Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement and quantum squeezing. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework. Together with quantum hypothesis testing, it represents an important theoretical model at the basis of quantum sensing.

A basic task of quantum metrology is estimating the parameter ${\displaystyle \theta }$ of the unitary dynamics

where ${\displaystyle \varrho _{0}}$ is the initial state of the system and ${\displaystyle H}$ is the Hamiltonian of the system. ${\displaystyle \theta }$ is estimated based on measurements on ${\displaystyle \varrho (\theta ).}$

Typically, the system is composed of many particles, and the Hamiltonian is a sum of single-particle terms

where ${\displaystyle H_{k}}$ acts on the kth particle. In this case, there is no interaction between the particles, and we talk about linear interferometers.

The achievable precision is bounded from below by the quantum Cramér-Rao bound as

where ${\displaystyle m}$ is the number of independent repetitions and ${\displaystyle F_{\rm {Q}}[\varrho ,H]}$ is the quantum Fisher information.

One example of note is the use of the NOON state in a Mach–Zehnder interferometer to perform accurate phase measurements. A similar effect can be produced using less exotic states such as squeezed states. In quantum illumination protocols, two-mode squeezed states are widely studied to overcome the limit of classical states represented in coherent states. In atomic ensembles, spin squeezed states can be used for phase measurements.