Exact Quantum Capacity of Decohering Channels in Arbitrary Dimensions

TL;DR

The paper addresses the challenge of determining the quantum capacity for a broad class of generalized dephasing channels in arbitrary dimensions. By proving degradability of channels of the form $\Lambda(\rho)=(1-x)\rho+x D(\rho)$, including fully, block-, and weakly decohering variants, the authors derive exact, closed-form single-letter capacities using diagonal-input optimization enabled by Schur-Horn majorization and channel covariance. The resulting formulas $Q(\Lambda)$, $Q(\Lambda_k)$, and $Q(\Phi_k)$ reveal that quantum information transmission can persist even under strong decoherence, depending on subspace coherence structure, with practical implications for encoding into decoherence-free subspaces. These results provide precise benchmarks for high-dimensional quantum channels and inform encoding strategies for quantum memories and fault-tolerant communication in realistic noisy settings.

Abstract

We derive exact analytical expressions for the quantum capacity of a broad subclasses of generalized dephasing channels of the form $Λ(ρ)=(1-x)ρ+ x D(ρ)$, where $D(ρ)$ represents a structured decoherence process. These channels are degradable for all noise parameters and in arbitrary dimensions, yielding closed-form, single-letter capacity formulas. Our analysis includes fully decohering, block-decohering, and weakly decohering channels, the latter involving coherence preservation within overlapping subspaces. Surprisingly, even under maximal decoherence, the channel may retain nonzero capacity due to residual coherence structure. These results provide quantitative role for decoherence-free and partially coherent subspaces in preserving quantum information, offering guidance for encoding strategies in quantum memories and fault-tolerant quantum communication systems.

Figures (3)

• Figure 1: The quantum capacities of the fully decohering $\Lambda$ (eq. \ref{['decorenceQcapa']}) for dimensions $d\in\{2,3,4,6,10\}$ as a function of the noise parameter $x$. Naturally in all dimensions, when complete decoherence occurs, the capacity vanishes.
• Figure 2: The quantum capacities of the block-decohering channels $\Lambda_k$ (eq. \ref{['e33']}) for $d=12$ for various values of $k$. For each $k$, the number of blocks is given by $r=\frac{12}{k}$. It is seen that even when $x=1$, full decoherence among subspaces, allows a nonzero quantum capacity. This is the capacity of quantum information transfer, when a suitable encoding into decoherence-free subspaces is used.
• Figure 3: Coherent information $Q\bigl(\Phi\bigr)$ (eq. \ref{['e40']}) versus $x$ for $d=12$ and $k\in\{1,2,3,4,6\}$. It is seen that compared with figure (\ref{['Capacities']}), here the capacity for each value of $k$, although still non-vanishing, is smaller than the correponding capacity of the block-decohering channel. This is due to the absence of any decoherence-free subspace in this case.

Theorems & Definitions (1)

• Definition 1