Quantum Fisher information in moving frames

Abstract

In the field of quantum metrology, a significant application is the quantum parameter estimation. As the fundamental theory of quantum parameter estimation, quantum Cramér-Rao inequality shows that the variance of parameter estimation is determined by the inverse of quantum Fisher information. A higher quantum Fisher information corresponds to a lower variance, and thereby improves the precision of parameter estimation. Quantum Fisher information has been extensively investigated in many aspects of non-relativistic quantum mechanics, including entanglement structure detection, quantum teleportation, quantum phase transition, quantum chaos, quantum computation and so on. However, there are few researches considering the impact of relativistic effects on quantum Fisher information, and therefore, we attempt to investigate this topic in this work. The relativistic transformation for particle states is employed, and the quantum Fisher information of amplitude parameter <i>θ</i> and phase parameter <i>φ</i> are investigated in moving frames. In this paper, the parameters to be estimated are encoded into the spin degree of freedom, and both the pure single-qubit state and the pure two-qubit state are considered. The quantum Fisher information for <i>θ</i> and <i>φ</i> of both single-qubit state and two-qubit state in moving frames are numerically calculated, respectively. It can be observed that the quantum Fisher information is associated with rapidity, amplitude parameters, and the ratio of the width to the particle mass <i>σ</i>_r/<i>m</i>. The quantum Fisher information of the estimated parameters decreases with rapidity increasing for both single-qubit state and two-qubit state. As rapidity approaches infinity (boost to the speed of light), the quantum Fisher information reaches to a constant which decreases as the ratio <i>σ</i>_r/<i>m</i> increases. More importantly, for the phase parameter <i>φ</i>, it is observed that the quantum Fisher information of two-qubit state reduces more significantly compared to that of single-qubit states. While, for the amplitude parameter <i>θ</i>, the quantum Fisher information of two-qubit state is greater than that of single-qubit state. These results are useful and valuable for improving the precision of parameter estimation under the influence of relativistic effects.