Erasure Modality: A Multi-Domain Framework

• Erasure modality is a framework for operational deletion that specifies which components to erase, how to intervene, and the guarantees required to maintain system function.
• In multimodal learning, it enforces certified deletion through techniques like parameter surgery and saliency analysis to effectively remove designated input channels.
• Across domains such as diffusion models, quantum information, thermodynamics, and type theory, erasure modality integrates structured control mechanisms to validate and monitor information removal.

Searching arXiv for the cited works and recent uses of “erasure modality” across domains. Search query: "erasure modality" Erasure modality is not a single standardized term across contemporary research. In current usage, it denotes a structured way of specifying what is being erased, where the erasure acts, which access channel or phase is affected, and what guarantee accompanies the intervention. In multimodal learning it can mean certified deletion of one sensed channel from model parameters; in diffusion models it can mean either a mode of intervention or an input attack surface; in quantum information it can mean recoverable coherence under restricted access or a flagged loss channel; in thermodynamics it can mean a controlled regime of memory reset; and in type theory it can mean a phase or grade distinction between runtime-relevant and runtime-irrelevant data (Fu et al., 18 Feb 2026, Lu et al., 22 May 2025, Miatto et al., 2014, Mir et al., 5 Dec 2025, Theocharis et al., 1 May 2026).

1. Terminological scope

The phrase appears in several technically distinct literatures. In each case, “erasure” is not merely absence. It is an operational condition: the system must continue to function, and the erased component must be made irrelevant in a specific sense defined by the surrounding formalism.

Domain	Erased object	Meaning of “modality”
Multimodal ML	One input channel	Certified deletion regime for model behavior and parameters
Diffusion models	A concept or its access path	Intervention locus or input/attack surface
Quantum information	Which-path information or a lost mode	Recoverable coherence or flagged-noise regime
Thermodynamics	Memory content or low-cost reset capability	Strong/weak erasure or exclusive control
Type theory	Runtime-irrelevant terms	Phase or grade governing compilation-time removal

A plausible unifying implication is that an erasure modality always couples an object of erasure to a control structure. The control structure may be a calibration batch and certificate, a text-conditioning interface, an accessible subsystem, a work budget, or a typing discipline. What varies from field to field is the formal criterion for saying that erasure has actually occurred.

2. Certified modality deletion in multimodal learning

In revocable multimodal sentiment analysis, erasure modality is formulated most explicitly as deletion of one synchronized input stream—such as $X^L$, $X^A$, or $X^V$—from a model that was previously trained on all of them. The central claim of Missing-by-Design is that deletion must exceed “not reading” a modality at inference time. It introduces two stronger requirements: parameter-level erasure and certifiability. The target is a surgically modified model $\mathcal{W}'$ that is hard to distinguish from a hypothetical model $\mathcal{W}^{-m^\star}$ trained without the deleted modality, up to a formal $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$-style bound (Fu et al., 18 Feb 2026).

The framework prepares for deletion during representation learning. Frozen feature extractors yield $d_L=768$, $d_A=74$, and $d_V=512$, missing modalities are zero-padded, and a fusion network produces a joint representation $Z_i$. For each modality $X^A$0, a generator $X^A$1 reconstructs $X^A$2 from the remaining modalities and a property embedding $X^A$3 with $X^A$4. A decomposition module splits each modality into a sample-specific component $X^A$5 and a sample-invariant component $X^A$6, constrained by orthogonality, invariance, and property-alignment losses. Back-translation networks $X^A$7 and an NCE objective ensure that the fused representation retains modality-specific detail.

Deletion itself is implemented as weight surgery. For each parameter $X^A$8, modality saliency is computed from reconstruction-loss gradients,

$X^A$9

while a SwiftPrune-inspired proxy

$X^V$0

approximates task-loss increase under removal. Candidates satisfy high saliency and low $X^V$1, after which only a fraction $X^V$2 is modified. The update is either deterministic zeroing for $X^V$3 or Gaussian perturbation with

$X^V$4

The framework outputs a Modality Deletion Certificate containing the target modality, modified parameter indices, seed and noise scale, privacy budget, SHA-256 digest, and diagnostic specifications. On CMU-MOSI audio deletion, the reported white-box and black-box attack success rates drop from approximately $X^V$5 and $X^V$6 without deletion to approximately $X^V$7–$X^V$8 after surgery, while Acc2 remains approximately $X^V$9–$\mathcal{W}'$0 versus a no-deletion baseline of approximately $\mathcal{W}'$1. Full retraining without audio takes $\mathcal{W}'$2 hours, whereas MBD surgical deletion takes $\mathcal{W}'$3 seconds total—$\mathcal{W}'$4 s for saliency computation, $\mathcal{W}'$5 s for sensitivity calibration, and $\mathcal{W}'$6 s for parameter surgery. In this literature, erasure modality is therefore a certified operating regime for revocable multimodal models, not merely a missing-input pattern.

