Information Invariance & Continuity

• Information Invariance & Continuity is a framework defining conditions where informational measures remain unchanged under redundant transformations and vary in a controlled manner under perturbations.
• It distinguishes between exact and approximate invariance, using tools like local isometries, gauge symmetry, and smoothing neighborhoods to ensure stability in diverse applications.
• Continuity conditions guarantee robustness in quantum states, channels, and dynamical systems, with practical implications for quantum information, PDEs, and machine learning.

Searching arXiv for recent and foundational papers on information invariance and continuity across quantum information, control, learning, and PDEs. Information invariance and continuity designate a recurrent pair of structural requirements on informational models, observables, and dynamics: invariance requires that the relevant quantity be unchanged under transformations deemed informationally redundant, while continuity requires controlled variation under perturbations of states, channels, priors, geometries, or trajectories. In the cited literature, these requirements appear as local isometric or unitary invariance of divergence-based quantum measures, local gauge invariance of nuclear energy density functionals, translation, rotation, and re-description invariance in crystal representations, motion invariance in video learning, garbling-based monotonicity in games, and reparameterization covariance of Fisher information; continuity appears as stability of entropies, quantum Fisher information, channel capacities, value functions, and Fisher-information fluxes under state, channel, geometric, and topological perturbations (Popp et al., 4 Sep 2025, Raimondi et al., 2011, Tone et al., 4 Feb 2025, Betti et al., 2019, Hogeboom-Burr et al., 2021, Hüpfl et al., 2023).

1. Conceptual architecture

A common formal pattern is that invariance is defined relative to a transformation class, whereas continuity is defined relative to a topology or metric on admissible descriptions. In quantum information, the transformation class is often local isometries or unitaries; in control it is exact or approximate viability under admissible controls; in games it is garbling or perturbation of information structures; in wave physics it is reparameterization and propagation through lossless, parameter-independent regions; in learning it is symmetry of the underlying data description rather than of a particular coordinate realization (Popp et al., 4 Sep 2025, Yao, 2 Jul 2026, Hogeboom-Burr et al., 2021, Hüpfl et al., 2023, Tone et al., 4 Feb 2025).

The corresponding continuity notions are comparably heterogeneous. They include trace-norm, Bures, and energy-constrained Bures continuity for quantum states and channels, Hausdorff perturbations of initial sets in invariance entropy, weak, setwise, and total variation convergence of information structures in games, and modulus-of-continuity preservation for nonlinear diffusions (Rezakhani et al., 2015, Shirokov, 2016, Yao, 2 Jul 2026, Hogeboom-Burr et al., 2021, Caillet et al., 2024). This suggests that “continuity” is not a single principle but a family of topology-dependent stability statements whose content depends on which variables are optimized over and which symmetries are quotiented out.

A second recurrent pattern is the distinction between exact and approximate formulations. Exact invariance may enforce symbolic matching, exact local gauge symmetry, or exact local isometric equivalence; approximate invariance typically introduces neighborhoods, smoothing balls, tolerances, or asymptotic limits (Yao, 2 Jul 2026, Popp et al., 4 Sep 2025, Raimondi et al., 2011). Much of the technical literature studies when these two regimes coincide and when they do not.

2. Quantum states, divergences, and metrological stability

In divergence-based quantum information theory, a generalized quantum divergence $D(\rho\Vert\sigma)$ is defined by the data-processing inequality. The central invariance mechanism is that any isometry admits a reversal channel, so the data-processing inequality can be applied twice to obtain

$$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$

which then propagates to mutual-information and conditional-entropy constructions derived from $D$ (Popp et al., 4 Sep 2025). On this basis, the quantities $I_1$, $I_2$, and $I_3^\varepsilon$ are invariant under local isometries, while $I_4^\varepsilon$, $H_1$, $H_2$, and $H_3^\varepsilon$ are invariant under local transformations of the form $$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$0, with $$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$1 unitary and $$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$2 isometric (Popp et al., 4 Sep 2025). The same framework also tracks smoothing neighborhoods through

$$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$3

so one-shot smoothing is compatible with local isometries (Popp et al., 4 Sep 2025).

A distinct continuity program replaces trace-distance control by purity control. For finite-dimensional systems, continuity bounds for the von Neumann entropy can be made to depend only on purity distance and system dimension, while conditional von Neumann entropy admits uniform continuity bounds in terms of purity distance that are free of the dimension of the conditioning subsystem. The same abstract reports uniform continuity bounds for relative entropy distance, quantum mutual information, and quantum conditional mutual information, together with applications to squashed entanglement and to states arbitrarily close to quantum Markov chains (Kumar et al., 2023). The importance of this formulation is that purity is unitary invariant, so the resulting bounds are naturally adapted to spectral perturbations rather than only to operational distinguishability.

