Extreme Capacities in Generalized Direct Sum Channels

TL;DR

The paper introduces generalized direct sum (GDS) channels, extending direct sum channels by allowing block-coherent coupling between subchannels through a Kraus-operator direct-sum structure. It establishes that degradability of a GDS channel is equivalent to degradability of all subchannels, and that the optimal coherent information for many GDS channels inherits a block-diagonal form, enabling a single-letter quantum capacity formula for a broad class. It provides analytic bounds on quantum, private, and Holevo capacities, and identifies a new class of channels with a single-letter quantum capacity given by $Q(\mathcal{N}) = Q^{(1)}(\mathcal{N}) = \log[\sum_i 2^{Q^{(1)}(\mathcal{N}_i)}]$, including cases where all subchannels are antidegradable or PPT. As a concrete illustration, the authors construct GDS channels from completely depolarizing subchannels, proving arbitrarily small quantum capacity while private and classical capacities remain large (up to $\log(n+1)$), yielding an unbounded separation between quantum and private/classical capacities and demonstrating pronounced superadditivity within a conceptually simple framework. Overall, the GDS framework provides a clearer, explicit avenue to study capacity additivity, superadditivity, and the interplay between block coherence and information transmission in quantum channels.

Abstract

Quantum channel capacities play a central role in quantum Shannon theory, a formalism built upon rigorous coding theorems for noisy channels. Evaluating exact capacity values for general quantum channels remains intractable due to superadditivity. As a step toward understanding this phenomenon, we construct the generalized direct sum (GDS) channel, extending conventional direct sum channels through a direct sum structure in their Kraus operators. This construction forms the basis of the GDS framework, encompassing classes of channels with single-letter formula for quantum capacities and others exhibiting striking capacity features. The quantum capacity can approach zero yet display unbounded superadditivity combined with erasure channels. Private and classical capacities coincide and can become arbitrarily large, resulting in an unbounded gap with the quantum capacity. Providing a simpler and more intuitive approach, the framework deepens our understanding of quantum channel capacities.

Figures (2)

• Figure 1: Structural contrast and communication advantage of generalized direct sum (GDS) over direct sum (DS) channels. The upper panel illustrates the structural distinction: for simplicity, only two subchannels, $\mathcal{N}_1$ and $\mathcal{N}_2$, are shown. The DS channel exhibits a block-diagonal structure, whereas the GDS channel possess block coherent components $\mathcal{M}$ and $\mathcal{M}^\dagger$ linking the two subchannels. The lower panel highlights the resulting communication advantage: this structural distinction enables GDS channels to perform quantum communication tasks beyond the reach of DS channels.
• Figure 2: Channel capacities analysis for the GDS channel. (left) Solid lines denote upper bounds on the quantum capacity of the GDS channel $\mathcal{N}$ for various $p$. The dash line indicates the exact private capacity $\log(n+1)$. The gap between quantum and private capacity diverges as $n$ or $p$ increases. (right) Maximal value of $\lambda$ for which the quantum capacity of $\mathcal{N}$ combined with $\mathcal{E}_\lambda$ exhibits superadditivity.

Theorems & Definitions (36)

• Lemma 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• proof
• Example 1
• Proposition 6
• proof