }{z}_{2}^{{\prime\prime} }{z}_{4}^{\prime\prime} \\ & & -{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{z}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{z}_{3}^{{\prime\prime} }{z}_{4}^{\prime} {y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime\prime} }{z}_{4}^{\prime\prime} \\ & & -{y}_{1}^{{\prime} }{z}_{2}^{{\prime} }{z}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{z}_{3}^{{\prime\prime} }{x}_{4}^{\prime} {x}_{1}^{\prime\prime\prime }{z}_{2}^{{\prime\prime} }{x}_{4}^{\prime\prime} \\ & & -{y}_{1}^{{\prime} }{z}_{2}^{{\prime} }{x}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{z}_{4}^{\prime} {x}_{1}^{\prime\prime\prime }{z}_{2}^{{\prime\prime} }{z}_{4}^{\prime\prime} \\ & & -{y}_{1}^{{\prime} }{x}_{2}^{{\prime} }{z}_{3}^{{\prime} }{y}_{1}^{{\prime\prime} }{z}_{3}^{{\prime\prime} }{z}_{4}^{\prime} {x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime\prime} }{z}_{4}^{\prime\prime} \\ & & +{x}_{1}^{{\prime} }{x}_{2}^{{\prime} }{x}_{3}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{3}^{{\prime\prime} }{x}_{4}^{\prime} {x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime\prime} }{x}_{4}^{\prime\prime} ,\end{array}\end{eqnarray}$
where xi, zj, yk are measurement operators on the i, j, kth particles, respectively. It is easy to check the total state ∣Φ⟩ is an eigenstate of the operator, i.e. O∣Φ⟩ = 10∣Φ⟩.
In the realm of local realism, with the elements of +1 or −1, it follows that ∣O∣ ≤ 8. This can be applied to witness noisy entanglement. In particular, consider the Werner state of the EPR state as ${\varrho }_{v}=v| \varphi \rangle \langle \varphi | +\tfrac{1-v}{4}{{\rm{I}}}_{4}$, where v ∈ (0, 1] and I4 is the identity operator on the two-qubit state space. Meanwhile, consider the Werner state of the GHZ state as ${\rho }_{w}=w| \phi \rangle \langle \phi | +\tfrac{1-w}{8}{{\rm{I}}}_{8}$, where w ∈ (0, 1]. We obtain that
$\begin{eqnarray}\mathrm{Tr}({\rm{O}}{\rho }_{w}{\varrho }_{v}^{(2)}{\varrho }_{v}^{(3)}{\varrho }_{v}^{(4)})=10{{wv}}^{3}\gt 8,\end{eqnarray}$
if wv3 > 4/5. This implies that the present method can witness the noisy GHZ state and EPR states simultaneously. One special case is v = w > 0.9283. The present result can be easily extended for a triangle network consisting of multiple GHZ and EPR states.