For parameter estimation, the quantum Fisher information $$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$4 depends on both the state and its parameter derivative. The continuity result established for differentiable density operators is that two close states with close first derivatives have close QFIs, with an explicit two-argument estimate

$$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$5

in finite dimension (Rezakhani et al., 2015). The same work introduces a regularized symmetric logarithmic derivative for incomplete-rank states, proves a reduced continuity form when two initial states evolve through one channel, and recovers the general form when one initial state evolves through two channels, a setting explicitly linked there to open-system metrology (Rezakhani et al., 2015). Reparameterization covariance and invariance under $$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$6-independent unitaries reinforce the interpretation of $$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$7 as a metric quantity rather than a representation-dependent observable (Rezakhani et al., 2015).

3. Channels, capacities, and dimension or energy constrained continuity

For quantum channels, the continuity problem is sharpened by asking how informational characteristics vary with the channel when the relevant resource is input dimension rather than output dimension. Using conditional mutual information techniques, continuity bounds were derived for the output Holevo quantity under simultaneous perturbations of a channel and an input ensemble, and then lifted to capacities. In finite input dimension $$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$8, representative estimates include

$$\mathbf{D}(V(\rho)\Vert V(\sigma))=\mathbf{D}(\rho\Vert\sigma),$$9

together with analogous bounds for classical, quantum, and private capacities, all depending on $D$0 and complementing Leung–Smith bounds that scale with output dimension (Shirokov, 2016).

The same program extends to infinite-dimensional inputs by replacing dimensional control with energy control. There the relevant metric is the energy-constrained Bures distance $D$1, which generates the strong convergence of channels and yields tight or close-to-tight continuity bounds for basic capacities of energy-constrained channels (Shirokov, 2016). The entropy–energy function $D$2 controls the modulus, and a special emphasis is placed on multi-mode quantum oscillators, for which explicit upper bounds are available (Shirokov, 2016). In this regime, continuity is no longer merely a finite-dimensional compactness phenomenon; it is tied to physically meaningful constraints on admissible inputs.

Taken together, these results separate two stability mechanisms. The first is finite-dimensional logarithmic control in $D$3; the second is energy-constrained control through $D$4 and $D$5. This division is structurally important because it shows that channel-information continuity is resource-relative rather than absolute (Shirokov, 2016, Shirokov, 2016).

4. Exact invariance, order of limits, and information structures in control and games

In control theory, the 2026 Cantor-dust construction resolves two questions of Kawan negatively. For a continuous-time control system on $D$6 with coordinates $D$7, a frozen Cantor coordinate stores an infinite symbolic code, an exponentially contracting coordinate $D$8 suppresses late mismatches, and a compact graph $D$9 forces exact symbolic agreement (Yao, 2 Jul 2026). The resulting separation theorem states

$I_1$0

so finite strict invariance entropy need not equal ordinary invariance entropy (Yao, 2 Jul 2026). The same example shows that strict invariance entropy is not lower semicontinuous under Hausdorff perturbations of the initial set: there exist $I_1$1 with $I_1$2 for every $I_1$3 while $I_1$4 (Yao, 2 Jul 2026). The source of complexity is not dynamical expansion; the paper explicitly notes the absence of positive Lyapunov exponents and identifies the complexity as arising from exact viability constraints in thin invariant geometry (Yao, 2 Jul 2026).

This order-of-limits phenomenon has a close analogue in strategic settings. For stochastic zero-sum games and team problems, value functions are continuous under total variation convergence of information structures when costs are bounded, but continuity may fail under setwise or weak convergence (Hogeboom-Burr et al., 2021). For teams, upper semicontinuity holds under weak and setwise convergence, whereas for zero-sum games one obtains upper or lower semicontinuity under weak or setwise convergence when the sequence is ordered by Blackwell garbling (Hogeboom-Burr et al., 2021). If the individual channels are independent, fixed, and total-variation continuous in the state, then value functions are continuous under weak convergence of priors (Hogeboom-Burr et al., 2021). By contrast, in general non-zero-sum games, equilibrium payoffs for players may fail to be continuous even under total variation convergence of information structures (Hogeboom-Burr et al., 2021).

These results show that continuity depends jointly on topology and order structure. Invariance under garbling yields monotonicity, and monotonicity can rescue semicontinuity where topology alone does not. Conversely, optimization over policies can destroy continuity even when pointwise integrals are stable (Hogeboom-Burr et al., 2021, Yao, 2 Jul 2026).

5. Local conservation laws, gauge symmetry, and information flow

A different meaning of information continuity appears in local balance laws. In wave scattering, Fisher information is promoted from a global estimation-theoretic quantity to a local density and flux. For quasi-monochromatic electromagnetic fields, the paper introduces a Fisher information density $I_1$5 and a Fisher information flux $I_1$6 satisfying the continuity equation

$I_1$7

with explicit source and absorption terms (Hüpfl et al., 2023). In parameter-independent, lossless regions,

$I_1$8

so Fisher information is locally conserved, and its net outward flux is invariant under propagation through such regions (Hüpfl et al., 2023). The same work gives a matrix-valued multi-parameter generalization and an experimental verification at microwave frequencies, including a case where energy flow and Fisher-information flow are directed differently (Hüpfl et al., 2023).

