Quantum Fisher Information Overview

• Quantum Fisher Information is a measure of the sensitivity of quantum states to parameter changes, setting ultimate precision limits in estimation tasks.
• Its definition relies on operator inner products and monotonicity under completely positive maps to ensure physically consistent and optimal metrics.
• The framework incorporates specialized measures like skew information and quantum χ²-divergence, linking quantum geometry with practical metrology applications.

Quantum Fisher information (QFI) generalizes the classical Fisher information to the setting of quantum statistical models, characterizing the ultimate precision limits achievable in quantum parameter estimation and furnishing a family of monotone Riemannian metrics on the manifold of quantum states. The transition from classical to quantum Fisher information requires careful treatment of operator noncommutativity, the introduction of monotonicity constraints, and the choice of inner product structures on the operator space. In quantum theory, QFI plays a central role in metrology, quantum state estimation, quantum information geometry, and the formulation of quantum uncertainty relations, and is closely connected to abstract covariances, monotone metrics, and quantities such as the Wigner–Yanase–Dyson skew information and the quantum χ²-divergence (Petz et al., 2010).

1. Transition from Classical to Quantum Fisher Information

Classically, the Fisher information for a parameterized family of probability densities $f_\theta(x)$ is

$$J(p_\theta; 0) = \int [\partial_\theta \log f_\theta(x)|_{\theta=0}]^2 f_0(x) dx,$$

which quantifies the sensitivity of the likelihood with respect to changes in $\theta$ and sets the Cramér–Rao bound for unbiased estimators.

In the quantum setting, probability distributions are replaced by density matrices $\rho(\theta)$—positive semidefinite, unit trace operators on a Hilbert space. Parameter estimation translates to measuring observables $A$ (self-adjoint operators), with mean value $\operatorname{Tr}[\rho(\theta)A]$. The unbiasedness condition is typically encoded as a centering constraint, e.g., $\operatorname{Tr}[\rho(0)A] = 0$.

A key challenge in quantum estimation is operator noncommutativity. Instead of ordinary derivatives of logarithms, quantum Fisher information relies on a "quantum score" operator $L$ and an associated inner product $[\cdot,\cdot]_\rho$ on observables. If $L$ satisfies

$$\partial_\theta \operatorname{Tr}[\rho(\theta) B] |_{\theta=0} = [B, L]_\rho,$$

for all $B$, one arrives at a quantum Cramér–Rao inequality: $[A,A]_\rho \geq 1/[L,L]_\rho,$ where $[L,L]_\rho$ is interpreted as the quantum Fisher information. In cases where $[\rho, \partial_\theta \rho] = 0$ (commuting scenario), this reduces to $$F(\rho; B) = \operatorname{Tr}[\rho^{-1}(\partial_\theta\rho)^2]$$.

Unlike the classical case, the choice of inner product $[\cdot,\cdot]_\rho$ is not unique due to operator structure. A standard construction is $[A,B]_\rho = \operatorname{Tr}[A J_\rho(B)]$ where $J_\rho$ is a positive linear operator determined by a function $f$ (subject to monotonicity and normalization requirements).

2. Monotonicity and the Family of Quantum Fisher Informations

Monotonicity is the principal requirement distinguishing physically meaningful quantum Fisher informations. For a completely positive, trace-preserving (CPTP) map $\beta$ (such as a quantum channel or coarse-graining operation), one requires

$J_{\beta(\rho)}(\beta(L)) \leq J_\rho(L),$

guaranteeing that statistical distinguishability does not increase under physical processing.

This monotonicity requirement, together with the operator structure, leads to a continuous family of quantum Fisher informations and associated monotone metrics. Each Fisher information is determined by a standard operator monotone function $f$, which must satisfy:

• $f$ is operator monotone: for $A \leq B$, $f(A) \leq f(B)$,
• $x f(x^{-1}) = f(x)$,
• $f(1) = 1$.

Classical choices include the arithmetic mean $f(x) = (1+x)/2$ (yielding the symmetric logarithmic derivative [SLD] metric), harmonic mean $f(x) = 2x/(1+x)$, and geometric mean. The operator $J_\rho$ acts by

$$(J_\rho(B))_{ij} = m_f(\lambda_i, \lambda_j) B_{ij}$$

where $m_f$ is the mean corresponding to $f$.

Theorem 1.2 in (Petz et al., 2010) establishes the formal equivalence between the monotonicity of $f$ and the matrix monotonicity of the induced Fisher information.

