Collective uncertainty in partially polarized and partially decohered spin-(1)/(2) systems

It has become common practice to model large spin ensembles as an effective pseudospin with total angular momentum $J=\mathit{Nj}$, where $j$ is the spin per particle. Such approaches (at least implicitly) restrict the quantum state of the ensemble to the so-called symmetric Hilbert space. Here, we argue that symmetric states are not generally well preserved under the type of decoherence typical of experiments involving large clouds of atoms or ions. In particular, symmetric states are rapidly degraded under models of decoherence that act identically but locally on the different members of the ensemble. Using an approach [Phys. Rev. A 78, 052101 (2008)] that is not limited to the symmetric Hilbert space, we explore potential pitfalls in the design and interpretation of experiments on spin-squeezing and collective atomic phenomena when the properties of the symmetric states are extended to systems where they do not apply.