  Following on our previous work arXiv:2204.07593 and arXiv:2306.01043 studying
the orbits of quantum states under Clifford circuits via `reachability graphs',
we introduce `contracted graphs' whose vertices represent classes of quantum
states with the same entropy vector. These contracted graphs represent the
double cosets of the Clifford group, where the left cosets are built from the
stabilizer subgroup of the starting state and the right cosets are built from
the entropy-preserving operators. We study contracted graphs for stabilizer
states, as well as W states and Dicke states, discussing how the diameter of a
state's contracted graph constrains the `entropic diversity' of its $2$-qubit
Clifford orbit. We derive an upper bound on the number of entropy vectors that
can be generated using any $n$-qubit Clifford circuit, for any quantum state.
We speculate on the holographic implications for the relative proximity of
gravitational duals of states within the same Clifford orbit. Although we
concentrate on how entropy evolves under the Clifford group, our double-coset
formalism, and thus the contracted graph picture, is extendable to generic gate
sets and generic state properties.
