Complementarity in generic open quantum systems

TL;DR

This paper presents a unified information-theoretic framework for understanding number-phase complementarity in quantum systems, applicable to both finite and infinite dimensions, and explores the impact of noise on these variables.

Contribution

It introduces a new entropy excess-based uncertainty relation for quantum complementarity, applicable to diverse quantum systems, with a novel weighting factor for phase knowledge.

Findings

The entropy excess bound is tight with a phase knowledge weight  in finite systems.

The weight  approaches 1 as system dimension increases.

Noise affects the complementarity and entropy bounds in both atomic and oscillator systems.

Abstract

We develop a unified, information theoretic interpretation of the number-phase complementarity that is applicable both to finite-dimensional (atomic) and infinite-dimensional (oscillator) systems, with number treated as a discrete Hermitian observable and phase as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as a lower bound on {\it entropy excess}, $X$, the difference between the entropy of one variable, typically the number, and the knowledge of its complementary variable, typically the phase, where knowledge of a variable is defined as its relative entropy with respect to the uniform distribution. In the case of finite dimensional systems, a weighting of phase knowledge by a factor $\mu$ ($> 1$) is necessary in order to make the bound tight, essentially on account of the POVM nature of phase as defined here. Numerical and analytical evidence suggests that $\mu$ tends to 1 as system dimension becomes infinite. We study the effect of non-dissipative and dissipative noise on these complementary variables for oscillator as well as atomic systems.