Parameter estimation in memory-assisted noisy quantum interferometry

We demonstrate that memory in an $N$-qubit system subjected to decoherence, is a potential resource for the slowdown of the entanglement decay. We show that this effect can be used to retain the sub-shot-noise sensitivity of the parameter estimation in quantum interferometry. We calculate quantum Fisher information, which sets the ultimate bound for the precision of the estimation. We also derive the sensitivity of such a noisy interferometer, when the phase is either estimated from the measurements of the population imbalance or from the one-body density.

Figure 1

The temporal behavior of the damping factor $exp(-{\Gamma }_{+})$ for $\Omega {\tau }_c=0$ (solid black), $\Omega {\tau }_c=2.5$ (dashed red), and $\Omega {\tau }_c=10$ (dotted blue). The value of ${\Gamma }_{+}$ is calculated for the Ornstein-Uhlenbeck process [32], for which $\kappa \left(t\right)={\omega }_0^{2}exp(-\frac{\left|t\right|}{{\tau }_c})$ with ${\omega }_0{\tau }_c=1$. The inset focuses on early times and compares the colored-noise cases with the white-noise limit $exp(-{\omega }_0^2{\tau }_{c}t)$ (dot-dashed green).

Figure 2

The QFI in units of ${\tau }^2$ for different input states ${\widehat{\varrho }}_S\left(0\right)$ of $N=100$ particles parametrized with $\gamma$ of the Bose-Hubbard Hamiltonian (see text for details) and with ${\omega }_z\left(t\right)=0$. The figure compares the QFI at $\tau =0.5{\tau }_c$ in the absence of noise (solid black), in the presence of memory (dashed red), and in the white-noise limit (dotted blue). The gray horizontal dot-dashed line denotes the shot-noise limit $F_Q=100$. The inset shows the QFI for the isotropic case in the presence of memory (dashed red) and in the white-noise limit (dotted blue). As in Fig. 1 the colored noise is the Ornstein-Uhlenbeck process with ${\omega }_0{\tau }_c=1$ and $\Omega {\tau }_c=10$.

Figure 3

Illustration of the structure of the initial density matrices ${\widehat{\varrho }}_S\left(0\right)$ generated with the Bose-Hubbard Hamiltonian. The contour plot shows the distribution $K_m={\sum }_{j=m}^N{\left|\mathrm{Tr}\left(\widehat{\varrho }\left(0\right){\widehat{T}}_m^{\left(j\right)}\right)\right|}^2$. For $\gamma \simeq 0$, the density matrix consist of close-to-diagonal terms, so only the tensors with $m\simeq 0$ contribute. When $\gamma \lesssim -1$, there is a growing contribution from large $m$'s.

Figure 4

The comparison of Fisher information for the population imbalance estimation, Eq. (35), and QFI plotted for different input states ${\widehat{\varrho }}_S\left(0\right)$ of $N=100$ particles parametrized with $\gamma$ of the Bose-Hubbard Hamiltonian (see text for details). The evolution time is $\tau =0.5{\tau }_c$ and the noise is the Ornstein-Uhlenbeck process with ${\omega }_0{\tau }_c=1$ and ${\omega }_z\left(t\right)=0$. The upper (lower) solid curve is QFI (Fisher) information for $\Omega {\tau }_c=10$, the upper (lower) dotted curve is QFI (Fisher) information for $\Omega {\tau }_c=9$, and the upper (lower) dashed curve is QFI (Fisher) information for $\Omega {\tau }_c=11$. The horizontal dot-dashed line represents the shot-noise limit $F=100$.

Figure 5

The Fisher information from Eq. (43) for different input states calculated with the Bose-Hubbard Hamiltonian (31). The solid black line is the $\tilde{F}$ for the presence of noise. We use the same model of the Ornstein-Uhlenbeck process as in Fig. 1 with $\Omega {\tau }_c=10$ for $\tau =0.5{\tau }_c$. The dotted line is the maximal value of $\tilde{F}$ in the absence of noise. For reference, the ultimate bound of the QFI is shown with the red dashed line. The dot-dashed gray line represents the shot noise limit $\tilde{F}=100$.