Susceptibility-Based Open-System Theory

• Susceptibility-based open-system theory is a framework that decomposes the stationary response tensor into symmetric (metric) and antisymmetric (curvature) sectors to capture local and nonreciprocal responses.
• It quantifies how generalized forces change under parameter variations on a control manifold, linking equilibrium thermodynamic geometry to nonequilibrium steady states.
• The framework provides an operational route to measure geometric work in systems like driven dissipative qubits by analyzing cross-susceptibilities without full state tomography.

Susceptibility-based open-system theory is a response-geometric framework for open quantum systems in which the central object is the stationary-state response tensor $\chi_{\mu\nu}$, defined on a control manifold of Hamiltonian and bath parameters. In this formulation, the symmetric sector of $\chi_{\mu\nu}$ defines a metric-like susceptibility tensor governing local response, while the antisymmetric sector defines a curvature two-form associated with nonreciprocal response and geometric work. The framework extends equilibrium thermodynamic geometry to nonequilibrium steady states, where reciprocal and nonreciprocal responses coexist on the same control manifold. In equilibrium the antisymmetric sector vanishes by reciprocity, whereas open quantum systems admit a broader structure with both metric and symplectic sectors. A driven dissipative qubit under pure dephasing provides the canonical example: finite curvature emerges from the misalignment between the Hamiltonian eigenbasis and the pointer basis selected by the environment, and geometric work appears as a measurable signature of nonreciprocal response (Bittner et al., 21 Jun 2026).

1. Stationary-state susceptibility as the central object

The theory begins with a family of stationary density matrices

$$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$

where the real parameters $\lambda^\mu$ are control parameters on a control manifold. These parameters may include Hamiltonian parameters such as fields, detunings, couplings, and drive strengths, as well as bath parameters such as dephasing rates and temperatures. For each parameter one defines a generalized force

$$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$

with stationary expectation value

$$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$

The stationary-state response tensor is then

$$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$

equivalently,

$$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$

Operationally, if $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$, then

$$\delta\langle O_\mu\rangle_{\text{ss}}=\chi_{\mu\nu}\,\delta\lambda^\nu+O(\delta\lambda^2).$$

Thus $\chi_{\mu\nu}$0 is a static susceptibility matrix on the stationary manifold, defined without assuming equilibrium or detailed balance (Bittner et al., 21 Jun 2026).

In equilibrium, the stationary state is Gibbs,

$\chi_{\mu\nu}$1

and the generalized forces derive from the thermal free energy,

$\chi_{\mu\nu}$2

It follows that

$\chi_{\mu\nu}$3

The equilibrium theory is therefore purely reciprocal. In open systems, by contrast, $\chi_{\mu\nu}$4 need not derive from any scalar potential, and $\chi_{\mu\nu}$5 can be non-integrable.

2. Decomposition into metric and curvature sectors

The response tensor is decomposed as

$\chi_{\mu\nu}$6

with

$\chi_{\mu\nu}$7

The symmetric tensor $\chi_{\mu\nu}$8 is interpreted as a metric-like susceptibility tensor. It defines a real symmetric bilinear form on parameter variations $\chi_{\mu\nu}$9 and quantifies the local cost or sensitivity of quasistatic driving. The antisymmetric tensor $$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$0 is identified with a curvature sector (Bittner et al., 21 Jun 2026).

The geometric construction proceeds from the work connection

$$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$1

For a quasistatic closed path $$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$2 in control space,

$$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$3

Its curvature is

$$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$4

Using $$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$5 and the definition of $$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$6,

$$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$7

In the notation of the paper,

$$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$8

The curvature two-form is therefore exactly the antisymmetric sector of the stationary response tensor. The control manifold carries both a metric $$\rho_{\text{ss}}(\lambda),\qquad \lambda=(\lambda^1,\lambda^2,\dots),$$9 and a closed two-form

$\lambda^\mu$0

Where $\lambda^\mu$1 is nondegenerate, its inverse defines a local Poisson tensor,

$\lambda^\mu$2

so the response geometry is described as almost Kähler-like: it has both metric and symplectic structures, although a compatible complex structure is not explicitly constructed. In equilibrium, reciprocity implies $\lambda^\mu$3, so only the metric sector survives.

3. Fluctuation-response structure in nonequilibrium steady states

For Hamiltonian controls with fixed dissipator, the framework introduces the retarded stationary response function

$\lambda^\mu$4

with time evolution generated by the Liouvillian of the open system and averages taken in the stationary state. The zero-frequency response is

$\lambda^\mu$5

Using the quantum regression theorem for Markovian dynamics, the stationary response tensor may be written as

$\lambda^\mu$6

For a smooth Hamiltonian,

$\lambda^\mu$7

so this explicit-parameter term is symmetric and drops out of the antisymmetric sector (Bittner et al., 21 Jun 2026).

The curvature is therefore purely dynamical: $\lambda^\mu$8 or explicitly,

$\lambda^\mu$9

This is the fluctuation-response relation for curvature: the antisymmetric response is determined by antisymmetric stationary correlation functions in the steady state. The symmetric sector is governed by reciprocal parts of the Kubo response and, in equilibrium, connects to the usual fluctuation-dissipation structure; the antisymmetric sector extends that geometric structure to nonequilibrium steady states.

The role of reciprocity is decisive. In equilibrium, detailed balance and time-reversal symmetry impose Onsager reciprocity,

$$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$0

hence $$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$1. Breaking detailed balance and reciprocity—by driving, coupling to non-thermal baths, or Hamiltonian–dissipator basis incompatibility—allows $$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$2 and activates a nontrivial Poisson sector of response.

