Memory Effects in Quantum Metrology

Quantum metrology concerns estimating a parameter from multiple identical uses of a quantum channel. We extend quantum metrology beyond this standard setting and consider the estimation of a physical process with quantum memory, here referred to as a parametrized quantum comb. We present a theoretic framework of metrology of quantum combs, and derive a general upper bound of the comb quantum Fisher information. The bound can be operationally interpreted as the quantum Fisher information of a memoryless quantum channel times a dimensional factor. We then show an example where the bound can be attained up to a factor of 4. With the example and the bound, we show that memory in quantum sensors plays an even more crucial role in the estimation of combs than in the standard setting of quantum metrology.

Figure 1

Estimating a $K$ comb with a sensor. In quantum comb metrology, the goal is to estimate a parameter $\theta$ from a $K$ comb ${\mathcal{N}}_{\theta }$ consisting of $K$ phases (plotted in black). For this purpose, one uses a sensor $\mathcal{S}$ (plotted in blue), which consists of preparing an input state and applying quantum gates in between the phases.

Figure 2

Bounding the memory effect of ${\mathcal{N}}_{\theta }$. The wires connecting the sensor to the comb ${\mathcal{N}}_{\theta }$ can be winded using probabilistic teleportation, as plotted in Fig. 2. The QFI of ${\mathcal{N}}_{\theta }$ can thus be bounded in terms of the QFI of the quantum state ${\mathrm{\Phi }}_{{\mathcal{N}}_{\theta }*{\mathrm{\Psi }}_1}$, which is depicted in Fig. 2.

Figure 3

Estimating a protected parameter. An experimenter is to estimate a protected parameter $\theta$ from a $K$ comb, consisting of $K$ blue boxes (whose latent structure is invisible) with input and output ports. The first box performs a parametrized gate ${\mathcal{V}}_{\theta }$, and mixes the resultant state with a fixed state by a random unitary $\mathcal{U}$ that serves as the shield. Accessing the remaining $K-1$ boxes jointly will result in a key unitary ${\mathcal{U}}^{†}$, which recovers the information on $\theta$. If the experimenter fails to do so, the information on $\theta$ will be deteriorated.