Characterize zero-private-capacity channels with activating support asymmetry

Determine which zero-private-capacity quantum channels admit the support asymmetry exhibited by the constructed four-level channel and can consequently convert a failure of joint simulation into an explicit private-communication protocol.

Background

The paper constructs a four-level quantum channel with zero private capacity that, when used jointly with a qubit erasure channel of suitable erasure probability, enables positive-rate private communication, despite each factor having zero private capacity individually. The construction relies on a support asymmetry: a receiver measurement event has zero probability for the background state but positive probability for the signal state, while the corresponding environmental signal support is contained within the environmental background support.

The authors leave unresolved the broader structural characterization of zero-private-capacity channels possessing this support asymmetry. Resolving this problem would clarify which failures of joint simulation can be leveraged to construct explicit private-communication protocols beyond the particular channel exhibited in the paper.