Separability criteria for genuine multiparticle entanglement

We present a method to derive separability criteria for the different classes of multiparticle entanglement, especially genuine multiparticle entanglement. The resulting criteria are necessary and sufficient for certain families of states. Further, the criteria are superior to all known entanglement criteria for many other families; also they allow the detection of bound entanglement. We next demonstrate that they are easily implementable in experiments and discuss applications to the decoherence of multiparticle entangled states.


I. INTRODUCTION
Entanglement is relevant for many effects in quantum optics or condensed matter physics and its characterization is of eminent importance for studies in quantum information processing [1,2]. Concerning entanglement between two particles, many questions are still open, but there exist at least various criteria which can be used to test whether a given quantum state is entangled or separable. For more than two particles, however, the situation is significantly more complicated, as several inequivalent classes of multiparticle entanglement exist and it is difficult to decide to which class a given state belongs. Entanglement witnesses and Bell inequalities can sometimes distinguish between the different classes [2,3]. However, it would be desirable to have useful criteria which allow to detect the different classes of multipartite entanglement directly from a given density matrix; a general method to derive such criteria is missing [4].
In this paper we present such a systematic way to develop multiparticle entanglement criteria. The resulting criteria solve the separability problem for certain families of states (notably, the well-studied N -qubit GHZ states mixed with white noise) and improve known results in many other cases. Also, they allow to detect bound entangled states which are separable under each partition, but not fully separable. Moreover, our criteria can easily be used in todays experiments and they improve the understanding of decoherence in multiparticle quantum systems.
Let us recall the main definitions for multipartite entanglement. For three particles, a pure state is fully separable if it is of the form |ψ fs = |a |b |c and a mixed state is fully separable if it can be written as a convex combination of fully separable pure states where the p k form a probability distribution. A pure state is called biseparable if it is separable under some bipartition. An example is |ψ bs = |a |φ bc where |φ bc is a possibly entangled state on the particles B and C. This state is biseparable under the A|BC-partition, other bipartitions are the B|AC-or C|AB-partition. A mixed state is biseparable if it can be written as ̺ bs = k p k |ψ bs k ψ bs k | where the |ψ bs k might be biseparable under different partitions. Finally a state is genuine multipartite entangled, if it is not biseparable. This class of entanglement one usually aims to generate and verify in experiments [5] and we mainly consider entanglement criteria for this type of entanglement. Note that generalizations and further classifications can be found e.g. in Refs. [2,[6][7][8].

II. THREE QUBITS
We explain our main ideas using three qubits, the generalization to more particles (or higher dimensions) is straightforward and will be discussed later. For a threequbit density matrix ̺ we denote its entries by ̺ i,j , where 1 ≤ i; j ≤ 8, here and in the following we always use the standard product basis {|000 , |001 , ..., |111 }. Then we have: Observation 1. Let ̺ be a biseparable three-qubit state. Then its matrix entries fulfill and violation implies genuine three-qubit entanglement.
This criterion has also been derived in the context of quadratic Bell inequalities [6], however, our proof is considerably shorter and, most importantly, it can be generalized to derive other characterizations of the different entanglement classes. Note that Eq. (2) is independent of the normalization of the state, simplifying many calculations below. Eq. (2) is maximally violated by the GHZ state, For other states, one may first change the local basis (leading, e.g., to the criterion |̺ 2,7 | ≤ √ ̺ 1,1 ̺ 8,8 + √ ̺ 3,3 ̺ 6,6 + √ ̺ 4,4 ̺ 5,5 ), but these will not be considered as independent criteria.
To discuss the strength of Observation 1, we consider states which are diagonal in the GHZ basis. This basis consists of the eight states |ψ 1} and x j =x j . States which are diagonal in this basis are of the form with real λ i and µ i , fulfilling λ i = λ 9−i for i = 1, ..., 4, and N denotes a normalization. We can state: Observation 2. For GHZ-diagonal states, the criterion from Observation 1 constitutes a necessary and sufficient criterion for genuine multipartite entanglement. Proof. The proof is given in the Appendix.
This shows that the criterion of Observation 1 is a strong criterion in the vicinity of GHZ states, indeed its later generalization solves the problem of classifying Nqubit GHZ states mixed with white noise (see Fig. 1).
So far, we have only considered criteria for biseparable states. Our approach also allows to derive criteria for other entanglement classes: Observation 4. (i) For fully separable three-qubit states, the following inequalities hold: (ii) Eq. (5) and Eq. (6) (5) is a necessary and sufficient criterion for full separability for GHZ states mixed with white noise. Proof. The proof is essentially the same as before, using the concavity of more generalized functions [9]. The inequalities (5,7) are equalities for pure fully separable states. The substitutions as in Eq. (6) can be made, since ̺ 2,2 ̺ 3,3 = ̺ 1,1 ̺ 4,4 , etc. holds for any pure fully separable state. Concerning (iv), note that Eq. (5) detects noisy GHZ states for p < 4/5, and this value is known to mark the border of the fully separable states [7].

