Quantum metrological bounds for vector parameters

Precise measurement is crucial to science and technology. However, the rule of nature imposes various restrictions on the precision that can be achieved depending on specific methods of measurement. In particular, quantum mechanics poses the ultimate limit on precision which can only be approached but never be violated. Depending on analytic techniques, these bounds may not be unique. Here, in view of prior information, we investigate systematically the precision bounds of the total mean-square error of vector parameter estimation which contains $d$ independent parameters. From quantum Ziv-Zakai error bounds, we derive two kinds of quantum metrological bounds for vector parameter estimation, both of which should be satisfied. By these bounds, we show that a constant advantage can be expected via a simultaneous estimation strategy over the optimal individual estimation strategy, which solves a long-standing problem. A general framework for obtaining the lower bounds in a noisy system is also proposed.

Figure 1

Vector parameter estimation procedure. $d$ independent dynamic maps corresponding to $d$ parameters, with $\mathit{x}$ transforming the initial state $\rho$ to the final state ${\rho }_{\mathit{x}}$. The vector estimator $\mathit{X}\left(\xi \right)$ is obtained according to results $\mathit{\xi }$ from the measurements performed on the final state ${\rho }_{\mathit{x}}$.

Figure 2

SE vs IE for the optimal state. Four bounds, multiplied by $N^2/d^3$, of the optimal probe state: ML bound ${\Delta }_1^{\mathrm{SE}}$ and MT bound ${\Delta }_2^{\mathrm{SE}}$ for the SE strategy together with ML bound ${\Delta }_1^{\mathrm{IE}}$ and MT bound ${\Delta }_2^{\mathrm{IE}}$ for the IE strategy are compared against $d$. The hatched area represents the forbidden value of precision for the SE strategy.

Figure 3

SE vs IE for multimode squeezed vacuum states. (a) ML bound ${\Delta }_1^{\mathrm{SE}}$ and MT bound ${\Delta }_2^{\mathrm{SE}}$ for the SE strategy using the $(d+1)$-mode squeezed vacuum state against $d$ and total average particle number $N$. (b) ML bound ${\Delta }_1^{\mathrm{IE}}$ and MT bound ${\Delta }_2^{\mathrm{IE}}$ for the IE strategy using $d$-identical two-mode squeezed vacuum states against $d$ and total average particle number $N$. (c) The combined bounds for SE strategy and the one for IE strategy are compared against $d$ and total average particle number $N$.

Figure 4

Noisy ML bound vs MT bound for optimal probe state. Green curved surface is for ${\Delta }_2^{\mathrm{SE}}$ and the yellow one is for ${\Delta }_1^{\mathrm{SE}}$. (a)–(d) are for the photon loss model: two bounds for state (5) are compared against particle number $N$ and intensity of the transmissivity $\eta$. (e)–(h) are for the phase diffusion model: two bounds for state (5) are compared against particle number $N$ and diffusion parameter $\eta$.