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We study invertible states of 1d bosonic quantum lattice systems. We showthat every invertible 1d state is in a trivial phase: after tensoring with someunentangled ancillas it can be disentangled by a fuzzy analog of a finite-depthquantum circuit. If an invertible state has symmetries, it may be impossible todisentangle it in a way that preserves the symmetries, even after addingunentagled ancillas. We show that in the case of a finite unitary symmetry Gthe only obstruction is an index valued in degree-2 cohomology of $G$. We showthat two invertible $G$-invariant states are in the same phase if and only iftheir indices coincide.

A classification of phases of bosonic quantum lattice systems in one dimension