# What is the difference between "Shot-Noise-Limit" and "Standard Quantum Limit"?

It seems that in a lot of papers in the field of quantum metrology, there are two terms Shot-Noise-Limit and Standard Quantum Limit which are frequently referred to. What's the difference between them, because it seems they all refer to the limit $$\Delta \theta\ge\frac{1}{\sqrt{N}}$$.

You are correct that both terms reference the central limit theorem (CLT), which states that[1]

...the average of a large number $$n$$ of independent measurements (each having standard deviation $$\Delta \sigma$$) will converge to a Gaussian distribution with standard deviation $$\Delta \sigma / \sqrt{n}$$, so that the error on average scales with $$n^{-1/2}$$.

The CLT on its own poses nothing more than a statistical relationship. Any physical implications of the CLT depend entirely on its application, i.e. the context in which it is invoked.

Metrology and optics both leverage the CLT. In quantum metrology, the statistical scaling of errors with $$n^{-1/2}$$ is referred to as the 'standard quantum limit'. In this more theoretical context, the exact definition of $$n$$ is left ambiguous. In quantum (and classical) optics, the precision limit in optical phase sensing (i.e. the minimum uncertainty) is the shot-noise limit. In this more experimental context, $$n$$ is given an exact definition, e.g. the number of photons.

However, despite these 'technical' distinctions, the subjects of quantum metrology and quantum optics have so much overlap that the two naming conventions are often used interchangeably in scientific literature.

[1] Advances in Quantum Metrology, V. Giovannetti, et al., 2011. https://www.nature.com/articles/nphoton.2011.35