Heisenberg versus standard scaling in quantum metrology with Markov generated states and monitored environment

Finding optimal and noise robust probe states is a key problem in quantum metrology. In this paper we propose Markov dynamics as a possible mechanism for generating such states, and show how the Heisenberg scaling emerges for systems with multiple “dynamical phases” (stationary states), and noiseless channels. We model noisy channels by coupling the Markov output to “environment” ancillas, and consider the scenario where the environment is monitored to increase the quantum Fisher information of the output. In this setup we find that the survival of the Heisenberg limit depends on whether the environment receives “which phase” information about the memory system.

Figure 1

Quantum metrology with $n$ noisy channels acting on a pure state input $|{\Psi }^n\rangle$. The channel $T_{\theta }^{\left(i\right)}$ is monitored by measuring the associated “environment” and the result $c_i$ is obtained. Conditional on the measurement record $\underline{c}:=(c_1,\cdots ,c_n)$, the final probe state has quantum Fisher information $F({\Psi }^n\left(\theta \right|\underline{c}))$.

Figure 2

Model of discrete dynamics with Markov generated probe state. The memory system $A$ interacts successively (from right to left) with the probe systems $B$ via the unitary $U^{AB}$. The channel $T_{\theta }$ on $B$ is implemented by unitary coupling of the probe systems with ancillas $C$.

Figure 3

Distribution of the “total generator” $\overline{G}$ in $|{\Psi }_{ABC}^n\rangle$ is a mixture of (approximately) Gaussian distributions centered at $n{g}_0$ and $n{g}_1$; cf. (4). When $g_0\ne g_1$, the distance between the two peaks is $n(g_1-g_0)$ and the quantum Fisher information [$F=4\mathrm{Var}\left(\overline{G}\right)$] scales as $n^2$. If phase purification occurs due to “which phase” information leaking to the environment, one of the peaks decays and the variance scales as $n$.

Figure 4

Phase purification and quantum Fisher information for noise in $x$ direction with parameters ${\alpha }_0=0.1,{\beta }_0=0.7$, and $p=0.6$. Main plot: the “scaled” Fisher information $F(|{\Psi }_{AB}^n\left(\theta \right|\underline{c})\rangle )/n$ as function of “time” $n$, for two trajectories with different limiting phases (continuous blue and dotted red curves); the corresponding averages are plotted in black. Inset plot: the weights of the limiting phase $p_1^n\left(\underline{c}\right)$ and $p_0^n\left(\underline{c}\right)$ for the two given trajectories converge to 1 due to phase purification.