An exponential strong converse for private communication over degradable quantum channels

2026-09-26

AI Summary

This study addresses the long-standing open problem of establishing an exponential strong converse for the unassisted private capacity of degradable quantum channels. Under a single error criterion, this work proposes a privacy test overlap bound combined with a symbol-averaged construction. By integrating Gaussian filtering techniques with smooth entropy estimation to handle correlated channel inputs, the approach overcomes the dimensional dependence on the receiver’s system inherent in traditional methods. The authors rigorously prove that when the transmission rate exceeds the maximum coherent information, the fidelity decays exponentially with the number of channel uses. Furthermore, these results are extended to the full parameter regimes of anti-degradable and quantum erasure channels, thereby strengthening the theoretical foundations of quantum secure communication.

Abstract

We prove an exponential strong converse for the unassisted private capacity of every finite-dimensional degradable quantum channel. We use a single error criterion: the infidelity between the actual message-estimate-environment state and an ideal state consisting of uniformly distributed, perfectly correlated messages that are independent of the environment. At every rate strictly above the channel's maximum coherent information, the optimized fidelity to this ideal state decays exponentially in the number of channel uses. The same bound holds when the ideal environment state is fixed to the actual environment marginal. The proof applies to arbitrary mixed-state encoders and arbitrary collective decoders. Its main operator ingredient is an overlap bound for privacy tests: two receiver placements with a common subsystem of dimension $d_C$ have overlap at most $d_C/M$, where $M$ is the key size, independently of both shield dimensions. We combine this bound with the signed averaging construction presented in arXiv:2608.01308 and a single-filter reduction to symmetric states. A Gaussian filter replaces the common receiver subsystem by a system of controlled dimension from which the original correlations can be approximately recovered using quantum side information. The filter preserves exact exchange symmetry, and a uniform smooth-entropy estimate bounds its output dimension for arbitrary correlated channel inputs. We also obtain an exponential strong converse at zero for antidegradable channels under the same joint criterion and, consequently, for the private capacity of the quantum erasure channel throughout its parameter range.

Problem

• private communication
• degradable quantum channels
• exponential strong converse
• coherent information
• quantum erasure channel

Innovation

• exponential strong converse
• degradable quantum channels
• private capacity
• privacy test overlap bound
• Gaussian filter