Absolute separability witnesses for symmetric multiqubit states

Entanglement is a valuable resource for quantum applications, and a well-established method for creating entangled multiqubit symmetric states in
a controlled manner is the application of a global unitary operation. However, certain states, called symmetric absolutely separable (SAS), remain
unentangled after any unitary gate preserving permutation invariance in the constituents of the system. In this work, we develop criteria for detecting
SAS states of any number of qubits [1, 2]. Our approach is based on the Glauber-Sudarshan P representation for finite-dimensional quantum systems.
We introduce families of linear and non-linear SAS witnesses formulated respectively as algebraic inequalities or a quadratic optimization problem. These
witnesses are capable of identifying more SAS states than previously known counterparts [3]. [1] E. Serrano-Ensástiga and J. Martin, SciPost Phys. 15, 120 (2023).
[2] E. Serrano-Ensástiga, J. Denis, and J. Martin, Phys. Rev. A 109, 022430 (2024).
[3] F. Bohnet-Waldraff, O. Giraud, and D. Braun, Phys. Rev. A 95, 012318 (2017).