Quantum Resource Complementarity in Finite-Dimensional Systems

Abstract

Quantum resources such as entanglement, information redundancy, and coherence enable revolutionary advantages but obey fundamental tradeoffs. We present a unified geometric constraint governing three core operational tasks: teleportation ($q_1$), cloning ($q_2$), and coherence-based metrology ($q_3$). For any tripartite quantum state $\rho_{ABC}$, we show the tight inequality $q_1^2 + q_2^2 + q_3^2 \leq 1$ confines all physically achievable resources to the positive octant of the unit ball. This Quantum Information Resource Constraint (QIRC) reflects an exclusion principle intrinsic to Hilbert space: optimizing one task necessitates sacrificing others. Crucially, $q_1, q_2, q_3$ are experimentally measurable, making QIRC falsifiable in quantum platforms. Unlike abstract quantum resource theories (QRT) that quantify resources through entropy or monotones, our framework is fundamentally operational, deriving tight constraints from measurable task fidelities in teleportation, cloning, and metrology. The emergent \(\ell^2\)-norm exclusion is irreducible to existing QRT axioms. Remarkably, we demonstrate the resource norm $\mathcal{I} = q_1^2 + q_2^2 + q_3^2$ is conserved under symmetry-preserving unitaries (quantum resource covariance principle) but contracts irreversibly under decoherence. This work establishes a fundamental link between quantum information geometry, symmetry, and thermodynamics.