No Free Quantum Fisher Information: Limitations and Opportunities in Broadband Signal Estimation

Quantum systems are exquisite sensors, with applications from magnetometry to gravitational wave detection. We consider the problem of estimating the magnitude of a signal that couples to a two-level system via $H = B \cos(\omega t)Z$. For any detection protocol, the precision achieved depends on the signal's frequency and can be quantified via the quantum Fisher information. We find a limitation on having a high sensitivity across a wide range of frequencies. In particular, we show perturbatively that for small $B$ and $T$, the quantum Fisher information accumulated over time $T$ of a single spin integrated over $\omega$ cannot exceed $2 \pi T + \mathcal{O}(B^2T^3)$ and is $\mathcal{O}(T^2)$ for long times. As a result, there is a fundamental limit on the broadband sensitivity of quantum sensors. We interpret this as a form of standard quantum limit, which applies to separable strategies but can be exceeded by entangled strategies - indeed, for $n$ particles, we give strategies that accumulate QFI $n$ times faster. We give several examples where, for $BT$ $\gtrsim$ 1 we find that the integrated QFI is sometimes substantially more than $2\pi T$, which may allow the very rapid detection of a signal with unknown frequency over a very wide bandwidth.