Nonclassicality as a Quantifiable Resource for Quantum Metrology

We establish the nonclassicality of continuous-variable states as a resource for quantum metrology. Based on the quantum Fisher information of multimode quadratures, we introduce the metrological power as a measure of nonclassicality with a concrete operational meaning of displacement sensitivity beyond the classical limit. This measure belongs to the resource theory of nonclassicality, which is nonincreasing under linear optical elements. Our Letter reveals that a single copy, highly nonclassical quantum state is intrinsically advantageous when compared to multiple copies of a quantum state with moderate nonclassicality. This suggests that metrological power is related to the degree of quantum macroscopicity. Finally, we demonstrate that metrological resources useful for nonclassical displacement sensing tasks can be always converted into a useful resource state for phase sensitivity beyond the classical limit.

Figure 1

(a) Linear optical unitary and (b) linear optical map.

Figure 2

(a) Nonclassicality measure $\mathcal{Q}$ achieves the maximum value (solid line) for NOON, cat, squeezed, and Fock states. Also, superposition between Fock state and coherent state $|n⟩+|\alpha ⟩$ with $n=|\alpha {|}^2$ (dotted line), squeezed coherent states $\widehat{S}(\xi )|\alpha ⟩$ for $\xi =1$ (dot-dashed line), and photon-added coherent states ${\widehat{a}}^{†}|\alpha ⟩$ (dashed line) are evaluated. (b) Metrological power $\mathcal{M}$ for decohered cat states ${\widehat{\rho }}_{\mathrm{\Gamma }}$ (solid lines) and squeezed thermal states $\widehat{S}(\xi )\widehat{\tau }{\widehat{S}}^{†}(\xi )$ (dashed lines) with the parameters $\mathrm{\Gamma }=0.01$, 0.3, 0.7 and ${\overline{n}}_{\mathrm{th}}=0.01$, 0.1, 0.5, 1.0 (both starting from above).

Figure 3

Relationship between nonclassical resources for the displacement and phase estimation tasks.