---
title: Subsystem Quantum Fisher Information
url: https://www.emergentmind.com/topics/subsystem-quantum-fisher-information-qfi
type: topic
---

# Subsystem Quantum Fisher Information

Searching arXiv for recent and foundational papers on subsystem quantum Fisher information.
Subsystem quantum Fisher information (QFI) is the quantum Fisher information associated with a reduced state of a composite quantum system, or, equivalently in the operational language of local accessibility, the maximum Fisher information obtainable when measurements are restricted to a given subsystem. For a bipartite family \(\rho_\theta\) on \(H^a\otimes H^b\), the reduced states are \(\rho_\theta^a=\operatorname{Tr}_b\rho_\theta\) and \(\rho_\theta^b=\operatorname{Tr}_a\rho_\theta\); the foundational operational result is that the Fisher information accessible by measurements on subsystem \(a\) alone is exactly the ordinary SLD-QFI of \(\rho_\theta^a\), and likewise for \(b\) [1208.0104]. In that sense, subsystem QFI is both a metrological quantity for reduced density operators and one layer in a broader hierarchy of locally, adaptively, and globally accessible Fisher informations.

## 1. Definition and formal scope

For a parameterized quantum state \(\rho_\theta\), the symmetric logarithmic derivative \(L_\theta\) is defined by
\[
\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),
\]
and the corresponding SLD-QFI is
\[
F_Q(\rho_\theta)=\operatorname{Tr}(\rho_\theta L_\theta^2).
\]
When the state is composite, subsystem QFI ordinarily means the QFI of a reduced density operator such as \(\rho_\theta^a\) or \(\rho_\theta^b\). In explicitly metrological settings, one also fixes a phase-imprinting generator on the reduced subsystem. For example, in the Dicke-model analysis of reduced atomic and field states, the field subsystem is encoded with generator \(\hat G=\hat b^\dagger \hat b\), while the atomic subsystem is encoded effectively with generator \(\hat G=\hat J_x\) after the Ramsey \(\pi/2\) pulse [1312.1426].

The same quantity admits an operational reinterpretation through measurement-induced Fisher information. If a POVM \(M=\{M_i\}\) is measured on \(\rho_\theta\), the induced probabilities are
\[
p_\theta(i)=\operatorname{Tr}(\rho_\theta M_i),
\]
and the corresponding classical Fisher information is
\[
F(\rho_\theta\mid M)=\sum_i p_\theta(i)\left(\frac{\partial \ln p_\theta(i)}{\partial \theta}\right)^2.
\]
The Braunstein–Caves relation
\[
F_Q(\rho_\theta)=\sup_M F(\rho_\theta\mid M)
\]
identifies QFI as the maximal classical Fisher information over all POVMs [1208.0104]. For subsystems, the relevant question becomes not only how much QFI the global state contains, but which part of that information remains accessible after partial trace or under measurement restrictions.

## 2. Measurement-induced hierarchy and local accessibility

The central operational framework for subsystem QFI is the hierarchy of restricted measurement classes on a bipartite state \(\rho_\theta\) [1208.0104]. For subsystem \(a\), local measurements are
\[
\mathcal M^a=\{M_i^a\otimes \mathbf 1^b\},
\]
and similarly for subsystem \(b\),
\[
\mathcal M^b=\{\mathbf 1^a\otimes M_i^b\}.
\]
The framework also includes independent local product measurements \(\mathcal M^{a,b}\), adaptive sequential measurements \(\mathcal M^{a\to b}\) and \(\mathcal M^{a\leftarrow b}\), and unrestricted global measurements \(\mathcal M^{ab}\).

For any measurement set \(\mathcal M\), the induced Fisher information is defined by
\[
F(\rho_\theta\mid\mathcal M)=\sup_{M\in\mathcal M}F(\rho_\theta\mid M).
\]
This produces the hierarchy
\[
F(\rho_\theta\mid \mathcal M^a)\le F(\rho_\theta\mid \mathcal M^{a,b}) \le F(\rho_\theta\mid \mathcal M^{a\to b})\le F(\rho_\theta\mid \mathcal M^{ab}),
\]
and
\[
F(\rho_\theta\mid \mathcal M^b)\le F(\rho_\theta\mid \mathcal M^{a,b}) \le F(\rho_\theta\mid \mathcal M^{a\leftarrow b})\le F(\rho_\theta\mid \mathcal M^{ab}).
\]

The decisive bridge to subsystem QFI is the identity
\[
F(\rho_\theta\mid M^a\otimes \mathbf 1^b)=F(\rho_\theta^a\mid M^a),
\]
which yields
\[
F(\rho_\theta\mid \mathcal M^a)=F_Q(\rho_\theta^a),\qquad
F(\rho_\theta\mid \mathcal M^b)=F_Q(\rho_\theta^b),\qquad
F(\rho_\theta\mid \mathcal M^{ab})=F_Q(\rho_\theta).
\]
Accordingly, reduced-state QFI is exactly the locally accessible Fisher information under non-adaptive measurements on that subsystem alone.

