Quantum Fisher Information With General Quantum Coherence in multi-dimensional quantum systems

TL;DR

The paper defines General Quantum Coherence (GQC) to fuse quantum coherence with the energy-gap structure of the parametrization Hamiltonian and proves a universal relation F_Q = M^2 between QFI and GQC for both pure and mixed states in multi-dimensional quantum systems. It provides explicit GQC forms for qubits and qudits, and derives the corresponding QFI expressions, illustrating how coherence and energy differences jointly set metrological precision. An experimental demonstration via linear optics confirms the predicted scaling, showing F_Q scaling as the square of GQC when moving from single-qubit to two-qubit entangled probes. The work offers a practical guideline for engineering probe states to maximize precision in quantum parameter estimation and clarifies the quantum-to-classical transition in metrological performance.

Abstract

Quantum metrology is a science about quantum measurements and it plays a key role in precision of quantum parameter estimation. Meanwhile, quantum coherence is an important quantum feature and quantum Fisher information (QFI) is an important indicator for precision of quantum parameter estimation. In this paper, we explore the relationship between QFI and quantum coherence in multi-dimensional quantum systems. We introduce a new concept referred to as General Quantum Coherence (GQC), which characterizes the quantum coherence and the eigenenergies of the Hamiltonian in the interaction processes. GQC captures quantum nature of high-dimensional quantum states and addresses shortcomings in coherence measurement. Additionally, we observe a stringent square relationship between GQC and QFI. This finding provides a crucial guideline for improving the precision of parameter estimation.

Figures (3)

• Figure 1: The illustration of calculation process for mixed state case.
• Figure 2: Experiment setup: (a) Quantum parameter estimation process when the parametrization system is a single qubit system. The single photons source is created through a degenerated spontaneous parametric down-conversion (SPDC) process by pumping a type-I phase nonlinear $\beta$-barium-borate (BBO) crystal in the aqua area. The preparation of the probe state is shown in the yellow area. The parametrization process is shown in the blue area. The measurement of the output state is shown in the gray area. (b) Quantum parameter estimation process when the parametrization system is a two-qubit system. A pair of entangled photons are generated through degenerated (SPDC) process by pumping two type-II phase perpendicular each other nonlinear $\beta$-BBO crystals in the aqua area. The parametrization process is shown in the blue area. The measurement of output state is shown in the gray area. In the yellow area, the target quantum state is prepared by the local operation on the entangled photon source with a maximum entangled state. IF: Interference filter, QWP: quarter-wave plate, HWP: half-wave plate, PBS: polarization beam splitter.
• Figure 3: The QFI varies with the estimated parameter $\theta$. The subfigures (a) denotes the QFI results when $|\psi\rangle_1$ undergoes an evolution under $\hat{H}_1$, while (b) denotes the QFI results when $|\psi\rangle_2$ undergoes an evolution under $\hat{H}_2$.