Classical Second-Order Moments and Tensor Squeezing in Spin-1 Systems

We give a compact, frame-independent characterization of the set of classical second-order moments for a single spin-1 particle. Defining the moment matrix M = 2Q + (1/3) I, we show that a moment pair (s, Q) arises from a positive mixture of spin-coherent states if and only if M is positive semidefinite, M minus ss^T is positive semidefinite, and the trace of M equals one. These necessary and sufficient matrix conditions delimit the classical moment region and yield simple, basis-free witnesses of higher-order tensor nonclassicality, such as bounds on Tr(Q^2). A constructive proof of sufficiency is given in the appendix.

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