Entanglement Susceptibility in Quantum Systems

• Entanglement susceptibility is a family of response-based measures that quantify how entanglement or its witnesses change in response to perturbations like boundary couplings and geometric deformations.
• It links the variations in entanglement entropy, such as the 2-Rényi entropy, to experimentally observable quantities including quantum Fisher information and magnetic susceptibility.
• This framework is applied across theoretical models and experiments in quantum many-body systems, aiding in the analysis of area laws, critical scaling, and phase transitions.

Searching arXiv for recent and foundational papers on entanglement susceptibility and susceptibility-based entanglement diagnostics. Entanglement susceptibility denotes a family of response-based constructions in which entanglement, or an entanglement witness, is organized as the response to a perturbation, a boundary coupling, a geometric deformation, or a measurable susceptibility. In the current literature the phrase does not refer to a single canonical observable. It may denote the coefficient of the $2$-Rényi entropy under boundary stitching (Zanardi et al., 2012), the kernels governing the variation of entanglement entropy under deformations of the entangling surface (Witczak-Krempa, 2018), the relative-entropy information metric of a reduced density matrix (Sarkar, 2024), or the use of dynamical or magnetic susceptibility to reconstruct entanglement witnesses such as the quantum Fisher information (QFI) (Hauke et al., 2015, Bippus et al., 16 Mar 2025). This plurality of usage is central to the subject.

1. Conceptual scope and terminology

The literature uses “entanglement susceptibility” in several technically distinct senses. The common structure is a response principle: an entanglement quantity, or a rigorous witness of entanglement, is made into a differential or perturbative object.

Usage in the literature	Core object	Representative setting
Boundary-coupling response	$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$	Area laws and gapless violations
Shape-response theory	$\chi^{(1)},\chi^{(2)},\dots$ under $\zeta(r)$ deformations	Corners, cones, trihedral vertices
Reduced-state information geometry	$$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$	Finite-size criticality in spin chains
Response-based entanglement witness	$$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$	Thermal many-body systems
Thermodynamic witness from magnetic data	$EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$ and related bounds	Spin chains, dimers, molecular magnets

Several papers also delimit the term by negation. Studies of fidelity susceptibility, first derivatives of entanglement entropy, or susceptibility–entanglement mappings often state explicitly that they do not define a quantity called entanglement susceptibility, even though they analyze closely related response diagnostics (Ren et al., 2015, Ren et al., 2010, Arisa et al., 2020). This suggests that “entanglement susceptibility” is best treated as a family of response constructions rather than a uniquely standardized observable.

2. Boundary-coupling entanglement susceptibility and area laws

A foundational definition appears in the perturbative study of bipartite entanglement for local Hamiltonians (Zanardi et al., 2012). One considers a bipartition $A\cup B=\Lambda$ and a one-parameter Hamiltonian

$$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$

At $\lambda=0$, the ground state factorizes as $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$0. Using the $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$1-Rényi entropy,

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$2

the leading entanglement generated by turning on the boundary coupling is

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$3

with

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$4

Only processes that excite both subsystems contribute. In this sense, $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$5 measures the amplitude for the boundary Hamiltonian to create correlated virtual excitations across the cut (Zanardi et al., 2012).

The same work establishes the comparison

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$6

where $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$7 is the fidelity susceptibility of the stitched ground state. For local gapped Hamiltonians with spectral gap $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$8, exponential clustering, and boundary term $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$9, the resulting bound is

$\chi^{(1)},\chi^{(2)},\dots$0

This is an area-law statement at the level of entanglement response. By contrast, for gapless free fermions with a Fermi surface the same framework yields

$\chi^{(1)},\chi^{(2)},\dots$1

which reproduces the familiar logarithmic violation of the area law (Zanardi et al., 2012).

The formalism extends beyond second order. The $\chi^{(1)},\chi^{(2)},\dots$2-Rényi entropy can be re-expressed as a ground-state fidelity in a doubled and twisted theory, which leads to an exact linked-cluster expansion in connected imaginary-time correlators of a twisted boundary perturbation. This identifies entanglement susceptibility as the first nontrivial term in a full response expansion of Rényi entanglement (Zanardi et al., 2012).

3. Geometric entanglement susceptibilities

A second meaning of entanglement susceptibility concerns the response of entanglement entropy to deformations of the entangling surface (Witczak-Krempa, 2018). If a point $\chi^{(1)},\chi^{(2)},\dots$3 on $\chi^{(1)},\chi^{(2)},\dots$4 is displaced along the local unit normal by $\chi^{(1)},\chi^{(2)},\dots$5,

$\chi^{(1)},\chi^{(2)},\dots$6

then the entropy admits the expansion

$\chi^{(1)},\chi^{(2)},\dots$7

The kernels $\chi^{(1)},\chi^{(2)},\dots$8 are the entanglement susceptibilities. Only normal deformations are physically meaningful; tangential deformations merely reparameterize the surface (Witczak-Krempa, 2018).

