Quantum Fisher Information Matrix

Updated 14 November 2025

Quantum Fisher Information Matrix is a Riemannian metric on quantum state spaces that quantifies the sensitivity of states to infinitesimal parameter changes.
It is constructed via symmetric logarithmic derivatives and evaluated using techniques like vectorization and block methods for efficient computation.
The QFIM underpins precision quantum metrology, entanglement witnessing, and holographic connections, setting ultimate bounds through the quantum Cramér–Rao inequality.

The quantum Fisher information matrix (QFIM) is the central Riemannian metric on manifolds of quantum states parameterized by continuous variables, providing the upper bound—via the matrix quantum Cramér–Rao inequality—for the achievable precision in multi-parameter quantum estimation. Its entries quantify the ultimate sensitivity of a quantum state to infinitesimal changes in physical parameters, making the QFIM a fundamental object not only in quantum metrology but also in quantum information geometry, critical phenomena, entanglement witnessing, quantum control, and resource theories.

1. Formal Definitions and General Construction

Given a smooth family of quantum states $\rho(\boldsymbol\theta)$ on a finite- or infinite-dimensional Hilbert space, with real parameter vector $\boldsymbol\theta=(\theta_1,\dots,\theta_n)$ , the QFIM $(F_{ij})$ is defined in terms of the symmetric logarithmic derivatives (SLDs) $L_i$ as

$\frac{\partial\rho}{\partial\theta_i} = \frac{1}{2}\big( L_i\rho + \rho L_i \big)$

$F_{ij}(\boldsymbol\theta) = \frac12\,\mathrm{Tr}\big[\,\rho(\boldsymbol\theta)\,\{L_i,L_j\}\,\big]$

which holds for arbitrary (full-rank) states (Šafránek, 2018, Petz et al., 2010, Liu et al., 2019). For pure states $|\psi(\boldsymbol\theta)\rangle$ , the QFIM reduces to

$F_{ij}(\boldsymbol\theta) = 4\,\mathrm{Re}\Bigl[\,\langle \partial_i\psi | \partial_j\psi \rangle - \langle \partial_i\psi|\psi\rangle\langle\psi|\partial_j\psi\rangle\,\Bigr]$

i.e., it is four times the Fubini–Study metric (Fiderer et al., 2020, Petz et al., 2010).

In an eigenbasis where $\rho=\sum_k p_k|k\rangle\langle k|$ , the SLDs can be expressed as

$\langle k|L_i|l\rangle = \frac{2\,\langle k|\partial_i\rho|l\rangle}{p_k + p_l}$

for $p_k+p_l > 0$ , and thus

$F_{ij} = 2 \sum_{k,l: p_k+p_l>0} \frac{ \mathrm{Re}\left[\langle k | \partial_i \rho | l \rangle \langle l | \partial_j \rho | k \rangle\right]}{p_k + p_l}$

(Šafránek, 2016, Liu et al., 2019).

The QFIM defines a Riemannian metric on quantum state space that, for mixed states, connects directly to the infinitesimal Bures distance: $D_B^2(\rho(\boldsymbol\theta), \rho(\boldsymbol\theta+d\boldsymbol\theta)) = \frac{1}{4} d\boldsymbol\theta^\top F(\boldsymbol\theta) d\boldsymbol\theta$ (Liu et al., 2019, Šafránek, 2016).

2. Quantum Fisher Metrics, Monotonicity, and Matrix Families

The QFIM can be understood within Petz’s theory of monotone quantum metrics. Any quantum Fisher information matrix associated with a monotone Riemannian metric is parameterized by an operator-monotone function $f$ : $g^f_\rho(A,B) = \mathrm{Tr}[A (J^f_\rho)^{-1}(B)]$ for tangent Hermitian operators $A,B$ , where $J^f_\rho$ is built from modular operators of $\rho$ (Petz et al., 2010).

The most widely used QFIM is “minimal,” i.e., the SLD metric corresponding to $f(x) = (1+x)/2$ . Other notable cases include the right-logarithmic derivative (RLD) and Kubo–Mori (Bogoliubov) metrics, with the QFIM constructed as the Hessian of Rényi or other contractive divergences (Wilde, 2 Oct 2025). Hybrid matrix families interpolating these metrics can possess advantageous properties for data-processing or numerical stability, e.g., the $\alpha$ – $z$ metric for generalized entropic quantities.

