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
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
This work analyzes the operational status of the exact zero-temperature limit in quantum statistical mechanics under finite energy-resolution constraints. Working entirely within the standard formalism of completely positive trace-preserving (CPTP) maps, finite energy resolution is modeled through a physically motivated class of quantum channels acting as coarse-graining operations in the energy spectrum. We show that the mathematical closure defining the zero-temperature limit does not commute with operational projection. In particular, taking the limit T → 0 and applying finite-resolution coarse-graining produce inequivalent quantum states for any nonzero resolution scale. As a consequence, the exact ground state cannot be operationally represented within finite-resolution measurement regimes. The result is quantitative and operationally explicit. A trace-distance lower bound separates the ground state from the set of states obtainable through finite-resolution channels, with the bound controlled by the effective degeneracy of the low-energy subspace contained within the resolution window. This obstruction persists even as the resolution scale becomes arbitrarily small, unless infinite precision, infinite time, or unbounded physical resources are assumed. The analysis provides a precise operational formulation consistent with the third law of thermodynamics and establishes intrinsic limitations on ground-state preparation and quantum thermometry in realistic quantum platforms. The construction is minimal: it requires no modification of standard quantum mechanics and introduces no interpretational assumptions. The work is formulated within the conceptual framework of Finite-Closure Geometry (FCG), a structural program investigating the incompatibility between ideal mathematical closures and physically admissible descriptions under finite resolution. In the present case, FCG appears as an explicit non-commutativity between mathematical closure and operational projection within quantum statistical mechanics. Related work The present result is structurally complemented and generalized in Resource-Induced Non-Representability of Exact Quantum Limits (Menéndez, 2025), where the non-representability of the zero-temperature limit is derived from explicit finite physical resources (energy budget, implementation time, and control precision), independently of any specific coarse-graining model. That work shows that the obstruction identified here is not merely a consequence of finite energy resolution but a universal feature of finite-resource quantum operations. Together, these results place the phenomenon within a broader resource-theoretic and dynamical framework. Conceptual index Conceptual index of the Finite-Closure Geometry (FCG) research program: Finite Closure Geometry (FCG): A Structural Program for Global Non-Closure DOI: 10.5281/zenodo.18001440 Keywords zero-temperature limit, quantum statistical mechanics, finite energy resolution, CPTP maps, quantum coarse-graining, trace distance, ground-state preparation, third law of thermodynamics, quantum thermometry, non-commutativity, finite closure, operational limits, quantum channels, effective descriptions, finite-closure geometry
This work establishes a sharp and fully explicit geometric bound on the simultaneous representability of quantum states under finite measurement resources. Working within standard quantum information theory, we consider general measurement protocols constrained by finite ancilla dimension, a finite number of measurement rounds, and a finite classical outcome alphabet. Using trace-distance distinguishability, the Helstrom bound, and Fano’s inequality, we derive a universal packing bound that limits the number of mutually distinguishable quantum states that can be represented within any such finite-closure protocol. The bound scales polynomially with measurement resources rather than exponentially with Hilbert space dimension, implying a drastic reduction of the accessible geometric complexity of quantum state space under finite closure. The result is purely structural and does not rely on assumptions about dynamics, decoherence, noise models, or interpretation. It applies to arbitrary state sets, including energy-bounded subsets, and holds independently of any specific physical implementation. This construction provides a quantitative obstruction to the simultaneous representability of large families of quantum states and clarifies fundamental limits in quantum tomography, simulation, and effective classical descriptions. It shows that finite measurement resources induce an intrinsic coarse-graining of state space that cannot be eliminated by protocol optimization. The present work forms the geometric and information-theoretic foundation of the Finite-Closure Geometry (FCG) research program, which studies how finite closure procedures constrain physical descriptions and give rise to incompatible effective limits. Here, FCG is instantiated as a precise packing limitation within standard CPTP-based measurement theory. Conceptual index of the Finite-Closure Geometry (FCG) research program: Finite Closure Geometry (FGC):} A Structural Program for Global Non-Closure (DOI: 18001440) Keywords finite closure, quantum state representability, geometric packing, quantum measurements, CPTP maps, trace distance, Helstrom bound, Fano inequality, quantum tomography, information-theoretic bounds, quantum state space, distinguishability, measurement resources, adaptive protocols, effective descriptions, quantum information theory, mathematical physics, finite-resolution limits
The third law of thermodynamics forbids cooling a physical system to absolute zero in a finite number of operational steps. Although this unattainability principle has been quantified for specific state-to-state transitions, a universal, dynamics-independent bound for implementing a state-agnostic reset map remains elusive. In this work, we unveil the fundamental limits of physical map implementation by deriving a trade-off relation based on geometric complexity. By analyzing continuous paths of maps on a geometric manifold, we prove that the geometric complexity of any classical stochastic map or quantum channel is bounded from below by its execution error. As a consequence, we show that achieving zero error in a state-reset operation requires a divergent geometric complexity -- a unified measure that naturally incorporates disparate physical resources, including infinite time, energetic cost, or control bandwidth. This unattainability principle holds universally across both classical and quantum regimes, establishing a strict geometric limit on the physical realization of reset operations in thermodynamic control and quantum computation.
