Kronecker states: a powerful source of multipartite maximally entangled states in quantum information

• URL: http://arxiv.org/abs/2504.16256v1
• Date: Tue, 22 Apr 2025 20:30:52 GMT
• Title: Kronecker states: a powerful source of multipartite maximally entangled states in quantum information
• Authors: Walther Gonzalez,
• Abstract summary: In quantum information theory, maximally entangled states, specifically locally maximally entangled (LME) states, are essential for quantum protocols.<n>While many focus on bipartite entanglement, applications such as quantum error correction and multiparty secret sharing rely on multipartite entanglement.<n>We first propose a graphical construction called W-state Stitching, where multiqubit entangled states are represented as tensor networks built from $W$ states.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: In quantum information theory, maximally entangled states, specifically locally maximally entangled (LME) states, are essential for quantum protocols. While many focus on bipartite entanglement, applications such as quantum error correction and multiparty secret sharing rely on multipartite entanglement. These LME states naturally appear in the invariant subspaces of tensor products of irreducible representations of the symmetric group $S_n$, called Kronecker subspaces, whose dimensions are the Kronecker coefficients. A Kronecker subspace is a space of multipartite LME states that entangle high-dimensional Hilbert spaces. Although these states can be derived from Clebsch-Gordan coefficients of $S_n$, known methods are inefficient even for small $n$. A quantum-information-based alternative comes from entanglement concentration protocols, where Kronecker subspaces arise in the isotypic decomposition of multiple copies of entangled states. Closed forms have been found for the multiqubit $W$-class states, but not in general. This thesis extends that approach to any multiqubit system. We first propose a graphical construction called W-state Stitching, where multiqubit entangled states are represented as tensor networks built from $W$ states. By analyzing the isotypic decomposition of copies of these graph states, corresponding graph Kronecker states can be constructed. In particular, graph states of generic multiqubit systems can generate any Kronecker subspace. We explicitly construct bases for three- and four-qubit systems and show that the W-stitching technique also serves as a valuable tool for multiqubit entanglement classification. These results may open new directions in multipartite entanglement resource theories, with bipartite and tripartite $W$ states as foundational elements, and asymptotic analysis based on Kronecker states.

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