  We investigate the bipartite fidelity $\mathcal F_d$ for a lattice model
described by a logarithmic CFT: the model of critical dense polymers. We define
this observable in terms of a partition function on the pants geometry, where
$d$ defects enter at the top of the pants lattice and exit in one of the legs.
Using the correspondence with the XX spin chain, we obtain an exact closed-form
expression for $\mathcal F_d$ and compute the leading terms in its $1/N$
asymptotic expansion as a function of $x = N_A/N$, where $N$ is the lattice
width at the top of the pants and $N_A$ is the width of the leg where the
defects exit. We find an agreement with the results of St\'ephan and Dubail for
rational CFTs, with the central charge and conformal weights specialised to
$c=-2$ and $\Delta = \Delta_{1,d+1} = d(d-2)/8$.
  We compute a second instance $\mathcal {\tilde F}_2$ of the bipartite
fidelity for $d=2$ by imposing a different rule for the connection of the
defects. In the conformal setting, this choice corresponds to inserting two
boundary condition changing fields of weight $\Delta = 0$ that are logarithmic
instead of primary. We compute the asymptotic expansion in this case as well
and find a simple additive correction compared to $\mathcal F_2$, of the form
$-2\log((1+x)/(2\sqrt{x}))$. We confirm this lattice result with a CFT
derivation and find that this correction term is identical for all logarithmic
theories, independently of $c$ and $\Delta$.
