  We propose the convex-roof extension of quantum conditional mutual
information ("co(QCMI)") as a diagnostic of long-range entanglement in a mixed
state. We focus primarily on topological states subjected to local decoherence,
and employ the Levin-Wen scheme to define co(QCMI), so that for a pure state,
co(QCMI) equals topological entanglement entropy (TEE). By construction,
co(QCMI) is zero if and only if a mixed state can be decomposed as a convex sum
of pure states with zero TEE. We show that co(QCMI) is non-increasing with
increasing decoherence when Kraus operators are proportional to the product of
onsite unitaries. This implies that unlike a pure state transition between a
topologically trivial and a non-trivial phase, the long-range entanglement at a
decoherence-induced topological phase transition as quantified by co(QCMI) is
less than or equal to that in the proximate topological phase. For the 2d toric
code decohered by onsite bit/phase-flip noise, we show that co(QCMI) is
non-zero below the error-recovery threshold and zero above it. Relatedly, the
decohered state cannot be written as a convex sum of short-range entangled pure
states below the threshold. We conjecture and provide evidence that in this
example, co(QCMI) equals TEE of a recently introduced pure state. In
particular, we develop a tensor-assisted Monte Carlo (TMC) computation method
to efficiently evaluate the R\'enyi TEE for the aforementioned pure state and
provide non-trivial consistency checks for our conjecture. We use TMC to also
calculate the universal scaling dimension of the anyon-condensation order
parameter at this transition.
