Quantifying Nonclassicality of Mixed Bosonic States

We present progress towards quantifying the nonclassicality of a bosonic mode in a mixed state. In particular, we calculate 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, for density matrices that are diagonal in the Fock basis. Generally speaking, evaluating the ORT measure for mixed states is challenging, since it involves finding a convex roof. However, our particular problem can be reduced to a linear optimization problem. By analyzing the results of numerical optimization, we are able to extract exact, analytical results for the case where three neighboring Fock states have nonzero population. Interestingly, we find that such a mode can be in three distinct phases, depending on the populations; these results also provide a means to examine the conservation of nonclassicality when a two-photon state enters a beamsplitter and splits into two entangled modes, which has not been studied before. Lastly, we demonstrate how our method is generalizable to density matrices containing four or more nonzero populations.