2026/02/17 by Aabhas Gulati, Ion Nechita, Clément Pellegrini · 1 voice
Computer Science · Mathematics · #Algebraic structures and combinatorial models #Diagonal #Dimension (graph theory) #Dual polyhedron #Multipartite #Multipartite entanglement #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum entanglement #Semidefinite programming #Separable state #Squashed entanglement #Subspace topology #math-ph #quant-ph
paper · pdf · doi:10.22331/q-2026-07-01-2150
published in Quantum 10, 2150 (Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften)
openalex publication_date 2026/07/01 · openalex created_date 2026/07/02 · openalex updated_date 2026/08/01
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors.This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. We establish that PPT entanglement exists for all multipartite systems with local dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> <mml:mo>&#x2265;</mml:mo> <mml:mn>3</mml:mn> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>&#x2265;</mml:mo> <mml:mn>3</mml:mn> </mml:math> parties, disproving a recent conjecture. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions.We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.