Bilinear Observation Operator

• Bilinear observation operators are mappings where measured outcomes depend multiplicatively on both measurement operators and state variables, fundamental to various scientific applications.
• They exhibit distinctive singular value plateaus and sector-localized nullspaces, indicating inherent structural rank deficits that resist resolution via simple numerical refinements.
• Modifying measurement protocols and employing structured learning methods are crucial for achieving full-rank recovery and accurate system identification.

A bilinear observation operator is a mapping in which the observed quantities depend multiplicatively on two distinct sets of variables—commonly, a measurement operator and a state variable—arising in a variety of physical, information-theoretic, control, and learning contexts. Such operators underlie rank diagnostics in quantum tomography, input-output system identification, operator learning, and modern control formulations, where their structural and statistical properties govern reconstructibility, identifiability, and effectiveness of measurement protocols.

1. Formal Definition and Algebraic Structure

Let $E_i\in\mathbb C^{d\times d}$ denote a collection of measurement operators and $\rho_j\in\mathbb C^{d\times d}$ a set of states (which may correspond to density matrices). The bilinear observation operator is constructed as follows:

$$\varphi(E_i,\rho_j) = \mathrm{vec}\bigl(E_i\otimes \rho_j^{\mathsf T}\bigr) \in \mathbb C^{d^4}$$

where “vec” denotes column-wise stacking and $\otimes$ is the Kronecker product. The design matrix $A$ is assembled with rows $\varphi(E_i,\rho_j)^{\mathsf T}$, yielding

$A \in \mathbb C^{(n_E\,n_\rho)\times d^4}$

which defines a linear map

$A:\mathbb C^{d^4} \to \mathbb C^{n_E n_\rho}$

acting on a vectorized operator $x = \mathrm{vec}(X)$ as

$A\,x = [\langle\varphi(E_i,\rho_j),x\rangle]_{i,j}$

This encodes all pairwise bilinear measurements, each corresponding to $\mathrm{tr}(E_i X \rho_j)$ up to vectorization. For discrete-time dynamics and input-output systems, related forms appear, with observations expressed as $y_k = C_0 x_k + \sum_{i=1}^p (u_k)_i C_i x_k + v_k$, where the observation operator $C(u_k) = C_0 + \sum_{i=1}^p (u_k)_i C_i$ is affine in control and linear in the state, thus bilinear overall (Sattar et al., 15 Apr 2025, Liu et al., 21 Feb 2025, Sattar et al., 2024).

2. Singular Value Spectrum, Rank, and Nullity

The analysis of a bilinear observation operator $A$ proceeds via the singular value decomposition (SVD). For a tolerance grid $\tau$, the τ–rank is defined as

$$\mathrm{rank}_\tau(A) = \#\{k: \sigma_k > \tau\sigma_{\max}\}$$

with nullity $$\mathrm{nullity}_\tau(A) = d^4 - \mathrm{rank}_\tau(A)$$. A salient phenomenon in such problems is the appearance of extended rank plateaus: as $\tau$ varies over several orders of magnitude, the τ–rank and nullity remain constant across broad intervals. This reflects the clustering of the singular spectrum into well-separated groups, with large gaps between clusters that cannot be surmounted by numerical adjustment of $\tau$. These plateaus indicate true, structurally determined deficits in the observable subspace, in contrast to numerical rank deficiencies found in generic linear systems (Choi, 13 Jan 2026).

3. Sectoral Organization of the Nullspace

The nullspace of a bilinear observation operator exhibits pronounced internal structure. Decomposing the ambient space $\mathbb C^{d^4}$ into sectors $V_s$ according to block structure (e.g., block-diagonal vs. block-off-diagonal with respect to $E$ and $\rho$), and projecting the nullspace basis onto these sectors, one defines the sector weights

$$w_s(\tau) = \frac{\sum_{\ell=1}^{m(\tau)}\|P_s n_\ell\|_2^2}{\sum_{\ell=1}^{m(\tau)}\|n_\ell\|_2^2}$$

At typical $\tau$, the nullspace exhibits pronounced localization; for instance, $w_{\mathrm{off\diag}}(\tau)\approx 0.7$ (Choi, 13 Jan 2026). This reveals that rank loss is not random but is highly concentrated in specific algebraic directions, impeding information flow only in identifiable sectors.

