Energy-Based Uncertainty Principle

• Energy-Based Uncertainty Principle is a methodological class that replaces traditional localization with energy-like quantities capturing hidden additive, dynamical, and control structures.
• It connects additive energy in harmonic analysis, Hamiltonian dispersion in quantum mechanics, and L2-energy in control theory to refine classical uncertainty bounds.
• The framework offers deterministic recovery and improved signal reconstruction by integrating precise structural measures and energy corrections into uncertainty assessments.

In current usage across these works, the expression Energy-Based Uncertainty Principle denotes a family of mathematically distinct constructions in which an energy-like quantity sharpens an uncertainty statement. In finite-group harmonic analysis, the relevant quantity is additive energy and the principle constrains simultaneous sparsity of a signal and its discrete Fourier transform; in quantum measurement theory and quantum speed limits, it is energy fluctuation, energy width, or self-energy; in entropic and geometric formulations, it is encoded by Rényi or Tsallis entropies, map-energy identities, or deformation-theoretic nontriviality; and in control theory it links transient specifications, bandwidth, and control energy (Bortnovskyi et al., 30 Oct 2025, Miyadera, 2015, Rastegin, 2018, King, 2014, Gauthier, 17 Jul 2025). This suggests that the topic is best understood as a methodological class rather than a single canonical theorem.

1. Terminological scope and baseline distinctions

A first distinction concerns what “energy” means. In additive-combinatorial uncertainty principles on $\mathbb Z_N^d$, energy is the additive quadruple count

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$

or its normalized form $\mathcal E(A)=\Lambda_2(A)/|A|^3$, which measures additive structure rather than physical energy (Bortnovskyi et al., 30 Oct 2025). In quantum-mechanical time–energy formulations, by contrast, energy typically means Hamiltonian dispersion $\Delta H$, an effective energy width $AE(H,\psi,X)$, or a self-energy correction entering an effective dispersion relation (Asch et al., 2015, Jusufi et al., 2023). In control theory, energy is the $L^2$-energy $E=\|h\|_2^2$ of the impulse response of a stable LTI closed loop (King, 2014).

A second distinction concerns the status of time. Several of the quantum sources emphasize that time is not represented by a universal Hermitian operator in nonrelativistic quantum mechanics and is instead treated operationally, as the duration needed for a process or as the scale extracted from dynamics (Campaioli, 2020, Rastegin, 2018). This separates time–energy uncertainty from position–momentum uncertainty, even when both admit Robertson-type or entropy-based formulations.

The standard baselines are likewise domain-specific. In finite harmonic analysis, the baseline is the Donoho–Stark support-product inequality $|S||T|\ge N^d$ and the deterministic recovery threshold $|E||S|<N^d/2$ (Bortnovskyi et al., 30 Oct 2025, Aldahleh et al., 20 Apr 2025). In quantum theory, the familiar baselines are the Robertson inequality, the Mandelstam–Tamm relation, and the Margolus–Levitin bound (Gauthier, 17 Jul 2025, Campaioli, 2020). The energy-based variants do not discard these results; they refine them by inserting additional structure.

2. Additive energy and finite-group uncertainty

The most systematic use of the phrase in recent harmonic analysis is the additive-energy uncertainty principle on finite abelian groups. Both papers work on $G=\mathbb Z_N^d$, but they adopt different Fourier normalizations: (Bortnovskyi et al., 30 Oct 2025) uses the unitary discrete Fourier transform

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$0

whereas (Aldahleh et al., 20 Apr 2025) uses

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$1

In both settings, the classical support-product principle states that for nonzero $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$2, with $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$3 and $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$4,

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$5

The key refinement is to replace raw support size by additive structure. Aldahleh–Iosevich–Iosevich–Jaimangal–Mayeli–Pack proved

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$6

and the 2025 refinement strengthens this to

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$7

with explicit nonnegative correction terms $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$8 and $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$9 that vanish precisely in the extremal coset case (Bortnovskyi et al., 30 Oct 2025). The correction terms are built from a product-gap factor $\mathcal E(A)=\Lambda_2(A)/|A|^3$0 and an energy-gap factor that detects deviation from maximal normalized additive energy. Accordingly, the refined principle is strict whenever $\mathcal E(A)=\Lambda_2(A)/|A|^3$1 or the normalized energies are below $\mathcal E(A)=\Lambda_2(A)/|A|^3$2.

