2023/03/23 by Wang, Cai-Hong, Yuan, Jiang-Tao, Ma, Zhi-Hao +2
#81P18 #81P40 #FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2303.13304
It is known that every (single-qudit) Clifford operator maps the full set of generalized Pauli matrices (GPMs) to itself under unitary conjugation, which is an important quantum operation and plays a crucial role in quantum computation and information. However, in many quantum information processing tasks, it is required that a specific set of GPMs be mapped to another such set under conjugation, instead of the entire set. We formalize this by introducing local Clifford operator, which maps a given n-GPM set to another such set under unitary conjugation. We establish necessary and sufficient conditions for such an operator to transform a pair of GPMs, showing that these local Clifford operators admit a classical matrix representation, analogous to the classical (or symplectic) representation of standard (single-qudit) Clifford operators. Furthermore, we demonstrate that any local Clifford operator acting on an n-GPM (n≥ 2) set can be decomposed into a product of standard Clifford operators and a local Clifford operator acting on a pair of GPMs. This decomposition provides a complete classical characterization of unitary conjugation mappings between n-GPM sets. As a key application, we use this framework to address the local unitary equivalence (LU-equivalence) of sets of generalized Bell states (GBSs). We prove that the 31 equivalence classes of 4-GBS sets in bipartite system ℂ6⊗ ℂ6 previously identified via Clifford operators are indeed distinct under LU-equivalence, confirming that this classification is complete.