--- title: 'Local Activability: Theory & Applications' url: https://www.emergentmind.com/topics/local-activability type: topic --- # Local Activability: Theory & Applications Searching arXiv for recent and foundational uses of “local activability” and closely related terms. I’m retrieving relevant arXiv records on local activity, local activation, and excitability across the main research areas where the term is used. Local activability denotes a family of technically distinct notions organized around a common question: whether a latent property of a system can be induced, revealed, or operationalized by interventions that remain local in ports, subsystems, spatial regions, activation sites, or field algebras. In control theory, local activity is the negation of local passivity and is defined by the sign of a work functional. In quantum information, local activability includes catalytic Bell-nonlocality activation and the conversion of locally distinguishable state sets into locally indistinguishable or incompletable ones by LOCC-type procedures. In stochastic transport and active matter, it refers to dynamics whose parameters change only at designated sites or on designated subregions. A cognate but distinct usage appears in glassy dynamics, where “local activity” is a space- and time-integrated count of rearrangement-like jumps [1610.01294], [2504.02042], [2311.10647], [1705.00855], [2604.19861]. ## 1. Conceptual range and recurrent structure The arXiv literature does not use a single universal definition of local activability. Instead, the term and its close variants—local activity, local activation, hidden nonlocality activation, and local excitability—are domain-specific formalizations of locality-constrained access to a resource, instability, or dynamical mode. | Domain | Local object | Operational meaning | |---|---|---| | Linear systems | Ports and work functional \(W_T(u)\) | Negative dissipation for some local excitation | | Quantum information | LOCC, local CPTP maps, catalysts, OPLMs | Activation of Bell nonlocality or local indistinguishability | | Stochastic dynamics | Activation site or active segment | History-dependent jump rates, persistence, or transport | | Quantum field theory | Local field algebra | One state excitable from another in a GNS sector | A common structural pattern is that the property of interest is not necessarily manifest in the unprocessed system. In the control setting, instability or non-dissipativity may only become visible through suitable port excitation. In Bell scenarios, a Bell-local state may become Bell nonlocal after a deterministic local catalytic protocol. In state discrimination, a locally distinguishable set may become locally indistinguishable only after an orthogonality-preserving local measurement. In locally activated random walks, memory is generated only by repeated visits to an activation site. In algebraic field theory, local excitability is not a statement about arbitrary state conversion but about realization within the local operator algebra [1610.01294], [2504.02042], [2507.10698], [2311.10647], [2604.19861]. This suggests that “local activability” is best understood as a locality-constrained transition principle rather than a single resource-theoretic invariant. What changes from field to field is the relevant notion of locality: ports, tensor factors, spatial hotspots, or von Neumann algebras. ## 2. Local activity and passivity in linear systems In the linear-systems formulation, the basic model is \[ \dot x(t)=Ax(t)+Pu(t), \] with \(A\in \mathbb{R}^{n\times n}\) and \(P\) a projection matrix. For \(x(0)=0\), the work functional is \[ W_T(u):=\int_0^T (x(t),Pu(t))\,dt. \] The system is locally active if there exist \(T>0\) and a continuous input \(u\in C([0,T],\mathbb{R}^n)\) such that \(W_T(u)<0\); it is locally passive if \(W_T(u)\ge 0\) for all \(T>0\) and all continuous \(u\). Chua introduced local activity as an indicator of the possible emergence of complex behavior [1610.01294]. The