Quantifying the Coherence between Coherent States

In this Letter, we detail an orthogonalization procedure that allows for the quantification of the amount of coherence present in an arbitrary superposition of coherent states. The present construction is based on the quantum coherence resource theory introduced by Baumgratz, Cramer, and Plenio and the coherence resource monotone that we identify is found to characterize the nonclassicality traditionally analyzed via the Glauber-Sudarshan $P$ distribution. This suggests that identical quantum resources underlie both quantum coherence in the discrete finite dimensional case and the nonclassicality of quantum light. We show that our construction belongs to a family of resource monotones within the framework of a resource theory of linear optics, thus establishing deeper connections between the class of incoherent operations in the finite dimensional regime and linear optical operations in the continuous variable regime.

Figure 1

$\alpha$ coherence for photonic states. (a) Even (solid line) and odd (dotted line) cat states $|\alpha ⟩\pm |-\alpha ⟩$, (b) Fock states $|n⟩$, and (c) squeezed states $S(\xi )|0⟩$ are compared.

Figure 2

$\alpha$ coherence plotted against mean photon numbers $⟨n⟩=⟨a^{†}a⟩$. Even (solid line) and odd (dotted line) cat states $|\alpha ⟩\pm |-\alpha ⟩$, Fock states (circular points) $|n⟩$, and squeezed states (dashed-dotted line) $S(\xi )|0⟩$ are compared.