Quantum parameter estimation on coherently superposed noisy channels

A generic qubit unitary operator affected by quantum noise is duplicated and inserted in a coherently superposed channel, superposing two paths offered to a probe qubit across the noisy unitary, and driven by a control qubit. A characterization is performed of the transformation realized by the superposed channel on the joint state of the probe-control qubit pair. The superposed channel is then specifically analyzed for the fundamental metrological task of phase estimation on the noisy unitary, with the performance assessed by the Fisher information, classical or quantum. A comparison is made with conventional estimation techniques and also with a quantum switched channel with indefinite causal order recently investigated for a similar task of phase estimation. In the analysis here, a first important observation is that the control qubit of the superposed channel, although it never directly interacts with the unitary being estimated, can nevertheless be measured alone for effective estimation, while discarding the probe qubit that interacts with the unitary. This property is also present with the switched channel but is inaccessible with conventional techniques. The optimal measurement of the control qubit here is characterized in general conditions. A second important observation is that the noise plays an essential role in coupling the control qubit to the unitary, and that the control qubit remains operative for phase estimation at very strong noise, even with a fully depolarizing noise, whereas conventional estimation and the switched channel become inoperative in these conditions. The results extend the analysis of the capabilities of coherently controlled channels which represent novel devices exploitable for quantum signal and information processing.

Figure 1

A quantum state $\rho$ as an input signal can be sent across either channel (1) or channel (2), according to the state ${\rho }_c$ of a control qubit.

Figure 2

A qubit channel formed by the unitary transformation ${\mathsf{U}}_{\xi }$ of Eq. (16) and the general qubit noise $\mathcal{N}(·)$ of Eq. (18). As a whole this channel is an instance of channel (1) or (2) considered in Fig. 1.

Figure 3

The vectors in ${\mathbb{R}}^3$ from the coherently superposed channel ensuring maximum estimation efficiency $F_c^{\mathrm{con}}\left(\xi \right)$ in Eq. (43), with the rotation axis $\vec{n}$ orthogonal to $\vec{s}$ of Eq. (28), an input probe of unit Bloch vector $\vec{r}//\vec{s}$, and a rotated Bloch vector ${\vec{r}}_1\left(\xi \right)=U_{\xi }\vec{r}$ remaining orthogonal to $\vec{n}$ for any phase angle $\xi$ to be estimated.

Figure 4

Phase-averaged Fisher information ${\overline{F}}_c^{\mathrm{con}}\left(\xi \right)={\overline{F}}_q^{\mathrm{con}}\left(\xi \right)$ of Eq. (54) via Eq. (64) for the control qubit of the coherently superposed channel (solid line), ${\overline{F}}_c\left(\xi \right)$ of Eq. (65) for the standard probing qubit (dotted line), and ${\overline{F}}_c^{\mathrm{swi}}\left(\xi \right)={\overline{F}}_q^{\mathrm{swi}}\left(\xi \right)$ of Eq. (66) for the control qubit of the switched channel with indefinite order of [9] (dashed line), as a function of the contraction factor $\alpha$ of the depolarizing noise of Eq. (59).