Steerability Indices across Disciplines

• Steerability indices are quantitative measures that capture the degree to which systems can be controlled along prescribed dimensions.
• They combine measurable quantities, normalization procedures, and computational frameworks to contrast observed behaviors against unsteered models.
• Applications span quantum information, language model alignment, soft robotics, and causal inference, highlighting both empirical insights and open challenges.

Steerability indices are quantitative measures designed to capture the extent to which a system—quantum, algorithmic, robotic, or linguistic—can be “steered” or reliably controlled along prescribed dimensions. Originating in quantum information research to formalize Einstein–Podolsky–Rosen (EPR) steering as intermediate between entanglement and nonlocality, analogous indices have since been introduced to various domains, including LLMs, soft-body robotics, and causal inference in dynamical systems. Steerability indices consistently possess three structural elements: (i) an operationally meaningful, measurable quantity; (ii) a normalization or calibration procedure for comparability; and (iii) methodological frameworks for estimation or computation. This article surveys the mathematical formulations, physical and algorithmic interpretations, and practical implementation of steerability indices across quantum information, LLM alignment, dynamical system control, and robotics, summarizing their mathematical definitions, empirical properties, and limitations.

1. Foundational Definitions and Mathematical Formalism

Steerability indices typically quantify the deviation of observed joint outcomes or behaviors from what could be produced by noncontextual, local, or unsteered models. In quantum information, most indices operate on an assemblage—a set of conditional or marginal states or probability distributions produced by measurement—and compare it, either via optimization or distance metrics, to convex sets of unsteerable or local-hidden-state (LHS) models.

General Structure:

• Continuous-Variable Quantum Steering:

Let $$S = \frac{1}{2}\left[ P(b_{\beta} \mid a_{\alpha_1}) + P(b_{-\beta} \mid a_{\alpha_2}) \right]$$, where $P(b_{\beta} \mid a_{\alpha_1})$ is the conditional probability for Bob's parity measurement given Alice's displacement. The steerability threshold is $S > 3/4$ or $S < 1/4$; violation certifies steering (Chowdhury et al., 2015).

• Two-Qubit and General Assemblage Settings:
  • Measurement-Device-Independent (MDI) Steerability:

  $$S^{\mathrm{MDI}}(\{\sigma_{a|x}\}) = \sup_{F_{a|x}\succeq 0} \frac{1}{\beta(F)} \sum_{a,x} \mathrm{tr}[F_{a|x} \sigma_{a|x}] - 1,$$

  with $\beta(F)$ the maximal value over LHS assemblages. $S^{\mathrm{MDI}}$ coincides with the steering robustness (Ku et al., 2018), a convex steering monotone. - Critical Radius:

  For a map $\Lambda_{A \to B}$ (Alice to Bob's projections) and a class of probability ansätze $u$,

  $$R(\Lambda_{A \to B}) = \sup_{u} r_u(\Lambda_{A \to B}),$$

  with $R < 1$ iff the state is steerable for projective measurements (Nguyen et al., 2016). - Steering Fraction:

  $$\Gamma_s(\{\sigma_{a|x}\}, F) = \frac{1}{\beta(F)} \sum_{a,x} \mathrm{tr}[F_{a|x} \sigma_{a|x}],$$

  and the optimal steerability index is $$S(\sigma) = \max\{0, \sup_{F_{a|x}} \Gamma_s(\sigma, F) - 1\}$$ (Hsieh et al., 2016).

• Robustness and Maximal Extractable Steerability:

For an assemblage $\sigma_{a|x}$, maximal extractable steerability under filtering is $\mathrm{IR}(B)$, where $B_{a|x}$ are the steering-equivalent observables (SEO), obtainable via efficient SDP (Ku et al., 2022).

• LLM Prompt Steerability:

Given a model distribution $p_X^i$ under varying prompt modifications, steerability indices along persona dimension $d_i$ are:

$$\gamma_{i,k}^+ = \frac{ W(p_X^i, \tilde{p}_X^{i+}) - W(p_{X,k}^{i+}, \tilde{p}_X^{i+}) }{ W(\tilde{p}_X^{i+}, \tilde{p}_X^{i-}) }, \quad \gamma_{i,k}^- = ...$$

where $W$ is the Wasserstein distance and normalization guarantees values in $[-1,1]$ (Miehling et al., 19 Nov 2024).

• LLM Goal-Space Steerability:

  • Coverage: $$\Pr(\| \hat{\mathbf{z}} - \mathbf{z}^* \|_2 \le \epsilon)$$
  • Miscalibration: $$\frac{ \langle \mathbf{e}, \Delta\mathbf{z} \rangle }{ \| \Delta\mathbf{z} \|^2_2 }$$
  • Side-effects: $$\frac{ \| \mathbf{e} - \mathrm{proj}_{\Delta\mathbf{z}}(\mathbf{e}) \|_2 }{ \| \Delta\mathbf{z} \|_2 }$$
  • where $\mathbf{e} = \hat{\mathbf{z}} - \mathbf{z}_0$ and $\Delta\mathbf{z} = \mathbf{z}^* - \mathbf{z}_0$ (Chang et al., 27 May 2025).
• Soft Robot Steerability:

Characterized by a composite index $S = \kappa_{\max} \cdot L_n$, where $\kappa_{\max}$ is the maximal achievable planar curvature and $L_n$ the characteristic length under self-support (McFarland et al., 26 Oct 2025).

