# Measure of not-completely-positive qubit maps: the general case

## TL;DR

This paper investigates the properties of not-completely-positive (NCP) qubit maps, showing that their set is unbounded and that their spectral properties lead to issues in defining a volume measure for these maps.

## Contribution

It introduces a new definition of valid NCP maps and demonstrates their unbounded spectral spectrum, highlighting challenges in quantifying these maps.

## Key findings

- NCP maps are unbounded unless additional assumptions are made.
- The eigenvalue spectrum of the dynamical matrix for valid NCP maps is unbounded.
- The volume measure for qubit maps, including NCP maps, is not well defined.

## Abstract

We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map, which is nontrivial because NCP maps may lack a full positivity domain. The definition is motivated by specific examples. We prove that for valid NCP maps, the eigenvalue spectrum of the corresponding dynamical matrix is not bounded. Based on this, we argue that in general the volume measure of qubit maps, including NCP maps, is not well defined.