Quantum separability criteria based on realignment moments

Abstract

Quantum entanglement is the core resource in quantum information processing and quantum computing. It is an significant challenge to effectively characterize the entanglement of quantum states. Recently, elegant separability criterion is presented in [Phys. Rev. Lett. 125, 200501 (2020)] by Elben et al. based on the first three partially transposed (PT) moments of density matrices. Then in [Phys. Rev. Lett. 127, 060504 (2021)] Yu et al. proposed two general powerful criteria based on the PT moments. In this paper, based on the realignment operations of matrices we propose entanglement detection criteria in terms of such realignment moments. We show by detailed example that the realignment moments can also be used to identify quantum entanglement.

Entanglement detection in arbitrary dimensional bipartite quantum systems through partial realigned moments

“…So inequality ( 21) is violated and Corollary-2 implies that the state ρ p0,q0 is PPT entangled. We further observe that the criterion [26] given in (7) does not detect the BES described by the density operator ρ p0,q0 belonging to the ρ p,q family. On the other hand, when (p, q) = (p 0 , q 0 ), ρ p,q represents an NPT entangled state for which the detection range by employing our R-moment criterion is given by (0.00659601 < q < 0.153105) and (0.26477 < q ≤ 1/2), which is comparitively larger than the range (0.425035 < q ≤ 1/2) detected by employing the D 3 citerion (3).…”

“…Our R-moment criterion for m ⊗ n systems is further significant since it is able to detect certain NPT entangled states that are undetected by the criteria based on partial tranpose moments. Moreover, our separability criterion is able to detect certain bound entangled states that are not detected by another recently proposed criterion [26] using realigned moments.…”

“…al. [26] proposed another entanglement detection criteria in terms of the quantities called realignment moments. To derive their entanglement detection criterion, they have defined the realignment moments for a bipartite state ρ as…”

“…A stronger separability criterion based on Hankel matrices and involving higher order r k has been derived in [26]. For r = (r 0 , r 1 , r 2 , ...r n ), Hankel matrices can be constructed as [H k (r)] ij = r i+j and [B l (r)] ij = r i+j+1 for i, j = 0, 1, 2, ..., k. The criterion may be stated as follows.…”

“…The motivation of our work is to constuct computable entanglement conditions that can detect NPT entangled states as well as bound entangled states (BES) through partial knowledge of the density matrix. Recently, detection of bound entangled states using a moment based criterion has been studied in [26,27]. Our criterion based on moments is stronger in the sense that lower-order realigned moments are sufficient to detect entanglement.…”