For pure states and a planar bipartition, odd-order susceptibilities vanish. For the von Neumann entropy, strong subadditivity implies

$\chi^{(1)},\chi^{(2)},\dots$9

For translation- and scale-invariant states with a planar cut, the leading nonlocal second-order susceptibility is fixed up to local/contact terms: $\zeta(r)$0 In conformal field theories, the coefficient is tied to the stress-tensor two-point normalization,

$\zeta(r)$1

For Rényi entropies, the same form holds with $\zeta(r)$2, but the positivity constraint need not survive because strong subadditivity is unavailable for $\zeta(r)$3 (Witczak-Krempa, 2018).

This shape-response theory yields universal singular contributions for non-smooth regions. In $\zeta(r)$4, a nearly smooth corner with opening angle $\zeta(r)$5 gives

$\zeta(r)$6

In $\zeta(r)$7, a nearly flat cone gives

$\zeta(r)$8

For a nearly smooth symmetric trihedral vertex,

$\zeta(r)$9

Accordingly, smooth-shape susceptibilities control universal corner, cone, and vertex terms in geometric entanglement entropy (Witczak-Krempa, 2018).

4. Reduced-state information geometry and the susceptibility of entanglement entropy

A more recent construction defines the susceptibility of entanglement entropy through the information geometry of reduced density matrices (Sarkar, 2024). Starting from the relative entropy

$$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$0

one expands between nearby states $$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$1 and $$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$2: $$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$3 with

$$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$4

The diagonal component $$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$5 is then interpreted as a susceptibility with respect to the control parameter $$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$6 (Sarkar, 2024).

In the finite $$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$7D XY chain and the transverse-field Ising model (TFIM), the construction is implemented using the single-site reduced density matrix

$$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$8

which yields the explicit one-parameter formula

$$\Sigma_{ij}=\frac12\mathrm{Tr}\!\left[\rho\,\partial_i(\ln\rho)\,\partial_j(\ln\rho)\right]$$9

This object is not simply $$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$0 or $$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$1; it is the Fisher–Rao-type metric of the reduced state (Sarkar, 2024).

Its utility is finite-size critical scaling. For the XY chain at $$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$2, with control parameter $$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$3, the turning points and global maxima of $$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$4 converge to

$$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$5

as

$$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$6

The corresponding maximum saturates numerically near

$$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$7

For the TFIM, with $$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$8 and $$F_Q=\frac{4}{\pi}\int_0^\infty d\omega\,\tanh(\omega/2T)\chi''(\omega,T)$$9, the turning points converge to $EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$0 with

$EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$1

while the maximum scales as

$EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$2

The analytical derivations use perturbation theory for the XY case and, for the TFIM, an exact route involving Chebyshev polynomials, hypergeometric functions, and complete elliptic integrals (Sarkar, 2024).

5. Dynamical susceptibility, quantum Fisher information, and multipartite entanglement depth

A separate and experimentally consequential line of work identifies entanglement through dynamical response functions. For a thermal many-body state and a Hermitian generator $EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$3, the QFI is exactly related to the imaginary part of the Kubo dynamical susceptibility (Hauke et al., 2015): $EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$4 This relation isolates the finite-frequency quantum response. It is not equivalent to the static isothermal susceptibility, which contains an elastic or Curie contribution and a classical Fisher-information piece,

$EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$5

Accordingly, a divergence of static thermal susceptibility at a thermal transition need not imply enhanced entanglement (Hauke et al., 2015).

The QFI then functions as a multipartite entanglement witness. For $EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$6 spin-$EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$7 particles and, when $EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$8 divides $EW=1-\frac{6k_BT\chi}{g^2\mu_B^2N}$9, the density

$A\cup B=\Lambda$0

the criterion

$A\cup B=\Lambda$1

certifies at least $A\cup B=\Lambda$2-partite entanglement (Hauke et al., 2015). This framework carries over directly to correlated-electron systems. In the pseudogap regime of cuprate superconductors, the relevant observable is the antiferromagnetic susceptibility at the leading wave vector $A\cup B=\Lambda$3, and the QFI can be estimated either from neutron-scattering data or, more robustly in theory, through the quantum variance

$A\cup B=\Lambda$4

In that setting the entanglement criterion is stated as

$A\cup B=\Lambda$5

and likewise $A\cup B=\Lambda$6 suffices because $A\cup B=\Lambda$7 (Bippus et al., 16 Mar 2025).

