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arXiv:2303.13304 · Physical Review A

Generalized symplectic representation of local Clifford operators

Caihong Wang, Jiangtao Yuan, Zhihao Ma, Shao-Ming Fei, Shang-Quan Bu

Published Feb 4, 2026• 0 citations
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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.

DOI
10.1103/xwyr-sr6l

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Local unitary classification of sets of generalized Bell states in Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd
Two sets of quantum entangled states that are equivalent under local unitary transformations may exhibit identical effectiveness and versatility in various quantum information processing tasks. Consequently, classification under local unitary transformations has become a fundamental issue in the theory of quantum entanglement. The primary objective of this work is to establish a practical LU-classification for all sets of l (≥2)l\ (\geq 2)l (≥2) generalized Bell states (GBSs), high-dimensional generalizations of Bell states, in a bipartite system Cd⊗Cd\mathbb{C}^{d}\otimes \mathbb{C}^{d}Cd⊗Cd with d≥3d\geq 3d≥3 . Based on this classification, we determine the minimal cardinality of indistinguishable GBS sets in C6⊗C6\mathbb{C}^{6}\otimes \mathbb{C}^{6}C6⊗C6 under one-way local operations and classical communication (one-way LOCC). We first propose two classification methods based on LU-equivalence for all sets of l GBSs (l-GBS sets). We then establish LU-classification for all 2-GBS, 3-GBS, 4-GBS and 5-GBS sets in C6⊗C6\mathbb{C}^{6}\otimes \mathbb{C}^{6}C6⊗C6 . Since LU-equivalent sets share identical local distinguishability, it suffices to examine representative GBS sets from equivalent classes. Notably, we identify a one-way LOCC indistinguishable 4-GBS set among these representatives, thereby resolving the case of d=6d = 6d=6 for the problem of determining the minimum cardinality of one-way LOCC indistinguishable GBS sets in (Yuan et al. in Quantum Inf Process. 18:145, 2019) or (Zhang et al. in Phys Rev A 91:012329, 2015).
In this paper, we study the local unitary (LU) classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we first introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all pairs in the $d\ensuremath{\bigotimes}d$ quantum system into ${\ensuremath{\prod}}_{j=1}^{n}({k}_{j}+1)\ensuremath{-}1$ LU-inequivalent pairs when the prime factorization of $d={\ensuremath{\prod}}_{j=1}^{n}{p}_{j}^{{k}_{j}}$. Moreover, all triples in the ${p}^{\ensuremath{\alpha}}\ensuremath{\bigotimes}{p}^{\ensuremath{\alpha}}$ quantum system for prime ppp can be partitioned into $\frac{(\ensuremath{\alpha}+3)}{6}{p}^{\ensuremath{\alpha}}+O(\ensuremath{\alpha}{p}^{\ensuremath{\alpha}\ensuremath{-}1})$ LU-inequivalent triples; especially, when $\ensuremath{\alpha}=2$ and p>2p>2p>2, there are exactly $\ensuremath{\lfloor}\frac{5}{6}{p}^{2}\ensuremath{\rfloor}+\ensuremath{\lfloor}\frac{p\ensuremath{-}2}{6}+{(\ensuremath{-}1)}^{\ensuremath{\lfloor}\frac{p}{3}\ensuremath{\rfloor}}\frac{p}{3}\ensuremath{\rfloor}+3$ LU-inequivalent triples.
In order to study a large number of generalized Bell states (GBSs) efficiently, it is important to classify them by local unitary (LU) equivalence. For example, there are C254(=12650) sets of four GBSs in C5⊗C5, which greatly exceed the C164(=1820) sets of four GBSs in C4⊗C4, and it is inconceivable to explore these sets directly without prior classification. In this paper, we consider the classification of GBS sets in C5⊗C5. First, two simple and useful characterizations of LU-equivalence of GBS sets are given. Then, one characterization is used to divide all 12 650 sets of four GBSs into eight equivalence classes including six locally distinguishable classes and two classes that cannot be discriminated by one-way local operations and classical communication. The second characterization is powerful and can be used to write a program, by using which all C255(=53130) sets of five GBSs in C5⊗C5 are divided into 21 equivalence classes including 9 locally distinguishable classes and 12 locally indistinguishable classes.
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47 , 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2 , which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.
The Clifford hierarchy of unitary operators is a foundational concept for universal quantum computation. It was introduced to show that universal quantum computation can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. [S. X. Cui et al., Phys. Rev. A 95, 012329 (2017)] recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal unitaries whose action on a computational basis qudit state is described by a 2kth{2}^{k}\mathrm{th}2kth root of unity raised to some polynomial function of the state, and they established the level of such unitaries in the hierarchy as a function of kkk and the degree of the polynomial. For qubit systems, we consider kthk\mathrm{th}kth-level diagonal unitaries that can be described just by quadratic forms of the state over the ring Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ of integers modulo 2k{2}^{k}2k. The quadratic forms involve symmetric matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ that can be used to efficiently describe all two-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal unitaries from lower levels. The result involves symplectic matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ and hence our perspective unifies a subgroup of diagonal gates in the Clifford hierarchy with the binary symplectic framework for gates in the Clifford group. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to expensive magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+T\mathrm{Clifford}+TClifford+T set of gates, perhaps leading to shorter depth circuits. Our results suggest that some nondiagonal gates in the hierarchy might also be understood by generalizing other binary symplectic matrices to integer rings.
Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs. Graph states are used in many applications, such as measurement-based quantum computation, as multipartite entangled resources. It is thus crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations. In this case, we say that the graph states are LU-equivalent (local unitary). If the local operations are restricted to the so-called Clifford group, we say that the graph states are LC-equivalent (local Clifford). Interestingly, a simple graph rule called local complementation fully captures LC-equivalence, in the sense that two graph states are LC-equivalent if and only if the underlying graphs are related by a sequence of local complementations. While it was once conjectured that two LU-equivalent graph states are always LC-equivalent, counterexamples do exist and local complementation fails to fully capture the entanglement of graph states. We introduce in this thesis a generalization of local complementation that does fully capture LU-equivalence. Using this characterization, we prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence. This also leads to the design of a quasi-polynomial algorithm for deciding whether two graph states are LU-equivalent, and to a proof that two LU-equivalent graph states are LC-equivalent if they are defined on at most 19 qubits. Furthermore, we study graph states that are universal in the sense that any smaller graph state, defined on any small enough set of qubits, can be induced using only local operations. We provide bounds and an optimal, probabilistic construction.