Dephasing-Induced Distribution of Entanglement in Tripartite Quantum Systems

Preserving multipartite entanglement amidst decoherence poses a pivotal challenge in quantum information processing. However, assessing multipartite entanglement in mixed states amid decoherence presenting a formidable task. Employing reservoir memory offers a means to attenuate the decoherence dynamics impacting multipartite entanglement, thereby slowing its degradation. One of the important measures which can be implemented to quantify entanglement is the relative entropy of entanglement. Although this measure is not monogamous \cite{horodeckirev2009}, it can universally be applied to both pure and mixed states. Based on this fundamental novelty, in this work, therefore, we introduce a quantifier which will investigate how entanglement remain distributed among the qubits of multipartite states when these states are exposed to multipartite dephasing setting. For our study we use various pure and mixed tripartite states subjected to finite temperature in both Markovian and non-Markovian local/common bath. Here, we consider situations where the three qubits interact with a common reservoir as well as a local bosonic reservoir. We also show that the robustness of a quantum system to decoherence depends on the distribution of entanglement and its interaction with various configurations of the bath. When each qubit has its own local environment, the system exhibits different distribution dynamics compared to when all three qubits share a common environment with one exception regarding a mixed state.

💡 Research Summary

The paper investigates how entanglement is distributed and preserved in three‑qubit systems subjected to dephasing noise, focusing on both local (independent) and common (collective) reservoir configurations. Because multipartite entanglement in mixed states is notoriously difficult to quantify, the authors adopt the relative entropy of entanglement (E_R) as a universal measure that applies to pure and mixed states alike. Building on E_R, they define a “distribution of entanglement” quantity D = E_{A:BC} − E_{A:B} − E_{A:C}, which captures the excess of genuine tripartite entanglement over the sum of pairwise contributions. Positive D indicates dominance of true three‑party entanglement, while negative D signals that bipartite correlations dominate.

Two open‑system models are considered. In the first (local dephasing), each qubit couples to its own bosonic bath via a σ_z · (B + B†) interaction, leading to pure dephasing. In the second (common dephasing), all three qubits couple collectively to a single bosonic reservoir through the collective operator S_z = ∑_i σ_z^{(i)}. Both baths are modeled with an Ohmic spectral density J(ω)=η ω exp(−ω/Λ) at a finite temperature T (k_B T = ħ ω₀/4π). The authors derive time‑local master equations: for Markovian dynamics the rates become time‑independent (γ₀ = 4π η k_B T/ħ), whereas for non‑Markovian dynamics the dephasing rate γ(t) and an additional coefficient α(t) retain explicit time dependence, reflecting reservoir memory.

The study evaluates D(t) for several representative three‑qubit states: the GHZ state (|000⟩+|111⟩)/√2, the W state (|001⟩+|010⟩+|100⟩)/√3, the superposition |W̅W⟩ = (|W⟩+|W̅⟩)/√2 (where |W̅⟩ is the spin‑flipped W), and the “Star” state (|000⟩+|100⟩+|101⟩+|111⟩)/2. GHZ exhibits genuine tripartite entanglement (initial D > 0) but loses all entanglement rapidly when any qubit decoheres. W and |W̅W⟩ possess only bipartite entanglement (initial D < 0); their D remains negative and changes only modestly under dephasing, reflecting robustness of pairwise correlations. The Star state shows mixed behavior: it starts with positive D but decays quickly under Markovian noise.

A central finding is the role of reservoir memory. In the common‑bath scenario, non‑Markovian dynamics (finite bath correlation time) produce oscillatory γ(t) and α(t) that significantly slow the decay of D(t). In many parameter regimes D(t) even exhibits partial revivals, indicating that memory can temporarily restore lost coherence. For local baths, memory effects are weaker; the difference between Markovian and non‑Markovian cases is modest, though lower temperatures and larger cutoff frequencies Λ can modestly extend entanglement lifetimes.

The authors also explore mixed initial states, such as GHZ mixed with white noise. As the mixing proportion increases, D crosses zero at a critical noise level, signaling a transition from tripartite‑dominated to bipartite‑dominated entanglement. This critical point shifts depending on whether the environment is local or common and on the degree of non‑Markovianity.

Overall, the paper demonstrates three key insights: (1) the relative‑entropy‑based distribution measure D is an effective tool for distinguishing how different multipartite states allocate entanglement under decoherence; (2) non‑Markovian (memory‑bearing) reservoirs, especially common dephasing baths, can markedly enhance the persistence of multipartite entanglement; (3) the intrinsic pattern of entanglement distribution (genuine tripartite vs. pairwise) fundamentally determines a state’s resilience to dephasing. These results have practical implications for quantum network design, distributed quantum computing, and reservoir‑engineering strategies aimed at prolonging useful entanglement in realistic noisy environments.