# Geometric Multiaxial Representation of N-qubit Mixed Symmetric Separable States

## TL;DR

This paper investigates N-qubit mixed symmetric separable states using geometric multiaxial representation, revealing their classicality and visualizing them as uniaxial systems through the P-distribution function.

## Contribution

It introduces a geometric multiaxial framework for characterizing symmetric separable states and links separability with classicality, simplifying analysis via the P-distribution function.

## Key findings

- Separable states can be visualized as uniaxial systems with distribution independence.
- Statistical tensor parameters are expansion coefficients of the distribution function.
- Discrete case analysis of a two-qubit symmetric separable state.

## Abstract

Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2^N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use geometric multiaxial representation(MAR) of density matrix to characterize the mixed symmetric separable states. Since separability problem is NP hard, we choose to study it in the continuum limit where mixed symmetric separable states are characterized by the P-distribution function \lambda(\theta; \phi). We show that the N-qubit mixed symmetric separable state can be visualized as a uniaxial system if the distribution function is independent of \theta and \phi. We further choose distribution function to be the most general positive function on a sphere and observe that the statistical tensor parameters characterizing the N-qubit symmetric system are the expansion coefficients of the distribution function. As an example for the discrete case, we investigate the MAR of a uniformly weighted two qubit mixed symmetric separable state. We also observe that there exists a correspondence between separability and classicality of states.