  Nonclassical states of bosonic modes are important resources for
quantum-enhanced technologies. Yet, quantifying nonclassicality of these
states, in particular mixed states, can be a challenge. Here we present results
of quantifying the nonclassicality of a bosonic mode in a mixed Fock state via
the operational resource theory (ORT) measure [W. Ge, K. Jacobs, S. Asiri, M.
Foss-Feig, and M. S. Zubairy, Phys. Rev. Res. 2, 023400 (2020)], which relates
nonclassicality to metrological advantage. Generally speaking, evaluating the
ORT measure for mixed states is challenging, since it involves finding a convex
roof. However, we show that our problem can be reduced to a linear programming
problem. By analyzing the results of numerical optimization, we are able to
extract exact, analytical results for the case where three or four neighboring
Fock states have nonzero population. Interestingly, we find that such a mode
can be in distinct phases, depending on the populations. Lastly, we demonstrate
how our method is generalizable to density matrices of higher ranks. Our
findings suggests a viable method for evaluating nonclassicality of arbitrary
mixed bosonic states and potentially for solving other convex roof optimization
problems.
