Physics-Aware Learnability: From Set-Theoretic Independence to Operational Constraints

TL;DR

Finite-precision coarse-graining reduces continuum EMX to a countable problem, via an exact pushforward/pullback reduction that preserves the EMX objective, making the independence example provably learnable with explicit $(\epsilon,\delta)$ sample complexity.

Abstract

Beyond binary classification, learnability can become a logically fragile notion: in EMX, even the class of all finite subsets of $[0,1]$ is learnable in some models of ZFC and not in others. We argue the paradox is operational. The standard definitions quantify over arbitrary set-theoretic learners that implicitly assume non-operational resources (infinite precision, unphysical data access, and non-representable outputs). We introduce physics-aware learnability (PL), which defines the learnability relative to an explicit access model -- a family of admissible physical protocols. Finite-precision coarse-graining reduces continuum EMX to a countable problem, via an exact pushforward/pullback reduction that preserves the EMX objective, making the independence example provably learnable with explicit $(ε,δ)$ sample complexity. For quantum data, admissible learners are exactly POVMs on $d$ copies, turning sample size into copy complexity and yielding Helstrom(-type) lower bounds. For finite no-signaling and quantum models, PL feasibility becomes linear or semidefinite and is therefore decidable.

Figures (4)

• Figure 1: Schematic of physics-aware learnability (PL). A learning task is specified by $(\Theta,\mathcal{H},U)$: environments $\theta\in\Theta$, a representable (finitely nameable) hypothesis set $\mathcal{H}$, and a utility $U(\theta,h)\in[0,1]$. A physical access model fixes, for each resource budget $d$, a set $\mathfrak{L}_d$ of admissible input--output behaviors, represented as Markov kernels $Q(\cdot|\theta)\in\Delta(\mathcal{H})$. Any concrete protocol---possibly randomized, adaptive, quantum, or distributed---induces exactly one such kernel; convexity and closure under classical post-processing capture free randomization and classical relabeling of outcomes. PL asks whether there exist $d$ and $Q\in\mathfrak{L}_d$ such that, uniformly for all $\theta$, the output $H\sim Q(\cdot|\theta)$ achieves near-optimal performance with high probability, $\Pr[U(\theta,H)\ge \mathrm{opt}_{\mathcal{H}}(\theta)-\epsilon]\ge 1-\delta$, where $\mathrm{opt}_{\mathcal{H}}(\theta)=\sup_{h\in\mathcal{H}}U(\theta,h)$. This relocates the existential quantifier from arbitrary set-theoretic maps on samples to operational behaviors at the laboratory boundary: the objective $U$ (what counts as success) is held fixed while the admissible physics enters only through $\mathfrak{L}$. Different choices of $\mathfrak{L}$ recover classical i.i.d. sampling, finite-precision coarse-graining interfaces, $d$-copy quantum access (POVM-induced kernels), or finite no-signaling models; accordingly, $d$ becomes a genuine resource complexity (samples/copies/queries) that can both eliminate unphysical independence phenomena and expose genuine information-theoretic limits.
• Figure 2: Schematic of physics-aware learnability (PL). A learning task $(\Theta,\mathcal{H},U)$ is paired with a physically specified access model that determines, for each sample budget $d$, a set $\mathfrak{L}_d$ of admissible conditional output laws $Q(\cdot \mid \theta)$. PL learnability asks whether some $Q \in \mathfrak{L}_d$ achieves near-optimal utility with high probability uniformly over environments.
• Figure 3: Quantum instantiation of PL. Under $d$-copy access to an unknown state $\hat{\rho}_\theta$, any admissible protocol reduces to a POVM $\{\hat{M}_h\}_{h \in \mathcal{H}}$ on $\hat{\rho}_\theta^{\otimes d}$ followed by classical post-processing ( Theorem \ref{['thm:povm_representation']}). The copy budget $d$ is a physical resource due to the no-cloning theorem ( Theorem \ref{['thm:no_cloning']}).
• Figure 4: Coarse-graining reduction. A finite-precision interface $\pi:\mathcal{X} \to \mathcal{Y}$ induces a pushforward distribution $Q=\pi_\#P$ on the countable alphabet and a pulled-back hypothesis class $\pi^{-1}(\mathcal{G})$ on $\mathcal{X}$. The identity $P(g\circ\pi)=Q(g)$ (Eq. \ref{['eq:utility_preserved']}) implies that learning on $\mathcal{Y}$ yields PL learning on $\mathcal{X}$ under coarse-grained access (Theorem \ref{['thm:coarse_graining_reduction']}).

Theorems & Definitions (62)

• Definition 1: Learning task and optimal value
• Definition 2: Admissible protocol family
• Definition 3: Physics-aware learnability (PL)
• Theorem 1: Coarse-graining reduction for EMX
• proof : Proof sketch
• Corollary 1: Finite-precision EMX is provably learnable
• proof : Proof sketch
• Theorem 2: Quantum PL kernels are POVM-induced
• proof : Proof sketch
• Theorem 3: Decidability of PL feasibility in finite operational models