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Modern Physics Letters A

Multipartite states with restricted unitary symmetry

Shu Li, Jie Wang, Binfeng Wang, Lin Chen

Published Apr 6, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732326501257

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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.
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⊗U†U \otimes U^{\dagger}U⊗U† and U⊗VU \otimes VU⊗V can be expressed as 1nIn\frac{1}{n}I_nn1​In​, where UUU and VVV are arbitary n×nn\times nn×n unitary matrices. Furthermore, in tripartite systems, any quantum states commuting with U⊗U⊗U†U \otimes U \otimes U^{\dagger}U⊗U⊗U† must necessarily adopt the form: W=xIn3+y(∑i,j=1n(∣i⟩⟨j∣)⊗(∣j⟩⟨i∣))⊗InW = xI_{n^3} + y\left(\sum_{i,j=1}^n (|i\rangle \langle j|) \otimes (|j\rangle \langle i|)\right) \otimes I_nW=xIn3​+y(∑i,j=1n​(∣i⟩⟨j∣)⊗(∣j⟩⟨i∣))⊗In​, where FnF_nFn​ represents the canonical swap operator. These results provide theoretical tools for characterizing multipartite entanglement constraints and designing symmetry-protected quantum protocols.
We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of [Formula: see text] subsystems with [Formula: see text] levels each to the set of complex Hadamard matrices of order [Formula: see text]. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ([Formula: see text] or [Formula: see text]), two-unitary ([Formula: see text] and [Formula: see text] are unitary), or [Formula: see text]-unitary. Here [Formula: see text] denotes reshuffling of a matrix [Formula: see text] describing a bipartite system, and [Formula: see text] its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in [Formula: see text] dimensions.
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 derive the entangled thermal mixed states by using the formalism of phase states for a finite-dimensional algebra of a multi-qubit system in contact with an independent thermal environment of absolute temperature [Formula: see text]. Thermal entangled states describing the multi-qubit system in equilibrium with the thermal bath are a special kind of mixed states that exhibit genuine multipartite correlation. We define the unitary phase operators for a multipartite system of non-interacting qubits. Entangled density matrices are derived for qubits interacting via an Hermitian Hamiltonian of Heisenberg type [Formula: see text]. By assuming that the noisy interaction of the entangled qubit ensemble with the bath is governed by a local Hamiltonian [Formula: see text], we show that the entangled phase states can be decohered. When the multi-qubits entangled system reaches the equilibrium with the thermal bath, the decohered mixed states are identified with entangled thermal states. The thermal mixed states for bipartite and multipartite systems are explicitly expressed and their bipartite entanglement properties are investigated.
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.
Local unitary equivalence plays a crucial role in quantifying and categorizing the entanglement of quantum states. When it comes to two isospectral diagonal bipartite quantum states, determining whether they are locally unitary equivalent turns out to be a rather challenging problem. Intuitively speaking, we conjecture that for two diagonal quantum states, local unitary equivalence and local permutation equivalence between them are the same. By taking into account the cosets of the fixed point subgroup and the tensor decomposability of permutation matrices, we present the classifications of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text] under local permutation equivalence, respectively. Furthermore, relying on the classification results of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text], we demonstrate that if two diagonal quantum states are not locally permutation equivalent, then they cannot be locally unitary equivalent. This offers significant backing for the conjecture that two general bipartite diagonal quantum states are locally unitary equivalent if and only if they are locally permutation equivalent.