III. MANY QUBITS
Let us start with introducing a compact notation. First, we label the diagonal elements of ̺ by the corresponding product vector in the standard basis. That is, if I = (i 1 , i 2 , ..., i N ) is a tuple consisting of N indices i k ∈ {0, 1} then ̺ I = ̺ (i1,i2,...,iN ) is the diagonal entry corresponding to |i 1 , i 2 , ..., i N i 1 , i 2 , ..., i N |. For example, for three qubits ̺ (000) = ̺ 1,1 and ̺ (001) = ̺ 2,2 etc. For a given I one can define I as the tuple arising from I if zeroes and ones are exchanged, e.g., (001) = (110) Furthermore, let |I| denote the number of i k = 1 in I, then |I|=n denotes a sum over all I with |I| = n. Second, let σ = |ψ ψ| be a target state and ̺ be a different state. We abbreviate with O |ψ (̺) the sum of the absolute values of the off-diagonal elements of ̺ in the upper triangle, which correspond to matrix entries where σ does not vanish. For instance, for the three-qubit GHZ state we have O |GHZ3 (̺) = |̺ 1,8 | and Eq. (4) can now be conveniently rewritten as O |W3 (̺) ≤ |I|=2 √ ̺ (000) ̺ I + 1 2 |I|=1 ̺ I . The idea behind this notation is to estimate all offdiagonal elements similarly as in Observation 1. Explicitely, we have for four qubits: Observation 5. (i) From the four-qubit GHZ state, |GHZ 4 = (|0000 + |1111 )/ √ 2, a necessary condition for biseparability of a general state ̺ is This condition is necessary and sufficient for biseparability of GHZ-diagonal states in the sense of Observation 2.
(ii) From the four-qubit W state, |W 4 = (|0001 + |0010 + |0100 + |1000 )/2, a criterion is derived as (iii) From the four-qubit Dicke state, (10) Proof. (i) is proved as in Observations 1 and 2. The factor 1/2 takes into account that each possible term occurs twice in the sum. (ii) and (iii) follow as in Observation 3. Here, estimating an off-diagonal element can be simplified by the following rule: If the off-diagonal element η corresponds to |i 1 i 2 i 3 i 4 j 1 j 2 j 3 j 4 | and the state is separable under the A|BCD-bipartition, one has η ≤ √ ̺ (i1j2j3j4) ̺ (j1i2i3i4) while one has η ≤ √ ̺ (i1i2j3j4) ̺ (j1j2i3i4) for the AB|CD-bipartition, etc. Further, one needs that for a positive n × n matrix P the bound i<j |P ij | ≤ n−1 2 T r(P ) holds [13]. Again, these criteria improve known conditions: For the four-qubit W state mixed with white noise, Eq. (9) detects genuine multipartite entanglement for p < 4/9 ≈ 0.444, while the fidelity based witness detects it only for p < 4/15 ≈ 0.267 and the improved witness [10] for p < 16/45 ≈ 0.356. A four-qubit Dicke state mixed with white noise is detected by Eq. (10) for p < 8/21 ≈ 0.381, while the best known witness detects it for p > 16/45 ≈ 0.356 [14].
For arbitrary states similar entanglement criteria can be derived as follows: In a given basis and for a fixed partition, any off-diagonal element can be estimated as in the proof of Observation 5. Then, all these estimates can be summarized to an estimate of the sum of all off-diagonal elements. This might be further improved by considering a weighted sum. For instance, for N -qubit GHZ states, the criterion reads O |GHZN (̺) ≤ 1 2 N −1 |I|=1 √ ̺ I ̺ I , and is again necessary and sufficient for GHZ diagonal states as the proof of Observation 2 can directly be generalized (see Fig. 1). Further criteria for cluster states or the four-qubit singlet state will be presented elsewhere.

