Detection of the quantum states containing at most k?1 unentangled particles?

Yan Hong(宏艷) Xianfei Qi(祁先飛) Ting Gao(高亭) and Fengli Yan(閆鳳利)

1School of Mathematics and Science,Hebei GEO University,Shijiazhuang 050031,China

2School of Mathematics and Statistics,Shangqiu Normal University,Shangqiu 476000,China

3School of Mathematical Sciences,Hebei Normal University,Shijiazhuang 050024,China

4College of Physics,Hebei Normal University,Shijiazhuang 050024,China

Keywords: multipartite entanglement,k ?1 unentangled particles,Cauchy–Schwarz inequality

1. Introduction

As a fundamental concept of quantum theory, quantum entanglement plays a crucial role in quantum information processing.[1]It has been successfully identified as a key ingredient for a wide range of applications, such as quantum cryptography,[2]quantum dense coding,[3]quantum teleportation,[4,5]factoring,[6]and quantum computation.[7,8]

One of the significant problems in the study of quantum entanglement theory is to decide whether a quantum state is entangled or not. For bipartite systems, quantum states consist of separable states and entangled states.Many well-known separability criteria have been proposed to distinguish separable from entangled states.[9,10]In multipartite case, the classification of quantum states is much more complicated due to the complex structure of multipartite quantum states. A reasonable way of classification is based on the number of partitions that are separable. According to that,N-partite quantum states can be divided intok-separable states andknonseparable states with 2 ≤k≤N. The detection ofknonseparability has been investigated extensively, many efficient criteria[11–21]and computable measures[22–26]have been presented. Different from the above classification,N-partite quantum states can also be divided intok-producible states and(k+1)-partite entangled states by consideration of the number of partitions that are entangled. It is worth noting that the(k+1)-partite entanglement and thek-nonseparability are two different concepts involving the partitions of subsystem inNpartite quantum systems,and they are equivalent only in some special cases.

In this paper, we focus on another characterization of multipartite quantum states which is based on the number of unentangled particles. We first present the definition of quantum states containing at leastkunentangled particles,and then derive several criteria to identify quantum states containing at mostk ?1 unentangled particles by using some well-known inequalities.Several specific examples illustrate the advantage of our results in detecting quantum states containing at mostk ?1 unentangled particles.

The organization of this article is as follows:In Section 2,we review the basic knowledge which will be used in the rest of the paper. In Section 3,we provide our central results,several criteria that can effectively detect quantum states containing at mostk?1 unentangled particles,and then their strengths are exhibited by several examples. Finally,a brief summary is given in Section 4.

2. Preliminaries

In this section, we introduce the preliminary knowledge used in this paper. We consider a multiparticle quantum system with state space?=?1??2?···??N,where?i(i=1,2,...,N) denotedi-dimensional Hilbert spaces. For convenience, we introduce the following concepts. AnN-partite pure state|ψ〉∈?1??2?···??Ncontainskunentangled particles,if there isk+1 partitionγ1|γ2|···|γk+1such that

wherep,q>1,and 1/p+1/q=1.

3. Main results

Now let us state our criteria identifying quantum states containing at mostk?1 unentangled particles for arbitrary dimensional multipartite quantum systems.

Theorem 1 If anN-partite quantum stateρcontains at leastkunentangled particles for 1 ≤k≤N ?1,then it satisfies

where we have used the absolute value inequality(2),inequality(6)for pure states, the H¨older inequality(5)and Cauchy–Schwarz inequality(4). The proof is complete.

Theorem 2 For anyN-partite density matrix acting on Hilbert spaceρ ∈?1??2?···??Ncontaining at leastkunentangled particles,where 1≤k ≤N ?2,we have

for anym/=n. Ifρdoes not satisfy the above inequality(10),then it is entangled.

Proof The proof of this result is quite similar to Theorem 2. Note that there is only one case thatm,nbelong to differentγlifρ=|ψ〉〈ψ| is fully separable pure state, which ensures that inequality(10)is true for fully separable pure state.Hence, inequality (10) also holds for fully separable mixed states.

The following examples shows that the power of our results by comparison with observation 5 in Ref.[29].

Example 1 For the family of quantum states

Table 1. For ρ(p)=p|G〉〈G|+ 1,the thresholds of pN?2, p′N?2 for the quantum states containing at most N ?3 unentangled particles detected by Theorem 1 and observation 5 in Ref.[29]for 9 ≤N ≤15,respectively,are illustrated. When pN?2 < p ≤1 and p′N?2 < p ≤1,ρ(p)contains at most N ?3 unentangled particles by Theorem 1 and observation 5 in Ref. [29],respectively. Clearly, Theorem 1 can detect more states containing at most N ?3 unentangled particles than observation 5 in Ref.[29]for 9 ≤N ≤15.

Table 1. For ρ(p)=p|G〉〈G|+ 1,the thresholds of pN?2, p′N?2 for the quantum states containing at most N ?3 unentangled particles detected by Theorem 1 and observation 5 in Ref.[29]for 9 ≤N ≤15,respectively,are illustrated. When pN?2 < p ≤1 and p′N?2 < p ≤1,ρ(p)contains at most N ?3 unentangled particles by Theorem 1 and observation 5 in Ref. [29],respectively. Clearly, Theorem 1 can detect more states containing at most N ?3 unentangled particles than observation 5 in Ref.[29]for 9 ≤N ≤15.

N 9 10 11 12 13 14 15 pN?2 0.1263 0.0824 0.0519 0.0317 0.0189 0.0111 0.0064 p′N?2 0.1547 0.1350 0.1197 0.1076 0.0977 0.0894 0.0824

Table 2. For ρ(q)=qσx? N|WN〉〈WN|σx? N+ 1?Nq 1 when N =8,the thresh-2 olds of qk, q′k for the quantum states containing at most k ?1 unentangled particles detected by Theorem 2 and observation 5 in Ref.[29]for 1 ≤k ≤6,respectively,are illustrated. When qk

Fig.1. Detection quality of Theorem 2 for the state ρ(p,q)=p|WN〉〈WN|+1 for k=2 when N =6,7,8,9. Theareaenclosed by magenta a (green line b, blue line c, red line d), p axis, line q=1?p and q axis corresponds to the quantum states containing at most 1 unentangled particles when N=6(N=7,N=8,N=9),respectively.

4. Conclusion

In this paper,we have investigated the problem of detection of quantum states containing at mostk ?1 unentangled particles. Several criteria for detecting states containing at mostk ?1 unentangled particles were presented for arbitrary dimensional multipartite quantum systems. It turned out that our results were effective by some specific examples.We hope that our results can contribute to a further understanding of entanglement properties of multipartite quantum systems.