Standard Quantum Limit and Heisenberg Limit in Function Estimation

Unlike well-established parameter estimation, function estimation faces conceptual and mathematical difficulties despite its enormous potential utility. We establish the fundamental error bounds on function estimation in quantum metrology for a spatially varying phase operator, where various degrees of smooth functions are considered. The error bounds are identified in the cases of the absence and the presence of interparticle entanglement, which correspond to the standard quantum limit and the Heisenberg limit, respectively. Notably, these error bounds can be reached by either position-localized states or wave-number-localized ones. In fact, we show that these error bounds are theoretically optimal for any type of probe states, indicating that quantum metrology on functions is also subject to the Nyquist-Shannon sampling theorem, even if classical detection is replaced by quantum measurement.

Figure 1

Schematic illustration of quantum estimation of functions. First, some multiparticle state is prepared as an input probe state. It then passes through a phase-shift gate ${\widehat{U}}_{\phi }$ generated by a spatially varying field $\phi (x)$ that we want to know. Finally, the output probe state is measured, whence the estimated field $\tilde{\phi }(x)$ is computed.