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arXiv:2504.09232

Commutators with multiple unitary symmetry

Shu Li, Jie Wang, Binfeng Wang, Lin Chen

Published Apr 12, 2025• 0 citations
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Abstract

Commutators are essential in quantum information theory, influencing quantum state symmetries and information storage robustness. This paper systematically investigates the characteristics of bipartite and multipartite quantum states invariant under local unitary group actions. The results demonstrate that any quantum states commuting with $U \otimes U^{\dagger}$ and $U \otimes V$ can be expressed as $\frac{1}{n}I_n$, where $U$ and $V$ are arbitary $n\times n$ unitary matrices. Furthermore, in tripartite systems, any quantum states commuting with $U \otimes U \otimes U^{\dagger}$ must necessarily adopt the form: $W = xI_{n^3} + y\left(\sum_{i,j=1}^n (|i\rangle \langle j|) \otimes (|j\rangle \langle i|)\right) \otimes I_n$, where $F_n$ represents the canonical swap operator. These results provide theoretical tools for characterizing multipartite entanglement constraints and designing symmetry-protected quantum protocols.

DOI
10.48550/arxiv.2504.09232

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We investigate the bipartite and tripartite quantum states invariant under the so-called finite-order unitary matrix U. We show that bipartite states that commute with [Formula: see text] are maximally mixed states. For the unitary matrix V being the finite-order matrix or an exceptional unitary matrix, the tripartite state commuting with [Formula: see text] adopts the form [Formula: see text] with the restriction of [Formula: see text], where [Formula: see text] are real numbers and [Formula: see text] represents the swap operator. These results can be applied to characterize multipartite states restricted by similar unitary symmetry.
We propose a natural generalization of bipartite Werner and isotropic states to multipartite systems consisting of an arbitrary even number of ddd-dimensional subsystems (qudits). These generalized states are invariant under the action of local unitary operations. We study basic properties of multipartite invariant states and present necessary and sufficient separability criteria.
We investigate the relation between local unitary symmetries and entanglement invariants of multi-qubit systems. The Hilbert space of such systems can be stratified in terms of states with different types of symmetry. We review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties. On a stratum of a non-trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement. Finding these invariants is often a simpler problem than finding the invariants of the local unitary group. The conditions, as given by the Luna–Richardson theorem, for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed. As an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non-trivial. On the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations. This canonical form and the invariants are directly determined by the symmetry group. Further, we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties.
This dissertation describes the application of symmetry to multipartite quantum systems from the viewpoint of quantum information theory. It is written in the framework of abstract quantum information theory, i.e. without distinguishing the physical nature of the d-level systems considered. The results are mathematically rigorous and aim at investigating entanglement properties as well as operational properties of states of multipartite systems. The first chapter gathers the mathematical tools and concepts needed in the following chapters and can be omitted by the experienced reader. In chapter two we apply the concept of symmetry to multipartite systems and show how to construct families of symmetric states that can be described by few parameters only regardless of the size of single systems. Chapter three is devoted to the states of tripartite systems having Werner symmetry. We characterize their separability properties and analyze the strength of the known separability criteria. Furthermore we investigate their entanglement properties in terms of the relative entropy of entanglement, the entanglement monotone induced by the trace norm distance and the maximal violation of Bell inequalities. As a third aspect we analyze the possibility of embedding bipartite Werner states and explore the inner geometry of the manifold given by the statistical distance of two neighbouring tripartite Werner states. In the fourth chapter we introduce multipartite quantum data hiding as an application of multipartite Werner states. We prove the security of this protocol for any coalition and give various examples. The last chapter is concerned with the shared fidelity, its frustration and how it can be maximized in certain multipartite systems.
Local unitary (LU) equivalent states are crucial for understanding the structure of quantum entanglement. We investigate the LU equivalence based on the LU Bargmann invariants (LUBIs). We consider for which states the LUBIs are sufficient to ensure the LU equivalence. We focus on a class of multipartite states with complete local commutativity, called CLC states. We first show that for multipartite CLC states, all the LUBIs are sufficient to ensure the local permutation equivalence, if their single-party marginals are diagonal and non-degenerate. By virtue of the fact that the CLC property remains unchanged under local operations, we verify that the LUBIs are also sufficient to ensure the LU equivalence of multipartite CLC states whose single-party marginals are non-degenerate. Furthermore, when the single-party marginals are degenerate, we propose concrete examples of two-qudit and three-qubit systems, showing that there are states which share all the LUBIs but are not LU equivalent.
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.