Nonclassicality of Mixed States with Photon Number Coherence*

Nonclassicality measures can be used to characterize exotic states for quantum-enhanced sensing. One such measure is the operational resource theory (ORT) measure, which is both resource-theoretic (weakly monotonic under classical operations) and lower-bounded by metrological power. Calculating the ORT measure for mixed states is challenging, but was previously achieved for states that are diagonal in the photon-number basis. Here we calculate the ORT measure for mixed states that have nonzero coherence in the photon-number basis. We obtain exact results for a partially coherent mixture of two neighboring Fock states |n> and |n+1>. Interestingly, we find that, below a certain threshold, the nonclassicality is invariant under changes to the photon number coherence (and thus equal to that of a completely incoherent mixture). Above this threshold, the nonclassicality increases with the photon number coherence, reaching a maximum for pure superpositions. Lastly, we show that, given certain symmetries, we can also numerically calculate the nonclassicality for partially coherent mixtures of three neighboring Fock states |n>, |n+1>, and |n+2>.