Quantum Fisher information


The quantum Fisher information is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information. The quantum Fisher information of a state with respect to the observable is defined as
where and are the eigenvalues and eigenvectors of the density matrix respectively.
When the observable generates a unitary transformation of the system with a parameter from initial state,
the quantum Fisher information constrains the achievable precision in statistical estimation of the parameter via the quantum Cramér–Rao bound:
It is often desirable to estimate the magnitude of an unknown parameter that controls the strength of a system's Hamiltonian with respect to a known observable during a known dynamical time. In this case, defining, so that, means estimates of can be directly translated into estimates of.

Relation to the Symmetric Logarithmic Derivative

The quantum Fisher information equals the expectation value of, where is the
Symmetric Logarithmic Derivative.

Convexity properties

The quantum Fisher information equals four times the variance for pure states
For mixed states it is convex in that is,
The quantum Fisher information is the largest function that is convex and that equals four times the variance for pure states.
That is, it equals four times the convex roof of the variance
where the infimum is over all decompositions of the density matrix
Note that are not necessarily orthogonal to each other.

Inequalities for composite systems

We need to understand the behavior of quantum Fisher information in composite system in order to study quantum metrology of many-particle systems.
For product states,
holds.
For the reduced state, we have
where.

Relation to entanglement

There are strong links between quantum metrology and quantum information science. For a multiparticle system of spin-1/2 particles
holds for separable states, where
and is a single particle angular momentum component. The maximum for general quantum states is given by
Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology.
Moreover, for quantum states with an entanglement depth,
holds, where is the remainder from dividing by. Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.

Similar quantities

The Wigner–Yanase skew information is defined as
It follows that is convex in
For the quantum Fisher information and the Wigner–Yanase skew information, the inequality
holds, where there is an equality for pure states.