IV. EXPERIMENTAL CONSEQUENCES
Obviously, these criteria can be applied to experiments where the full density matrix has been determined [15]. However, often this can not be done. Still, our results may be directly applied. For example, let us consider Eq. (4) for the detection of entanglement around the three-qubit W state. Using the fidelity F = T r(̺|W 3 W 3 |) one may rewrite Eq. (4) as The fidelity of the W state can be measured experimentally with five local measurements [16] and the diagonal elements can also be determined from measurement of σ z ⊗ σ z ⊗ σ z , which is already included in the measurements needed for the fidelity. This shows that Eq. (4) (and similarly all other criteria presented) is experimentally easily testable. For the usual error models in photon experiments one can also check that criterion (4) detects entanglement with a higher statistical significance than the witness, unless the fidelity is close to one and the significance of both methods is high.
This state is not diagonal in the GHZ basis, but applying on each qubit a filter ̺ → F ̺F with F = α|0 0| + (1/α)|1 1| and α 4 = x/(1 − x) maps it to a state that differs from a GHZ diagonal state only in the element ̺ 1,1 . This filtering keeps all entanglement properties, but finally Observations 1 and 2 can be used. From this one can conclude that GHZ states coupled to a bath with zero temperature are genuine multipartite entangled, if and only if t < − ln[1 − (2 N −1 − 1) −2/N ]/γ.

VI. CONCLUSION
We presented a method to derive separability criteria for different classes of multipartite entanglement directly in terms of density matrix elements. The resulting criteria are strong and can be used in experiments, as well as for the investigation of decoherence. It would be interesting to use our approach to discriminate between more special entanglement classes (such as the W and GHZ class for three qubits [8]) and to connect it to the quantification of entanglement with entanglement measures.
We thank J. Uffink for fruitful discussions. This work has been supported by the FWF (START prize) and the EU (OLAQUI, QICS, SCALA). MPS acknowledges the hospitality of the Centre for Time, University of Sydney.
Note added: Half a year after submission of our manuscript to the arxiv, a preprint [19] has appeared, in which criteria for multipartite entanglement have been presented. This method is analog to ours, by using estimates for off-diagonal terms [cf. Eq. (I) in Ref. [19] with the proof of our Observation 5] and using convexity arguments. Consequently, the obtained separability criterion used for N-qubit GHZ states [Eq. (II) in Ref. [19]] is the same as our criterion in Eqs. (2,8) combined with local filtering operations, and the criterion for W states [Eq. (III) in Ref. [19]] is for three qubits the same as our Eq. (4), while for four qubits it is weaker than our Eq. (9).
(i) First, we consider the extremal case when µ i = λ i for all i and by assumption the separability condition implies that we have µ i = λ i ≤ k =i λ k , where for the index 1 ≤ k ≤ 4. If λ 1 = λ 2 + λ 3 + λ 4 we can directly write ̺ (dia) = k=2,3,4 ̺ (1k) (λ k ) hence ̺ (dia) is biseparable. Otherwise, the idea is to write for some parameters χ k such that the rest ̺ (r) (which is then characterized by parameters λ ). Then, ̺ (r) can be iteratively further decomposed and finally a decomposition of ̺ (dia) into biseparable states can be found.
The idea is to choose the λ (r) k , k = 2, 3, 4 as equal as possible (they have to fulfill λ (r) k ≤ λ k ), but monotonically decreasing. For that, we define α 4 := λ 2 + λ 3 +  directly, but the previous scheme can directly be extended to more qubits.
(iv) The previous arguments prove the claim if all µ i ≥ 0. If some µ i are negative, one can prove it as follows: Let ̺ be a GHZ diagonal state, with some µ i < 0, which fulfills the condition of biseparability. The state ̺ which arises from ̺ when all µ i are replaced by |µ i | fulfills the same condition, and is biseparable due to points (i)-(iii). It can be decomposed into several ̺ (kl) , maybe in some of them we have µ i (̺ (kl) ) < λ i (̺ (kl) ) according to points (ii) and (iii). Nevertheless, we can built out of this decomposition of ̺ a decomposition of ̺, if we flip the signs of all the µ i (̺ (kl) ) appropriately. An arbitrary flipping of the signs of the µ i of a given ̺ (kl) can be done for each k, l by local operations, hence ̺ is also biseparable.