Detection of genuine n-qubit entanglement via the proportionality of two vectors

TL;DR

This paper introduces a new criterion for detecting genuine n-qubit entanglement based on the proportionality of coefficient vectors, avoiding complex decompositions and extending previous polytope-based methods.

Contribution

It establishes a novel proportionality-based condition for identifying genuinely entangled states of n qubits, improving detection efficiency and generalizing prior approaches.

Findings

A pure product state coefficient vector cannot be decomposed into a tensor product after any permutation.

A state is genuinely entangled if its coefficient vector has non-proportional block vectors under all permutations.

The paper provides a full decomposition theorem for product states of n qubits.

Abstract

In [Science 340, 1205, 7 June (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, we indicate that generally, the coefficient vector of a pure product state of $n$ qubits cannot be decomposed into a tensor product of two vectors, and show that a pure state of $n$ qubits is a product state if and only if there exists a permutation of qubits such that under the permutation, its coefficient vector arranged in ascending lexicographical order can be decomposed into a tensor product of two vectors. The contrapositive of this result reads that a pure state of $n$ qubits is genuinely entangled if and only if its coefficient vector cannot be decomposed into a tensor product of two vectors under any permutation of qubits. Further, by dividing a coefficient vector into $2^{i}$ equal-size block vectors, we show that the coefficient vector can be decomposed into a tensor product of two vectors if and only if any two non-zero block vectors of the coefficient vector are proportional. In terms of \textquotedblleft proportionality\textquotedblright , we can rephrase that a pure state of $n$ qubits is genuinely entangled if and only if there are two non-zero block vectors of the coefficient vector which are not proportional under any permutation of qubits. Thus, we avoid decomposing a coefficient vector into a tensor product of two vectors to detect the genuine entanglement. We also present the full decomposition theorem for product states of n qubits.