Resource-Induced Non-Representability of Exact Quantum Limits

This work studies the operational limitations on the physical realization of exact mathematical limits in quantum theory. Focusing on the zero-temperature limit, we analyze the extent to which ground-state preparation can be achieved using quantum operations constrained by finite physical resources. We introduce a general class of physically realizable quantum channels characterized by bounded interaction strength, finite implementation time, and finite control resolution. Within this resource-theoretic setting, we derive bounds on spectral localization using quantum speed limits and variance propagation arguments. These bounds imply that arbitrarily sharp spectral concentration cannot be achieved by finite-resource operations in finite time. As a consequence, the preparation of states arbitrarily close to the exact ground state is subject to resource-dependent limitations. In particular, the achievable trace distance to the ground state is bounded from below by a quantity determined by the available energy, control precision, and operational time. The result is independent of specific coarse-graining models or measurement schemes. Instead, the limitation follows from general dynamical and informational constraints associated with finite resources in quantum operations. This perspective provides a resource-theoretic interpretation of the third law of thermodynamics. Within this framework, the unattainability of absolute zero can be understood not only as a dynamical constraint but also as a consequence of operational bounds on spectral localization. These considerations also imply practical limits for ground-state preparation, quantum annealing protocols, and low-temperature quantum metrology, where finite resources impose irreducible accuracy floors. The analysis is interpreted within the broader program of Finite-Closure Geometry (FCG), which distinguishes between mathematical closure operations (such as exact limits) and the physically realizable projections produced by finite-resource dynamics. In this view, exact limits function as ideal mathematical constructs that guide theoretical descriptions but cannot be implemented exactly by finite physical processes. Related work This work complements and extends the results of Operational Non-Representability of the Zero-Temperature Limit under Finite Energy Resolution (Menéndez, 2025). The earlier paper demonstrates non-representability through explicit finite-resolution coarse-graining channels. The present work instead analyzes the problem from a more general resource-theoretic perspective, where the obstruction arises from dynamical bounds on quantum operations implemented with finite resources. In this sense, finite-resolution models appear as particular realizations within a broader class of resource-constrained dynamics. Together, the two works provide both an explicit operational example and a general dynamical perspective on the limitations associated with exact quantum limits. Conceptual index of the Finite-Closure Geometry (FCG) research program Finite-Closure Geometry (FCG): A Structural Program for Global Non-ClosureDOI: 10.5281/zenodo.18001440 Keywords quantum channels, finite resources, zero-temperature limit, third law of thermodynamics, quantum speed limits, spectral localization, ground-state preparation, quantum thermodynamics, operational limits, trace distance, resource theory, finite-closure geometry, quantum foundations, mathematical limits