Interplay between teleportation fidelity and basis-independent coherence in maximally sliced states under decoherence

Technical Summary: Interplay between Teleportation Fidelity and Basis-Independent Coherence in Maximally Sliced States under Decoherence

Problem Statement
Quantum teleportation relies heavily on the quality of shared entangled resources. While the efficiency of teleportation protocols is typically measured by teleportation fidelity, this metric does not directly quantify the underlying quantum resources, such as coherence, responsible for that performance. Furthermore, in realistic environments, quantum systems inevitably interact with their surroundings, leading to decoherence that degrades both entanglement and coherence. Although previous studies have analyzed teleportation fidelity and quantum coherence separately under various noise channels, an explicit analytical relationship linking basis-independent coherence and teleportation fidelity for Maximally Sliced (MS) states under fundamental decoherence mechanisms has been lacking. This work addresses this gap by investigating how amplitude damping (AD) and phase damping (PD) channels affect the interplay between these quantities in three-qubit MS states.

Methodology
The authors employ the Kraus operator formalism to model the evolution of a three-qubit Maximally Sliced (MS) state under two distinct decoherence channels:

1. Amplitude Damping (AD): Models irreversible energy dissipation (spontaneous emission) from the system to the environment.
2. Phase Damping (PD): Models the loss of quantum coherence without energy exchange (pure dephasing).

The study proceeds through the following analytical steps:

• Resource Definition: The MS state, parameterized by $\theta$, is defined as $|MS(\theta)\rangle = \frac{1}{\sqrt{2}}(|000\rangle + \cos\theta|110\rangle + \sin\theta|111\rangle)$.
• Channel Modeling: The noisy density matrices are derived by applying Kraus operators to each of the three qubits independently.
• Reduced State Analysis: The third qubit is traced out to obtain the reduced bipartite density matrix ($\rho_{AB}$) shared between the sender (Alice) and receiver (Bob), which serves as the quantum channel.
• Fidelity Calculation: Teleportation fidelity ($F_{Tel}$) is calculated using the maximal singlet fraction ($f_{max}$) of the reduced bipartite state, utilizing the relation $F_{Tel} = \frac{2f_{max} + 1}{3}$.
• Coherence Quantification: Basis-independent coherence ($C_{BI}$) is calculated for the reduced bipartite state using the measure proposed by Radhakrishnan et al., which is derived from the von Neumann entropy.
• Entanglement Parametrization: The results are re-expressed in terms of the Coffman-Kundu-Wootters (CKW) three-tangle ($\tau$), a measure of genuine tripartite entanglement, to connect the resource properties directly to the state's initial entanglement structure.

Key Contributions and Results

1. Analytical Relations for Ideal States:
   For the ideal (noiseless) MS state, the authors derive a closed-form expression linking basis-independent coherence ($C_{BI}$) and teleportation fidelity ($F_{Tel}$). They establish that $C_{BI}$ is a monotonic, nonlinear function of $F_{Tel}$, ranging from approximately 0.558 at the classical teleportation threshold ($F_{Tel} = 2/3$) to 0.741 for perfect teleportation ($F_{Tel} = 1$).

2. Distinct Effects of Decoherence Channels:

   • Amplitude Damping (AD): The derived expression for fidelity under AD is $F_{AD} = \frac{2 + (1-p)(\cos\theta - p)}{3}$, where $p$ is the damping probability. Crucially, the authors find that $C_{BI}$ under AD depends explicitly on both $F_{AD}$ and the damping parameter $p$. This indicates that teleportation fidelity alone cannot uniquely determine the coherence in the presence of energy dissipation. Furthermore, a state-dependent threshold exists: quantum teleportation is only possible if $p < \cos\theta$. Beyond this threshold, the resource loses its quantum advantage.
   • Phase Damping (PD): The fidelity under PD is given by $F_{PD} = \frac{2 + (1-\lambda)^2 \cos\theta}{3}$, where $\lambda$ is the dephasing probability. Remarkably, the authors show that the decoherence parameter $\lambda$ can be analytically eliminated from the coherence-fidelity relation. Consequently, $C_{BI}$ remains a unique function of $F_{PD}$, identical in functional form to the noiseless case. This implies that for phase damping, the coherence-fidelity correspondence is preserved regardless of the noise strength.

3. Connection to Genuine Tripartite Entanglement:
   By expressing the results in terms of the three-tangle ($\tau = \sin^2\theta$), the paper provides a unified framework connecting genuine tripartite entanglement, coherence, and teleportation performance. The analysis reveals that while the three-tangle characterizes the initial resource, the teleportation fidelity is determined by the maximal singlet fraction of the reduced bipartite state.

Significance and Claims
The paper claims to provide a unified analytical framework for understanding the interplay among multipartite entanglement, quantum coherence, and teleportation in noisy three-qubit MS states. The primary significance lies in the following findings:

• Resource-Performance Correspondence: The study demonstrates that the relationship between a fundamental quantum resource (basis-independent coherence) and operational performance (teleportation fidelity) is robust under phase damping but breaks down under amplitude damping.
• Differential Robustness: The MS state exhibits greater robustness against phase damping than against amplitude damping. While phase damping allows the state to maintain a quantum advantage (fidelity > 2/3) until complete dephasing, amplitude damping introduces a critical threshold beyond which the quantum advantage is lost entirely.
• Analytical Benchmarking: The derived closed-form expressions serve as analytical benchmarks for understanding how different noise mechanisms degrade quantum resources. The work suggests that basis-independent coherence can serve as an effective indicator of teleportation capability, provided the specific nature of the decoherence channel is accounted for.

The authors conclude that these results offer physical insight into how different decoherence mechanisms modify the resource-performance correspondence, moving beyond fidelity alone to characterize the robustness of MS states in realistic noisy environments.