sharpest characterization occurs in the full-port case \(P=I\). Then local activity is equivalent to non-dissipativity of \(A\), namely the existence of \(x\neq 0\) such that \((Ax,x)>0\). Equivalently, \(A+A^\top\) has an eigenvalue \(>0\). Local passivity is therefore equivalent to dissipativity, \((Ax,x)\le 0\) for all \(x\), or \(A+A^\top\le 0\). This is the clean matrix criterion of the theory [1610.01294]. For general projections \(P\neq I\), the situation is subtler. A necessary condition is the existence of \(x\) such that \((PAx,x)>0\), but the paper states that a full characterization in terms of \(A\) and \(P\) remains open. Nonetheless, a generic result is available: for every nontrivial projection \(P\neq 0\), there is an open and dense set \(\mathcal M\subset \mathbb R^{n\times n}\) such that if \(A\in\mathcal M\) and \(A\) has an eigenvalue with positive real part, then the system is locally active. The generic proof uses a distinguished dominant spectral component and a nondegeneracy condition \(g_{11}h_{11}\neq 0\) in the eigenvector matrix and its inverse [1610.01294]. The same paper proposes an abstract nonlinear pipeline. Starting from \[ \dot x(t)=f(x(t)) - P D(x(t)), \] one freezes the dissipation term at an equilibrium \(x_0\), obtaining a perturbed system \[ \dot x(t)=f(x(t)) - P D(x_0)+Pu(t), \] and then linearizes at \(x_0\) to \[ \dot x(t)=Df(x_0)\,x(t)+Pu(t). \] The intended interpretation is local: if the linearized system is locally active, this may indicate a route to complexity in the nonlinear dynamics near the equilibrium. The paper does not claim a complete nonlinear theory and explicitly leaves open questions on topological conjugacy, locally active yet asymptotically stable systems, and extensions to Hilbert-space projections and edge-of-chaos phenomena [1610.01294]. ## 3. Quantum-information activation phenomena In Bell nonlocality, local activability appears in catalytic form. A Bell-local entangled state \(\rho_{AB}\) can be transformed by local CPTP maps \(\mathcal E_{AC_A}\otimes \mathcal E_{BC_B}\) into a Bell-nonlocal state while a bipartite catalyst \(\omega_{C_A C_B}\) is returned exactly in its initial state. The protocol is deterministic and uses local operations only; no classical communication is needed. The central structural theorem states that if \(\rho_{AB}^{\otimes n}\) is Bell nonlocal for some \(n\), then one copy of \(\rho_{AB}\) exhibits catalytic Bell-nonlocality activation with an explicit catalyst built from mixtures of \(\rho_{AB}^{\otimes i}\sigma_{AB}^{\otimes(n-1-i)}\). If \(n\) is the smallest activating copy number, the catalyst itself can still be Bell local. The paper further derives catalytic activation for every state with singlet fraction \(F(\rho_{AB})>1/d\), including two-qubit isotropic states in the regime \(1/3\max(C(P_1),C(P_2))\). An explicit example uses the BS wiring on \(P_{\mathrm{ISO}(\xi)}\times P_{\mathrm{NLC}(\xi')}\), yielding \[ C(P')=\xi+\frac{\xi'(1-\xi)}{8}, \] which exceeds the non-local cost of either component for \(0<\xi'\le \xi<1\) [1009.4207]. A distinct, and now extensive, usage concerns orthogonal state discrimination by LOCC. Here a locally distinguishable set \(\mathcal S\) is locally activable if it can be transformed into a locally indistinguishable orthogonal set via orthogonality-preserving local measurements. This notion supports a hierarchy by the number of parties that must come together. The classification paper gives both activable and non-activable structures: \(\mathcal S_3\subset \mathbf C^6\otimes \mathbf C^6\) is locally distinguishable and can be transformed deterministically into the UPB in \(\mathbf C^3\otimes \mathbf C^3\), while \(\mathcal S_2\subset \mathbf C^4\otimes \mathbf C^2\otimes \mathbf C^2\) satisfies \(\mathcal H^{\mathrm{LOCC}_1}(\mathcal S_2)=0\) and \(\mathcal H^{\mathrm{LOCC}_2}(\mathcal S_2)=0\), giving a concrete instance of “no activation across bipartitions” [2507.10698]. The same discrimination program has