Dephasing-Induced Distribution of Entanglement in Tripartite Quantum Systems

1. Problem Statement

Quantum entanglement is a fundamental resource for quantum information processing tasks (e.g., teleportation, cryptography, computing). However, in realistic scenarios, quantum systems interact with their environments, leading to decoherence, which degrades entanglement.

• The Challenge: While entanglement decay in bipartite systems is well-studied, measuring and understanding the distribution of entanglement in multipartite (specifically tripartite) systems under noisy conditions is difficult.
• The Gap: There is a lack of robust measures for multipartite entanglement in mixed states, and the specific dynamics of how entanglement is distributed among qubits (rather than just total entanglement) under different environmental configurations (local vs. common, Markovian vs. non-Markovian) require further investigation.
• Objective: The authors aim to quantify the distribution of entanglement ($D$) in various tripartite pure and mixed states subjected to dephasing noise, analyzing how different environmental parameters affect the robustness of this distribution.

2. Methodology

A. Quantifier: Distribution of Entanglement ($D$)

The authors utilize the Relative Entropy of Entanglement ($E$) because it applies to both pure and mixed states of arbitrary dimensions.

• Definition: $E(\rho) = \min_{\sigma \in D} S(\rho || \sigma)$, where $S$ is the quantum relative entropy and $D$ is the set of separable states.
• Distribution Metric: To analyze how entanglement is shared, they define a quantity $D$ based on the monogamy of entanglement inequality ($E_{A:BC} \ge E_{A:B} + E_{A:C}$).
  $$D = E_{A:BC} - E_{A:B} - E_{A:C}$$
  • If $D > 0$, it indicates genuine multipartite entanglement (monogamous).
  • If $D \le 0$, it suggests the entanglement is distributed more locally (bipartite).
  • The authors use $R$ (the relative entropy expression) in place of $E$ for universality, denoting the distribution as $D = R_{A:BC} - R_{A:B} - R_{A:C}$.

B. System Models

The study investigates a three-qubit system interacting with a bosonic reservoir at finite temperature under two configurations:

1. Local Dephasing: Each qubit interacts with its own independent local environment.
2. Common Dephasing: All three qubits interact with a single shared environment.

C. Environmental Dynamics

The dynamics are modeled using Quantum Master Equations for both:

• Markovian Regime: Memoryless environment where the reservoir correlation time is negligible. The decay rate is constant ($\gamma_0$).
• Non-Markovian Regime: Environment with memory effects. The decay rate is time-dependent ($\gamma(t)$), governed by the spectral density $J(\omega)$ (Ohmic-type with a cutoff frequency $\Lambda$).

D. States Investigated

The study covers:

• Pure States: $|GHZ\rangle$, $|W\rangle$, $|W\bar{W}\rangle$ (superposition of $|W\rangle$ and spin-flipped $|W\rangle$), and $|Star\rangle$ (asymmetric state).
• Mixed States:
  • $\rho_{ghz}^{wer}$: Mixture of $|GHZ\rangle$ and white noise.
  • $\rho_{w}^{wer}$: Mixture of $|W\rangle$ and white noise.
  • $\rho_{w}^{ghz}$: Statistical mixture of $|GHZ\rangle$ and $|W\rangle$.

3. Key Results

A. Pure States

• Markovian Local Environment: All states ($|GHZ\rangle$, $|W\rangle$, $|W\bar{W}\rangle$, $|Star\rangle$) exhibit decay in the distribution of entanglement $D(t)$. $|GHZ\rangle$ decays fastest initially.
• Markovian Common Environment:
  • $|GHZ\rangle$ and $|Star\rangle$ decay to zero.
  • Crucial Finding: The $|W\rangle$ state shows zero decay in $D(t)$; its entanglement distribution is perfectly robust against common Markovian dephasing.
  • Contrast: The $|W\bar{W}\rangle$ state (superposition of $|W\rangle$ and $\bar{W}$) does decay, highlighting that the robustness of $|W\rangle$ is not simply inherited by its superpositions in this context.
• Non-Markovian Environments:
  • Local: No decay is observed for any of the pure states ($D(t)$ remains constant).
  • Common: $|W\rangle$, $|W\bar{W}\rangle$, and $|Star\rangle$ show no decay. $|GHZ\rangle$ remains stable until $\gamma_0 t > 0.6$, after which slight decay occurs.

B. Mixed States

• Werner-GHZ ($\rho_{ghz}^{wer}$) and Werner-W ($\rho_{w}^{wer}$):
  • In Non-Markovian environments (both local and common), these states exhibit no decay in $D(t)$ regardless of the mixing parameter $p$ (provided $p$ is high enough to sustain entanglement initially).
  • In Local Markovian environments, they decay.
  • In Common Markovian environments, $\rho_{w}^{wer}$ shows no decay, whereas $\rho_{ghz}^{wer}$ decays.
• GHZ-W Mixture ($\rho_{w}^{ghz}$):
  • In Common Markovian environments, decay occurs but stabilizes at a fixed non-zero value for certain mixing parameters ($p=0.9, 0.5$) after an initial drop.
  • In Non-Markovian environments, decay is minimal or absent depending on the specific bath type.

4. Key Contributions

1. Metric Application: Successfully applied the Relative Entropy of Entanglement to define and quantify the "distribution of entanglement" ($D$) in tripartite mixed states, a task often hindered by the lack of suitable measures for mixed multipartite systems.
2. Robustness Discovery: Identified that the $|W\rangle$ state and specific Werner-W mixed states possess exceptional robustness against common Markovian dephasing, maintaining their entanglement distribution without decay.
3. Non-Markovian Protection: Demonstrated that non-Markovian memory effects can completely suppress the decay of entanglement distribution for a wide range of states (both pure and mixed) in local environments.
4. State Sensitivity: Revealed that linear superpositions of robust states (like $|W\bar{W}\rangle$) do not necessarily inherit the robustness of their components, emphasizing the complexity of multipartite entanglement dynamics.

5. Significance and Conclusion

The paper provides critical insights for quantum engineering and reservoir engineering.

• Practical Implication: By identifying specific states (like $|W\rangle$) and environmental configurations (common non-Markovian baths) that preserve entanglement distribution, the authors suggest pathways to design more robust quantum networks and processors.
• Theoretical Insight: The study clarifies the distinction between "genuine" multipartite entanglement (fragile, e.g., $|GHZ\rangle$) and "distributed" bipartite entanglement (robust, e.g., $|W\rangle$) under noise.
• Future Work: The authors suggest that these findings open doors for experimental realization using photonic qubits and propose using "squashed entanglement" (a strictly monogamous measure) for future studies to further refine the understanding of multipartite entanglement distribution.

In summary, the work establishes that the survival of multipartite entanglement is not just a function of the state itself but is heavily dependent on the correlation structure of the noise (local vs. common) and the memory properties of the environment (Markovian vs. non-Markovian).