The companion paper gives a two-parameter interpolation form: $\mathcal E(A)=\Lambda_2(A)/|A|^3$3 and a restriction-enhanced variant involving

$\mathcal E(A)=\Lambda_2(A)/|A|^3$4

This formulation makes explicit that low additive energy forces more spread in the dual support, while high additive energy corresponds to structured sets such as subgroup cosets and arithmetic progressions (Aldahleh et al., 20 Apr 2025).

The structural interpretation is uniform across these papers. Large additive energy counts many additive quadruples and signals strong additive relations; normalized energy near $\mathcal E(A)=\Lambda_2(A)/|A|^3$5 identifies coset-like behavior, while values near $\mathcal E(A)=\Lambda_2(A)/|A|^3$6 indicate low structure (Bortnovskyi et al., 30 Oct 2025). Random-like sets therefore produce stronger uncertainty constraints than the classical support-product law, whereas cosets recover the extremal classical scale.

3. Partial Fourier data and deterministic recovery

The same additive-energy formalism yields recovery theorems from incomplete frequency information. The classical deterministic sufficient condition is the Donoho–Stark threshold

$\mathcal E(A)=\Lambda_2(A)/|A|^3$7

where $\mathcal E(A)=\Lambda_2(A)/|A|^3$8 is the signal support and $\mathcal E(A)=\Lambda_2(A)/|A|^3$9 is the set of unobserved frequencies (Bortnovskyi et al., 30 Oct 2025, Aldahleh et al., 20 Apr 2025). Energy-based principles strengthen this by replacing $\Delta H$0 with an additive-energy surrogate.

For binary signals, the direct recovery algorithm in (Aldahleh et al., 20 Apr 2025) proves exact recovery by rounding when

$\Delta H$1

For general signals, the same paper obtains a uniqueness criterion: $\Delta H$2 Since always $\Delta H$3, this condition is never weaker than Donoho–Stark and is strictly stronger when $\Delta H$4 is pseudorandom.

The paper also integrates additive energy into the Logan–Santosa–Symes $\Delta H$5- and $\Delta H$6-minimizing mechanisms. Under

$\Delta H$7

the $\Delta H$8-minimizer equals the original signal if

$\Delta H$9

Under the stronger nested hypothesis

$AE(H,\psi,X)$0

and for any $AE(H,\psi,X)$1,

$AE(H,\psi,X)$2

must satisfy the explicit threshold given in the paper for the $AE(H,\psi,X)$3-minimizer to recover the signal exactly (Aldahleh et al., 20 Apr 2025).

The refined 2025 result sharpens deterministic exact recovery in a related way. Assuming

$AE(H,\psi,X)$4

it gives a sufficient uniqueness condition depending explicitly on $AE(H,\psi,X)$5, $AE(H,\psi,X)$6, $AE(H,\psi,X)$7, $AE(H,\psi,X)$8, and $AE(H,\psi,X)$9 (Bortnovskyi et al., 30 Oct 2025). The emphasis there is that the missing-frequency set $L^2$0 need not be random: the guarantee is deterministic and structure-aware.

A recurrent conclusion is that low-energy masks permit substantially more erasures than sparsity-only theory predicts. For random missing sets $L^2$1 of size $L^2$2, (Aldahleh et al., 20 Apr 2025) states the heuristic $L^2$3, which improves the Donoho–Stark threshold by a factor of $L^2$4. By contrast, arithmetic progressions and subgroup cosets have $L^2$5, so the gain disappears.

