Detection and measure of genuine tripartite entanglement with partial transposition and realignment of density matrices

It is challenging task to detect and measure genuine multipartite entanglement. We investigate the problem by considering the average based positive partial transposition(PPT) criterion and the realignment criterion. Sufficient conditions for detecting genuine tripartite entanglement are presented. We also derive lower bounds for the genuine tripartite entanglement concurrence with respect to the conditions. While the PPT criterion and the realignment criterion are powerful for detecting bipartite entanglement and for providing lower bounds of bipartite concurrences, our results give an effective operational way to detect and measure the genuine tripartite entanglement.

Quantum entanglement is recognized as a remarkable resource in the rapidly expanding field of quantum information science, with various applications [1].A multipartite quantum state that is not separable with respect to any bi-partition is said to be genuinely multipartite entangled(GME) [2], which is one of the important type of entanglement, and offers significant advantage in quantum tasks comparing with bipartite entanglement [3].In particular, it is the basic ingredient in measurement-based quantum computation [4], and is beneficial in various quantum communication protocols, including secret sharing [5,6], extreme spin squeezing [7], high sensitivity in some general metrology tasks [8], quantum computing using cluster states [9], and multiparty quantum network [10].Although its significance, detecting and measuring such kind of entanglement turns out to be quite difficult.To certify GME, an abundance of linear and nonlinear entanglement witnesses [11][12][13][14][15][16][17][18][19], generalized concurrence for multi genuine entanglement [20][21][22][23], and Bell-like inequalities [24]entanglement witnesses were derived (see e.g.reviews [2,25]) and a characterisation in terms of semi-definite programs (SDP) was developed [26,27].Nevertheless, the problem remains far from being satisfactorily solved.
For bipartite systems, Peres in [28]has presented a much stronger separability criterion, which is called positive partial transpose (PPT) criterion.It says that if ρ AB is separable, then the partial transposition ρ T B AB with matrix elements defined as: (ρ T B AB ) ij,kl = ρ il,kj is a density operator (i.e. has nonnegative spectrum).It has interpretation as a partial time reversal [29].There is yet another strong class of criteria based on linear contractions on product states.They stem from the new criterion discovered in [30,31] called computable cross norm criterion or matrix realignment criterion(CCNR) which is operational and independent on PPT test [28].In terms of matrix elements it can be stated as follows: if the state ρ AB is separable then the matrix R(ρ) with elements R(ρ) ij,kl = ρ ik,jl has trace norm not greater than one, i.e. ||R(ρ)|| KF ≤ 1. Quite remarkably, the realignment criterion has been found to be able to detect some PPT entangled states [30,31] and to be useful for construction of some nondecomposable maps.It also provides nice lower bound on concurrence [32].Further more, a necessary and sufficient criterion of the local unitary equivalence for general multipartite states based on matrix realignment has been presented in [33].
In this manuscript, we investigate the detection of GME for arbitrary tripartite quantum systems.We will derive an effective criterion based on PPT and CCNR.A lower bound for GME concurrence will be also obtained.We then compute examples to show the effectiveness of our results.

Results
In the following, we present a criterion to detect GME for tripartite qudits systems by using the PPT and CCNR criteria.A lower bound for GME concurrence of tripartite quantum systems will be also obtained.We start with some definitions and notations.
Let Define that , where T i s are the partial transposition over the ith subsystem, i = 1, 2, 3 and R i|jk stands for the bipartite realignment with respect to subsystem i and subsystems jk, i, j, k = 1, 2, 3. • denotes the trace norm of a matrix.
To derive GME criterion, we first obtain the following lemma.
Then we are ready to show the theorems.
3 , then ρ is GME.See Methods for the proof of theorem 1.The GME concurrence for tripartite quantum systems, which is defined as follows, is proved to be a well defined measure [20,21].For a pure state |ψ ∈ where ρ i is the reduced matrix for the ith subsystem.For mixed state ρ ∈ H d 1 ⊗ H d 2 ⊗ H d 3 , the GME concurrence is then defined by the convex roof The minimum is taken over all pure ensemble decompositions of ρ.Since one has to find the optimal ensemble to do the minimization, the GME concurrence is hard to compute.In the following we derive an effective lower bound for GME concurrence in terms of the PPT criterion and the CCNR criterion.
See Methods for the proof of theorem 2.

Applications
The following two examples show that the criterion and the lower bound of GME concurrence above are much effective for detecting and measuring GME in tripartite quantum systems.
) is the GHZ state.By Theorem 1 in [13] we can detect GME for 0.894427 < x ≤ 1.Using the Theorem 1 in this manuscript, we compute max{M(ρ), N(ρ)} =   [13]) and by the theorem 1 in this manuscript(blue region).
The lower bound of GME concurrence in Theorem 2 for ρ is computed to be as ploted in Figure 2.
Discussions It is a basic and fundamental question in quantum information theory to detect and measure GME.In this manuscript we have presented a GME criterion based on the PPT and Realignment criteria.A lower bound of GME concurrence for tripartite quantum system has also been obtained.Examples show that our criterion is independent of Vicente criteria and can detect more genuine entangled quantum states.Our results are derived by average Figure 2: The lower bound of GME concurrence for ρ in example 2. g(x, y) stands for the lower bound.
based PPT and CCNR criteria.One can construct more effective criteria to detect GME and lower bounds of GME concurrence by taking the average of the correlation matrices or covariance matrices and so on.It is also of interesting to investigate the implementation of the criterion with measurements or to extend the results to systems consisting of more than three parties.

Proof of Theorem 2
Still we consider a pure state first.
be a pure quantum state.From the result in [32], we have One computes which ends the proof of the theorem.

Lemma:
Let d = min{m, n}.For a bipartite quantum state |ϕ ∈ H m A ⊗ H n B , we have (|ϕ ϕ|) T A ≤ d, and R A|B (|ϕ ϕ|) ≤ d.Proof.By Schmidt decomposition, we set |ϕ = d i=1 √ u i |ii with d i=1 u i = 1, u i ≥ 0. By the Cauchy-Schwarz inequality one computes

Figure 1 :
Figure 1: GME Detected by Vicente criterion (pink region by Theorem 1 and yellow region by Theorem 2 in[13]) and by the theorem 1 in this manuscript(blue region).