Digital Quantum Estimation

Quantum metrology calculates the ultimate precision of all estimation strategies, measuring what is their root-mean-square error (RMSE) and their Fisher information. Here, instead, we ask how many bits of the parameter we can recover; namely, we derive an information-theoretic quantum metrology. In this setting, we redefine “Heisenberg bound” and “standard quantum limit” (the usual benchmarks in the quantum estimation theory) and show that the former can be attained only by sequential strategies or parallel strategies that employ entanglement among probes, whereas parallel-separable strategies are limited by the latter. We highlight the differences between this setting and the RMSE-based one.

Figure 1

Sequential and parallel-entangled strategies. (a) Sequential strategy, where a single probe (large triangle) samples $N$ unitaries $U_{\phi }$ (black boxes) sequentially. Ancillary systems (small triangles) may interact through arbitrary intermediate unitaries (gray squares). (b) QPEA. To see that it is equivalent to a sequential strategy [26], where the last unitary is the inverse quantum Fourier transform (${\mathrm{QFT}}^{†}$), use intermediate unitaries that swap the state of the ancillas with the state of the probe. The output (measured in the computational basis) is a $t$-bit digital estimate of the parameter $\phi$ with $t={\log }_2(N+1)$. (c) Parallel QPEA, which uses entangled N00N states (dashed boxes) composed of $1,2,4,\dots ,2^{t-1}$ qubits. The circles represent CNOT gates that remove the entanglement, and the cups represent the discarding of qubits in the state $|0⟩$.

Figure 2

Heisenberg bound of the QPEA. (a) Plot of the mutual information $I(m:\phi )$ as a function of $N$ (blue line) and of the function ${\log }_{2}N$ (dashed red line). Note that $I$ quickly acquires the same linear dependence in a log scale as the Heisenberg bound. The inset shows the same behavior for large $N$. (b) Ratio between the mutual information and ${\log }_{2}N-1.22$, showing the rapid onset of the asymptotic behavior to this quantity.

Figure 3

Standard quantum limit for unentangled probes. (a) Plot of the mutual information $I(\varphi :\phi )$ relative to the optimal POVM (6), which uses entangled measurements, as a function of $N$ (blue line) and of the standard quantum limit ${\log }_2(N)/2$ (cyan dashed line). The fluctuations are due to the Monte Carlo integration used here. (b) Plot of the mutual information $I(\overrightarrow{m}:\phi )$ of (12), relative to the separable POVM that projects onto $|\pm ⟩$ each probe, as a function of $N$ (blue line) and of the standard quantum limit ${\log }_2(N)/2$ (cyan dashed line). The inset shows the large-$N$ scaling. Both cases asymptotically scale at the standard quantum limit (apart from a small additive constant).