Classical Commitment over Quantum Channels with Limited Entanglement Assistance
Information Theory
Summary
The authors explore how to securely commit to a classical message (string commitment) using quantum channels that share limited entanglement beforehand. They focus on a specific family of quantum channels that apply random quantum operations to the input and send extra classical information to the receiver. For simple, noninteractive methods, they find the maximum rate (capacity) at which commitment is possible, depending on the channel's randomness and available entanglement. They also show that for interactive methods, the capacity cannot be higher than a certain limit, and for some channels like the quantum erasure channel, this fully determines the commitment capacity.
Authors
Remi A. Chou
Abstract
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 $d^2$ 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 $H(Z|F)\geq\log_2d+E$, interactive communication does not increase the capacity. As a consequence, for interactive protocols, we determine the capacity of the quantum erasure channel.