The quantum Fisher information is a central quantity in
quantum metrology and is the quantum analogue of the classical
Fisher information.[1][2][3][4][5] It is one of the central quantities used to qualify the utility of an input state, especially in Mach–Zehnder (or, equivalently, Ramsey) interferometer-based phase or parameter estimation.[1][3][6] It is shown that the quantum Fisher information can also be a sensitive probe of a quantum
phase transition (e.g. recognizing the superradiant
quantum phase transition in the
Dicke model[6]). 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, and the summation goes over all and such that .
When the observable generates a
unitary transformation of the system with a parameter from initial state ,
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 .
Connection with Fisher information
Classical Fisher information of measuring observable on density matrix is defined as , where is the probability of obtaining outcome when measuring observable on the transformed density matrix . is the eigenvalue corresponding to eigenvector of observable .
Quantum Fisher information is the
supremum of the classical Fisher information over all such observables,[7]
where denotes
complex conjugate, and denotes
conjugate transpose. This formula holds for invertible density matrices. For non-invertible density matrices, the inverse above is substituted by the
Moore-Penrose pseudoinverse. Alternatively, one can compute the quantum Fisher information for invertible state , where is any full-rank density matrix, and then perform the limit to obtain the quantum Fisher information for . Density matrix can be, for example, in a finite-dimensional system, or a thermal state in infinite dimensional systems.
Generalization and relations to Bures metric and quantum fidelity
For any differentiable parametrization of the density matrix by a vector of parameters , the quantum Fisher information matrix is defined as
where denotes partial derivative with respect to parameter . The formula also holds without taking the real part , because the imaginary part leads to an antisymmetric contribution that disappears under the sum. Note that all eigenvalues and eigenvectors of the density matrix potentially depend on the vector of parameters .
This definition is identical to four times the
Bures metric, up to singular points where the rank of the density matrix changes (those are the points at which suddenly becomes zero.) Through this relation, it also connects with
quantum fidelity of two infinitesimally close states,[10]
where the inner sum goes over all at which eigenvalues . The extra term (which is however zero in most applications) can be avoided by taking a symmetric expansion of fidelity,[11]
For and unitary encoding, the quantum Fisher information matrix reduces to the original definition.
Quantum Fisher information matrix is a part of a wider family of quantum statistical distances.[12]
Relation to fidelity susceptibility
Assuming that is a ground state of a parameter-dependent non-degenerate Hamiltonian , four times the quantum Fisher information of this state is called fidelity susceptibility, and denoted[13]
Fidelity susceptibility measures the sensitivity of the ground state to the parameter, and its divergence indicates a quantum phase transition. This is because of the aforementioned connection with fidelity: a diverging quantum Fisher information means that and are orthogonal to each other, for any infinitesimal change in parameter , and thus are said to undergo a phase-transition at point .
Convexity properties
The quantum Fisher information equals four times the variance for pure states
.
For mixed states, when the probabilities are parameter independent, i.e., when , the quantum Fisher information is convex:
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[14][15]
where the infimum is over all decompositions of the density matrix
Note that are not necessarily orthogonal to each other. The above optimization can be rewritten as an optimization over the two-copy space as [16]
Later the above statement has been proved even for the case of a minimization over general (not necessarily symmetric) separable states.[17]
When the probabilities are -dependent, an extended-convexity relation has been proved:[18]
where is the classical Fisher information associated to the probabilities contributing to the convex decomposition. The first term, in the right hand side of the above inequality, can be considered as the average quantum Fisher information of the density matrices in the convex decomposition.
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.[19]
For product states,
holds, where is the largest integer smaller than or equal to and 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.[21][22] It is possible to obtain a weaker but simpler bound [23]
Hence, a lower bound on the entanglement depth is obtained as
Relation to the Wigner–Yanase skew information
The Wigner–Yanase skew information is defined as [24]
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.
Relation to the variance
For any decomposition of the density matrix given by and the relation [14]
holds, where both inequalities are tight. That is, there is a decomposition for which the second inequality is saturated, which is the same as stating that the quantum Fisher information is the convex roof of the variance over four, discussed above. There is also a decomposition for which the first inequality is saturated, which means that
the variance is its own concave roof [14]
Uncertainty relations with the quantum Fisher information and the variance
Knowing that the quantum Fisher information is the convex roof of the variance times four, we obtain the relation [25]
which is stronger than the Heisenberg
uncertainty relation. For a particle of spin- the following uncertainty relation holds
where are angular momentum components. The relation can be strengthened as [26][27]
References
^
abHelstrom, C (1976). Quantum detection and estimation theory. Academic Press.
ISBN0123400503.
^Holevo, Alexander S (1982). Probabilistic and statistical aspects of quantum theory (2nd English ed.). Scuola Normale Superiore.
ISBN978-88-7642-378-9.
^Zhou, Sisi; Jiang, Liang (18 Oct 2019). "An exact correspondence between the quantum Fisher information and the Bures metric".
arXiv:1910.08473 [
quant-ph].