Reducing the detection of genuine entanglement of n qubits to two qubits

TL;DR

This paper introduces a new criterion for detecting genuine multiqubit entanglement by reducing the problem to analyzing two-qubit states, simplifying the process of entanglement detection.

Contribution

It proposes the losing one qubit operator and a reduction method that simplifies genuine n-qubit entanglement detection to two-qubit cases.

Findings

All projected states of a pure product n-qubit state are pure product states.

A pure n-qubit state is genuinely entangled if it has at least two genuinely entangled (n-1)-qubit projected states.

The reduction process is implemented via a LISP program.

Abstract

We propose a criterion for the detection of genuine entanglement of pure multiqubit states. To this aim, we define an operator called the losing one qubit operator, which is different from the reduced density operator. The states obtained from a multiqubit state by applying the losing one qubit operator are referred to as its projected states. We show that all of the projected states of a pure product n-qubit state are pure product states provided that it cannot be written as a product of a single qubit state and a genuinely entangled (n-1)-qubit state. We also show that a pure n-qubit state is genuinely entangled provided that the state has at least two genuinely entangled (n-1)-qubit projected states. By repeating the losing process, we reduce the detection of entanglement of pure n-qubit states to the one of pure two-qubit states. Also we write a LISP program for the reduction process.