The pseudogap calculations use the two-dimensional Hubbard model treated by DMFT followed by ladder D$A\cup B=\Lambda$8A with $A\cup B=\Lambda$9 correction. The susceptibility is fitted to an overdamped Ornstein–Zernike form,

$$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$0

which yields the low-temperature asymptotic QFI

$$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$1

Hence the normal-state entanglement susceptibility diverges logarithmically,

$$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$2

until superconductivity cuts off the growth (Bippus et al., 16 Mar 2025). At $$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$3, the quantum variance in the theoretical pseudogap region exceeds the threshold for at least tripartite entanglement, $$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$4, and experimental neutron data in underdoped cuprates give $$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$5, certifying at least bipartite entanglement (Bippus et al., 16 Mar 2025).

A related response-based use of QFI appears in quantum dots. There the transverse dynamical susceptibility is used to compute the QFI through the same integral formula, and the reported trends are nontrivial: in Ising dots far below the Stoner instability, lowering temperature can reduce the QFI, whereas anisotropic Heisenberg dots show enhanced QFI near isotropic points (Abouie et al., 2024).

6. Thermodynamic witnesses, related diagnostics, and persistent distinctions

Long before the QFI–susceptibility relation was applied to correlated electrons, ordinary magnetic susceptibility was already used as a macroscopic witness of entanglement in low-dimensional spin systems. For the Heisenberg spin-$$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$6 chain $$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$7, the witness is

$$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$8

with entanglement certified whenever $$H(\lambda)=H_A+H_B+\lambda H_\partial,\qquad \lambda\in[0,1].$$9. The fit to the Bonner–Fisher model gives $\lambda=0$0, and the witness remains positive up to about $\lambda=0$1 (Chakraborty et al., 2012). In copper nitrate, $\lambda=0$2, susceptibility and magnetization data are used in the same spirit across a field-driven quantum phase transition near $\lambda=0$3, with entanglement persisting up to about $\lambda=0$4 (Das et al., 2011).

The same logic extends to dimers and few-spin systems. In the engineered molecular magnet $\lambda=0$5, the witness

$\lambda=0$6

is negative in the entangled regime, and extrapolation of the fitted dimer susceptibility gives an entanglement temperature $\lambda=0$7, with analytic estimate $\lambda=0$8 from $\lambda=0$9 and $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$00 (Reis et al., 2012). For $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$01, another Heisenberg spin-$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$02 dimer, the same witness gives $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$03, while the entanglement of formation extracted from susceptibility survives to about $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$04 (Chakraborty et al., 2019). In a three-qubit XXX model with field and anisotropy, separable states satisfy

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$05

so entanglement is witnessed when

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$06

and the critical temperature shifts as

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$07

for small anisotropy (Castorene et al., 2024).

A different development is the empirical mapping between susceptibility and entanglement rather than a witness inequality. In the one-dimensional attractive Hubbard model, the single-site von Neumann entropy and magnetic susceptibility become linearly related in conventional superfluids, with

$S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$08

for $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$09 and $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$10 (Arisa et al., 2020). At finite temperature in the two-dimensional Hubbard model, the average single-site entanglement and magnetic susceptibility are also linearly connected in some regimes, while the derivatives $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$11 and $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$12 act as markers of interaction-driven quantum phase transitions (Silva et al., 1 May 2025).

Several conceptual distinctions remain essential. First, susceptibility-based witnesses are generally sufficient but not necessary: failure to violate the bound does not imply separability (Chakraborty et al., 2012, Das et al., 2011, Castorene et al., 2024). Second, QFI and quantum variance are witnesses or lower bounds on multipartite entanglement depth, not the full entanglement content of the many-body state (Bippus et al., 16 Mar 2025). Third, some closely related diagnostics are not called entanglement susceptibility in the underlying papers: the first derivative of entanglement entropy with respect to rung coupling in a two-leg $S_2(\lambda)=2\lambda^2\chi_E+O(\lambda^3)$13 ladder (Ren et al., 2010), the combination of fidelity susceptibility, entanglement entropy, and Schmidt gap in the spin-1 XXZ chain (Ren et al., 2015), and quench-time entanglement entropy studied alongside fidelity susceptibility and dynamical quantum phase transitions in multiband topological insulators (Masłowski et al., 2019). The resulting picture is not terminological confusion so much as a stable division of labor: “entanglement susceptibility” now names several response-based constructions, each adapted to a different notion of entanglement and a different experimental or theoretical access route.