In nuclear time-dependent density functional theory, continuity equations are tied directly to local gauge invariance. For N3LO nuclear energy density functionals, four sets of constraints on the coupling constants were derived to guarantee continuity equations in all four spin-isospin channels. In particular, validity of the continuity equation in the scalar-isoscalar channel is equivalent to the standard $I_1$9 local-gauge-invariance conditions, while validity in vector and isovector channels requires invariance under local rotations in spin and isospin spaces (Raimondi et al., 2011). The paper’s concluding statement is explicit: validity of the continuity equation in the four spin-isospin channels is equivalent to local-gauge invariance of the energy density functional (Raimondi et al., 2011).

The common structure is that continuity equations encode invariance in differential form. In one case the conserved quantity is Fisher information carried by a wave field; in the other it is current conservation generated by local gauge symmetry. In both cases, continuity is not merely regularity of a functional but a local transport law (Hüpfl et al., 2023, Raimondi et al., 2011).

6. Diffusive dissipation and controlled discontinuity in physical models

For nonlinear but 1-homogeneous diffusive PDEs, information invariance takes the form of monotone nonincrease rather than exact conservation. Using the JKO scheme with generalized transport cost $I_2$0, the canonical evolution is written as

$I_2$1

and the Fisher information

$I_2$2

is shown to stay bounded or decrease across time (Caillet et al., 2024). More generally, functionals of the form $I_2$3 are non-increasing for convex radial $I_2$4, and as a corollary any concave modulus of continuity of $I_2$5 is conserved across time (Caillet et al., 2024). The framework includes relativistic costs as well as $I_2$6-Laplacian-type examples (Caillet et al., 2024). Here continuity is a propagated regularity estimate, while invariance is expressed through 1-homogeneity and scaling invariance of $I_2$7.

A markedly different conclusion arises in the black-hole moving-mirror model of spacetime continuity. There, continuity across a shock wave is relaxed at the Planck scale by introducing a controlled discontinuity through the giant tortoise coordinate. The paper states that a Planck scale sized spacetime discontinuity leads to unitarity, meaning constant asymptotic entanglement entropy, and also yields thermal equilibration and finite total evaporation energy, because the origin of coordinates remains timelike rather than becoming null (Good, 2022). In the uncompromised continuity case, by contrast, the origin approaches null future infinity and entropy and emitted energy diverge (Good, 2022).

These two examples pull in opposite directions. One studies how continuity of regularity is preserved by a diffusion; the other studies how relaxing geometric continuity can preserve information. This suggests that “continuity” is not intrinsically pro-informational: whether it stabilizes or destroys information depends on the role continuity plays in the underlying dynamical model (Caillet et al., 2024, Good, 2022).

7. Representation learning and symmetry-aware generative modeling

In machine learning, invariance and continuity are increasingly imposed at the objective or architecture level rather than treated as emergent properties. Continuous Invariance Learning addresses continuously indexed domains $I_2$8 and targets the conditional independence condition

$I_2$9

rather than domain-blindness $I_3^\varepsilon$0 (Lin et al., 2023). The practical proxy compares a regressor $I_3^\varepsilon$1, estimating $I_3^\varepsilon$2, with a regressor $I_3^\varepsilon$3, estimating $I_3^\varepsilon$4, and uses the min–max objective

$I_3^\varepsilon$5

The theoretical argument is that naive discretization of continuous domains can fail because empirical per-domain estimators become too noisy when domains are numerous, whereas the regression-gap criterion exploits the continuity of $I_3^\varepsilon$6 directly (Lin et al., 2023).

In crystal structure prediction, ContinuouSP formalizes translation invariance, rotation invariance, and strong re-description invariance at the level of solid materials and then proves continuity of the induced energy and density models (Tone et al., 4 Feb 2025). Its energy-based model uses a modified CGCNN with continuous attenuation

$I_3^\varepsilon$7

to remove cutoff discontinuities, and the resulting theorems state that the energy function is strongly re-description invariant while $I_3^\varepsilon$8 is translation- and rotation-invariant and continuous (Tone et al., 4 Feb 2025). The paper explicitly interprets this as enforcing a quotient structure in which the model depends on the crystal rather than on arbitrary coordinate choices (Tone et al., 4 Feb 2025).

The same conjunction appears in video learning. Betti, Gori, and Melacci formulate motion invariance by requiring that features be constant along optical-flow trajectories,

$I_3^\varepsilon$9

and incorporate this into a variational “principle of least cognitive action” together with information-based terms and space-time regularization (Betti et al., 2019). The result is an unsupervised theory of convolutional filter learning over the retina in which temporal continuity of the stream induces invariance constraints on learned representations (Betti et al., 2019).

Across these models, invariance removes representational redundancy, while continuity regularizes optimization, sampling, and temporal evolution. In this sense, information invariance is not only a property of a finished model but a design principle for the admissible description space itself (Lin et al., 2023, Tone et al., 4 Feb 2025, Betti et al., 2019).