Asymptotic role of entanglement in quantum metrology

Quantum systems allow one to sense physical parameters beyond the reach of classical statistics—with resolutions greater than $1/N$, where $N$ is the number of constituent particles independently probing a parameter. In the canonical phase-sensing scenario the Heisenberg limit $1/N^2$ may be reached, which requires, as we show, both the relative size of the largest entangled block and the geometric measure of entanglement to be nonvanishing as $N\to \infty$. Yet, we also demonstrate that in the asymptotic $N$ limit any precision scaling arbitrarily close to the Heisenberg limit ($1/N^{2-\varepsilon }$ with any $\varepsilon >0$) may be attained, even though the system gradually becomes noisier and separable, so that both the above entanglement quantifiers asymptotically vanish. Our work shows that sufficiently large quantum systems achieve nearly optimal resolutions despite their relative amount of entanglement being arbitrarily small. In deriving our results, we establish the continuity relation of the quantum Fisher information evaluated for a phaselike parameter, which lets us link it directly to the geometry of quantum states, and hence naturally to the geometric measure of entanglement.

Figure 1

Quantum phase-sensing protocol—designed to most precisely sense fluctuations of a phaselike parameter $\phi$ around its given value. The system is prepared in an $N$-particle entangled state ${\rho }^N$ obtained from ${|\psi \rangle }^{\otimes N}$ by the preparation map ${\mathrm{\Lambda }}^N$ that also incorporates noise. $\phi$ is encoded on each particle by a unitary $U_{\phi }$ and the final state ${\rho }_{\phi }^N$ is measured. The procedure is repeated sufficiently many times ($\nu \gg 1$) to construct most accurate parameter estimate $\tilde{\phi }$.

Figure 2

Hierarchy of convex sets ${\mathcal{S}}_k^N$ containing all $k$-producible states. The set ${\mathcal{S}}_N^N$ contains all states acting on ${\mathcal{H}}^{\otimes N}$, while ${\mathcal{S}}_1^N$ is its subset of separable states. Shaded region is the set ${\mathcal{S}}_l^N\setminus {\mathcal{S}}_{l-1}^N$ of states with the relative size of largest entangled block: $R_{\mathrm{LEB}}=l/N$.

Figure 3

Collapse of the hierarchy of sets ${\mathcal{S}}_k^N$ allowing two sequences $\left\{{\sigma }_{\left(l\right)}^N\right\}$ and $\left\{{\rho }^N\right\}$ to approach one another with $N$, despite their contrasting metrological properties. States ${\sigma }_{\left(l\right)}^N$ are of constant $\mathrm{LEB}=l$, which [due to Eq. (6)] constrains their QFI to scale at most linearly with $N$: $F_{\mathrm{Q}}\left[{\sigma }_{\left(l\right)}^N\right]\le lN$. ${\rho }^N$ are states with their LEB rising with $N$, so that their QFI is taken to scale as $N^{\alpha }$ with some $\alpha <2$. Still, it is possible to choose the states ${\sigma }_{\left(l\right)}^N$ and ${\rho }^N$ [e.g., ones of Eq. (10)] so that for sufficiently large $N$ they become arbitrarily close to each other. In particular, for the two $N^{\prime }>N$ depicted above, both ${\sigma }_{\left(l\right)}^N\in {\mathcal{S}}_{\left(l\right)}^N$ and ${\sigma }_{\left(l\right)}^{N^{\prime }}\in {\mathcal{S}}_{\left(l\right)}^{N^{\prime }}$, whereas ${\rho }^N\in {\mathcal{S}}_k^N$ but ${\rho }^{N^{\prime }}\notin {\mathcal{S}}_k^{N^{\prime }}$ for some $k>l$, although the trace distance ($T^{\prime }<T$) between ${\sigma }_{\left(l\right)}^N$ and ${\rho }^N$ decreases with $N$. This is possible because the sets ${\mathcal{S}}_l^N$ collapse faster than ${\rho }^N$ approach ${\sigma }_{\left(l\right)}^N$.

Figure 4

Geometric measure of entanglement (GME), $E_G$, as a function of the parameter $p$ for states given in Eq. (C1). The number of qubits, $N$, is chosen to be: 2 (solid line), 3 (dashed line), 4 (dotted line), and 10 (dot-dashed line).