Dimension-free entanglement detection in multipartite Werner states

Abstract

Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a complete characterization of their entanglement that is independent of the underlying local Hilbert space: for every entangled Werner state there exists a dimension-free entanglement witness. The construction of such a witness is formulated as an optimization problem. To solve it, two semidefinite programming hierarchies are introduced. The first one is derived using real algebraic geometry applied to positive polynomials in the entries of a Gram matrix, and is complete in the sense that for every entangled Werner state it converges to a witness. The second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables, and is a relaxation that involves smaller semidefinite constraints.

updated abstract

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• Positive maps from the walled Brauer algebra
• $su(d)$-squeezing and many-body entanglement geometry in finite-dimensional systems
• Refuting spectral compatibility of quantum marginals
• State-witness contraction
• Entanglement and distillation from symmetric positive maps
• Positivity of state, trace, and moment polynomials, and applications in quantum information
• Iterative construction of ${\mathfrak{S}}_p\times {\mathfrak{S}}_p$ group-adapted irreducible matrix units for the walled Brauer algebra