Geometric optimization for quantum communication

TL;DR

The paper tackles the challenging problem of quantifying fundamental quantum communication limits, namely the one-way distillable entanglement $D_{ o}$ and the quantum capacity $Q$, both hampered by superadditivity. It introduces a principled geometric framework that uses Riemannian optimization on state and channel extensions, representing extension spaces as complex Stiefel manifolds, to compute tight two-sided bounds. On the upper-bound side, it combines systematic extension search with an improved conditional-entropy continuity bound to produce state- and channel-extension based bounds that surpass previous benchmarks, including SOTA bounds for the qubit depolarizing channel. On the lower-bound side, it optimizes over unitary manifolds and interleaved local-unitary codes to reveal and quantify superadditivity in $D_{ o}^{(1)}$ and channel coherent information, while proving that amortization cannot improve $I_c$. Overall, the work establishes Riemannian optimization as a powerful tool for navigating quantum communication limits with broad practical implications for coding, error correction, and capacity estimation.

Abstract

Determining the ultimate limits of quantum communication, such as the quantum capacity of a channel and the distillable entanglement of a shared state, remains a central challenge in quantum information theory, primarily due to the phenomenon of superadditivity. This work develops Riemannian optimization methods to establish significantly tighter, computable two-sided bounds on these fundamental quantities. For upper bounds, our method systematically searches for state and channel extensions that minimize known information-theoretic bounds. We achieve this by parameterizing the space of all possible extensions as a Stiefel manifold, enabling a universal search that overcomes the limitations of ad-hoc constructions. Combined with an improved upper bound on the one-way distillable entanglement based on a refined continuity bound on quantum conditional entropy, our approach yields new state-of-the-art upper bounds on the quantum capacity of the qubit depolarizing channel for large values of the depolarizing parameter, strictly improving the previously best-known bounds. For lower bounds, we introduce Riemannian optimization methods to compute multi-shot coherent information. We establish lower bounds on the one-way distillable entanglement by parameterizing quantum instruments on the unitary manifold, and on the quantum capacity by parameterizing code states with a product of unitary manifolds. Numerical results for noisy entangled states and different channels demonstrate that our methods successfully unlock superadditive gains, improving previous results. Together, these findings establish Riemannian optimization as a principled and powerful tool for navigating the complex landscape of quantum communication limits. Furthermore, we prove that amortization does not enhance the channel coherent information, thereby closing a potential avenue for improving capacity lower bounds in general.

Figures (10)

• Figure 1: An overview of our geometric optimization methods for computing two-sided bounds on the quantum capacity and the one-way distillable entanglement.
• Figure 2: Continuity bounds comparison, $\left| A \right| = 4$.
• Figure 3: Schematic illustration of an $\varepsilon$-degradable state $\rho_{AB}$ and its $\varepsilon$-degrading channel ${\cal M}$, along with their respective Stinespring dilations ${\cal V}(\cdot) = V(\cdot)V^\dag$.
• Figure 4: Bounds on the one-way distillable entanglement of the qubit isotropic states. The $x$-axis is the parameter $3/4p=1-f \in [0,0.1]$ of the state $I_2(f)$.
• Figure 5: Bounds on the quantum capacity of the qubit depolarizing channel. The $x$-axis is the parameter $3/4p=1-f \in [0.0925, 0.1875]$ of the qubit depolarizing channel ${\cal D}_p$.

Theorems & Definitions (38)

• Definition 1: State extension
• Definition 2: Channel extension
• proof
• proof
• Definition 3: Degradability parameter of bipartite states Leditzky2018usefulsates
• Lemma 1: Continuity of the conditional entropy winterTightUniformContinuity2016
• Lemma 2: Continuity of the conditional entropy Koenraad2024Mario2025
• Definition 4
• Theorem 1: Improved upper bound on the one-way distillable entanglement
• proof