been refined into a hierarchy of locality. In \((2m+1)\otimes 2\otimes (2m+1)\) systems, one family of sets can be activated by a single-party OPLM, while another cannot be activated locally but can be activated by joint measurements of two parties. On that basis the paper introduces “strong local” sets, namely locally distinguishable sets whose hidden nonlocality cannot be activated even with joint measurements. It also defines \(m\)-activable and strong-\(m\)-activable sets, thereby ordering locality and nonlocality by the minimal partition structure needed for activation [2408.01860]. An even stronger notion is activation of incompletability. A locally distinguishable orthogonal product set is incompletability activable if some OPLM/LOCC protocol maps it to an incompletable set; in the main examples the output is strictly incompletable. The principal example is a 10-state set \(\mathcal S_2\subset \mathbb C^6\otimes\mathbb C^6\) that is LOCC distinguishable, completable, and free from local redundancy, yet is transformed by a two-step local measurement protocol into the UPB in \(\mathbb C^3\otimes\mathbb C^3\) in each branch. The central theorem states \[ \text{activation of incompletability} \Rightarrow \text{activation of nonlocality}, \] whereas the converse fails. Under LICC, the paper further proves that if incompletability can be activated, then the original set can still be extended to a complete orthonormal basis, but the completed basis is no longer perfectly distinguishable by LOCC [2607.00797]. ## 4. Locally activated stochastic dynamics and partially active polymers Locally activated random walks provide an explicit microscopic realization of local activability in stochastic dynamics. The walker moves on an infinite \(d\)-dimensional lattice and ages only when it visits a designated activation site, taken to be the origin. The activation variable is \[ a(t)=\int_0^t \mathds{1}\!\left(x(t')\right)dt', \] the cumulative residence time at the activation site. The jump rate is \(1/\tau(a)\), so the walk is non-Markovian in position alone, though the joint process \((x(t),a(t))\) is Markovian. The theory covers both passive nearest-neighbor walks and persistent active walks; for the latter, large-scale behavior maps to a non-persistent walk with rescaled waiting time \[ \tilde{\tau}(a)=\frac{1-\varepsilon(a)}{1+\varepsilon(a)}\tau(a). \] The exact renewal framework yields the joint law \(P(x,a,t)\) and identifies a dimension-independent trapping criterion, \[ S\equiv e^{-\int_0^\infty \frac{da}{\tau(a)}}>0, \] under which the walker can remain at the activation site forever. The resulting phenomenology includes aging, non-Gaussian marginals, non-monotonic spatial profiles in accelerated cases, cusps and localization in decelerated cases, and recurrent-versus-transient distinctions between \(d=1,2\) and \(d=3\) [2311.10647]. Partially active Rouse polymers are a continuum counterpart in which activity is localized not at a point but on a segment \[ S_A=[f_1L,f_2L]. \] The overdamped dynamics \[ \gamma \frac{\partial \mathbf r(s,t)}{\partial t} = k \frac{\partial^2 \mathbf r(s,t)}{\partial s^2} + \mathbf n_T(s,t)+\mathbf o_A(s,t) \] combine thermal noise with active forcing whose spatial covariance is confined to \(S_A\) and whose temporal correlations are typically Ornstein–Uhlenbeck with persistence time \(\tau_A\). The location and size of the active segment control swelling and dynamics. End-localized activity swells the chain more strongly than a central active segment of the same length; \(\langle R_g^2\rangle\), \(\langle R_e^2\rangle\), local monomer–monomer distances, and tagged-monomer MSDs all depend strongly on the placement of \(S_A\). For \(\tau_1\ll \tau_A\), short chains show anomalous scaling, whereas in the long-chain regime \(\tau_1\gg \tau_A\) standard Rouse-like scaling is recovered. Reconfiguration and looping times also display crossover behavior, particularly when \(\tau_1\sim \tau_A\), reflecting competition between