Explicitly:

SLD QFIM: $\mathcal{J}_\rho(X) = \frac12(\rho X + X \rho)$ (minimal monotone metric).
RLD QFIM: $I_{ij}^{\text{RLD}} = \mathrm{Re} \,\mathrm{Tr}\,[ (\partial_i\rho)\,\rho^{-1} (\partial_j\rho)]$ (Wilde, 2 Oct 2025).
Kubo–Mori QFIM: $I_{ij}^{\text{KM}} = \mathrm{Tr}[(\partial_i\rho)\partial_j\ln\rho]$ .

For trace-monoid divergences $D(\rho\|\sigma)$ , any twice-differentiable quantum divergence induces a quantum information matrix through

$I_{ij}(\boldsymbol\theta) = \left. \frac{\partial^2}{\partial \epsilon_i \partial \epsilon_j} D(\rho(\boldsymbol\theta) \| \rho(\boldsymbol\theta+\epsilon)) \right|_{\epsilon=0}$

(Wilde, 2 Oct 2025).

3. Calculation Techniques and Efficient Algorithms

Closed-form evaluation of the QFIM is generally a major computational bottleneck, especially for high-dimensional or embedded parameterizations. Several techniques mitigate this:

Vectorization and Kronecker Sum Formulas: For full-rank $\rho$ of size $d$ , the “no-diagonalization” formula uses vectorization:

$F_{ij} = 2\,\text{vec}(\partial_i \rho)^\dagger\, [\bar\rho \otimes I + I \otimes \rho]^{-1} \, \text{vec}(\partial_j \rho)$

Efficient for small- to moderate-dimension (Šafránek, 2018, Fiderer et al., 2020).

Block Methods in Arbitrary Basis: Closed-form block-inverse expressions, compatible with non-orthogonal and rank-deficient representations, enable analytic computation in many practical scenarios, e.g., quantum imaging with coherent states (Fiderer et al., 2020).
Gaussian State Case: For multimode bosonic Gaussian states with covariance $\sigma$ and mean displacement $d$ , matrix and series expressions exist:

$H^{ij} = \frac12 \mathrm{vec}(\partial_i \sigma)^\dagger\,\mathfrak{M}^{-1}\,\mathrm{vec}(\partial_j \sigma) + 2\,\partial_i d^\dagger \sigma^{-1} \partial_j d$

with $\mathfrak{M} = \bar\sigma \otimes \sigma - K\otimes K$ (Šafránek, 2017).

Stein’s Identity for Variational Circuits: For high-dimensional variational quantum algorithms, Stein-based Monte Carlo estimators compute the whole QFIM at $O(1)$ quantum circuit cost per sample, enabling scalable quantum natural gradient methods (Halla, 24 Feb 2025):

$\hat{F}_{ij} = -\frac{c^2}{2 b^4 N}\sum_{k=1}^N \big( f(\theta + c Y_k) - f(\theta) \big) (Y_k Y_k^T - (b^2/c^2) I)$

where $f$ is the squared overlap, and $Y_k \sim \mathcal{N}(0, (b^2/c^2)I)$ .

Layered Quantum Circuits: In commuting-block quantum circuits, the off-block QFIM entries are measured with exponentially fewer unique circuit calls ( $O(L^2)$ for $L$ layers versus $O(m^2)$ for $m$ parameters) by leveraging the algebraic structure of the generators (Gómez-Lurbe, 14 May 2025).

4. Properties, Discontinuities, and Relation to the Bures Metric

The QFIM is real, symmetric, positive semi-definite, and covariant under reparameterization: $F_\varphi = J^T F_\theta J,\qquad J_{ij} = \frac{\partial \theta_i}{\partial\varphi_j}$ Convexity, unitary invariance, monotonicity under CPTP maps, and additivity under independent quantum systems all hold (Liu et al., 2019, Kudo et al., 2022, Petz et al., 2010).

The QFIM generically exhibits discontinuities at rank-changing points of $\rho$ , i.e., where an eigenvalue vanishes (Šafránek, 2016, Šafránek, 2017). The Bures metric provides the unique coordinate-wise continuous extension of the QFIM: $H_c^{ij}(\boldsymbol\theta) = F_{ij}(\boldsymbol\theta) + 2 \sum_{p_k(\theta) = 0}\partial_{ij}p_k(\theta)$ Regularization through convex combinations with invertible states, or by explicit addition of missing Hessian terms, restores continuity (Šafránek, 2016).