• Causal Steerability in Dynamical Systems:

For state $x_t$, control $u_t$, the steerability index is:

$$\mathrm{steer}_T(u, u') = \mathbb{E}[x_T | \mathrm{do}(u_{T-1} = u')] - \mathbb{E}[x_T | \mathrm{do}(u_{T-1} = u)],$$

estimated via adjustment formulas or two-stage regression, subject to identifiability conditions (Cheng et al., 2023).

2. Quantum Steerability Indices: Characterizations and Properties

Quantum steerability is formalized as the impossibility of modeling conditional assemblages by local hidden-state models. Key indices include:

• Partial Transpose Eigenvalue Index:

For two-qubit $\rho_{AB}$,

$$S = \lambda_1 + \lambda_2 - (\lambda_1 - \lambda_2)^2,$$

with $\lambda_1, \lambda_2$ the smallest eigenvalues of $\rho_{AB}^{T_B}$ (partial transpose). $S < 0$ detects steering; it is strictly stronger than any finite-setting linear steering inequality for two qubits (Chen et al., 2011).

• Geometric and SDP-based Indices:

The critical radius $R$ for two-qubit states is necessary and sufficient for projective-measurement steerability; $R<1$ implies steerability (Nguyen et al., 2016). The steering fraction, maximal extractable steerability, and steering robustness are convex monotones, and can be efficiently computed by semidefinite programming (Ku et al., 2018, Hsieh et al., 2016, Ku et al., 2022).

• Measurement-Device-Independent (MDI) Steering:

The MDI-steering index coincides numerically and operationally with the steering robustness, ensuring device independence up to trusting tomographically complete inputs (Ku et al., 2018).

• Joint Measurability Connection:

For two-setting scenarios, steerability is quantitatively characterized by the degree of incompatibility (non-joint measurability) of two unsharp observables induced on Bob, yielding tight necessary and sufficient conditions for "zero-states" and analytic formulas for Bell-diagonal and X-states (Chen et al., 2017).

3. Steerability in LLMs: Persona and Attribute Steering

Steerability indices for LLMs formalize the model's controllability under prompt modification or system-level steering interventions.

• Prompt Persona Steering (Miehling et al., 19 Nov 2024): Steerability indices $\gamma_{i,k}^{\pm}$ capture, for each persona dimension, the normalized reduction in 1D Wasserstein distance between the distribution of model responses under $k$ steering exemplars and the maximally/ minimally steered profile. Indices saturate near $\pm1$ for fully steerable dimensions and fluctuate depending on the baseline skew and asymmetry of model behaviors.
• OCEAN Behavioral Steerability (Noever et al., 2023): In the OCEAN psychometric framework, the index for each trait $T$ is the normalized sum $\hat{S}_T = (S_T-10)/40$. Distinctive trait targeting can be quantified by $$D_{T^*} = \hat{S}_{T^*} - \max_{T \neq T^*}\hat{S}_T$$. Statistical analysis (ANOVA, effect sizes, repeatability) highlights model-specific strengths, ambiguity regions (Openness), and overlaps (Extraversion–Agreeableness).
• Multi-dimensional Attribute Steering and Failures (Chang et al., 27 May 2025): By defining a vector-valued goal space over interpretable text attributes, steerability analysis decomposes model performance into coverage (fraction of reachable $\mathcal{Z}$), calibration (directional accuracy), and side-effects (orthogonal unintended drift). Empirical results demonstrate that side-effects are persistent even under advanced prompt engineering and RL fine-tuning.

4. Generalizations: Robotics, Networks, and Causal Systems

Non-quantum domains have adopted structure-preserving steerability indices, typically marking the distance to a limiting notion of uncontrollability or non-steerability.

• Vine Robotics (McFarland et al., 26 Oct 2025): Steerability is jointly captured by maximum planar curvature $\kappa$ and characteristic self-supporting length $L_n$, both empirically measured. Design variables (tip load, chamber pressure, length, fabrication method) influence $\kappa$ and $L_n$ nonlinearly; guidance is summarized by $S = \kappa_{\max} L_n$ for efficient design optimization.
• Network Quantum Steerability (Li et al., 24 Feb 2025): The network steerability index $\mathcal{S}^N$ minimizes the averaged trace distance between the observed network assemblage and the closest excursion within the network local hidden-state polytope, using neural architectures matched to the network topology. Analytical and numerical thresholds (e.g., $\nu\omega = 1/3$ for bilocal steering) are recovered.
• Causal Inference in Dynamical Systems (Cheng et al., 2023): The steerability of consumption under algorithmic action is measured by differential interventional outcomes. The identifiability of this index requires two system-exciting exogenous shocks and a surjective, "responsive" control map; two-stage regression and adjustment estimators are provided for linear and non-parametric models.

