A necessary and sufficient condition for genuinely entangled n-qubit states with six non-zero coefficients

TL;DR

This work delivers a concrete, checkable criterion for separability and genuine entanglement of pure $n$-qubit states with exactly six nonzero coefficients by exploiting the basis-state structure, namely three partially complementary pairs, and the proportional-row condition of the associated $2\times3$ coefficient matrices. It shows that separability occurs if and only if the six basis states partition into three partially complementary pairs with proportional rows in the coefficient matrices; otherwise the state is genuinely entangled. The paper also offers corollaries and examples that validate the method, including recovery of Osterloh–Siewert states as genuinely entangled cases and a corollary asserting genuine entanglement when the basis states form three complete complementary pairs. A discussion sketches a path toward generalization to states with $m=2p$ nonzero coefficients and outlines the practical impact for entanglement detection in few-nonzero-coefficient regimes.

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, assume that a state with six non-zero coefficients is not a trivially separable state. Then the state is separable if and only if its six basis states consist of the three partially complementary pairs and the corresponding coefficient matrix has proportional rows. The contrapositive of this result reads that the state is genuinely entangled if and only if its six basis states do not consist of the three partially complementary pairs or though the six basis states consist of the three partially complementary pairs, the corresponding coefficient matrix does not have proportional rows. We propose four corresponding coefficient 2 by 3 matrices and show that if the four coefficient matrices don't have proportional rows, then the state is genuinely entangled. It is trivial to know if two rows of a 2 by 3 coefficient matrix are proportional. The difference from the previous articles is that the structure of the basis states is used to detect entanglement in this paper. One can see that Osterloh and Siewert's states of five and six qubits are genuinely entangled because two rows for any one of the four corresponding coefficient 2 by 3 matrices are not proportional. These states were distinguished as the maximal entangled states by the complicated filters before. Keywords: entanglement, separability, entangled states, separable states, qubits.