# Squaring parametrization of constrained and unconstrained sets of quantum states

## TL;DR

This paper introduces a squaring parametrization method for representing quantum states, simplifying the positivity constraint, and explores its applications to symmetric and constrained quantum state sets, including Werner states.

## Contribution

It develops a new parametrization of quantum states using a squaring map, providing explicit boundary equations and applications to constrained state sets.

## Key findings

- Derived boundary equation for quantum state set
- Applied parametrization to symmetric and constrained states
- Illustrated with Werner states of qubits

## Abstract

A mixed quantum state is represented by a Hermitian positive semi-definite operator $\rho$ with unit trace. The positivity requirement is responsible for a highly nontrivial geometry of the set of quantum states. A known way to satisfy this requirement automatically is to use the map $\rho=\tau^2 / \mathrm {tr} \, \tau^2$, where $\tau$ can be an arbitrary Hermitian operator. We elaborate a parametrization of the set of quantum states induced by the parametrization of the linear space of Hermitian operators by virtue of this map. In particular, we derive an equation for the boundary of the set. Further, we discuss how this parametrization can be applied to a set of quantum states constrained by some symmetry, or, more generally, some linear condition. As an example, we consider the parametrization of sets of Werner states of qubits.