# Standard Quantum Limit and Heisenberg Limit in Function Estimation

## TL;DR

This paper establishes fundamental quantum error bounds for function estimation, revealing the limits imposed by quantum mechanics and the role of entanglement, with implications for quantum metrology and sampling theory.

## Contribution

It derives optimal error bounds for quantum function estimation, connecting quantum limits with classical sampling theorems, and compares entanglement's impact on estimation accuracy.

## Key findings

- Error bounds correspond to standard quantum limit and Heisenberg limit.
- Bounds are achievable with position- or wavenumber-localized states.
- Quantum metrology on functions adheres to Nyquist-Shannon sampling theorem.

## Abstract

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 both cases of absence and 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 wavenumber-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.