4. Distinction from information geometry

The theory explicitly distinguishes response geometry from information geometry. On the manifold of density matrices, information geometry is defined by the Bures metric $$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$3, equivalent to the symmetric logarithmic derivative quantum Fisher metric,

$$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$4

This metric quantifies distinguishability between neighboring stationary states and parameter-estimation sensitivity. By contrast, the response metric

$$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$5

measures how stationary generalized forces change under parameter variations; it is a susceptibility object rather than a distinguishability object (Bittner et al., 21 Jun 2026).

For the qubit example, the paper compares the determinant $$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$6 of the response metric with the determinant $$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$7 in the $$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$8-plane and finds that their spatial structures are markedly different. The conclusion is not an inequality between the two metrics, but conceptual and operational independence. Knowledge of the information geometry does not determine the response geometry, and knowledge of the response geometry does not determine the information geometry.

This distinction also clarifies the status of curvature. The response-geometric curvature $$\hat O_\mu \equiv -\frac{\partial H}{\partial \lambda^\mu},$$9 is tied to nonreciprocal susceptibility and geometric work, not to state distinguishability. The framework therefore separates two different structures living on the same stationary-state manifold: one generated by linear response of generalized forces, the other by statistical distinguishability of density matrices.

5. Driven dissipative qubit under pure dephasing

The explicit model is a driven qubit with Hamiltonian

$$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$0

where $$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$1 is the longitudinal splitting and $$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$2 is the transverse field. The effective field is $$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$3, so the Hamiltonian eigenbasis is aligned with $$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$4. Dissipation is pure dephasing in the $$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$5 pointer basis,

$$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$6

and the Lindblad master equation is

$$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$7

The misalignment angle between Hamiltonian eigenbasis and pointer basis is

$$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$8

When $$\langle O_\mu\rangle_{\text{ss}(\lambda)}=\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$9, the bases are aligned; when $$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$0, they are incompatible (Bittner et al., 21 Jun 2026).

Writing the stationary state in Bloch form,

$$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$1

the stationary Bloch components are

$$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$2

with

$$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$3

Here $$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$4 is a stationary polarization parameter specifying the population imbalance. For $$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$5, one has $$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$6, so the stationary state is diagonal in $$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$7. For $$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$8, $$\chi_{\mu\nu}(\lambda)=\partial_{\lambda^\nu}\langle O_\mu\rangle_{\text{ss}(\lambda)},$$9 and $$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$0 are nonzero, showing finite stationary coherence in the pointer basis.

Taking control parameters $$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$1 and $$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$2, the generalized forces are

$$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$3

The work-connection components are

$$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$4

and the response tensor is

$$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$5

In this two-dimensional control space, the independent curvature component is

$$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$6

Several properties are immediate. First, $$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$7 at $$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$8, corresponding to perfect alignment $$\chi_{\mu\nu}=\frac{\partial}{\partial\lambda^\nu}\mathrm{Tr}\!\big[\rho_{\text{ss}(\lambda)}\hat O_\mu\big].$$9. Second, $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$0 is odd in $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$1, so reversing the transverse field reverses the orientation of curvature. Third, curvature appears precisely when the Hamiltonian eigenbasis and pointer basis are misaligned. In this regime the stationary state no longer commutes with $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$2, stationary coherence appears, and nonreciprocal response is activated. The example is notable because finite curvature does not require strong driving, engineered reservoirs, population relaxation, or non-Markovian effects.

6. Geometric work, operational meaning, and broader significance

For a closed cycle $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$3 in control space, the quasistatic work is

$\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$4

By Stokes' theorem,

$\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$5

where $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$6. Since

$\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$7

the geometric part of the work is determined directly by antisymmetric susceptibility: $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$8 This work is geometric in the quasistatic regime: it depends on the loop in parameter space and the enclosed curvature, not on driving speed (Bittner et al., 21 Jun 2026).

This identification gives the framework an operational content. Measuring cross-susceptibilities such as $\lambda^\nu\to\lambda^\nu+\delta\lambda^\nu$9 and $$\delta\langle O_\mu\rangle_{\text{ss}}=\chi_{\mu\nu}\,\delta\lambda^\nu+O(\delta\lambda^2).$$0 independently determines the curvature,

$$\delta\langle O_\mu\rangle_{\text{ss}}=\chi_{\mu\nu}\,\delta\lambda^\nu+O(\delta\lambda^2).$$1

and thereby predicts nonzero geometric work for loops enclosing regions of finite curvature. The paper emphasizes that this is experimentally accessible without full state tomography; standard response measurements along different directions in control space are sufficient. Candidate platforms include driven spin systems in ESR or NMR, cavity or circuit QED, and excitonic or polaritonic systems in which Hamiltonian and bath act in different bases.

The broader conceptual placement is equally explicit. The symmetric sector recovers and extends thermodynamic geometry from equilibrium Gibbs states to nonequilibrium steady states, while the antisymmetric sector adds a Poisson or symplectic layer absent from equilibrium thermodynamic geometry. The curvature plays a role analogous to Berry curvature, but on the control manifold of open-system parameters and for stationary mixed states rather than instantaneous eigenstates of a closed Hamiltonian. The framework also differs from contemporaneous classical work noted by the paper—Beyen, Khodabandehlou, and Maes derive a Berry-like curvature for classical Markov jump processes from Liouvillian eigenvectors via counting fields—because the present construction derives curvature directly from the stationary response tensor and identifies basis incompatibility between Hamiltonian eigenbasis and pointer basis as the physical origin of curvature in the quantum setting.

Its scope is defined by clear assumptions. The explicit fluctuation-response relation uses Markovian dynamics and the quantum regression theorem; the geometry is developed for stationary states and quasistatic driving; and a full Kähler or para-Kähler structure is conjectured but not constructed. Within those bounds, susceptibility-based open-system theory treats geometry, nonreciprocal response, and geometric work as projections of a single stationary susceptibility tensor.