Classical Commitment over Quantum Channels with Limited Entanglement Assistance

We study classical string commitment over quantum channels with limited preshared entanglement. For noninteractive protocols, we determine the commitment capacity of a class of channels with input dimension d that, at each use, sample a pair of classical random variables (F,Z), apply one of the d² Heisenberg--Weyl operators indexed by Z to the input, and deliver the transformed quantum system together with F to the receiver. If E is the available entanglement rate in bits per channel use, then the capacity is min\{H(Z|F),log_2d+E\}. This class of channels encompasses quantum erasure and depolarizing channels, as well as families of Pauli channels. Additionally, for interactive protocols, we show that the commitment rate cannot exceed log_2d+E bits per channel use, so that when H(Z|F)geqlog_2d+E, interactive communication does not increase the capacity. As a consequence, for interactive protocols, we determine the capacity of the quantum erasure channel.