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Quantum Complementarity through Entropic Certainty Principles

Javier M. Magan, Diego Pontello

TL;DR

This work introduces an entropic certanity principle for pairs of complementary operator algebras (COA), proving that for connected finite-dimensional inclusions N ⊂ M there exists a dual conditional expectation ε' such that the sum of two relative entropies satisfies S_M(ω|ω∘ε) + S_N'(ω|ω∘ε') = log λ, where λ is the algebraic index [M:N]. This equality generalizes entropic uncertainty relations to the COA setting and links information-theoretic quantities to subfactor index theory, including its minimal realization ε0 with index [M:N]. In physical applications to quantum field theories, global symmetries yield an index λ_min = |G| (with |G| the group order), coinciding with topological entanglement entropy, while local symmetries involving Wilson and t Hooft loops give indices tied to the center. The framework captures order/disorder parameter physics and provides a route to new uncertainty relations via data-processing monotonicity, with open directions toward broader quantum channels and infinite-dimensional algebras.

Abstract

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.

Quantum Complementarity through Entropic Certainty Principles

TL;DR

This work introduces an entropic certanity principle for pairs of complementary operator algebras (COA), proving that for connected finite-dimensional inclusions N ⊂ M there exists a dual conditional expectation ε' such that the sum of two relative entropies satisfies S_M(ω|ω∘ε) + S_N'(ω|ω∘ε') = log λ, where λ is the algebraic index [M:N]. This equality generalizes entropic uncertainty relations to the COA setting and links information-theoretic quantities to subfactor index theory, including its minimal realization ε0 with index [M:N]. In physical applications to quantum field theories, global symmetries yield an index λ_min = |G| (with |G| the group order), coinciding with topological entanglement entropy, while local symmetries involving Wilson and t Hooft loops give indices tied to the center. The framework captures order/disorder parameter physics and provides a route to new uncertainty relations via data-processing monotonicity, with open directions toward broader quantum channels and infinite-dimensional algebras.

Abstract

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.

Paper Structure

This paper contains 11 sections, 7 theorems, 95 equations.

Table of Contents

  1. The Uncertainty Principle. Old and New.
  2. Relative entropy and Complementary Observable Algebras.
  3. The space of conditional expectations.
  4. Entropic certainties and the algebraic index.
  5. Applications.
  6. Discussion and future prospects.
  7. Acknowledgements.
  8. APPENDIX
  9. A. The space of conditional expectations
  10. B. Proof of the entropic certainty principle (theorem \ref{['cer_thm']})
  11. C. The algebraic index of inclusion of algebras

Key Result

Lemma 1

Let $\mathcal{N}\subset\mathcal{M}$ be a generic inclusion of finite dimensional algebras, and $\varepsilon\in C(\mathcal{M},\mathcal{N})$ a conditional expectation. Then, there exist unique conditional expectations $\varepsilon_k\in C(\mathcal{M}_k,\mathcal{N}_k)$ such that These conditional expectations are uniquely determined by states $\rho_{k}^{\varepsilon}=\bigoplus_{j=1}^{z_{\mathcal{M}}}p

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 5
  • proof
  • ...and 2 more