A Note on Holevo quantity of $SU(2)$-invariant states

TL;DR

This paper analytically calculates the Holevo quantity for $SU(2)$-invariant bipartite states involving spin-$j$ and spin-$rac{1}{2}$ subsystems, revealing how it varies with system parameters and dimension.

Contribution

It provides an explicit analytical expression for the Holevo quantity in $SU(2)$-invariant states and analyzes its dependence on system parameters and dimension.

Findings

Holevo quantity increases with parameter $F$ below $F_d$ and decreases above $F_d$.

Maximum Holevo quantity occurs at $F=1$ regardless of $j$.

Maximal Holevo quantity decreases as system dimension increases.

Abstract

The Holevo quantity and the $SU(2)$-invariant states have particular importance in quantum information processing. We calculate analytically the Holevo quantity for bipartite systems composed of spin-$j$ and spin-$\frac{1}{2}$ subsystems with $SU(2)$ symmetry, when the projective measurements are performed on the spin-$\frac{1}{2}$ subsystem. The relations among the Holevo quantity, the maximal values of the Holevo quantity and the states are analyzed in detail. In particular, we show that the Holevo quantity increases in the parameter region $F<F_d$ and decreases in region $F>F_d$ when $j$ increases, where $F$ is function of temperature in thermal equilibrium and $F_d=j/(2j+1)$, and the maximum value of the Holevo quantity is attained at $F=1$ for all $j$. Moreover, when the dimension of system increases, the maximal value of the Holevo quantity decreases.