Randomizing quantum states in Shatten $p$-norms

TL;DR

This paper develops a general method for randomizing quantum states across Schatten p-norms, unifying operator and trace norm cases, with theoretical proofs based on net constructions and probabilistic inequalities.

Contribution

It introduces a comprehensive randomization technique for quantum states applicable to all Schatten p-norms, extending previous results and unifying different norm cases.

Findings

Proves a general randomization theorem for all Schatten p-norms

Includes the operator and trace norm cases in a unified framework

Builds on and generalizes previous lemmas by Hayden and Winter

Abstract

In this paper, we formularize a method for randomizing quantum states with respect to the Schatten $p$-norm ($p\ge1$). Our theorem includes the Lemma 2.2 of Hayden and Winter [Commun. Math. Phys. {\bf 284}, 263--280 (2008)] for the norm case of $p>1$. We exploit the methods of a net construction on the unit sphere and McDiard's inequality in probability theory, and then we prove certain general cases (all $p$) of randomization tool for quantum states, which includes the operator norm and trace norm simultaneously in a single statement.