The same framework shows that reduced-state QFI is not the full story of local accessibility. For adaptive measurements \(a\to b\), the joint outcome distribution factorizes as
\[
p_\theta(i,j)=p_\theta^a(i)\,p_\theta^b(j\mid i),
\]
with conditional state
\[
\rho_\theta^{b\mid i}=\frac{\operatorname{Tr}_a[(M_i^a\otimes \mathbf 1^b)\rho_\theta]}{p_\theta^a(i)}.
\]
The induced Fisher information decomposes as
\[
F(\rho_\theta\mid M^{a\to b})=
F(\rho_\theta^a\mid M^a)+\sum_i p_\theta^a(i)\,F(\rho_\theta^{b\mid i}\mid M^{b\mid i}),
\]
and optimization gives
\[
F(\rho_\theta\mid \mathcal M^{a\to b})=\max_{M^a}F(\rho_{\theta,M^a}).
\]
Thus subsystem QFI is one layer of a larger accessibility structure: adaptive local protocols can reveal parameter dependence stored in correlations even when the reduced state by itself is insufficient.

## 3. Distribution of QFI across subsystems

Subsystem QFI is fundamentally about where parameter information resides inside a composite state. The measurement-induced framework distinguishes three extremal distribution types [1208.0104].

For product states,
\[
\rho_\theta=\rho_\theta^a\otimes \rho_\theta^b,
\]
QFI is additive,
\[
F_Q(\rho_\theta)=F_Q(\rho_\theta^a)+F_Q(\rho_\theta^b).
\]
This is the locally owned case: the parameter information is distributed across the subsystems in the ordinary additive way.

More distinctive is the locally inaccessible type,
\[
F_Q(\rho_\theta^a)=F_Q(\rho_\theta^b)=0,\qquad F_Q(\rho_\theta)\neq 0.
\]
A paradigmatic example is
\[
|\Psi_\theta\rangle=\frac{1}{\sqrt2}\bigl(|0^a0^b\rangle+e^{i\theta}|1^a1^b\rangle\bigr),
\]
for which
\[
F_Q(\rho_\theta)=1,\qquad F_Q(\rho_\theta^a)=F_Q(\rho_\theta^b)=0.
\]
Here the parameter is encoded purely in correlations: neither reduced state carries any QFI, although the global state does.

The fully shared type is characterized by
\[
F_Q(\rho_\theta)=F_Q(\rho_\theta^a)=F_Q(\rho_\theta^b)\neq 0.
\]
One example is the classically correlated family
\[
\rho_\theta=\sum_i p_\theta(i)\,|i^a\rangle\langle i^a|\otimes |i^b\rangle\langle i^b|,
\]
for which
\[
F_Q(\rho_\theta)=F_Q(\rho_\theta^a)=F_Q(\rho_\theta^b)
=\sum_i p_\theta(i)\left[\partial_\theta\ln p_\theta(i)\right]^2.
\]
Another is
\[
|\Psi_\theta\rangle=\cos\frac{\theta}{2}|0^a0^b\rangle+\sin\frac{\theta}{2}|1^a1^b\rangle,
\]
with
\[
F_Q(\rho_\theta)=F_Q(\rho_\theta^a)=F_Q(\rho_\theta^b)=1.
\]
In such cases, the same parameter information is duplicated across the subsystems rather than additively split.

These examples clarify the diagnostic role of subsystem QFI. The reduced-state quantities \(F_Q(\rho_\theta^a)\) and \(F_Q(\rho_\theta^b)\) reveal what each party can access individually; the gap between them and \(F_Q(\rho_\theta)\) measures information stored in correlations; and equality of global and local values indicates full sharing.

## 4. Dynamical transfer, hiding, and concentration

Subsystem QFI is not merely a static attribute of a state; it can be redistributed by dynamics. In the transfer examples of the measurement-induced framework, a joint unitary can move Fisher information from one subsystem to the other, from locally accessible form into correlations, or from correlations back into a measurable reduced state [1208.0104].

One class of examples begins with a product state \(\sigma_\theta^a\otimes \sigma^b\), so the parameter is initially only in subsystem \(a\). After a controlled unitary
\[
U=\sum_i |i^a\rangle\langle i^a|\otimes U_i^b,
\]
with
\[
\sigma_\theta^a=\sum_i p_\theta(i)|i^a\rangle\langle i^a|,
\]
the reduced state of \(a\) remains \(\rho_\theta^a=\sigma_\theta^a\), while
\[
\rho_\theta^b=\sum_i p_\theta(i)\,U_i^b\sigma^b U_i^{b\dagger}
\]
generally becomes \(\theta\)-dependent. The total QFI is unchanged by the unitary, but the distribution shifts from locally owned to shared.