polymer relaxation and active persistence [2412.04175]. Taken together, these models show that local activability in stochastic systems is a mechanism for generating memory without globally altering the dynamics. The source of non-Markovianity is geometrically sparse: a hotspot, a finite active segment, or repeated returns to a favored location. ## 5. Spatially localized activity in soft matter and materials In activated diffusiophoresis, a local quench of a conserved density field creates a diffusing perturbation governed by \[ \partial_t \rho(\mathbf r,t)=D_0\nabla^2\rho(\mathbf r,t), \] and the resulting density gradients generate phoretic motion of inclusions. In coarse-grained form, \[ \gamma\dot{\mathbf r}=\mathbf F(\mathbf r(t),t),\qquad \mathbf F=-\mu\nabla\rho. \] The paper shows that suitable activation protocols can extract work from an inclusion and, more strikingly, generate a conveyor by sequentially activating strips between two walls. It also proves a strong limitation: in a steady density profile \(\nabla^2\rho=0\), so \(\nabla\cdot \mathbf F=0\), and stable trapping is impossible, explicitly compared with Earnshaw’s theorem. Time dependence is therefore essential for controlled transport [1911.07367]. A different localization mechanism appears for particles moving on random arrays of parallel filaments. Here particles switch between diffusion and processive motion in a mixed-polarity quenched environment. In each region between filament start and end points, the local asymmetry \[ \alpha_i=\frac{b(n_i^+-n_i^-)}{n_i} \] sets an effective active bias. In the rapid attachment-detachment limit \(k_a,k_d\gg v/L_f\), the dynamics reduce to a one-dimensional advection-diffusion process with region-dependent \(v_{\mathrm{eff},i}\) and \(D_{\mathrm{eff},i}\), which can be rewritten as motion in a noisy effective energy landscape. Convergent filament orientations produce wells in that landscape, and localization is strongest at intermediate run lengths, where transport is long enough to sense filament polarity but not so long as to facilitate escape from traps [2606.00286]. In active solids, localizing activity across selected nodes functions as a control parameter for collective actuation. On triangular and disordered elastic lattices, only a fraction \[ \phi_{dn}=\frac{n_{dn}}{N} \] of nodes are driven by polar active units. The lattice displacement is decomposed into normal modes, and the modal energy distribution is tuned by changing which nodes are active. The paper introduces a Monte Carlo optimization over activity configurations \(|\sigma\rangle\), using the target-mode cost \(C_k^{|\sigma\rangle}=\overline{P_k^2}[\sigma]\) and Metropolis updates \[ P(|\sigma \rangle \rightarrow |\sigma' \rangle)= \min\left\{1,\exp\left(-\frac{C_k^{|\sigma' \rangle}-C_k^{|\sigma \rangle}}{T_e}\right)\right\}. \] For ordered lattices, a symmetry-aware manual choice can outperform the algorithm, whereas in disordered lattices the optimization outperforms manual trials. The ordered case further motivates a design principle based on mode susceptibility along activation paths, \[ S_k(L)=\sum_{l=0}^{|L|}\frac{|\hat{\mathbf e}_l\cdot(\boldsymbol\varphi_k)_l|}{|L|}, \] which measures how strongly a mode aligns with a proposed path of active nodes [2407.13682]. The paper on activated solids shifts the emphasis from mode selection to spontaneous deformation and failure. In a dense triangular solid of active Brownian particles, the local non-affinity is defined by the residual after best affine fitting of neighbor displacements, \[ \chi_0=\sum_i \left(\mathbf\Delta_i-\mathcal E^m[\mathbf R_i-\mathbf R_0]\right)^2, \] and the global non-affinity is \(X=\frac1N\sum_i \chi_i\). The central scaling law is \[ \chi=\mathcal G^{-1}\left(\tilde D_t+\frac{\Lambda^2}{2}\right), \] so, in the regime studied, \(\chi\sim \Lambda^2\) and increases as density approaches melting. Simulations verify \(\nu=2.112\pm0.006\) for \(\tilde D_t=1\) and \(\nu=2.00\pm0.01\) for \(\tilde