  Active microscopic objects, e.g., an enzyme molecule, are modeled by the
Langevin system with the odd elasticity, in which energy injection from the
substrate to the enzyme is described by the antisymmetric part of the elastic
matrix. By applying the Onsager--Machlup integral and large deviation theory to
the Langevin system with odd elasticity, we can calculate the cumulant
generating function of the irreversibility of the state transition. For an
$N$-component system, we obtain a formal expression of the cumulant generating
function and demonstrate that the oddness $\lambda$, which quantifies the
antisymmetric part of the elastic matrix, leads to higher-order cumulants that
do not appear in a passive elastic system. To demonstrate the effect of the
oddness under the concrete parameter, we analyze the simplest two-component
system and obtain the optimal transition path and cumulant generating function.
The cumulants obtained from expansion of the cumulant generating function
increase monotonically with the oddness. This implies that the oddness causes
the uncertainty of stochastic state transitions.