4. GHZ paradox on chain networks

In this section we extend the GHZ paradox to witness the entanglement of chain networks consisting of GHZ states.
Building upon the original GHZ paradox, we show it can be extended to witness the entangled network, as shown in figure 3. There are seven parties who share three GHZ states. Parties 3 and 5 each have two particles while the others have one particle. The goal is to witness the total network.
Figure 3. The GHZ paradox model for the nine nodes. It consists of three GHZ states shared by nine parties.
Let the GHZ state be $| \phi \rangle =\tfrac{1}{\sqrt{2}}(| 000\rangle +| 111\rangle )$. The total state is given by
$\begin{eqnarray}| {\rm{\Phi }}\rangle =| {\phi }^{(1)}\rangle | {\phi }^{(2)}\rangle | {\phi }^{(3)}\rangle .\end{eqnarray}$
By combining with equations (2)–(5), we obtain the following eigenvalue equations as
$\begin{eqnarray}{\sigma }_{{x}_{a}}^{(1)}{\sigma }_{{y}_{b}}^{(1)}{\sigma }_{{y}_{c}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{j}_{1}}^{(2)}{\sigma }_{{k}_{1}}^{(2)}{\sigma }_{{i}_{2}}^{(3)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{k}_{2}}^{(3)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}{\sigma }_{{x}_{b}}^{(1)}{\sigma }_{{y}_{c}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{j}_{1}}^{(2)}{\sigma }_{{k}_{1}}^{(2)}{\sigma }_{{i}_{2}}^{(3)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{k}_{2}}^{(3)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}{\sigma }_{{y}_{b}}^{(1)}{\sigma }_{{x}_{c}}^{(1)}{\sigma }_{{i}_{1}}^{(2)}{\sigma }_{{j}_{1}}^{(2)}{\sigma }_{{k}_{1}}^{(2)}{\sigma }_{{i}_{2}}^{(3)}{\sigma }_{{j}_{2}}^{(3)}{\sigma }_{{k}_{2}}^{(3)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{x}_{a}}^{(1)}{\sigma }_{{x}_{b}}^{(1)}{\sigma }_{{x}_{c}}^{(1)}{\sigma }_{{x}_{a}}^{(2)}{\sigma }_{{x}_{b}}^{(2)}{\sigma }_{{x}_{c}}^{(2)}{\sigma }_{{x}_{a}}^{(3)}{\sigma }_{{x}_{b}}^{(3)}{\sigma }_{{x}_{c}}^{(3)}| {\rm{\Phi }}\rangle =| {\rm{\Phi }}\rangle ,\end{eqnarray}$
where (i1, j1, k1), (i2, j2, k2) ∈ {(xa, yb, yc), (ya, xb, yc), (ya, yb, xc)}.
In local realism, each operator represents an observable quantity with values in {±1}. This implies that
$\begin{eqnarray}\begin{array}{l}{m}_{{x}_{a}}^{(1)}{m}_{{y}_{b}}^{(1)}{m}_{{y}_{c}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{j}_{1}}^{(2)}{m}_{{k}_{1}}^{(2)}{m}_{{i}_{2}}^{(3)}{m}_{{j}_{2}}^{(3)}{m}_{{k}_{2}}^{(3)}=-1,\\ {m}_{{y}_{a}}^{(1)}{m}_{{x}_{b}}^{(1)}{m}_{{y}_{c}}^{t(1)}{m}_{{i}_{1}}^{(2)}{m}_{{j}_{1}}^{(2)}{m}_{{k}_{1}}^{(2)}{m}_{{i}_{2}}^{(3)}{m}_{{j}_{2}}^{(3)}{m}_{{k}_{2}}^{(3)}=-1,\\ {m}_{{y}_{a}}^{(1)}{m}_{{y}_{b}}^{(1)}{m}_{{x}_{c}}^{(1)}{m}_{{i}_{1}}^{(2)}{m}_{{j}_{1}}^{(2)}{m}_{{k}_{1}}^{(2)}{m}_{{i}_{2}}^{(3)}{m}_{{j}_{2}}^{(3)}{m}_{{k}_{2}}^{(3)}=-1,\\ {m}_{{x}_{a}}^{(1)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{c}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{c}}^{(3)}=1,\end{array}\end{eqnarray}$
where (i1, j1, k1), (i2, j2, k2) ∈ {(xa, yb, yc), (ya, xb, yc), (ya, yb, xc)}. We further obtain that
$\begin{eqnarray}\begin{array}{l}{\left[{m}_{{y}_{a}}^{(1)}{m}_{{y}_{a}}^{(2)}{m}_{{y}_{a}}^{(3)}{m}_{{y}_{b}}^{(1)}{m}_{{y}_{b}}^{(2)}{m}_{{y}_{b}}^{(3)}{m}_{{y}_{c}}^{(1)}{m}_{{y}_{c}}^{(2)}{m}_{{y}_{c}}^{(3)}\right]}^{18}\\ \quad \times {\left[{m}_{{x}_{a}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{c}}^{(2)}{m}_{{x}_{c}}^{(3)}\right]}^{10}=-1\end{array}\end{eqnarray}$
which implies
$\begin{eqnarray}{m}_{{x}_{a}}^{(1)}{m}_{{x}_{b}}^{(1)}{m}_{{x}_{c}}^{(1)}{m}_{{x}_{a}}^{(2)}{m}_{{x}_{b}}^{(2)}{m}_{{x}_{c}}^{(2)}{m}_{{x}_{a}}^{(3)}{m}_{{x}_{b}}^{(3)}{m}_{{x}_{c}}^{(3)}=1.\end{eqnarray}$
This contradicts the last equation in equation (37). Therefore, the quantum features from the chain network cannot be derived from local realism.
Considering experimental verification with these inequalities, we extend it into one inequality to witness noisy networks. Define an operator as