4. Time–energy uncertainty, autonomous measurements, and quantum speed limits

In quantum theory, energy-based uncertainty principles often appear as lower bounds on physically meaningful times. The Mandelstam–Tamm relation,

$L^2$6

and the induced operational definition $L^2$7, remain a standard starting point (Gauthier, 17 Jul 2025, Campaioli, 2020). The same literature stresses that the time variable is extracted from dynamics rather than represented by a universal observable.

A particularly concrete operational result concerns autonomous quantum measurements. For closed measurements of a sharp observable, where the apparatus itself acts as a timing device and no external switch turns the interaction on, the apparatus energy fluctuation and measurement duration obey

$L^2$8

The same paper proves an interaction-strength trade-off,

$L^2$9

and for infinitely many outcomes,

$E=\|h\|_2^2$0

The bound is specific to autonomous, closed measurement models; the paper explicitly contrasts this with the standard externally switched model, where no nontrivial energy–time constraint of this kind appears (Miyadera, 2015).

A different energy-based time variable is Lavine’s energy width. For a self-adjoint Hamiltonian $E=\|h\|_2^2$1, normalized state $E=\|h\|_2^2$2, and parameter $E=\|h\|_2^2$3, the sojourn time

$E=\|h\|_2^2$4

satisfies

$E=\|h\|_2^2$5

Near perturbed embedded eigenstates, the energy width has the expansion

$E=\|h\|_2^2$6

with $E=\|h\|_2^2$7 the Fermi Golden Rule constant, so the lower bound on sojourn time scales like $E=\|h\|_2^2$8 (Asch et al., 2015). Here the “uncertainty principle” is not a variance bound but a resolvent-based lifetime estimate.

Quantum-speed-limit theory generalizes the same idea to isolated and open dynamics. The geometric thesis (Campaioli, 2020) restates Mandelstam–Tamm as

$E=\|h\|_2^2$9

and pairs it with the Margolus–Levitin bound

$|S||T|\ge N^d$0

For mixed-state unitary dynamics it introduces the tightened bounds $|S||T|\ge N^d$1 and $|S||T|\ge N^d$2, and for arbitrary CPTP dynamics the Hilbert–Schmidt-speed bound

$|S||T|\ge N^d$3

In this setting, an energy-based uncertainty principle is an operational lower limit on evolution time under finite energetic resources.

5. Entropic, geometric, and derived reformulations

A distinct line of work replaces variances by entropies of measurement statistics. For a finite-dimensional system with a non-degenerate, commensurate spectrum, the Pegg construction defines time-like states

$|S||T|\ge N^d$4

and from them a rank-one time POVM $|S||T|\ge N^d$5 with uniform overlaps

$|S||T|\ge N^d$6

This yields state-independent Rényi and Tsallis uncertainty relations

$|S||T|\ge N^d$7

and, in the continuous-time limit,

$|S||T|\ge N^d$8

The same framework includes detector inefficiencies and a Naimark extension in which energy and complement become mutually unbiased projective measurements (Rastegin, 2018).

Geometric quantum mechanics turns the Robertson–Schrödinger relation into an energy identity. On $|S||T|\ge N^d$9 with its Fubini–Study metric and symplectic form, maps $|E||S|<N^d/2$0 generated by two observables have pullback metric equal to the covariance tensor, and their energy density satisfies

$|E||S|<N^d/2$1

The positive term measures deviation from holomorphicity, so saturation of the Robertson–Schrödinger inequality occurs when the map is conformal and the off-diagonal covariance term vanishes (Sanborn, 2017).

An even more abstract reformulation places time–energy uncertainty inside homotopical algebraic geometry. There, energy is a realization functor $|E||S|<N^d/2$2, observables are stacks $|E||S|<N^d/2$3, energy dispersion is modeled by the derivation bifunctor $|E||S|<N^d/2$4, and the time scale by the relative derivation bifunctor $|E||S|<N^d/2$5. The analog of the Mandelstam–Tamm product is the bifunctor

$|E||S|<N^d/2$6

and the uncertainty principle is expressed as the non-contractibility of its geometric realization rather than as a commutator bound (Gauthier, 17 Jul 2025). This explicitly preserves the theme that time is not introduced as an operator.