D_t=0\), while the shear modulus softens as \[ \mathcal G=\mathcal G_0-\mathcal G_1\Lambda^2 \] with \(\mathcal G_0=109.84\) and \(\mathcal G_1=0.094\). A local activity patch, implemented by \(\Lambda_i(t)=\sigma_i(t)\Lambda\) inside a circular region, selectively increases non-affinity and nucleates defects there before these propagate outward. The activity-driven melting sequence is two-step, solid \(\to\) hexatic \(\to\) fluid [2504.06914]. A related but diagnostically distinct use of local activity appears in colloidal glasses. There the activity along a tracer trajectory is \[ K[\mathbf r_i](t_{\mathrm{obs}}) =\sum_{j=1}^{M=t_{\mathrm{obs}}/\Delta t} \Theta\!\left(\|\mathbf r_i(j\Delta t)-\mathbf r_i((j-1)\Delta t)\|-a\right), \] a space- and time-integrated count of rearrangement-like jumps. The mean activity is close to Brownian predictions and is therefore a poor discriminator, but the full distribution \(P(K)\) becomes broader and more skewed as the suspension is cooled, with a low-activity tail interpreted as evidence for inactive regions and as a precursor of a dynamical first-order transition picture. Although this work studies “activity” rather than “activability,” it supplies an experimentally accessible space-time extensive observable for inactive versus active dynamical phases [1705.00855]. ## 6. Local excitability in algebraic quantum field theory In algebraic quantum theory, local activability is formalized as excitability between states. Let \(A_0\) be a \(*\)-algebra of fields, \(\omega_1,\omega_2\) states, and \(A_{\omega_i}\) the associated von Neumann algebras in the GNS representations. The notation \[ \omega_2\prec \omega_1 \] means that \(\omega_2\) can be excited out of \(\omega_1\): there exists a density matrix on \(H_{\omega_1}\) reproducing the \(\omega_2\) correlation functions, or, more generally, a normal field-preserving homomorphism from \(A_{\omega_1}\) to \(A_{\omega_2}\). Mutual excitability is equivalent to quasiequivalence, \[ \omega_1\prec\omega_2\ \text{and}\ \omega_2\prec\omega_1 \iff \omega_1\sim\omega_2, \] so one-way excitability is, in principle, weaker than quasiequivalence [2604.19861]. The key technical instrument is canonical purification \(\hat\omega\), which allows the paper to prove \[ \omega_2\prec\omega_1 \iff \hat\omega_2\prec\hat\omega_1. \] For zero-mean Gaussian states in generalized free field theory, excitability becomes completely explicit in phase-space terms. If \(\mu_i\) denotes the symmetrized two-point form, \(L:K_{\mu_1}\to K_{\mu_2}\) the induced map, \(Q=L^\dagger L\), and \(X_i\) the corresponding two-point operators, the main criterion is \[ \omega_2\prec\omega_1 \iff \mu_2\prec\mu_1,\quad \sqrt{X_2}-\sqrt{X_1}\in\mathcal L^2,\quad \ker Q=0. \] For pure zero-mean Gaussian states this simplifies to the Hilbert–Schmidt condition \(Q-1\in\mathcal L^2\) together with \(\mu_2\prec\mu_1\) [2604.19861]. The most striking consequence is specific to the Gaussian setting: one-way excitability implies two-way excitability. Hence, for zero-mean Gaussian states, \[ \omega_2\prec\omega_1 \iff \omega_1\prec\omega_2 \iff \omega_1\sim\omega_2. \] The paper presents this as a generalization of the quasiequivalence theorems of Powers, Størmer, van Daele, Araki, and Yamagami. In this form, local excitability is an operational rephrasing of representation-theoretic equivalence, controlled by covariance domination and Hilbert–Schmidt regularity rather than by arbitrary state conversion [2604.19861]. Across these literatures, local activability is therefore not a single theorem but a recurring structural motif. It names the possibility that locality-preserving interventions—whether port excitations, OPLMs, local CPTP maps, activation sites, active patches, or local field operators—can expose latent instability, hidden nonlocality, memory, transport asymmetry, or sector equivalence that is not manifest at the level of the unprocessed system. Source: https://www.emergentmind.com/topics/local-activability