We describe explicitly how entanglement between quantum mechanical subsystems
can lead to emergent gauge symmetry in a classical limit. We first provide a
precise characterisation of when it is consistent to treat a quantum subsystem
classically in such a limit, namely: in any quantum state corresponding to a
definite classical state in the classical limit, the reduced density matrix of
the subsystem must be approximately proportional to a projection operator, and
the projection operators for different classical subsystem states must obey an
approximate mutual orthogonality condition. These are strong constraints on the
entanglement structure of classical states. They generically give rise to
fundamentally non-local classical degrees of freedom, which may nevertheless be
accounted for using a completely local kinematical description, if one gauges
this description in the right way. The mechanism we describe is very general,
but for concreteness we exhibit a toy example involving three entangled spins
at high angular momentum, and we also describe a significant group-theoretic
generalisation of this toy example. Finally, we give evidence that this
phenomenon plays a role in the emergence of bulk diffeomorphism invariance in
gravity.
