Classical representation of local Clifford operators

TL;DR

This work extends Clifford operator theory to local Clifford operators that map $n$-GPM sets to $n$-GPM sets under unitary conjugation. It establishes a classical (symplectic) representation for these local operators, proves a decomposition into a sequence of standard Clifford maps and a final $L_{(a,b)}$ acting on a 2-GPM pair, and provides concrete conditions for UC-equivalence in the 2-GPM case. This framework yields two procedures to compute U-equivalence classes and LU-equivalence of generalized Bell states, and it is applied to prove a complete LU-classification for 4-GBS sets in $\mathbb{C}^6\otimes\mathbb{C}^6$, confirming the 31 classes are LU-inequivalent. The results offer a computationally tractable route to analyze local discrimination tasks and quantum nonlocality by translating unitary conjugations into finite classical data, while leaving open the precise conditions under which local Clifford operators collapse to Clifford operators.

Abstract

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 ($n\geq 2$) 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 $\mathbb{C}^{6}\otimes \mathbb{C}^{6}$ previously identified via Clifford operators are indeed distinct under LU-equivalence, confirming that this classification is complete.

Figures (7)

• Figure 1: There exists a local Clifford operator mapping the GPM $X^sZ^t$ to $X^{s^{\prime}}Z^{t^{\prime}}$ if and only if their essential powers are the same.
• Figure 2: There exists a local Clifford operator mapping the GPM pair $\{X^a, Z^b\}$ to $\{X^{ua}, Z^b\}$ if and only if the two conditions in red ellipses hold.
• Figure 3: There exists a local Clifford operator mapping the GPM pair $\{X^a, Z^b\}$ to $\{X^{u_{1} a}Z^{v_{1} a}, X^{u_{2} b}Z^{v_{2} b}\}$ if and only if the three conditions in red ellipses hold.
• Figure 4: There exists a Clifford operator mapping the GPM pair $\{X, Z\}$ to $\{X^{u_{1}}Z^{v_{1}}, X^{u_{2}}Z^{v_{2}}\}$ if and only if $u_{1}v_{2}-u_{2}v_{1}\equiv 1 \pmod{d}$.
• Figure 5: Schematic diagram of the procedure for transforming a given $n$-GPM set, $\mathcal{M}$, into a GPM set, $\mathcal{M}^{\prime}_{m}$, satisfying Condition (I).

Theorems & Definitions (23)

• Definition 1
• Definition 2: wang2025epj
• Definition 3: wang2025epj
• Lemma 1: wang2025epj
• Lemma 2
• proof
• Theorem 1
• Remark 1
• Example 1
• Example 2