Multipartite entanglement criterion via generalized local uncertainty relations

We study the detection of multipartite entanglement based on the generalized local uncertainty relations. A sufficient criterion for the entanglement of four-partite quantum systems is presented in terms of the local uncertainty relations. Detailed examples are given to illustrate the advantages of our criterion. The approach is generalized to general multipartite entanglement cases.

In quantum theory, the observables of a quantum system are represented by a set of Hermitian operators {A i } . The uncertainty principle shows that it is impossible to predict the measurement results of all observables of the system at the same time. The variance of A i with respect to ρ is the uncertainty of an observable A i , defining as www.nature.com/scientificreports/ exits a constant U such that i (�A i ) 2 ρ ≥ U . This inequality gives a universally valid limitation of the measurement outcomes. Generally, it is difficult to determine the value U. For the case of Pauli matrices σ x , σ y and σ z , one has (�σ x ) 2 ρ + (�σ y ) 2 ρ + (�σ z ) 2 ρ ≥ 2 32 . In Ref. 33 , based on the local sum uncertainty relations, an entanglement criterion has been presented for tripartite systems.
Let {A i 1 } , {A i 2 } and {A i 3 } be the set of local observables associated to the subsystems H 1 , H 2 and H 3 , respectively. U 1 , U 2 , U 3 are lower bound of these local observables, such that For any separable tripartite states, the following inequalities hold under any permutations of {1, 2, 3} 33 : are the operators acting on the first, the second and the third subsystem with the rest subsystems as identity operators in the tripartite systems, respectively.
Generalizing the criterion (2) Theorem 1 provides a necessary condition of separable four-partite states. The violations of the inequalities in (1) suf f i c i e ntly imply e ntang l e m e nt . For th e four-qubit W state , ρ = |W 4 ��W 4 | w i t h ≤ 0 , so the state ρ 1 violates one of the inequalities (4). Therefore, the four-partite LUR criterion identifies the ρ 1 as an entangled state, see Fig. 1. While, ρ 1 is detected based on the witness W = 3 4 I − |W 4 ��W 4 | which is proposed in Ref. 27 when p < 0.267 , see Fig. 2. That is to say our result detects better the entanglement than the criterion of Ref. 27 .
For a more general case, we consider the set of local observable In order to simplify calculation, let i N represent {A i i N } and the bi-partition index For instance, if N = 4 , hence K = 2 , and k 1 |k 0 = {12|34, 13|24, 14|23} , which represents three classes of bi-partition index of local observable set in N-body quantum system. Similar to the derivation of the Theorem 1, we obtain the following lemma and theorem. Lemma 2 For multipartite separable states, the following inequalities must hold: in Theorem 1. We can see that when p ≤ 0.3605 , state ρ 1 violates one of the inequalities (4), hence ρ 1 is entangled for p ≤ 0.3605.

Figure 2.
For the four-partite W state mixed with the white noise ρ 1 . The the black line represents Tr(ρ 1 W) in Ref. 27 . We can see that ρ 1 is detected by the witness 3 Theorem 2 For any multipartite separable states, the following inequalities hold under any permutations of the subsystems,  2 mixed with the white noise ρ 2 . The the black line stands for Tr(ρW) in Ref. 27 . By using the witness W , we can see that ρ 2 is entangled for p ≤ 0.356.

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As a simple example, consider the five-qubit state ρ = |W 5 ��W 5 | , with

Conclusion
We have generalized the LUR criterion for three qubit quantum systems to multiqubit quantum systems, and obtained new entanglement criteria for four-partite quantum systems as well as for general multipartite systems.
By detailed examples we have shown that our criteria can detect better the entanglement than some existing criteria. It is further known that in certain situations they can provide a nonlinear refinement of linear entanglement witnesses 35 , and it can be measured in experimental settings similar to those of entanglement witnesses. The effectiveness of the LUR criteria relies heavily on certain notions of information content of quantum states and choice of observables. Quantum entanglement is fundamentally connected to the quantum steering, local uncertainty relations (LURs) are a common tool for entanglement detection, and the underlying idea can be directly generalized to steering detection 36 .
The considered system here is closed systems with no decoherence effects taken into account. Also, it would be interesting to find criteria for open quantum systems, since realistic quantum systems inevitably interact with the environment. It would be also interesting if our approach may highlight further investigations on the k-separability 37 of multipartite systems and genuine multipartite entanglement detection.

Proof of the Theorem 2
We denote the length of k 0 as |k 0 | . From above, one has |k 0 | + |k 1 | = N.