3. Concept erasure in diffusion models

In diffusion-model research, “erasure modality” has two distinct meanings. One is a mode of intervention inside the model. The other is an input modality through which an erased concept can re-enter generation.

The intervention-based meaning is made explicit in work that distinguishes destruction-based removal from guidance-based avoidance. Destruction-based removal changes the model so that outputs containing the target concept become low likelihood across prompts; guidance-based avoidance leaves concept knowledge in place but disrupts or redirects the conditional guidance that normally steers sampling toward it. This distinction maps onto intervention loci: global U-Net edits behave more destruction-like, while cross-attention or text-embedding edits are usually more guidance-like. Empirically, GA and STEREO are described as strongly destruction-based, UCE and ESD-x as guidance-based, and RECE as mixed (Lu et al., 22 May 2025).

A second line of work argues that erasure must be benchmarked across input modalities. M-ErasureBench evaluates text prompts, learned embeddings via Textual Inversion, and inverted latents via DDIM inversion, with white-box and black-box variants for the latter two. The benchmark reports that current methods often appear strong under text prompts yet fail badly under learned embeddings and inverted latents, with Concept Reproduction Rate exceeding $\mathcal{W}'$7 in the white-box setting. To mitigate this, IRECE localizes target concepts via cross-attention and perturbs the associated latents during denoising, reducing CRR by up to $\mathcal{W}'$8 under the hardest white-box latent inversion scenario while preserving visual quality (Weng et al., 28 Dec 2025).

A third formulation is Instant Concept Erasure, which defines erasure directly in semantic text-conditioning space and then compiles the edit into static weight updates. ICE constructs erase and preserve subspaces from text embeddings, uses anisotropic energy-weighted scaling to form $\mathcal{W}'$9 and $\mathcal{W}^{-m^\star}$0, computes an explicit overlap projector

$\mathcal{W}^{-m^\star}$1

and solves a convex Spectral Unlearning Objective whose analytical solution yields a dissociation operator $\mathcal{W}^{-m^\star}$2. The permanent edit is

$\mathcal{W}^{-m^\star}$3

Because this acts at early text-conditioning layers, the method is presented as training-free, one-shot, runtime-free, and modality-agnostic across T2I and T2V models (Biswas et al., 24 Nov 2025).

Taken together, these works make “erasure modality” in diffusion models fundamentally relational. It may denote where erasure is imposed, or which pathway an attacker uses to reconstitute the supposedly erased concept.

4. Quantum-information formulations

In quantum information, erasure modality has at least three non-equivalent meanings: limited-access coherence recovery, flagged erasure noise, and continuous erasure detection.

In the quantum-erasure problem with limited environmental access, the task is to maximize the average visibility of a signal qubit $\mathcal{W}^{-m^\star}$4 by performing a non-selective measurement only on an accessible subsystem $\mathcal{W}^{-m^\star}$5 of the environment, while another subsystem $\mathcal{W}^{-m^\star}$6 remains inaccessible. The operational quantity is the limited-access coherence

$\mathcal{W}^{-m^\star}$7

For a two-dimensional accessible environment, this becomes

$\mathcal{W}^{-m^\star}$8

where $\mathcal{W}^{-m^\star}$9 is the sub-fidelity of the conditional states of the inaccessible environment. Here the erasure modality is the maximal operational capability to recover coherence given restricted control; its limit is set by what the inaccessible environment still “remembers” (Miatto et al., 2014).

In continuous-variable communication, erasure is instead a noise modality: a mode is either transmitted or completely lost and replaced by vacuum,

$(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$0

with the erasure assumed detectable. A three-mode CV code is built precisely for this flagged replacement channel. It protects one signal mode by encoding it with a bipartite entangled resource, and the paper states that in realistic deployments it can almost completely reverse a single erasure and improve fidelities for two erasures relative to direct transmission. The code is presented as the simplest known code protecting a single mode against erasures, and performance gains are reported up to erasure probabilities of approximately $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$1 (Villasenor et al., 2022).

In superconducting dual-rail qubits, erasure is a hardware-level loss from the logical manifold to $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$2, and the “erasure modality” is a mode of operation in which such loss is monitored continuously with minimal disturbance of logical dynamics. A single symmetrically coupled resonator enables single-shot erasure detection in $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$3 ns, with residual error per check $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$4, induced dephasing per check $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$5, and erasure error per check $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$6. The same architecture also realizes time-continuous erasure detection during gates, with median $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$7 error per gate and less than $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$8 added by detection itself. In this setting, erasure modality denotes an always-on monitoring regime compatible with fault-tolerant control (Hung et al., 17 Apr 2026).

