Characterizing Fisher information of quantum measurement

Abstract

Informationally complete measurements form the foundation of universal quantum state reconstruction, while quantum parameter estimation is based on the local structure of the manifold of quantum states. Here we establish a general link between these two aspects, in the context of a single informationally complete measurement, by employing a suitably adapted operator frame theory. In particular, we bound the ratio between the classical and quantum Fisher information in terms of the spectral decomposition of the associated frame operator, and connect these bounds to the optimal and least optimal directions for parameter encoding. The geometric and operational characterization of information extraction thus obtained reveals the fundamental tradeoff imposed by informational completeness on local quantum parameter estimation.

Figures (1)

• Figure 1: In this example, we consider a SIC POVM in a two-dimensional Hilbert space. When no prior information about the quantum state is available (i.e. $\rho = \mathbb{I}/2$), the measurement is Fisher-symmetric, so every direction on the Bloch sphere yields the same ratio $I_C/I_Q$. We then choose an arbitrary mixed state ($\rho = \frac{\mathbb{I}}{2} + 0.3\,\sigma_x + 0.25\,\sigma_y + 0.4\,\sigma_z$),and evaluate the local parameter-estimation performance for all directions on the Bloch sphere, colour-coding the resulting values of $I_C/I_Q$ (dark blue = lower ratio, yellow-green = higher ratio), as shown in the legend. The solid arrow indicates the numerically obtained best estimation direction (maximal $I_C/I_Q$), while the dashed arrow marks the worst direction (minimal $I_C/I_Q$). For comparison, we compute the eigenvectors of the scaled frame operator $\mathcal{F}$ corresponding to its second-largest and smallest eigenvalues, which—according to our theory—identify the optimal and least informative parameter-encoding directions, respectively. The analytical predictions align precisely with the numerical results, demonstrating full consistency with the theoretical characterization.

Theorems & Definitions (6)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof