Entropic uncertainty and certainty relations for complementary observables

Jorge Sánchez-Ruiz

Alexey E. Rastegin

Javier M. Magán, Diego Pontello

We approach the physical implications of the noncommutative nature of complementary observable algebras (COAs) from an information theoretic perspective. In particular, we derive a general 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 COAs 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 COAs are formed by Wilson and t'Hooft loops. In general, the entropic certainty principle naturally captures the physics of order and disorder parameters, a feature that makes it a generic handle for the information theoretic characterization of quantum phases.

Iwo Białynicki‐Birula

M. D. Srinivas

K. Kraus

Two observables A and B of an n-level system (i.e., a quantum system with n-dimensional state space) are called complementary, if knowledge of the measured value of A implies maximal uncertainty of the measured value of B, and vice versa. Such observables exist for all n, but no classification (up to equivalence) of all possible pairs of complementary observables is known except for n\ensuremath{\le}4. Complementary observables are conjectured to satisfy an ``entropic'' uncertainty relation of the strongest possible form. This relation has been verified for n\ensuremath{\le}4 by explicit calculations. A recent attempt of substantiating the widespread interpretation of uncertainty relations in terms of mutual disturbances between measurements is criticized.