# The norms of Bloch vectors and classification of four qudits quantum states

## TL;DR

This paper derives tight bounds on the norms of Bloch vectors for four-qudit quantum states, providing new criteria for entanglement and separability, and offers a comprehensive classification of such states.

## Contribution

It introduces tight bounds on Bloch vector norms for four-qudit states and uses these to classify states and analyze entanglement and separability.

## Key findings

- Derived tight upper bounds for Bloch vector norms.
- Established necessary conditions for separability in four-partite systems.
- Provided a complete classification scheme for four-qudit quantum states.

## Abstract

We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. Theses upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four qudits quantum systems.