A more striking example uses the controlled-NOT
\[
U=|0^a\rangle\langle 0^a|\otimes \mathbf 1^b
+|1^a\rangle\langle 1^a|\otimes \bigl(|0^b\rangle\langle 1^b|+|1^b\rangle\langle 0^b|\bigr).
\]
Applied to
\[
|\Omega_\theta\rangle=\frac{1}{\sqrt2}\bigl(|0^a\rangle+e^{i\theta}|1^a\rangle\bigr)\otimes |0^b\rangle,
\]
it produces
\[
|\Psi_\theta\rangle=\frac{1}{\sqrt2}\bigl(|0^a0^b\rangle+e^{i\theta}|1^a1^b\rangle\bigr),
\]
thereby converting a locally accessible parameter into a locally inaccessible one. The reverse transformation concentrates correlation-encoded QFI back into a single subsystem.

A different operational concentration scheme appears in the auxiliary-system protocol for large systems. There one adjoins an auxiliary subsystem \(b\), applies
\[
U=\sum_{i=1}^N \Pi_i^a\otimes O_i^b,
\]
and measures the reduced auxiliary state
\[
\rho^b(x)=\operatorname{Tr}_a\!\left[U\bigl(\rho^a(x)\otimes \sigma^b\bigr)U^\dagger\right]
=\sum_i p_i(x)\rho_i^b.
\]
The protocol proves
\[
F(\rho^a)=F(U\rho^{ab}U^\dagger)\ge F(\rho^b)\ge F^{(\mathrm{sub})}(\rho^b),
\]
and under orthogonality and optimal-measurement conditions one can achieve
\[
F^{(\mathrm{sub})}(\rho^b)=F(\rho^b)=F(\rho^a).
\]
This is a concentration result for subsystem QFI in a precise operational sense: a reduced auxiliary subsystem can be engineered to carry the full original QFI [2408.12918].

## 5. Many-body realizations

Subsystem QFI has become a practical diagnostic in many-body physics because reduced states remain accessible even when full-system QFI is difficult to evaluate. In the Dicke model, the reduced atomic state \(\rho_A\) and reduced field state \(\rho_B\) of the ground state both exhibit enhanced QFI near the superradiant critical coupling
\[
\lambda_{\mathrm{cr}}=\sqrt{\omega\omega_0}/2.
\]
For finite and sufficiently large \(N\), the scaled quantities \(F_A/N\) and \(F_B/(4\bar n)\) can exceed the corresponding shot-noise or coherent-state limits near criticality, reflecting reduced spin squeezing and field quadrature squeezing rather than merely large occupations [1312.1426].

In the thermodynamic analysis of the same model, the reduced atomic QFI obeys
\[
F_A\xi^2=N\mu^2,
\]
while in the superradiant phase the field QFI satisfies the approximate relation
\[
F_B(\Delta \hat X_{\pi/2})^2\approx \beta_s^2\approx \bar n.
\]
For each subsystem, the QFI remains finite but its first derivative is singular at the critical point, so subsystem QFI acts simultaneously as a metrological witness, a squeezing witness, and a criticality indicator. Deep in the superradiant phase, however, reduced-state mixedness suppresses the advantage: \(F_A\to 0\) for the atomic subsystem, while \(F_B\) returns to the classical limit.

Out-of-equilibrium many-body work extends this perspective from phase transitions to localized multipartite entanglement. For a compact subsystem \(A\) with reduced density matrix \(\rho_A\) and extensive observable
\[
O_A=\sum_{\ell\in A}O_\ell,
\]
the normalized subsystem QFI is
\[
\chi(\rho_A,O_A)=\frac{F(\rho_A,O_A)}{4\|O_A\|^2}.
\]
In equilibrium local many-body systems, clustering implies a volume law,
\[
F(\rho_A,O_A)\propto |A|,
\qquad
\frac{F(\rho_A,O_A)}{|A|^2}\to 0,
\]
even though critical ground states can produce superlinear behavior such as
\[
F(\rho_A,X_A)\sim |A|^{7/4}
\]
in the critical Ising chain. By contrast, a localized kick in a low-temperature ordered phase can transiently generate
\[
F(\rho_A,O_A)\propto |A|^2
\]
for \(|A|\sim t\), so that \(\chi(\rho_A,O_A)=O(1)\); after a single kick the effect is transient, while periodic localized kicking can keep \(\chi\) nonzero indefinitely for subsystem sizes set by the driving period [2503.21905].

