Optimal quantum estimation in spin systems at criticality

It is a general fact that the coupling constant of an interacting many-body Hamiltonian does not correspond to any observable and one has to infer its value by an indirect measurement. For this purpose, quantum systems at criticality can be considered as a resource to improve the ultimate quantum limits to precision of the estimation procedure. In this paper, we consider the one-dimensional quantum Ising model as a paradigmatic example of a many-body system exhibiting criticality, and derive the optimal quantum estimator of the coupling constant varying size and temperature. We find the optimal external field, which maximizes the quantum Fisher information of the coupling constant, both for few spins and in the thermodynamic limit, and show that at the critical point a precision improvement of order $L$ is achieved. We also show that the measurement of the total magnetization provides optimal estimation for couplings larger than a threshold value, which itself decreases with temperature.

Figure 1

The ratio ${\gamma }_J$ as a function of the external field $h$ for $L=2$ [(a), (b)], 3 [(c), (d)], 4 [(e), (f)] and $J=0.5$ [(a), (c), (e)], $J=5$ [(b), (d), (f)]. The curves refer to different values of $\beta =1$ (black dashed), $\beta =10$ (gray dashed), $\beta =100$ (solid gray), and $\beta =1000$ (solid black).

Figure 2

The ratio $F_J(\beta ,J,h^{\sim })∕G_J(\beta ,J,h^{*})$ as a function of $J$ for $L=2$ (solid lines), $L=3$ (dotted lines), and $L=4$ (dashed lines). The bottom group of lines (gray) is for $\beta =3$, whereas the top group (black) is for $\beta =10$.

Figure 3

Estimation of the coupling constant by magnetization measurement and Bayesian analysis. The plot is for $L=2$ and $\beta =1$ with true value of the coupling equal to $J=3$. We report the variance of the Bayesian estimator for Monte Carlo simulated experiments ($M=500$, black dots), the corresponding variance evaluated using the asymptotic a posteriori distribution (dotted black line), and the (classical) Cramer-Rao bound (solid gray).