Jan 15 Huangjun Zhu. "Discrete Wigner function and Clifford group"

QIP 2016, Banff, 10-16 January 2016

Date: Jan 15 2016

Title: "Discrete Wigner function and Clifford group"
Authors: Huangjun Zhu.

Title: "The Clifford group forms a unitary 3- design"
Authors: Zak Webb.

Abstract: The Wigner function and Clifford group play fundamental roles in physics and quantum information science. Many desirable properties of the Wigner function are intimately connected to the high symmetry of the underlying operator basis composed of phase point operators. In each odd prime power dimension, this phase point basis is covariant with respect to the Clifford group. When the dimension is a power of 2, it is believed that no such basis can exist; however, no rigorous proof is known before despite the significance of this problem and intensive efforts of many researchers in the last three decades. We prove that in each odd prime power dimension, the phase point basis is the unique operator basis up to scaling that is covariant with respect to the Clifford group, while no such basis can exist when the dimension is a power of 2. In addition, the phase point basis is almost uniquely determined by the double transitivity of its symmetry group, that is, any pair of phase point operators can be transformed to any other pair by a unitary transformation in the symmetry group. Furthermore, we prove that the multiqubit Clifford group is a unitary 3-design and no operator basis can be covariant with respect to a unitary 3-design, thereby providing a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group. As another implication, any orbit of pure states of the Clifford group forms a 3-design; in particular, the set of stabilizer states forms a 3-design. Our work has profound implications for a number of research areas, such as quantum computation, quantum tomography, quantum foundation, and signal processing.

http://arxiv.org/abs/1504.03773
http://arxiv.org/abs/1510.02619