Quantum Metrology for Non-Markovian Processes

Quantum metrology is a rapidly developing branch of quantum technologies. While various theories have been established on quantum metrology for Markovian processes, i.e., quantum channel estimation, quantum metrology for non-Markovian processes is much less explored. In this Letter, we establish a general framework of non-Markovian quantum metrology. For any parametrized non-Markovian process on a finite-dimensional system, we derive a formula for the maximal amount of quantum Fisher information that can be extracted from it by an optimally controlled probe state. In addition, we design an algorithm that evaluates this quantum Fisher information via semidefinite programming. We apply our framework to noisy frequency estimation, where we find that the optimal performance of quantum metrology is better in the non-Markovian scenario than in the Markovian scenario and explore the possibility of efficient sensing via simple variational circuits.

Figure 1

Markovian versus non-Markovian quantum metrology. (a): The standard, Markovian setting of quantum metrology where the goal is to estimate $\theta$ from a sequence of quantum channels (orange), and the general strategy is to prepare a probe state and apply adaptive quantum control (blue). (b): Non-Markovian quantum metrology where $\theta$ is encoded in a non-Markovian process with inaccessible memory (orange).

Figure 2

Quantum combs and probes. A generic quantum comb $C\in \text{Comb}[({\mathcal{H}}_1,{\mathcal{H}}_2),\dots ,({\mathcal{H}}_{2N-1},{\mathcal{H}}_{2N})]$ (orange) consists of $N$ teeth, each representing one time step. A probe $T\in \text{Comb}[(\varnothing ,{\mathcal{H}}_1),({\mathcal{H}}_2,{\mathcal{H}}_3),\dots ,({\mathcal{H}}_{2N-2},{\mathcal{H}}_{2N-1}\otimes {\mathcal{H}}_{\mathrm{aux}})]$ for $C$ (blue) is a comb that “eats” $C$ and “spits out” a quantum state on ${\mathcal{H}}_{2N}\otimes {\mathcal{H}}_{\mathrm{aux}}$.

Figure 3

Link product. The link product between a comb $C\in \text{Comb}[({\mathcal{H}}_1,{\mathcal{H}}_2),({\mathcal{H}}_3,{\mathcal{H}}_4)]$ and a quantum channel $E\in \text{Comb}[({\mathcal{H}}_2,{\mathcal{H}}_3)]$ results in a channel $E*F$ from ${\mathcal{H}}_1$ to ${\mathcal{H}}_4$.

Figure 4

Noisy non-Markovian frequency estimation using a variational probe. As shown in (a), the task is to estimate $\omega$ given $N$ accesses ($N=2$ in the figure), each of duration $t$, to a qubit system $S$ with $H=\omega |1⟩⟨1|$. The system is coupled to a qubit environment $E$ by an interaction $U_{\mathrm{int}}(\tau )$, where the interaction time $\tau$ depends on $t$. To estimate $\omega$ we construct a variational probe which consists of two-qubit control operations $U({\overrightarrow{\varphi }}^{(i)})$ for $i=1,2,\dots ,N+1$ and projective measurements in the computational basis. In (b) we consider a specific variational probe: The rotations are $R_{\sigma }(\varphi )=e^{-\stackrel{\circ }{𝚤}(\varphi /2)\sigma }$ for $\sigma =X$, $Z$ and $R_{ZZ}(\varphi )=e^{-\stackrel{\circ }{𝚤}(\varphi /2)Z\otimes Z}$.

Figure 5

QFIs as a function of the total sampling time $t_{\mathrm{tot}}$ for noisy frequency estimation. We consider the comb depicted in Fig. 4 with the SWAP interaction for different value of $N$. As a comparison, we also consider the corresponding Markovian scenario, where the environment state is traced out and reset to $|0⟩$ after each interaction. The non-Markovian scenario with optimal control (red, solid) allows for the highest QFI. In particular, the QFI is higher than those of the scenario without control (red, dashed) and its Markovian counterpart (blue, solid). The Markovian scenario without control (blue, dashed) sees the worst performance. Notice that $N=1$ corresponds to the task of single channel estimation, where all scenarios coincide. When the interaction is trivial ($U_{\mathrm{int}}(\tau )=\pm \mathbb{1}$), the noiseless scaling (black, dashed) is achieved in all scenarios.

Figure 6

Performance of the variational probe. Performance of the variational probe (blue part of Fig. 4) is evaluated in terms of the Fisher information of the output probability distribution. Here the green dots are the Fisher information of the variational probe’s output and the red, solid lines are the maximal attainable Fisher information.