This work presents a minimal and fully explicit result showing that quantum coarse-graining procedures need not commute, even in the limit of arbitrarily fine resolution. Working entirely within the framework of completely positive trace-preserving (CPTP) maps, we construct two natural pinching (dephasing) channels acting on a two-dimensional effective subspace of an infinite-dimensional Hilbert space. These channels represent distinct coarse-graining operations corresponding to incompatible measurement bases. We prove that the order of these coarse-graining operations affects the final quantum state in trace norm, compute the exact optimal incompatibility over all input states, and show that this noncommutativity persists under CPTP refinements. The result is quantitative, stable, and operationally meaningful. The construction is minimal in dimension, requires no interpretational assumptions, and highlights a structural obstruction to the commutativity of coarse-graining procedures in quantum theory. It provides a concrete benchmark for discussions of decoherence, measurement order, and the emergence of effective classical descriptions from quantum systems. This result is part of the broader research program of Finite-Closure Geometry (FCG), which investigates how finite-resolution closure procedures give rise to effective descriptions and incompatible classical limits. The present work isolates a concrete and fully operational instance of this phenomenon within standard quantum information theory. Keywords quantum coarse-graining, CPTP maps, quantum channels, noncommutativity, pinching channels, trace norm, decoherence, quantum measurement, Bloch sphere, quantum foundations, operator algebras, information loss, dephasing, effective descriptions, quantum state space, incompatibility, refinement stability, mathematical physics, quantum information theory, finite-resolution limits
The definition of entropy in both classical and quantum contexts relies on a coarse-graining of the underlying microscopic description. While this dependence is widely acknowledged, the physical constraints that determine which coarse-grainings are admissible are rarely made explicit. This issue becomes particularly acute when the universe is treated as a closed system, where no external observer or privileged level of description can be assumed. Any characterization of entropy — including claims about the low-entropy initial conditions of the universe — must then be grounded in descriptions that can be physically instantiated from within the system itself. In this work, we introduce a rigorous operational framework that establishes explicit physical selection principles over the space of admissible descriptions. We organize these criteria into necessary conditions (local accessibility and physical implementation via internal interactions) and practical sufficiency conditions (finite resource requirements anchored in Landauer’s principle, dynamical stability, and effective consistency). To demonstrate the validity of our framework, we introduce a concrete one-dimensional spin-chain toy model, illustrating how mathematically valid global partitions fail to be physically realizable due to dynamic propagation limits and informational constraints. We analyze the far-reaching implications of this restriction across three domains. In classical statistical mechanics, it limits the arbitrariness of macrostate definitions. In quantum theory, it identifies environment-induced decoherence (einselection) not merely as a formal tool, but as the objective dynamical mechanism that forces the emergence of admissible internal partitions. In cosmology, it resolves foundational circularities in the quasi-classical histories framework and reframes the thermodynamic arrow of time as a "doubly relational" property—dependent both on cosmic boundary conditions and on the material limits of internal subsystems. Our framework clarifies the operational status of entropy as an emergent property of physically instantiated descriptions within a universe that observes itself.
We study the maximum speed of quantum computation and how it is affected by limitations on physical resources. We show how the resulting concepts generalize to a broader class of physical models of computation within dynamical systems and introduce a specific algebraic structure representing these speed limits. We derive a family of quantum speed limit results in resource-constrained quantum systems with pure states and a finite dimensional state space, by using a geometric method based on right invariant action functionals on [Formula: see text]. We show that when the action functional is bi-invariant, the minimum time for implementing any quantum gate using a potentially time-dependent Hamiltonian is equal to the minimum time when using a constant Hamiltonian, thus constant Hamiltonians are time optimal for these constraints. We give an explicit formula for the time in these cases, in terms of the resource constraint. We show how our method produces a rich family of speed limit results, of which the generalized Margolus–Levitin theorem and the Mandelstam–Tamm inequality are special cases. We discuss the broader context of geometric approaches to speed limits in physical computation, including the way geometric approaches to quantum speed limits are a model for physical speed limits to computation arising from a limited resource.