$\begin{eqnarray*}\begin{array}{rcl}{\rm{O}} & = & -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl} & & -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl} & & -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{rcl} & & -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }-{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\\ & & -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\\ & & +{x}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime },\end{array}\end{eqnarray}$
where xi, zj, yk are measurement operators on the i, j, kth particles, respectively. The total state ∣Φ⟩ is an eigenstate of the operator, i.e. O∣Φ⟩ = 28∣Φ⟩. Instead, in the local realism, since the elements can only take values of +1 or −1, we get ∣O∣ ≤ 26. This can be used to witness the noisy network. In fact, consider the Werner state of the GHZ state as ${\rho }_{w}=w| \phi \rangle \langle \phi | +\tfrac{1-w}{8}{{\rm{I}}}_{8}$, where w ∈ (0, 1]. It follows that $\mathrm{Tr}({\rm{O}}{\rho }_{w}^{(1)}{\rho }_{w}^{(2)}{\rho }_{w}^{(3)})=28{w}^{3}\gt 26$ if w > 0.9756. This means we can witness the entangled network with noise.
Now, we extend the present result in the subsection above to witness a long chain network, as shown in figure 4. Here, the network consists of 2n + 1 parties who share n GHZ states. All the parties 2n − 1 with n = 2, ⋯ ,n have two particles while the other has one particle.
Figure 4. The generalized GHZ paradox. It consists of n GHZ states shared by 2n + 1 parties.
Let the GHZ state be ∣φ(1)⟩, ∣φ(2)⟩, ⋯ ,∣φ(n)⟩. The total state is given by
$\begin{eqnarray}| {\rm{\Phi }}\rangle =| {\phi }^{(1)}\rangle | {\phi }^{(2)}\rangle \cdots | {\phi }^{(n)}\rangle .\end{eqnarray}$
By combining with equations (26), define the following equalities
$\begin{eqnarray}\begin{array}{l}{\vec{\sigma }}_{{Q}^{(1)}}={\sigma }_{{y}_{b}}^{(3P+1)}{\sigma }_{{y}_{c}}^{(3P+1)}{\sigma }_{{i}_{(2P+1)}}^{(3P+2)}{\sigma }_{{j}_{(2P+1)}}^{(3P+2)}\\ {\sigma }_{{k}_{(2P+1)}}^{(3P+2)}{\sigma }_{{i}_{(2P+2)}}^{(3P+3)}{\sigma }_{{j}_{(2P+2)}}^{(3P+3)}{\sigma }_{{k}_{(2P+2)}}^{(3P+3)}{\sigma }_{{x}_{a}}^{(3P+4)},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\vec{\sigma }}_{{Q}^{(2)}}={\sigma }_{{x}_{b}}^{(3P+1)}{\sigma }_{{y}_{c}}^{(3P+1)}{\sigma }_{{i}_{(2P+1)}}^{(3P+2)}{\sigma }_{{j}_{(2P+1)}}^{(3P+2)}{\sigma }_{{k}_{(2P+1)}}^{(3P+2)}\\ {\sigma }_{{i}_{(2P+2)}}^{(3P+3)}{\sigma }_{{j}_{(2P+2)}}^{(3P+3)}{\sigma }_{{k}_{(2P+2)}}^{(3P+3)}{\sigma }_{{y}_{a}}^{(3P+4)},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\vec{\sigma }}_{{Q}^{(3)}}={\sigma }_{{y}_{b}}^{(3P+1)}{\sigma }_{{x}_{c}}^{(3P+1)}{\sigma }_{{i}_{(2P+1)}}^{(3P+2)}{\sigma }_{{j}_{(2P+1)}}^{(3P+2)}\\ {\sigma }_{{k}_{(2P+1)}}^{(3P+2)}{\sigma }_{{i}_{(2P+2)}}^{(3P+3)}{\sigma }_{{j}_{(2P+2)}}^{(3P+3)}{\sigma }_{{k}_{(2P+2)}}^{(3P+3)}{\sigma }_{{y}_{a}}^{(3P+4)},\end{array}\end{eqnarray}$
where P = 0, 1 ⋯ n − 2, (iM, jM, kM) ∈ {(xa, yb, yc), (ya, xb, yc), (ya, yb, xc)}, M = 1, 2, ⋯ ,2n. We obtain the following eigenvalue equations as
$\begin{eqnarray}{\sigma }_{{x}_{a}}^{(1)}({\otimes }_{P=0}^{n-2}{\vec{\sigma }}_{{Q}^{(1)}}){\sigma }_{{y}_{b}}^{(3n)}{\sigma }_{{y}_{c}}^{(3n)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}({\otimes }_{P=0}^{n-2}{\vec{\sigma }}_{{Q}^{(2)}}){\sigma }_{{x}_{b}}^{(3n)}{\sigma }_{{y}_{c}}^{(3n)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{{y}_{a}}^{(1)}({\otimes }_{P=0}^{n-2}{\vec{\sigma }}_{{Q}^{(3)}}){\sigma }_{{y}_{b}}^{(3n)}{\sigma }_{{x}_{c}}^{(3n)}| {\rm{\Phi }}\rangle =-| {\rm{\Phi }}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{\otimes }_{i=1}^{3n}{\sigma }_{{x}_{a}}^{(i)}{\sigma }_{{y}_{b}}^{(i)}{\sigma }_{{y}_{c}}^{(i)}| {\rm{\Phi }}\rangle =| {\rm{\Phi }}\rangle ,\end{eqnarray}$
where (iM, jM, kM) ∈ {(xa, yb, yc), (ya, xb, yc), (ya, yb, xc)}, M = 1, 2, ⋯ ,2n.
In local realism, each operator represents an observable quantity with values in {±1}. This implies that
