Quantum capacity amplification via privacy

TL;DR

This work addresses the nonadditivity of quantum capacity by exploiting private-state induced channels, revealing how privacy structure and the shield can boost coherent transmission when paired with an assisting channel. Using a CJ-based channel perspective, the authors develop a general amplification framework with a Holevo-information–driven condition, and they derive a single-letter quantum-capacity formula for a broad family of private channels under the Spin Alignment Conjecture. They also demonstrate a fundamental gap where the private capacity can diverge while the quantum capacity vanishes, and they extend the analysis to approximate private channels, yielding new proofs of superactivation, metric separations from anti-degradable channels, and insights into the computability of the regularized quantum capacity. Collectively, the results illuminate how private-state structure governs capacity, provide explicit thresholds for amplification, and point to important open problems in quantum-capacity computability and universality beyond private-state constructions.

Abstract

We investigate superadditivity of quantum capacity through private channels whose Choi-Jamiolkowski operators are private states. This perspective links the security structure of private states to quantum capacity and clarifies the role of the shield system: information encoded in the shield system that would otherwise leak to the environment can be recycled when paired with an assisting channel, thereby boosting capacity. Our main contributions are threefold: Firstly, we develop a general framework that provides a sufficient condition for capacity amplification, which is formulated in terms of the assisting channel's Holevo information. As examples, we give explicit, dimension and parameter dependent amplification thresholds for erasure and depolarizing channels. Secondly, assuming the Spin alignment conjecture, we derive a single-letter expression for the quantum capacity of a family of private channels that are neither degradable, anti-degradable, nor PPT; as an application, we construct channels with vanishing quantum capacity yet unbounded private capacity. Thirdly, we further analyze approximate private channels: we give an alternative proof of superactivation that extends its validity to a broader parameter regime, and, by combining amplification bounds with continuity estimates, we establish a metric separation showing that channels exhibiting capacity amplification have nonzero diamond distance from the set of anti-degradable channels, indicating that existing approximate (anti-)degradability bounds are not tight. We also revisit the computability of the regularized quantum capacity and modestly suggest that this fundamental question still remains open.

Figures (5)

• Figure 1: Illustration of a perfect private state showing the key and shield subsystems for Alice and Bob. The shared key $a_0b_0$ is protected by shield subsystems $A_0$ and $B_0$, ensuring privacy against an eavesdropper (Eve).
• Figure 2: Quantum capacity amplification with the help of shield system.
• Figure 3: 2D plots of the quantity (LHS – RHS) for the inequality $\log \left((d^2-1)(r_0 + |r_1|) + |r_0 + d r_1| + |r_1 + d r_0|\right) + \max\{(1-2\lambda)\log d, 0\} < 1 - \lambda h(q)$ in Corollary \ref{['corollaryIII5']}, where LHS and RHS denote the left- and right-hand sides of the inequality, respectively. The plots show (LHS – RHS) as functions of erasure channel parameter $\lambda$ and private channel parameter $q$ for different dimension $d$, with the yellow solid line indicating the contour where $\text{LHS} - \text{RHS} = 0$.
• Figure 4: 2D plots of the quantity (LHS – RHS) for the inequality in Corollary \ref{['corollaryIII6']}, where LHS and RHS denote the left- and right-hand sides of the inequality, respectively. The plots display (LHS – RHS) as functions of depolarizing probability $p$ and private channel parameter $q$ for different values of dimension $d$. The yellow solid line marks the contour where $\text{LHS} - \text{RHS} = 0$, while the two regimes are separated by the black dashed line indicating the case boundary.
• Figure 5: Superactivation with the help of many copy use of channels

Theorems & Definitions (41)

• Theorem 1.1: Informal; see Theorem \ref{['thm:amplification']}
• Theorem 1.2: Section \ref{['subsec:construction']}
• Lemma 2.1
• proof
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Theorem 3.1
• Proposition 3.2