Multisetting protocol for Bell correlated states detection with spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> systems

We propose a multisetting protocol for the detection of two-body Bell correlations and apply it to spin-nematic squeezed states realized in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>f</a:mi> </a:math> pairs of SU(2) subsystems within spin- <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>f</b:mi> </b:math> atomic Bose-Einstein condensates. Experimental data for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>f</c:mi> <c:mo>=</c:mo> <c:mn>1</c:mn> </c:mrow> </c:math> , together with numerical simulations using the truncated Wigner method for <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>f</d:mi> <d:mo>=</d:mo> <d:mn>1</d:mn> <d:mo>,</d:mo> <d:mspace width="0.16em"/> <d:mn>2</d:mn> <d:mo>,</d:mo> <d:mspace width="0.16em"/> <d:mn>3</d:mn> </d:mrow> </d:math> , demonstrate the protocol's effectiveness. We derive Bell inequalities tailored to collective observables of these subsystems and show how tuning interaction parameters controls the number of independent settings. Our findings extend multisetting Bell tests to high-spin ultracold ensembles, enabling scalable, flexible measurement configurations. This establishes a pathway to quantum metrology and information processing in tunable many-body platforms.