Quantum $f$ -divergences via Nussbaum-Szkoła Distributions in Semifinite von Neumann AlgebrasQuantum $f$-divergences via Nussbaum-Szkoła Distributions in Semifinite von Neumann Algebras

Theodoros Anastasiadis, George Androulakis

In this article, we prove that the quantum $f$ -divergence between two normal states on a semifinite von~Neumann algebra is equal to the classical $f$ -divergence between two corresponding classical states, which are called Nussbaum-Szkoła distributions. This result has been proved by the second named author and T.C.~John for normal states on the von~Neumann algebra $\mathbb{B}(\mathscr{H})$ of all bounded operators on a Hilbert space $\mathscr{H}$ . We extend their result for normal states on any semifinite von~Neumann algebra, not only $\mathbb{B}(\mathscr{H})$ .In this article, we prove that the quantum $f$-divergence between two normal states on a semifinite von~Neumann algebra is equal to the classical $f$-divergence between two corresponding classical states, which are called Nussbaum-Szkoła distributions. This result has been proved by the second named author and T.C.~John for normal states on the von~Neumann algebra $\mathbb{B}(\mathscr{H})$ of all bounded operators on a Hilbert space $\mathscr{H}$. We extend their result for normal states on any semifinite von~Neumann algebra, not only $\mathbb{B}(\mathscr{H})$.

quant-pharXiv:2604.19853Apr 22, 2026

Quantum

f

-divergences via Nussbaum-Szkoła Distributions in Semifinite von Neumann Algebras

In this article, we prove that the quantum

f

-divergence between two normal states on a semifinite von~Neumann algebra is equal to the classical

f

-divergence between two corresponding classical states, which are called Nussbaum-Szkoła distributions. This result has been proved by the second named author and T.C.~John for normal states on the von~Neumann algebra

\mathbb{B}(\mathscr{H})

of all bounded operators on a Hilbert space

\mathscr{H}

. We extend their result for normal states on any semifinite von~Neumann algebra, not only

\mathbb{B}(\mathscr{H})