The quantum Cramér–Rao bound is the quantum analogue of the classical Cramér–Rao bound.
It bounds the achievable precision in parameter estimation with a quantum system:
θ
is the number of independent repetitions, and
[ ϱ ,
is the quantum Fisher information.
ϱ
is the state of the system and
is the Hamiltonian of the system.
When considering a unitary dynamics of the type
ϱ ( θ ) = exp ( − i
θ )
ϱ
ϱ
is the initial state of the system,
θ
is the parameter to be estimated based on measurements on
ϱ ( θ ) .
Let us consider the decomposition of the density matrix to pure components as
ϱ =
The Heisenberg uncertainty relation is valid for all
From these, employing the Cauchy-Schwarz inequality we arrive at [3]
θ
θ
is the error propagation formula, which roughly tells us how well
can be estimated by measuring
Moreover, the convex roof of the variance is given as[5][6]
[ ϱ ,
[ ϱ ,
is the quantum Fisher information.