Wave-Particle-Mixedness Relationships Based on lp-norm Coherence

TL;DR

The paper develops a unified framework to relate wave, particle, and mixedness in quantum states via $l_p$-norm coherence, introducing a particle-property measure based on the gap between maximal and general $l_2$-norm coherence. It first proves an exact wave-particle-mixedness equality for $p=2$, then extends to $1\le p<2$ by defining two quantities, $X$ and $Y$, that quantify deviations between $l_p$ and $l_2$ coherence; The resulting Theorems 2 and 3 give two complementary trade-offs linking $C_{l_p}^2(\rho)$, mixedness $M_l(\rho)$, particle measure $P(\rho)$, and the new variables. The work recovers and subsumes several existing results as special cases (e.g., $p=1$), and provides geometric and physical interpretations of how coherence distributes across wave and particle aspects under mixedness. Overall, it offers a cohesive lp-coherence perspective on wave-particle-mixedness that sharpens understanding of quantum coherence distribution and its relation to complementarity.

Abstract

We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the lp-norm coherence. By conforming that the lp-norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal l2-norm coherence and the general l2-norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For 1<= p <2, we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the lp-norm coherence given by the l2-norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.

Figures (2)

• Figure 1: The tetrahedron representation equality \ref{['lp - 4']}.
• Figure 2: The behavior of $X$, $Y$ and $\frac{1}{3}C_{l_1}^2$ versus the off-diagonal elements of the quantum state.