Kronecker states: a powerful source of multipartite maximally entangled states in quantum information (2024)
In quantum information theory, maximally entangled states are essential for well-known protocols like quantum teleportation or quantum key distribution. While many of these protocols focus on bipartite entanglement, other applications, such as quantum error correction or multiparty quantum secret sharing, are based on multipartite entanglement, precisely, on the so-called locally maximally entangled (LME) multipartite states, where each part is maximally entangled with their complement. Such LME states appear naturally in the invariant subspaces of tensor products of irreducible representations of the symmetric group Sn, which we term Kronecker subspaces, given that their dimensions are the so-called Kronecker coefficients. A Kronecker subspace is a vector space of LME multipartite states that we call Kronecker states, which entangle Hilbert spaces of large dimensions. Although such states can in principle be obtained from the Clebsch-Gordan coefficients of the symmetric group, the known methods to compute these coefficients tend to be inefficient even for small values of n. An alternative quantum-information-based approach is inspired by entanglement concentration protocols, where Kronecker subspaces appear naturally in the isotypic decomposition of tensor products of copies of multipartite entangled states. In this context, closed expressions have been obtained for a limited class of Kronecker states, associated with states in the so-called multiqubit Wclass. Our aim in this thesis is to extend this approach to build bases for Kronecker subspaces associated with any multiqubit system. For developing our method we first propose a graphical construction that we call “W-state Stitching”, where multiqubit entangled states are obtained as tensor networks built from W states. Analyzing the isotypic decomposition of copies of the graph state, an analogous set of graph Kronecker states, made from W-Kronecker states, can be obtained. In particular, the graph states of generic multiqubit states can generate any Kronecker subspace completely. Using this method, we show how to build any Kronecker subspace corresponding to systems of three and four qubits. Independently of the Kronecker state construction, the W-stitching technique has proven to be a powerful method for multiqubit entanglement classification. We hope the results of this work motivate the study of applications of Kronecker states in quantum information, and serve as a starting point for a resource theory of multipartite entanglement, with bipartite states and tripartite W states as building blocks, where the asymptotic analysis is based on Kronecker states.