# N-qubit system in a pure state: a necessary and sufficient condition for unentanglement

## TL;DR

This paper establishes a comprehensive set of conditions involving equalities among coefficients that precisely determine whether an N-qubit pure state is unentangled, extending known bipartite criteria to multipartite systems.

## Contribution

It introduces a necessary and sufficient set of algebraic equalities for unentanglement in arbitrary N-qubit pure states, classified into manageable subsets.

## Key findings

- Set of (2^N - (N+1)) equalities characterizes unentanglement.
- Number of equalities grows rapidly with N, but can be organized into (N-1) subsets.
- Results applicable to quantum source separation and process tomography.

## Abstract

If a pure state of a qubit pair is developed over the four basis states, it is known that an equality between the four coefficients of that development exists if and only if that state is unentangled. This paper considers an arbitrary pure state of an N-qubit system, developed over the 2^N basis states. It is shown that the state is unentangled if and only if a well-chosen collection of (2^N-(N+1)) equalities between the 2^N coefficients of that development is verified. The number of these equalities is large a soon as N = 10, but it is shown that this set of equalities may be classified into (N-1) subsets, which should facilitate their manipulation. This result should be useful e.g. in the contexts of Blind Quantum Source Separation (BQSS) and Blind Quantum Process Tomography (BQPT), with an aim which should not be confused with that found when using the concept of equivalence of pure states through local unitary transformations.