$\begin{eqnarray}\begin{array}{l}{m}_{{x}_{a}}^{(1)}\left(\displaystyle \prod _{P=0}^{n-2}{\vec{\sigma }}_{{Q}^{(1)}}\right){m}_{{y}_{b}}^{(3n)}{m}_{{y}_{c}}^{(3n)}=-1,\\ {m}_{{y}_{a}}^{(1)}\left(\displaystyle \prod _{P=0}^{n-2}{\vec{\sigma }}_{{Q}^{(2)}}\right){m}_{{x}_{b}}^{(3n)}{m}_{{y}_{c}}^{(3n)}=-1,\\ {m}_{{y}_{a}}^{(1)}\left(\displaystyle \prod _{P=0}^{n-2}{\vec{\sigma }}_{{Q}^{(3)}}\right){m}_{{y}_{b}}^{(3n)}{m}_{{x}_{c}}^{(3n)}=-1,\\ \displaystyle \prod _{i=1}^{3n}{m}_{{x}_{a}}^{(i)}{m}_{{y}_{b}}^{(i)}{m}_{{y}_{c}}^{(i)}=1,\end{array}\end{eqnarray}$
where (iM, jM, kM) ∈ {(xa, yb, yc), (ya, xb, yc), (ya, yb, xc)}, M = 1, 2 ⋯ , 2n.
Multiplying the first three equations in equation (49) implies that
$\begin{eqnarray}{\left[\displaystyle \prod _{i=1}^{3n}{m}_{{y}_{a}}^{(i)}{m}_{{y}_{b}}^{(i)}{m}_{{y}_{c}}^{(i)}\right]}^{18}{\left[\displaystyle \prod _{i=1}^{3n}{m}_{{x}_{a}}^{(i)}{m}_{{x}_{b}}^{(i)}{m}_{{x}_{c}}^{(i)}\right]}^{10}=-1.\end{eqnarray}$
Considering experimental verification with these inequalities, we extend it into one inequality to witness noisy networks. Define an operator as
$\begin{eqnarray*}\begin{array}{l}{\rm{O}}=-{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{x}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}{y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{y}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {y}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{y}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {y}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{y}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {y}_{3n}\prime\prime\prime \\ -{y}_{1}^{{\prime} }{y}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{y}_{2}^{{\prime} }{y}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{y}_{3}^{{\prime} }{y}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {y}_{3n-1}^{\prime} {y}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {y}_{3n-2}^{\prime} {y}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {y}_{3n}^{\prime} {y}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime \\ +{x}_{1}^{{\prime} }{x}_{1}^{{\prime\prime} }{x}_{1}^{\prime\prime\prime }{x}_{2}^{{\prime} }{x}_{2}^{{\prime\prime} }{x}_{2}^{\prime\prime\prime }{x}_{3}^{{\prime} }{x}_{3}^{{\prime\prime} }{x}_{3}^{\prime\prime\prime }\cdots \\ {x}_{3n-1}^{\prime} {x}_{3n-1}^{\prime\prime} {x}_{3n-1}\prime\prime\prime {x}_{3n-2}^{\prime} {x}_{3n-2}^{\prime\prime} {x}_{3n-2}\prime\prime\prime {x}_{3n}^{\prime} {x}_{3n}^{\prime\prime} {x}_{3n}\prime\prime\prime ,\end{array}\end{eqnarray}$
where xi, zj, yk are measurement operators on the i, j, kth particles, respectively. The total state ∣Φ⟩ is an eigenstate of the operator, i.e. O∣Φ⟩ = 28∣Φ⟩. Instead, in the local realism, since the elements can only take values of +1 or −1, we get ∣O∣ ≤ 26. This can be used to witness the noisy network. In fact, consider the Werner state of the GHZ state as ρv. It follows that $\mathrm{Tr}({\rm{O}}{\otimes }_{i=1}^{3n}{\rho }_{w}^{(i)})=28{w}^{3n}\gt 26$ if $w\gt {\left(13/14\right)}^{1/3n}$. This means we can witness the entangled network with noise.

5. Conclusion

In conclusion, we have extended the GHZ paradox to witness quantum networks consisting of GHZ states and EPR states. We firstly extended the GHZ paradox to witness triangle networks consisting of the GHZ state and EPR states. We then extended the GHZ paradox to witness the entanglement of chain networks consisting of multiple GHZ states. These results show new insights into the GHZ paradox for verification of quantum networks beyond previous results.

This work was supported by the National Natural Science Foundation of China (Nos. 62172341, 12204386), and Sichuan Natural Science Foundation (Nos. 2024NSFSC1365, 2024NSFSC1375 and 2023NSFSC0447).

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Outlines

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