Subsystem Quantum Fisher Information

• Subsystem QFI is defined as the Fisher information derived from the reduced state of a composite quantum system, capturing locally accessible parameter sensitivity.
• It distinguishes between locally accessible, hidden, and shared information through a hierarchy of measurement protocols, including adaptive local measurements.
• The framework supports metrological applications and many-body physics by enabling dynamic redistribution and computational approximations such as truncation and Krylov techniques.

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 $b$ (Lu et al., 2012). 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

$H^a\otimes H^b$0

When the state is composite, subsystem QFI ordinarily means the QFI of a reduced density operator such as $H^a\otimes H^b$1 or $H^a\otimes H^b$2. 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 $H^a\otimes H^b$3, while the atomic subsystem is encoded effectively with generator $H^a\otimes H^b$4 after the Ramsey $H^a\otimes H^b$5 pulse (Wang et al., 2013).

The same quantity admits an operational reinterpretation through measurement-induced Fisher information. If a POVM $H^a\otimes H^b$6 is measured on $H^a\otimes H^b$7, the induced probabilities are

$H^a\otimes H^b$8

and the corresponding classical Fisher information is

$H^a\otimes H^b$9

The Braunstein–Caves relation

$\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$0

identifies QFI as the maximal classical Fisher information over all POVMs (Lu et al., 2012). 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^a=\operatorname{Tr}_b\rho_\theta$1 (Lu et al., 2012). For subsystem $\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$2, local measurements are

$\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$3

and similarly for subsystem $\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$4,

$\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$5

The framework also includes independent local product measurements $\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$6, adaptive sequential measurements $\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$7 and $\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$8, and unrestricted global measurements $\rho_\theta^a=\operatorname{Tr}_b\rho_\theta$9.

For any measurement set $\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$0, the induced Fisher information is defined by

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$1

This produces the hierarchy

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$2

and

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$3

The decisive bridge to subsystem QFI is the identity

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$4

which yields

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$5

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 $\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$6, the joint outcome distribution factorizes as

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$7

with conditional state

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$8

The induced Fisher information decomposes as

$\rho_\theta^b=\operatorname{Tr}_a\rho_\theta$9

and optimization gives

$a$0

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 (Lu et al., 2012).

For product states,

$a$1

QFI is additive,

$a$2

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,

$a$3

A paradigmatic example is

$a$4

for which

$a$5

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

$a$6

One example is the classically correlated family

$a$7

for which

$a$8

Another is

$a$9

with

$\rho_\theta^a$0

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 $\rho_\theta^a$1 and $\rho_\theta^a$2 reveal what each party can access individually; the gap between them and $\rho_\theta^a$3 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 (Lu et al., 2012).

One class of examples begins with a product state $\rho_\theta^a$4, so the parameter is initially only in subsystem $\rho_\theta^a$5. After a controlled unitary

$\rho_\theta^a$6

with

$\rho_\theta^a$7

the reduced state of $\rho_\theta^a$8 remains $\rho_\theta^a$9, while

$b$0

generally becomes $b$1-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

$b$2

Applied to

$b$3

it produces

$b$4

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$5, applies

$b$6

and measures the reduced auxiliary state

$b$7

The protocol proves

$b$8

and under orthogonality and optimal-measurement conditions one can achieve

$b$9

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 (Liu, 2024).

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_\theta$0 and reduced field state $\rho_\theta$1 of the ground state both exhibit enhanced QFI near the superradiant critical coupling

$\rho_\theta$2

For finite and sufficiently large $\rho_\theta$3, the scaled quantities $\rho_\theta$4 and $\rho_\theta$5 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 (Wang et al., 2013).

In the thermodynamic analysis of the same model, the reduced atomic QFI obeys

$\rho_\theta$6

while in the superradiant phase the field QFI satisfies the approximate relation

$\rho_\theta$7

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: $\rho_\theta$8 for the atomic subsystem, while $\rho_\theta$9 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 $L_\theta$0 with reduced density matrix $L_\theta$1 and extensive observable

$L_\theta$2

the normalized subsystem QFI is

$L_\theta$3

In equilibrium local many-body systems, clustering implies a volume law,

$L_\theta$4

even though critical ground states can produce superlinear behavior such as

$L_\theta$5

in the critical Ising chain. By contrast, a localized kick in a low-temperature ordered phase can transiently generate

$L_\theta$6

for $L_\theta$7, so that $L_\theta$8; after a single kick the effect is transient, while periodic localized kicking can keep $L_\theta$9 nonzero indefinitely for subsystem sizes set by the driving period (Ferro et al., 27 Mar 2025).

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, $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$0 is exactly the locally accessible Fisher information under measurements on $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$1 alone, but it does not exhaust what can be revealed by adaptive local protocols or by global measurements (Lu et al., 2012).

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

$$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$2

derived from superfidelity. In that setting there are three logically distinct quantities: the original full-system QFI $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$3, the ordinary QFI of the reduced auxiliary state $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$4, and the auxiliary reduced-state sub-QFI $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$5 (Liu, 2024).

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 (Benavides-Riveros et al., 2023).

A further qualification is generator dependence. The Dicke-model subsystem QFIs are explicitly tied to the chosen generators $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$6 and $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$7, and the many-body subsystem quantity $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$8 is defined relative to an extensive observable $$\partial_\theta \rho_\theta=\frac12\bigl(L_\theta\rho_\theta+\rho_\theta L_\theta\bigr),$$9. 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

$H^a\otimes H^b$00

with generalized fidelity $H^a\otimes H^b$01 on subnormalized states. TQFI satisfies

$H^a\otimes H^b$02

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

$H^a\otimes H^b$03

as a plausible computational strategy when full reduced-state QFI is inaccessible (Sone et al., 2020).

A complementary computational development rewrites QFI as a resolvent moment of the superoperator

$H^a\otimes H^b$04

with

$H^a\otimes H^b$05

This Krylov framework is formulated for general density matrices rather than subsystems specifically, but it transfers directly by replacing $H^a\otimes H^b$06 with $H^a\otimes H^b$07 and $H^a\otimes H^b$08 with $H^a\otimes H^b$09. 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 (Alishahiha et al., 23 Feb 2026).

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.