Estimating phase with a random generator: Strategies and resources in multiparameter quantum metrology

Quantum metrology aims to exploit quantum phenomena to overcome classical limitations in the estimation of relevant parameters. We consider a probe undergoing a phase shift $\phi$ whose generator is randomly sampled according to a distribution with unknown concentration $\kappa$, which introduces a physical source of noise. We then investigate strategies for the joint estimation of the two parameters $\phi$ and $\kappa$ given a finite number $N$ of interactions with the phase imprinting channel. We consider both single qubit and multipartite entangled probes, and identify regions of the parameters where simultaneous estimation is advantageous, resulting in up to a twofold reduction in resources. Quantum enhanced precision is achievable at moderate $N$, while for sufficiently large $N$ classical strategies take over and the precision follows the standard quantum limit. We show that full-scale entanglement is not needed to reach such an enhancement, as efficient strategies using significantly fewer qubits in a scheme interpolating between the conventional sequential and parallel metrological schemes yield the same effective performance. These results may have relevant applications in optimization of sensing technologies.

Figure 1

General scheme for quantum metrology. Each qubit of an $M$-partite entangled state is subjected to a sequence of $N/M$ applications of a channel $\mathrm{\Lambda }$, which imprints the parameters to be estimated onto the probe. For $M=1$ this reduces to a sequential scheme relying on single qubit coherence and $N$ iterations of the channel, while for $M=N$ this reduces to a fully parallel scheme relying on $N$-qubit entanglement and a single application of the channel on each qubit. All the plotted quantities are dimensionless.

Figure 2

Bloch ball representation of a qubit noisy evolution, modeled by a phase-covariant unital channel, whose action consists of a rotation around the $z$ axis by an angle $g\left(\phi \right)$ and a contraction in the horizontal plane by factor ${\lambda }_{\perp }\left(\phi \right)$, as well as a contraction in the $z$ direction by factor ${\lambda }_{\parallel }\left(\phi \right)$. All the plotted quantities are dimensionless.

Figure 3

Multiparameter estimation with single qubit probes. Top: the variation of the optimal state for simultaneous estimation with the parameters $\phi$ and $\kappa$. The purple dashed line separates the two possible optimal initial states to estimate $\kappa$ individually: below the line, any equatorial state, characterized by $\theta =\pi /2$, is optimal, while above the optimal state becomes the one with $\theta =0$. The equatorial state is always optimal when estimating $\phi$ individually instead. Note, moreover, that the regions where the equatorial state is optimal for the individual estimation of $\kappa$ and for the joint estimation of both parameters simultaneously are not the same. Bottom: ratio $R={\mathrm{\Delta }}_{\mathrm{ind}}/{\mathrm{\Delta }}_{\mathrm{sim}}$ against the parameters to be estimated. This shows $R>1$, and thus the superiority of the simultaneous scheme, over all parameter space. All the plotted quantities are dimensionless.

Figure 4

Optimal bipartite probe states to individually estimate (left) $\phi$ and (right) $\kappa$, as specified by the parameter $\alpha$ in the two qubit state $|\psi \rangle$ of Eq. (14), with $\beta =\sqrt{\frac{1}{2}-{\alpha }^2}$. The thick boundaries delimit the region where the maximally entangled state ($\alpha =1/\sqrt{2}$) ceases to be optimal and is eventually superseded by a product state $(\alpha =1/2)$. The corresponding plot in the case of simultaneously estimating both parameters is almost indistinguishable from the one of estimating $\kappa$ alone (right), and is not shown here. All the plotted quantities are dimensionless.

Figure 5

Performance ratio $R={\mathrm{\Delta }}_{\mathrm{ind}}/{\mathrm{\Delta }}_{\mathrm{sim}}$ vs the parameters to be estimated, using optimal bipartite probe states. Here $R$ is always above 1 and usually close to its maximum, 2, showing the superiority of estimating both parameters simultaneously. The boundaries are the same as in Fig. 4. All the plotted quantities are dimensionless.

Figure 6

Error associated to each estimation strategy plotted as a function of the number of probe qubits, $N$, showing examples of different metrological regimes. Left: case A, produced with $\phi =0.150, \kappa =3.00$. Middle: case B, produced with $\phi =0.312, \kappa =4.31$. Right: case C, produced with $\phi =0.500, \kappa =5.50$. All the plotted quantities are dimensionless.

Figure 7

Left: the optimal number $N_{\mathrm{opt}}$ of entangled probes as a function of the phase parameter $\phi$ and noise parameter $\kappa$. The solid boundary ${\mathrm{\Delta }}_{\min }^{\mathrm{sim}}={\mathrm{\Delta }}_{\min }^{\mathrm{ind}}$ separates case A, where the $N_{\mathrm{opt}}=N_{\mathrm{opt}}^{\mathrm{ind}}$, and case B, where $N_{\mathrm{opt}}=N_{\mathrm{opt}}^{\mathrm{sim}}$. The dashed boundary $R_{\min }=1$ separates case B, where simultaneous and individual measurements can each give better performance for some $N$, and case C, where the simultaneous strategy is always preferred, until the classical strategy takes over with large $N$. Right: contour plot of the corresponding optimal error ${\mathrm{\Delta }}_{min}$ on the estimation of $\phi$ and $\kappa$ at $N=N_{\mathrm{opt}}$ for the strategies described above. Once more, the solid boundary separates case A, where the plotted ${\mathrm{\Delta }}_{min}$ corresponds to the error in the individual strategy, from cases B and C, where the plotted ${\mathrm{\Delta }}_{min}$ corresponds to the error in the simultaneous strategy. All the plotted quantities are dimensionless.

Figure 8

Estimation error against $M/N_{\mathrm{opt}}$ for the metrological scheme of Fig. 1. Top: an example of the behavior of the precision in the case of the individual estimation scheme being optimal. Here, $\phi =0.11$ and $\kappa =1.7$, which gives $N_{\mathrm{opt}}=120$. The precision falls within 5% of that of a GHZ state of $N_{\mathrm{opt}}$ qubits after using a GHZ state of only six qubits (5% of $N_{\mathrm{opt}}$). Bottom: here the simultaneous scheme is optimal and $\phi =0.27$ and $\kappa =4.6$ are used, again giving $N_{\mathrm{opt}}=120$. The precision reaches 5% of that of the $N_{\mathrm{opt}}$-qubit GHZ state after a GHZ state of only 12 qubits (10% of $N_{\mathrm{opt}}$) is used. All the plotted quantities are dimensionless.