Multiple phase estimation in quantum cloning machines

Since the initial discovery of the Wootters-Zurek no-cloning theorem, a wide variety of quantum cloning machines have been proposed aiming at imperfect but optimal cloning of quantum states within its own context. Remarkably, most previous studies have employed the Bures fidelity or the Hilbert-Schmidt norm as the figure of merit to characterize the quality of the corresponding cloning scenarios. However, in many situations, what we truly care about is the relevant information about certain parameters encoded in quantum states. In this work, we investigate the multiple phase estimation problem in the framework of quantum cloning machines, from the perspective of quantum Fisher information matrix (QFIM). Focusing on the generalized $d$-dimensional equatorial states, we obtain the analytical formulas of QFIM for both universal quantum cloning machine (UQCM) and phase-covariant quantum cloning machine (PQCM), and prove that PQCM indeed performs better than UQCM in terms of QFIM. We highlight that our method can be generalized to arbitrary cloning schemes where the fidelity between the single-copy input and output states is input-state independent. Furthermore, the attainability of the quantum Cramér-Rao bound is also explicitly discussed.

Figure 1

Confirmation of the inequality (33). ${\mathcal{F}}_{\mu \mu }^{\mathrm{in}}$ (orange solid line) and ${\mathcal{F}}_{\mu \mu }^{\mathrm{out}}$ (purple dashed line) represent the diagonal entries of the QFIM for the input state $\left|\psi \right(\mathit{\varphi })\rangle \langle \psi (\mathit{\varphi }\left)\right|$ and output state ${\rho }^{\mathrm{out}}\left(\mathit{\varphi }\right)$, respectively.

Figure 2

Comparison of the diagonal entries of QFIMs. ${\mathcal{F}}_{\mu \mu }^{\mathrm{UQCM}}$ (orange solid line) and ${\mathcal{F}}_{\mu \mu }^{\mathrm{PQCM}}$ (purple dashed line) correspond to UQCM and PQCM, respectively.

Figure 3

Total variances (errors) for multiple phase estimation. $E^{\mathrm{in}}$ (orange dot-dashed line), $E^{\mathrm{UQCM}}$ (green solid line), and $E^{\mathrm{PQCM}}$ (purple dashed line) represent the total errors for quantum simultaneous estimation of all the phases using the initial pure state, the output reduced state of UQCM and PQCM, respectively. The inset picture clearly shows that $E^{\mathrm{UQCM}}>E^{\mathrm{PQCM}}$.