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Quantum Complementarity through Entropic Certainty Principles

We approach the physical implications of the non-commutative nature of Complementary Observable Algebras (COA) from an information theoretic perspective. In particular, we derive a general \textit{entropic certainty principle} stating that the sum of two relative entropies, naturally related to the COA, is equal to the so-called algebraic index of the associated inclusion. Uncertainty relations then arise by monotonicity of the relative entropies that participate in the underlying entropic certainty. Examples and applications are described in quantum field theories with global symmetries, where the COA are formed by the charge-anticharge local operators (intertwiners) and the unitary representations of the symmetry group (twists), and in theories with local symmetries, where the COA are formed by Wilson and 't Hooft loops. In general, the entropic certainty principle naturally captures the physics of order/disorder parameters, a feature that makes it a generic handle for the information theoretic characterization of quantum phases.