Flagged Extensions and Numerical Simulations for Quantum Channel Capacity: Bridging Theory and Computation

TL;DR

This work develops a unified analytical–numerical framework to bound and estimate quantum and private channel capacities. It introduces flagged-extension techniques that produce computable single-letter upper bounds whenever the non-unitary branch is degradable, and couples this with a two-stage Bloch-sphere optimization to numerically maximize coherent information. The approach is applied to amplitude-damping, generalized amplitude-damping, and depolarizing channels, confirming degradable/anti-degradable regimes, zero-capacity transitions at high noise, and a persistent gap between hashing lower bounds and max-Rains upper bounds for non-additive channels. The study bridges theory and computation, offering practical bounds and insights for quantum network design and error mitigation, with results consistent with known degradability thresholds and capacity conjectures and demonstrated via a publicly available codebase.

Abstract

I will investigate the capacities of noisy quantum channels through a combined analytical and numerical approach. First, I introduce novel flagged extension techniques that embed a channel into a higher-dimensional space, enabling single-letter upper bounds on quantum and private capacities. My results refine previous bounds and clarify noise thresholds beyond which quantum transmission vanishes. Second, I present a simulation framework that uses coherent information to estimate channel capacities in practice, focusing on two canonical examples: the amplitude damping channel (which we confirm is degradable and thus single-letter) and the depolarizing channel (whose capacity requires multi-letter superadditivity). By parameterizing input qubit states on the Bloch sphere, I numerically pinpoint the maximum coherent information for each channel and validate the flagged extension bounds. Notably, I capture the abrupt transition to zero capacity at high noise and observe superadditivity for moderate noise levels.

Figures (6)

• Figure 1: (Amplitude Damping) Optimal coherent information versus the damping parameter $\gamma$. As $\gamma$ grows, the maximum coherent information gradually decreases and fluctuates around lower values, indicating stronger noise.
• Figure 2: (Depolarizing) Optimal coherent information versus the depolarizing probability $p$. Negative values on the y-axis are expected for single-letter coherent information, demonstrating that even moderate noise can overwhelm reliable quantum transmission.
• Figure 3: (Flagged Depolarizing) Optimal coherent information versus $p$ for a flagged extension of the depolarizing channel. The extra "flag" system can mitigate noise for small $p$, but as $p$ increases, coherent information decreases sharply.
• Figure 4: Amplitude damping (AD): comparison of $I_c$, reverse coherent information (RCI), entanglement-assisted capacity $C_E$, and an SDP/max-Rains upper bound, as a function of damping parameter $\gamma$. The vertical line marks the anti-degradable threshold $\gamma=\tfrac{1}{2}$ (Prop. \ref{['prop:ad-degradability']}).
• Figure 5: Panel comparison for AD (left) and GADC at $N_{\mathrm{th}}\!\in\!\{0.0,0.1,0.2\}$ (right panels): $I_c$, RCI, $C_E$, and an SDP/max-Rains-style upper bound. The dashed line at $\gamma=\tfrac{1}{2}$ marks the AD anti-degradable threshold.

Theorems & Definitions (19)

• Theorem 2.1: Flagged extension is degradable when the noise branch is degradable
• proof
• Corollary 2.2: Amplitude damping
• Remark 2.3
• proof
• Theorem 2.4: Flagged-extension upper bound under degradable-noise assumption
• proof
• Proposition 2.5: Zero-Capacity Noise Threshold
• proof
• Theorem A.1: Quantum capacity of degradable channels