6. Deformations, control-theoretic analogues, and methodological extensions

Several papers use energy corrections themselves to deform uncertainty relations. In the non-extensive-entropy framework, the generalized commutator

$|E||S|<N^d/2$7

implies

$|E||S|<N^d/2$8

The $|E||S|<N^d/2$9 branch ($G=\mathbb Z_N^d$0) yields a minimum time interval of order the Planck time, while the $G=\mathbb Z_N^d$1 branch ($G=\mathbb Z_N^d$2) yields a maximum energy uncertainty of order the Planck energy; the same deformation modifies dispersion relations and Hawking temperature (Bizet et al., 2024).

A related energy-based route starts from T-duality–regularized self-energy. For bosons, the corrected relativistic energy leads to the quadratic GUP

$G=\mathbb Z_N^d$3

with

$G=\mathbb Z_N^d$4

For fermions, the leading correction is linear,

$G=\mathbb Z_N^d$5

with

$G=\mathbb Z_N^d$6

The paper identifies this spin dependence as a consequence of how the regularized self-energy enters the Klein–Gordon, Proca, and Dirac equations (Jusufi et al., 2023).

Outside quantum mechanics proper, the same language appears in Lorentzian holographic gravity and control theory. The holographic construction identifies “the law of Lorentzian holographic gravity” with a time–energy uncertainty principle and proposes the on-shell equation

$G=\mathbb Z_N^d$7

where $G=\mathbb Z_N^d$8 is the proper-time expansion of a bulk area element (Konishi, 2024). In LTI control, the central Slepian–Landau–Pollak relation becomes

$G=\mathbb Z_N^d$9

with the simplified bound

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$00

Here $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$01 is control energy, $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$02 is bandwidth, and $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$03 encodes the transient. For the Gaussian optimal monotonic step response, the paper gives the explicit products

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$04

(King, 2014).

The term also names proof strategies and variational methods. Hardy’s uncertainty principle has been rederived by a real-variable method based on weighted energies, positive-viscosity structure, log-convexity, and elliptic $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$05-estimates, with the sharp threshold $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$06 under the unitary Fourier normalization (Cowling et al., 2010). A separate variational paper minimizes $$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$07 under Robertson–Schrödinger constraints and recovers the exact ground-state energies of the harmonic oscillator and hydrogen, together with the corresponding Gaussian and exponential wavefunctions (Khitrin, 2012). In the trapped unitary Fermi gas, by contrast, the decisive uncertainty relation is explicitly the position–momentum one, not the time–energy one: the zero-point energy is interpreted through collective-mode squeezing and a suppressed Pauli contribution, while “time–energy uncertainty is not used to set the ground-state energy” (Watson, 23 Feb 2026). In cosmology, GUP-corrected agegraphic dark energy yields

$$\Lambda_2(A)=\big|\{(a_1,a_2,a_3,a_4)\in A^4:\ a_1+a_2=a_3+a_4\}\big|,$$08

so that quantum-gravity corrections enter an energy density directly through a modified uncertainty relation (Ghosh et al., 2011).

Taken together, these developments show that the phrase Energy-Based Uncertainty Principle is not a single doctrine but a recurring structural move: replace bare localization data by an energy-like quantity that captures hidden geometry, combinatorial structure, dynamical cost, or deformation scale. In harmonic analysis that quantity is additive energy; in recovery theory it becomes a recoverability certificate; in quantum measurement it is the energetic resource required to generate timing internally; in geometric and entropic formalisms it is an area, entropy, or derivation bifunctor; and in deformation-based models it is a self-energy correction that modifies the uncertainty law itself (Bortnovskyi et al., 30 Oct 2025, Miyadera, 2015, Sanborn, 2017, Bizet et al., 2024, Jusufi et al., 2023).