Separability of 3-qubits density matrices, related to l(1) and l(2)norms and to unfolding of tensors into matrices

TL;DR

This paper investigates the separability of 3-qubit maximally disordered states using tensor unfolding, singular value decomposition, and Frobenius norms, providing new criteria for full and biseparability.

Contribution

It introduces novel separability criteria based on tensor unfolding and high order SVD, improving existing conditions for 3-qubit states.

Findings

Separable states satisfy the sum of absolute Hilbert-Schmidt parameters ≤ 1.

Biseparability is characterized by the sum of Frobenius norms of triads ≤ 1.

HOSVD can enhance the sufficiency of separability conditions.

Abstract

We treat 3-qubits states with maximally disordered subsystems, by using Hilbert-Schmidt decompositions.By using unfolding methods, the tensors are converted into matrices and by applying singular values decompositions to these matrices the number 27 of the Hilbert Schmidt parameters, is reduced to 9 and under the condition that the sum of absolute values of these parameters is not larger than 1, we conclude that the density matrix is fully separable . In another method we divide the 27 Hilbert Schmidt parameters into 9 triads where for each triad we calculate the Frobenius norm of 3 Hilbert-Schmidt parameters. If the sum of nine norms is not larger than 1 then we conclude that the density matrix is fully separable . The condition for biseparability of maximally disordered density matrices is obtained by the use of one qubit density matrix multiplied by Bell entangled states of the other two qubits. If the sum of the nine Frobenius norms for these different triads is not larger than 1, we conclude that the density matrix is biseparable. We analyze the relations between three qubits maximally disordered subsystems density matrices and the method of high order singular value decomposition (HOSVD) and show that this method may improve the sufficient condition for full separability. For three qubits states which include multiplication with the unit matrix we find a simple way to reduce the sum of the absolute values of the Hilbert Schmidt parameters absolute values and get better conditions for their full separability for special cases.