5. Applications in Quantum Estimation and Beyond

The QFIM forms the attainable limit for multiparameter estimation via the quantum Cramér–Rao bound: $\mathrm{Cov}(\hat{\boldsymbol\theta}) \geq F(\boldsymbol\theta)^{-1}$ Single-parameter attainability is generic if the SLD has a realizable projective measurement. In multiparameter quantum estimation, attainability requires the SLDs for all parameters commute on average. When this condition fails, the bound is not always saturable without collective measurements beyond product projective POVMs (Liu et al., 2019).

In the resource theory of asymmetry for general Lie group symmetries, the QFIM is a bona fide matrix-valued resource monotone with operational interpretation as the ultimate precision and as a measure of quantum fluctuations and correlations of the symmetry generators (Kudo et al., 2022).

In quantum thermodynamics, the QFIM for temperature coincides (up to powers of $T$ and specific heat $C_v$ ) with the variance of the Hamiltonian and determines the sensitivity of thermal probes (Liu et al., 2019).

The QFIM faithfully witnesses multipartite entanglement and quantifies entanglement dimensionality via operational criteria that relate sum and trace-norms of blocks to Schmidt numbers and multipartite entanglement vectors (Du et al., 24 Jan 2025). In open quantum systems, non-monotonic time-dependence of QFIM elements signals non-Markovian dynamics and quantum critical points (Parlato et al., 22 Aug 2025).

The maximal QFIM, $F^{\max}$ , which, if it exists, provides a universal probe-independent precision bound, is directly obtained from the channel family by maximizing the Bures distance over all input states and relates to channel distinguishability optimization (Chen et al., 2017).

6. Connections to Information Geometry, Quantum Speed Limits, and Holography

The QFIM is the quantum generalization of the classical Fisher information matrix and endows the quantum state space with the structure of a monotone Riemannian manifold, interpolating between several quantum metrics by varying the monotone function parameterization (Petz et al., 2010). These include the Fubini–Study metric (pure states) and Bures metric (mixed states). The QFIM appears as the (second) mixed derivative of the fidelity generating function, yielding Christoffel symbols and Berry curvature alongside the quantum metric (Chen, 7 Nov 2025).

Quantum speed limits are set by the Bures metric, with minimal time bound by $t \geq 2 D_B/\sqrt{F_{tt}}$ for time-parameterized Hamiltonians. QFIM also governs geodesic distances and the operational meaning of state distinguishability (Liu et al., 2019).

In the AdS/CFT context, the QFIM connects directly to canonical energy in the bulk: the second-order expansion coefficient of quantum relative entropy between nearby states in a CFT equals the canonical energy for the corresponding AdS perturbation. This identification upgrades the linearized Einstein equations (from first-order entropy positivity) to a second-order constraint (positivity of canonical energy) in the dual gravitational description (Lashkari et al., 2015).

7. Illustrative Examples and Special Constructions

For unitary processes, closed analytical QFIM forms for $SU(2)$ evolution utilize the Bloch-vector parametrization. For a probe state with Bloch vector $\vec{n}$ and process-defined Hermitian generators $M_{\theta_i}$ with Bloch vectors $\vec{m}_{\theta_i}$ : $F_{ij} = 4(p_0-p_1)^2\left[\vec{m}_{\theta_i}\cdot\vec{m}_{\theta_j} - (\vec{n}\cdot\vec{m}_{\theta_i})(\vec{n}\cdot\vec{m}_{\theta_j})\right]$ with $p_0$ , $p_1$ the eigenvalues of the probe (Shemshadi et al., 2018).

For Gaussian states, the QFIM admits explicit forms in terms of the symplectic eigenvalues and Williamson decomposition of the covariance matrix, as well as infinite series over powers of the quantum covariance matrix (Šafránek, 2017).

Quantum imaging and parameter estimation with non-orthogonal or rank-deficient density operators benefit from block-inverse and vectorization-based analytic formulas, bypassing full diagonalization and greatly facilitating both analytical and numerical work (Fiderer et al., 2020).

The QFIM, in its various formulations and specializations, underpins the modern quantum statistical estimation theory, multiplies as a resource-theoretic monotone, and imbues quantum geometry with precise, operational meaning. Numerous computational advances and new analytic techniques have broadened its applicability across quantum science, from metrological optimization, open systems, and multi-parameter estimation, to the emergent thermodynamic and holographic connections at the interfaces of quantum information and high energy physics.