3. Minimal Quantum Fisher Information and Physical Relevance

Among all quantum Fisher informations, the minimal metric—corresponding to $f(x) = (1+x)/2$ (the SLD Fisher information)—is of particular importance in physical applications:

• Quantum metrology: The minimal QFI sets the tightest quantum Cramér–Rao bound and governs precision in quantum parameter estimation tasks.
• State estimation and Riemannian geometry: The minimal metric defines the geometry of the quantum state space and underlies statistical distinguishability in quantum information geometry.

Explicitly, in the commuting case,

$$F_{\min}(\rho; B) = \operatorname{Tr}[\rho^{-1} B^2],$$

and in general,

$$F_{\min}(\rho; B) = \operatorname{Tr}[B J_\rho^{-1}(B)].$$

Many quantum estimation protocols, measurement optimizations, and uncertainty relations are formulated in terms of this minimal Fisher information.

4. Covariances, Monotone Metrics, and Duality

There is a one-to-one correspondence between monotone metrics (or quantum Fisher informations) and abstract covariances. For a density matrix $\rho$, the abstract covariance of two observables $A$, $B$ is

$$\operatorname{Cov}_\rho(A, B) = [A, B]_\rho - \operatorname{Tr}[\rho A]\operatorname{Tr}[\rho B],$$

with $[A,B]_\rho = \operatorname{Tr}[A J_\rho(B)]$. The Fisher information is obtained from the inverse of the mapping: $Y_\rho(A,B) = \operatorname{Tr}[A J_\rho^{-1}(B)].$ Hence the operator $J_\rho$ (and the function $f$) uniquely determines both the covariance and the Fisher information via

$$\text{Fisher Information} = (\text{Covariance})^{-1}.$$

Distinct choices of $f$ give rise to different monotone metrics and covariance structures on the quantum state manifold, with direct implications for quantum estimation and quantum Riemannian geometry.

5. Skew Information and Quantum χ²-Divergence as Specializations

Certain quantum information–theoretic quantities are retrieved as particular cases within this framework:

Skew Information

The Wigner–Yanase–Dyson skew information for a self-adjoint operator $A$ and state $\rho$ is

$$I_p(\rho, A) = -\operatorname{Tr}\!\left([\rho^p, A][\rho^{1-p}, A]\right), \quad 0 < p < 1.$$

This can be interpreted as the Fisher information on the subspace of observables generated by commutators with the state, i.e., $i[\rho, X]$. More generally, in terms of a monotone metric $Y_\rho$ and a function $f$, the skew information is

$$I_f(\rho, A) = \frac{1}{2} Y_\rho(i[\rho, A], i[\rho, A]).$$

The choice $f(x) = (1+x)/2$ recovers the standard (Wigner–Yanase) skew information up to a normalization. Skew information thus quantifies state asymmetry with respect to observables and is tightly linked to Fisher information geometry.

χ²-Divergence

A quantum analog of the χ²-divergence is given by

$$\chi^2(\rho, \sigma) = \operatorname{Tr}\!\left((\rho - \sigma) (J_\rho)^{-1}(\rho - \sigma)\right),$$

which is proportional to the Fisher information metric evaluated for the "distance" between $\rho$ and $\sigma$. In the classical (commuting density matrices) case, this quantity is independent of $f$. For noncommuting cases, it captures geometry determined by the choice of monotone metric.

6. Mathematical and Physical Implications

The passage from the classical to quantum Fisher information—by replacing probability densities with density matrices and introducing appropriate operator inner products—lays the foundation for quantum estimation theory and quantum information geometry. The requirement of monotonicity under CPTP maps is central both in classifying monotone Riemannian metrics and in guaranteeing consistency with quantum statistical mechanics.

Physically, the family of quantum Fisher informations encodes optimal statistical distinguishability under allowed quantum operations, underlies minimal error bounds (quantum Cramér–Rao), and defines the geometric structure of quantum state space. Skew information and χ²-divergence, as specialized forms of monotone metrics, provide refined measures of quantum asymmetry and statistical distinguishability with both operational and geometric significance.

The formal duality between monotone metrics and abstract covariances, and the embedding of skew information and related divergences into the monotone metrics framework, enable a unified mathematical treatment of quantum statistical inference, optimal parameter estimation, and the geometry of quantum theory (Petz et al., 2010).