5. Empirical and Operational Properties

Steerability indices are operationalized via optimization (robustness, fraction), analytical geometry (critical radius), distance minimization (trace norm in network scenarios), or distributional metrics (Wasserstein for LLM outputs).

Key empirical findings:

• In quantum contexts, maximal indices correspond to well-known maximally entangled or N00N-type states (e.g., $S_{\max}(N) = 1.00$ for all $N \ge 2$ in CV steering (Chowdhury et al., 2015)).
• LLMs exhibit inherent baselines far from neutrality, leading to structural asymmetries in steerability across dimensions (Miehling et al., 19 Nov 2024).
• Soft robots show nonmonotonic steerability as a function of design parameters, with steerability saturating after a threshold of actuator force (McFarland et al., 26 Oct 2025).
• Control and causal indices are typically nonzero only with sufficient excitation and responsive action policies; identification is generically unachievable in passively observed, near-deterministic settings (Cheng et al., 2023).

6. Limitations, Practicalities, and Open Questions

Steerability indices are subject to domain-specific limitations:

• Quantum: Most indices, though operationally meaningful, are computationally intensive outside symmetric state families; SDP-based approaches scale exponentially with system size; many provide only sufficiency and not necessity beyond qubits [(Chen et al., 2011), 16xx.xxxxx].
• LLMs: Attribute disentanglement in steerability remains open; measures can degrade rapidly as model families evolve (statistical drift, retraining, RLHF) (Noever et al., 2023, Chang et al., 27 May 2025). Empirical norms show persistent side-effects that are not eliminated by current interventions.
• Robotics: Composite indices depend on accurate environmental and material modeling, and empirical look-up remains necessary in many cases (McFarland et al., 26 Oct 2025).
• Causal Inference: Identification preconditions (exogenous shocks, platform responsiveness) are strict; nonparametric index estimation is sensitive to binning and support; system order and control structure are critical determinants (Cheng et al., 2023).
• Networks: ANN-based steerability estimations depend on correct causal-graph encoding; scaling to large heterogeneous networks is a computational challenge (Li et al., 24 Feb 2025).

Open questions, common across domains, concern the development of:

• Efficient, scalable computational tools for index evaluation in high dimension.
• Rigorous generalizations to multipartite, high-dimensional, and task-specific steering.
• Theoretical guarantees of necessary and sufficient criteria for steerability in complex, realistic systems.
• Techniques for effective suppression of steering side-effects and the design of architectures (quantum, algorithmic, or robotic) with high, robust, directionally symmetric steerability.

7. Comparative Table of Selected Steerability Indices Across Domains

Domain / Task	Steerability Index (Symbol)	Mathematical Definition / Principle
Quantum (2–qubit)	$S$ (PPT-eigenvalue)	$$S = \lambda_1 + \lambda_2 - (\lambda_1-\lambda_2)^2$$ (Chen et al., 2011)
Quantum (Assemblage, MDI, Robustness)	$S^{\rm MDI}$, $\mathsf{SR}$, $S(\sigma)$	SDP: $$\sup_{F \succeq 0}\frac{1}{\beta(F)}\sum_{a,x}\mathrm{tr}(F_{a|x}\sigma_{a|x})-1$$ (Ku et al., 2018, Hsieh et al., 2016)
Quantum (Critical radius)	$R$	$R = \sup_u r_u(\Lambda)$, $R<1 \Leftrightarrow$ steerable (Nguyen et al., 2016)
LLM (Prompt Persona)	$\gamma_{i,k}^{\pm}$	Normalized Wasserstein shift as in section 1 (Miehling et al., 19 Nov 2024)
LLM (Goal Space, Text Attribute)	miscalibration, side-effects, coverage	Vector decomp. of $\mathbf{e}$ wrt. $\Delta\mathbf{z}$ (Chang et al., 27 May 2025)
Vine Robots	$S = \kappa_{\max}\cdot L_n$	Product: max curvature $\times$ char. length (McFarland et al., 26 Oct 2025)
Dynamical Platform Steerability	$\mathrm{steer}_T(u,u')$	Interventional mean difference (Cheng et al., 2023)
Network Quantum Steering	$\mathcal{S}^{\mathrm{N}}$	Min. trace-distance to NLHS over $\bar a, \bar x$ (Li et al., 24 Feb 2025)

Steerability indices serve as the unifying vocabulary for precise quantification of controllability or deviation from “default” system behavior across physics, machine learning, robotics, and causal inference, underpinning both resource-theoretic analyses and practical engineering of steerable systems.