4. Recoverability, Refinement, and Problem Modification

Bilinear observation systems display distinctive limitations regarding rank recovery:

• Numerical refinement (adjusting $\tau$, rescaling, or reparameterizing within a fixed set of $E_i$, $\rho_j$) cannot resolve plateaued deficits; the singular-value gaps prevent recovery of additional rank.
• Problem modification (expanding the measurements $\{E_i\}$, altering coupling constraints, or fundamentally changing the families of operators) can shift the spectrum, fill gaps, and restore full rank ($d^4$), thus accessing the full observable space. This dichotomy distinguishes the bilinear setting from standard linear inverse problems, highlighting the need for structural, not numerical, interventions to enhance observability (Choi, 13 Jan 2026, Pacholska et al., 2020).

5. Solution Methods, Identifiability, and Learning

Bilinear observation operators are prevalent in system identification, operator learning, and estimation. For matrix recovery, the injectivity hinges on the vectorized design matrix $A$ spanning the ambient space; rank conditions on the measurement vectors and persistent excitation are essential for unique recovery (Pacholska et al., 2020, Sattar et al., 2024). Learning dynamics from bilinear observations naturally induces a Kronecker product design matrix, leading to heavy-tailed regression problems whose statistical rates depend on input distribution, noise covariances, and the spectrum of the design (Sattar et al., 2024). Probabilistic identification employs either maximum-likelihood or expectation-maximization methods, which remain well-posed and admit closed-form updates under standard Gaussian assumptions and mild excitation (Liu et al., 21 Feb 2025).

6. Infinite-dimensional Extensions and Schmidt Decomposition

In Hilbert space settings, a bilinear observation operator $T: H_1\times H_2 \to K$ admits a Schmidt representation under a compactness and orderability hypothesis. That is,

$$T(h_1,h_2) = \sum_{n=1}^{\infty} \sigma_n\, \langle h_1,u_n\rangle_{H_1}\, \langle h_2,v_n\rangle_{H_2}\, w_n$$

where $\{u_n\}$, $\{v_n\}$, $\{w_n\}$ are orthonormal in their respective spaces and $\sigma_n \downarrow 0$. The existence of such an expansion reveals the principal bilinear modes that dominate observability and provides a canonical form for regularization, model reduction, and inverse solution construction (Silva et al., 2021). In operator-learning, related frameworks recast operator regression as learning the associated bilinear form on pairs of input and dual-output coordinates, leveraging Kronecker-structured Gaussian process covariance for computationally tractable inference (Mora et al., 2024).

7. Illustrative Examples and Practical Implications

• In quantum tomography, measurement-state pairs structured as in (Choi, 13 Jan 2026) yield observable plateaus and sector-localized nullspaces, requiring modification of measurement protocols rather than finer numerical tuning for full-rank recovery.
• For control from bilinear observations, the Gramian becomes input-dependent, and the cost-to-go function nonconvex, invalidating the separation principle and requiring non-affine, jointly optimal estimation-control design (Sattar et al., 15 Apr 2025).
• Matrix recovery applications (e.g., mixed Time Encoding Machines, continuous localization) admit exact recovery from bilinear and quadratic measurements as soon as the vectorized measurement design achieves full rank, according to precise combinatorial and polynomial-rank criteria (Pacholska et al., 2020).
• In operator learning, encoding the solution operator as a bilinear form over input-function and dual output-function space enables scalable GP-based regression, with Kronecker product structure facilitating inversion and marginal likelihood training (Mora et al., 2024).

A distinguishing signature of bilinear observation operators, across domains, is the presence of structural rank barriers undetectable by simple rank counting or numerical adjustment, the necessity of algebraic or geometric intervention for full-dimensional recovery, and the deep connection between operator structure, identifiability, and practical recoverability of latent states or parameters.