On a Block Matrix Inequality quantifying the Monogamy of the Negativity of Entanglement

TL;DR

This paper converts a conjectured quantum information inequality into a block matrix form and proves it for a special case, advancing understanding of entanglement monogamy related to negativity.

Contribution

It reformulates a quantum inequality as a block matrix inequality and proves it for the case n=2 with one matrix as the identity, providing a partial proof of the conjecture.

Findings

Proved the block matrix inequality for n=2 with A_1 as identity.

Connected the inequality to the monogamy of negativity in entanglement.

Extended the understanding of matrix inequalities in quantum information theory.

Abstract

We convert a conjectured inequality from quantum information theory, due to He and Vidal, into a block matrix inequality and prove a special case. Given $n$ matrices $A_i$, $i=1,\ldots,n$, of the same size, let $Z_1$ and $Z_2$ be the block matrices $Z_1:=(A_jA_i^*)_{i,j=1}^n$ and $Z_2:=(A_j^*A_i)_{i,j=1}^n$. Then the conjectured inequality is \[ \left(||Z_1||_1-\trace Z_1\right)^2 + \left(||Z_2||_1-\trace Z_2\right)^2 \le \left(\sum_{i\neq j} ||A_i||_2 ||A_j||_2\right)^2. \] We prove this inequality for the already challenging case $n=2$ with $A_1$ equal to the identity matrix.