5. Thermodynamic and operational control of erasure

In thermodynamic work on memory reset, erasure modality is defined by the physical embedding of a logically many-to-one map. Strong erasure requires both a single state-independent procedure and an environment that ends in the same macrostate regardless of the initial logical value. Weak erasure keeps the single-procedure condition but allows the environment to end differently, so information may persist outside the designated memory cell. Phase-space analysis yields no minimum entropy cost for weak erasure and a positive minimum entropy cost for strong erasure; in the symmetric case of equal initial phase-space volumes, the lower bound is $(\varepsilon_{\mathrm{mod}},\delta_{\mathrm{mod}})$9. The same work argues that at molecular scales the dominant source of entropy creation is often the extra entropy required to suppress thermal fluctuations, rather than the many-to-one map itself (Norton, 24 Feb 2025).

A different thermodynamic formulation treats erasure as an assisted operational primitive whose cost can be distributed asymmetrically across agents. If Bob’s memory $d_L=768$0 is locally erased, the baseline cost is $d_L=768$1 in units with $d_L=768$2. If Alice holds a correlated system $d_L=768$3, the minimal assisted cost is

$d_L=768$4

Exclusive erasure is defined by a work budget $d_L=768$5 such that

$d_L=768$6

meaning Alice can erase within budget but Eve cannot. In the device-dependent regime, entanglement of formation exactly characterizes exclusivity; in a one-sided device-independent regime, exclusivity is certified when the assisted cost falls below a complementarity threshold $d_L=768$7. Here erasure modality means a controlled capability: “can erase at cost $d_L=768$8” versus “cannot,” and that capability can be exclusive (Mir et al., 5 Dec 2025).

These thermodynamic papers differ from the ML and quantum-channel literatures in one important respect: the central quantity is not only what disappears, but who is authorized to make it disappear and at what work cost.

6. Type-theoretic erasure

In type theory, erasure modality is a phase distinction internal to the logic. One line of work formulates type theory with erasure as a second-order generalized algebraic theory with two modes: runtime mode $d_L=768$9 and erased mode $d_A=74$0. The system includes a proposition $d_A=74$1 that can appear in a context and an isomorphism

$d_A=74$2

This gives erasure the form of an open modality: in the presence of $d_A=74$3, runtime and erased phases become isomorphic in a controlled way. The formulation has models based on categories with families, extends to Grothendieck toposes equipped with a tiny proposition, is conservative over MLTT in both phases, and supports code extraction by a presheaf model into untyped $d_A=74$4-calculus, with correctness proved by gluing (Theocharis et al., 1 May 2026).

A second line develops erasure as one instance of a graded modal dependent type theory parameterized by a modality structure $d_A=74$5. Grades track variable usage, and the grade $d_A=74$6 is interpreted as erasable. The system supports $d_A=74$7-types, weak and strong $d_A=74$8-types, natural numbers, an empty type, a universe, and extensions including weak and strong unit types and graded $d_A=74$9-types. The formalization proves subject reduction, normalization, decidability of definitional equality, a substitution theorem for grade assignment, and preservation of grades under reduction. Extraction removes content marked with grade $d_V=512$0, especially function arguments with the erasable grade, and is proved sound for natural number programs; soundness also extends to certain open programs when all variables in context are erasable, the context is consistent, and erased matches are disallowed for weak $d_V=512$1-types (Abel et al., 31 Mar 2026).

In both formulations, the modality is not a heuristic compiler pass. It is a semantic discipline that determines which terms may safely disappear from runtime code.

7. Cross-domain structure and open problems

Taken together, these works suggest four recurring coordinates for analyzing an erasure modality. The first is the carrier of erased information: a sensed channel, a concept direction, a quantum alternative, a logical value, or a term argument. The second is the locus of intervention: model parameters, conditioning interfaces, accessible environments, thermal operations, or typing rules. The third is the witness of success: a deletion certificate, a low CRR under multimodal attacks, a trace-norm or sub-fidelity bound, a work inequality, or a logical-relations proof. The fourth is the residual-utility constraint: sentiment accuracy, image or video quality, recoverable visibility, bounded work, or preservation of program outputs. This synthesis is inferential, but it matches the explicit structure of the surveyed papers.

The same synthesis also highlights recurrent limits. In revocable multimodal learning, residual leakage can persist in deeply shared layers and the sensitivity bounds are conservative (Fu et al., 18 Feb 2026). In diffusion models, no current method is presented as achieving fully representation-level erasure without side effects, and robustness can collapse once evaluation moves from text prompts to learned embeddings or inverted latents (Lu et al., 22 May 2025, Weng et al., 28 Dec 2025). In quantum hardware, erasure detection already supports a continuous operating modality, but readout-induced lifetime degradation and TLS-mediated instability remain material constraints (Hung et al., 17 Apr 2026). In type theory, extraction soundness depends on disciplined grading and, for some open-program settings, additional consistency conditions (Abel et al., 31 Mar 2026).

Erasure modality is therefore best regarded not as a single doctrine of deletion, but as a family of formal regimes for making designated information operationally irrelevant while preserving a controlled remainder of function, semantics, or coherence.