Characterizing Noise Effects on Multipartite Entanglement via Phase-Space Visualization

Abstract

This paper investigates the behavior of two fundamental types of multipartite entangled states, namely GHZ(3) and W(3) states under Gaussian-distributed amplitude perturbations and White noise model. The Uhlmann-Jozsa fidelity is taken to be the quantitative measure to show the overall degradation of the quantum states, and is implemented via TQIX : a tool specifically designed for quantum state measurement and related applications. While fidelity analysis captures the progressive decay of quantum states under noise, it offers only limited understanding regarding the state decay and doesn't provide a detailed analysis of how entanglement structures respond to noise models. To reveal the phase-space characteristics and nonclassical signatures of three-qubit entangled states, we employ the spin Wigner function using equal-angle projection. This approach reveals a continuous fading of quantum coherence with increasing noise strength, ultimately providing a clear picture of transition toward classical-like behavior in phase space. This combined qualitative-quantitative framework provides deeper understanding of how different entanglement structures respond to noise, offering practical applications for designing and implementing noise resilient protocols in quantum computing, and quantum information processing.

Figures (8)

• Figure 1: The probability distribution of the (a) ideal GHZ(3) state across the basis states exhibiting perfect correlations between $\ket{000}$ and $\ket{111}$. (b) the ideal W(3) state across the basis states $\ket{001}$, $\ket{010}$, and $\ket{100}$ with equal probabilities.
• Figure 2: Comparative analysis of GHZ(3) and W(3) states under gaussian-distributed amplitude perturbation for increasing noise strength $\sigma$: panels (a) and (b) show the probability distribution of GHZ(3) and W(3) states, respectively, at noise level $\sigma=0.4$, showing a noticeable probability redistribution into other basis states; panels (c) and (d) show the strong noise level $\sigma=1.0$ of GHZ(3) and W(3) states, respectively, showing suppression of the ideal states.
• Figure 3: Comparative analysis of GHZ(3) and W(3) states under white noise for increasing noise strength $p$: panels (a) and (b) show the probability redistribution of GHZ(3) and W(3) states, respectively, at noise level $p=0.4$, showing a noticeable uniform probability redistribution into other basis states; panels (c) and (d) show the strong noise level $p=1.0$ of GHZ(3) and W(3) states, respectively, showing suppression of the ideal states.
• Figure 4: Fidelity analysis of three-qubit GHZ and W states under Gaussian and white noise. Panel (a) and (b) show the comparison of fidelity of GHZ(3) and W(3) under gaussian-distributed amplitude perturbation and white noise, respectively; panel (c) compares the fidelity of the GHZ(3) state under gaussian-distributed amplitude perturbation; and panel (d) compares the fidelity of the W(3) state under Gaussian and white noise, with red and blue plots corresponding to GHZ(3) and W(3), respectively.
• Figure 5: Equal-angle Spin Wigner Function for three-qubit GHZ and W states. Panels (a) and (c) represent two dimensional contour plots of equal angle spin Wigner function for GHZ and W states respectively plotted over Bloch sphere with $\theta \in [0,\pi]$ and $\phi \in [0,2\pi]$. Panels (b) and (d) specify the corresponding three-dimensional surface plots obtained by mapping the angular coordinates $(\theta, \phi)$ to Cartesian coordinates $(\theta \cos\phi, \theta \sin\phi)$, with the Wigner function value along the vertical axis. The color scale indicates the magnitude and sign of the Wigner function.