Private communication via zero-private-capacity quantum channels
Chengkai Zhu, Xin Wang
quant-ph
Sep 9, 2026 · v1
cs.IT
TL;DR
The main superactivation result on private communication via zero-private-capacity quantum channels was formalized in Lean 4.
Abstract
Private communication over a noisy quantum channel requires reliable transmission to the receiver and secrecy from the environment. Whether two channels with zero private capacity can jointly enable private communication is a longstanding open problem in quantum information theory. Here we resolve this problem by exhibiting a four-level channel and a qubit erasure channel with half erasure probability, each with zero private capacity, whose joint use achieves more than 0.0001903 private bits per product use. The encoding gives the receiver a linear information gain with at most quadratic environmental leakage, enabling privacy through a fixed joint measurement and classical coding. This superactivation, impossible for independent classical memoryless wiretap channels, shows that a channel's private capacity alone does not determine its value for secure communication. The initial activation example was identified through interactions with large language models, and the result has been formalized in Lean 4.
Problem
Whether two quantum channels each with zero private capacity can jointly enable private communication is a longstanding open problem in quantum information theory.
Approach
Two independent finite-dimensional memoryless channels are constructed: a four-level channel with an explicit Kraus form and a qubit erasure channel with erasure probability at least 1/2, each shown to have zero private capacity via transpose-antidegradability certificates. An explicit encoding and fixed joint measurement plus classical coding achieve a positive private rate. A PPT-decoding converse shows the measurement must be joint. The formal result was verified in Lean 4.
Results
The joint use achieves more than 0.0001903 private bits per product use, resolving the private-capacity superactivation problem; a fixed joint measurement suffices while PPT decoders yield zero rate.