Stochastic channels are ubiquitous in the field of quantum information because they are simple and easy to analyze. In particular, Pauli channels and depolarizing channels are widely studied because they can be efficiently simulated in many relevant quantum circuits. Despite their wide use, the properties of general stochastic channels have received little attention. In this paper, we prove that the diamond distance of a general stochastic channel from the identity coincides with its process infidelity to the identity. We demonstrate with an explicit example that there exist multi-qubit stochastic channels that are not unital. We then discuss the relationship between unitary 1-designs and stochastic channels. We prove that the twirl of an arbitrary quantum channel by a unitary 1-design is always a stochastic channel. However, unlike with unitary 2-designs, the twirled channel depends upon the choice of unitary 1-design. Moreover, we prove by example that there exist stochastic channels that cannot be obtained by twirling a quantum channel by a unitary 1-design.
Designing Nonlinear Photonic Crystals for High-Dimensional Quantum State Engineering
Technion（以色列理工学院）；Tel Aviv University（特拉维夫大学）；Massachusetts Institute of Technology（麻省理工学院）；Physical Research Laboratory（物理研究实验室）；Verily Research（Verily研究院）
