Quantum metrology

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

${\displaystyle \varrho (\theta )=\exp(-iH\theta )\varrho _{0}\exp(+iH\theta ),}$

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

${\displaystyle H=\sum _{k}H_{k},}$

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

${\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,H]}},}$

where ${\displaystyle m}$ is the number of independent repetitions and ${\displaystyle F_{\rm {Q}}[\varrho ,H]}$ is the quantum Fisher information.[1][11]

One example of note is the use of the NOON state in a Mach–Zehnder interferometer to perform accurate phase measurements.[12] 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 classcial states represented in coherent states. In atomic ensembles, spin squeezed states can be used for phase measurements.

An important application of particular note is the detection of gravitational radiation in projects such as LIGO or the Virgo interferometer, where high-precision measurements must be made for the relative distance between two widely separated masses. However, the measurements described by quantum metrology are currently not used in this setting, being difficult to implement. Furthermore, there are other sources of noise affecting the detection of gravitational waves which must be overcome first. Nevertheless, plans may call for the use of quantum metrology in LIGO.[13]

A central question of quantum metrology is how the precision, i.e., the variance of the parameter estimation, scales with the number of particles. Classical interferometers cannot overcome the shot-noise limit. This limit is also frequently called standard quantum limit (SQL)

${\displaystyle (\Delta \theta )^{2}\geq {\tfrac {1}{mN}},}$

where is ${\displaystyle N}$ the number of particles. Shot-noise limit is known to be asymptotically achievable using coherent states and homodyne detection.[14]

Quantum metrology can reach the Heisenberg limit given by

${\displaystyle (\Delta \theta )^{2}\geq {\tfrac {1}{mN^{2}}}.}$

However, if uncorrelated local noise is present, then for large particle numbers the scaling of the precision returns to shot-noise scaling ${\displaystyle (\Delta \theta )^{2}\propto {\tfrac {1}{N}}.}$[15][16]

There are strong links between quantum metrology and quantum information science. It has been shown that quantum entanglement is needed to outperform classical interferometry in magnetrometry with a fully polarized ensemble of spins.[17] It has been proved that a similar relation is generally valid for any linear interferometer, independent of the details of the scheme.[18] Moreover, higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.[19][20] Additionally, entanglement in multiple degrees of freedom of quantum systems (known as "hyperentanglement"), can be used to enhance precision, with enhancement arising from entanglement in each degree of freedom.[21]