## 6. Conceptual distinctions and recurrent misconceptions

Several distinct notions are adjacent to subsystem QFI but should not be conflated. The first is the distinction between reduced-state QFI and measurement-induced accessibility. In the operational framework, \(F_Q(\rho_\theta^a)\) is exactly the locally accessible Fisher information under measurements on \(a\) alone, but it does not exhaust what can be revealed by adaptive local protocols or by global measurements [1208.0104].

The second is terminological. In the auxiliary-system literature, “sub-QFI” does not mean the QFI of a reduced subsystem in the ordinary sense. It denotes the superfidelity-based lower bound
\[
F^{(\mathrm{sub})}
=
8\lim_{\mathrm dx\to0}
\frac{1-\sqrt{g(\rho(x),\rho(x+\mathrm dx))}}{\mathrm dx^2},
\]
derived from superfidelity. In that setting there are three logically distinct quantities: the original full-system QFI \(F(\rho^a)\), the ordinary QFI of the reduced auxiliary state \(F(\rho^b)\), and the auxiliary reduced-state sub-QFI \(F^{(\mathrm{sub})}(\rho^b)\) [2408.12918].

The third distinction concerns reduced descriptors versus reduced states. One-body reduced density matrix functional theory shows that, for bosonic and fermionic ground states, the global many-body QFIM can be expressed as a universal functional of the 1-RDM and generated through derivatives with respect to interaction couplings. What it reconstructs, however, is the QFI or QFIM of the full ground state, not the QFI of the 1-RDM treated as a subsystem state [2311.12596].

A further qualification is generator dependence. The Dicke-model subsystem QFIs are explicitly tied to the chosen generators \(\hat b^\dagger\hat b\) and \(\hat J_x\), and the many-body subsystem quantity \(\chi(\rho_A,O_A)\) is defined relative to an extensive observable \(O_A\). This suggests that subsystem QFI is not an intrinsic scalar of the reduced density matrix alone unless the parameterization or generator class is fixed.

## 7. Approximation and computation

Because subsystem states are often mixed and high-dimensional, practical evaluation of subsystem QFI motivates approximation schemes. One route is the truncated quantum Fisher information (TQFI), defined for truncated subnormalized states by
\[
\mathcal I_*(\theta;\rho_\theta^{(m)})
=
8\lim_{\delta\to0}
\frac{1-F_*(\rho_\theta^{(m)},\rho_{\theta+\delta}^{(m)})}{\delta^2},
\]
with generalized fidelity \(F_*\) on subnormalized states. TQFI satisfies
\[
\mathcal I_*(\theta;\rho_\theta^{(m)})\le I(\theta;\rho_\theta),
\]
and becomes exact when the truncation includes the full support. The paper does not formulate subsystem QFI directly, but it explicitly identifies the most direct bridge: one may apply the truncation procedure to a reduced state itself, yielding a lower-bound surrogate
\[
\mathcal I_*(\theta;\rho_{A,\theta}^{(m)})\le I(\theta;\rho_{A,\theta}),
\]
as a plausible computational strategy when full reduced-state QFI is inaccessible [2010.02904].

A complementary computational development rewrites QFI as a resolvent moment of the superoperator
\[
\mathcal K_\rho(Q)=\frac12\{\rho,Q\},
\]
with
\[
\mathcal F
=
|\mathcal O_0|_\rho^2
\left\langle v_0,\mathcal K_\rho^{-2}v_0\right\rangle_\rho
=
|\mathcal O_0|_\rho^2
\int\frac{d\mu(\lambda)}{\lambda^2}.
\]
This Krylov framework is formulated for general density matrices rather than subsystems specifically, but it transfers directly by replacing \(\rho\) with \(\rho_A\) and \(\mathcal O_0\) with \(d\rho_A/d\theta\). The main caveat is that reduced density matrices frequently have many small eigenvalues or exact zero modes, so subsystem QFI is especially likely to fall into the hard-edge regime where Krylov convergence is algebraic rather than exponential [2602.19750].

Taken together, these methods indicate that subsystem QFI occupies a dual position. Conceptually, it is the reduced-state or locally accessible component of quantum statistical distinguishability in composite systems. Operationally, it is a quantity that can be hidden in correlations, shared redundantly, concentrated into an auxiliary subsystem, approximated by truncation, or computed through reduced-state Krylov techniques. In that combined sense, subsystem QFI is not a marginal variant of global QFI, but a precise way of asking where metrological information resides and how much of it survives under locality, reduction, and dynamical redistribution.

Source: https://www.emergentmind.com/topics/subsystem-quantum-fisher-information-qfi