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Uncertainty Principle from Operator Asymmetry

Xingze Qiu

Abstract

The uncertainty principle is fundamentally rooted in the algebraic asymmetry between observables. We introduce a new class of uncertainty relations grounded in the resource theory of asymmetry, where incompatibility is quantified by an observable's intrinsic, state-independent capacity to break the symmetry associated with another. This ``operator asymmetry,'' formalized as the incompatibility norm, leads to a variance-based uncertainty relation for pure states that can be tighter than the standard Robertson bound. Most significantly, this framework resolves a long-standing open problem in quantum information theory: the formulation of a universally valid, product-form uncertainty relation for the Wigner-Yanase skew information. We demonstrate the practical power of our framework by deriving tighter quantum speed limits for the dynamics of nearly conserved quantities, which are crucial for understanding non-equilibrium phenomena such as prethermalization and many-body localization. This work provides both a new conceptual lens for understanding quantum uncertainty and a powerful, versatile toolkit for its application.

Uncertainty Principle from Operator Asymmetry

Abstract

The uncertainty principle is fundamentally rooted in the algebraic asymmetry between observables. We introduce a new class of uncertainty relations grounded in the resource theory of asymmetry, where incompatibility is quantified by an observable's intrinsic, state-independent capacity to break the symmetry associated with another. This ``operator asymmetry,'' formalized as the incompatibility norm, leads to a variance-based uncertainty relation for pure states that can be tighter than the standard Robertson bound. Most significantly, this framework resolves a long-standing open problem in quantum information theory: the formulation of a universally valid, product-form uncertainty relation for the Wigner-Yanase skew information. We demonstrate the practical power of our framework by deriving tighter quantum speed limits for the dynamics of nearly conserved quantities, which are crucial for understanding non-equilibrium phenomena such as prethermalization and many-body localization. This work provides both a new conceptual lens for understanding quantum uncertainty and a powerful, versatile toolkit for its application.

Paper Structure

This paper contains 4 theorems, 10 equations, 1 figure, 1 table.

Key Result

Theorem 1

For any observables $A, B$, quantum state $\rho$, and pairs of conjugate exponents $(p,\, q)$ and $(r,\, s)$, the following uncertainty relation holds: where $C = -i[A,B]$, $\mathcal{N}_p(B|A)$ and $\mathcal{N}_r(A|B)$ are the corresponding incompatibility norms.

Figures (1)

  • Figure 1: Schematic interpretation of the incompatibility norm. The set of operators that commute with observable $A$ forms a commutant algebra, $\mathcal{C}(A)$ (oval), representing all operators (black dots) compatible with $A$. The incompatibility of another observable $B$ (black diamond) is quantified by its minimal distance (red arrow) to this algebra. This distance, the incompatibility norm $\mathcal{N}_p(B|A)$ [Eq. \ref{['eq:IncompatibilityNorm_definition']}], measures the magnitude of the essential symmetry-breaking component of $B$ relative to $A$. This state-independent quantity forms the foundation of our asymmetry-bounded uncertainty relations.

Theorems & Definitions (4)

  • Theorem 1: General AURs: I
  • Corollary 1: Pure State AURs for Variances
  • Theorem 2: General AURs: II
  • Corollary 2: Product-Form WYSI-AUR