Detection of the Quantum States Containing at Most $ K-1

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In Sec. III, we provide our central results, several criteria that can effectively detect quantum states containing at most k 1 unentangled particles, and then their strengths are exhibited by several examples. Finally, − a brief summary is given in Sec. IV.

II. PRELIMINARIES

In this section, we introduce preliminary knowledge used in this paper. We consider a multiparticle quantum system with state space = , where (i =1, 2,...,N) denote d -dimensional Hilbert spaces. H H1 ⊗ H2 ⊗···⊗HN Hi i For convenience, we introduce the following concepts. An N-partite pure state ψ contains | i ∈ H1 ⊗ H2 ⊗···⊗HN k unentangled particles, if there is k + 1 partition γ γ γ such that 1| 2| · · · | k+1 k+1 ψ = φ , | i | liγl Ol=1 where φ is single-partite state for 1 6 l 6 k, while φ is a (N k)-particle state. A mixed state ρ | liγl | k+1iγk+1 − contains at least k unentangled particles, if it can be written as

ρ = p ψ(j) ψ(j) , j | ih | j X where p > 0 with p = 1, and ψ(j) is the pure state containing m unentangled particles with m > k [23–25]. j j | i j j Otherwise we say ρ contains at most k 1 unentangled particles. P − For N-partite quantum system , let H1 ⊗ H2 ⊗···⊗HN N N AiBi ρ = tr ( AiBi)ρ (1) hi=1 i i=1 N N where ρ is the quantum state in , A ,B are operators acting on the i-th subsystem , and H1 ⊗ H2 ⊗···⊗HN i i Hi “tr” stands for trace operation. Inequalities play an important role in quantum information theory. In the following, we list some inequalities that will be used throughout the paper. Absolute value inequality:

n n a a . (2) i ≤ | i| i=1 i=1 P P

Cauchy-Schwarz inequality: x y 2 x x y y , (3) |h | i| ≤ h | ih | i

n 2 n n a b a2 b2 . (4) i i ≤ i i i=1 i=1 i=1 P P P Extending the Cauchy-Schwarz inequality is an important inequality known as the H¨older inequality:

1 1 n n p n q a b a p b q , (5) | i i|≤ | i| | i| i=1 i=1 i=1 P P P 1 1 where p,q > 1, + = 1. p q

III. MAIN RESULTS

Now let us state our criteria identifying quantum states containing at most k 1 unentangled particles for − arbitrary dimensional multipartite quantum systems. 3

Theorem 1. If an N-partite quantum states ρ contains at least k unentangled particles for 1 6 k 6 N 1, then − it satisﬁes

N AiBi ρ hi=1 i k+1 1 (6) N † † † † 2k+2 ( A A ) ( B B ) ( B B ) ( A A ) i i i i ρ i i i i ρ ≤ γ l=1 i∈γl ⊗ i/∈γl i∈γl ⊗ i/∈γl P Q N N N N where A ,B are operators acting on the i-th subsystem, the sum runs over all possible partitions γ γ = i i { | γ γ γ of N particles in which the number of particles in γ is 1 for 1 l k and is N k for i = k + 1 . 1| 2| · · · | k+1} l ≤ ≤ − If ρ violates inequality (6), then it contains at most k 1 unentangled particles. − Proof. First, we consider pure state ρ = ψ ψ containing k unentangled particles. Suppose that the pure state k+1 | ih |

ψ = ψl γl under the partition γ1 γ2 γk+1, where γl contains one particle for 1 l k, and γk+1 contains | i l=1 | i | | · · · | ≤ ≤ N k Nparticles. Then for any subsystems γl, we have − N A B h i iiρ i=1 NN N † † = AiBi ρ Bi Ai ρ shi=1 i hi=1 i k+1N N k+1 A B B†A† = i i ργt i i ργt st=1hi∈γt i t=1hi∈γt i kQ+1 N kQ+1 N A A† B†B i i ργt i i ργt ≤ st=1hi∈γt i t=1hi∈γt i = Q NA A† Q ANA† B†B B†B i i ργl i i ργt i i ργl i i ργt hi∈γl i t6=lhi∈γt i hi∈γl i t6=lhi∈γt i r N † Q N † N † Q N† = ( AiAi ) ( Bi Bi) ρ ( Bi Bi) ( AiAi ) ρ, h i∈γl ⊗ i/∈γl i h i∈γl ⊗ i/∈γl i r N N N N where ρ = ψ ψ . Here we have used the Cauchy-Schwarz inequality (3). Thus, γt | tiγt h t|

N AiBi ρ hi=1 i k+1 1 N † † † † k+1 ( A A ) ( B B ) ( B B ) ( A A ) i i i i ρ i i i i ρ ≤ l=1 i∈γl ⊗ i/∈γl i∈γl ⊗ i/∈γl r 1 Qk+1 N N N N k+1 A A† B†B B†B A A† ( i i ) ( i i) ρ ( i i) ( i i ) ρ ≤ γ l=1 i∈γl ⊗ i/∈γl i∈γl ⊗ i/∈γl r 1 P kQ+1 N N N N 2k+2 A A† B†B B†B A A† . = ( i i ) ( i i) ρ ( i i) ( i i ρ γ l=1 i∈γl ⊗ i/∈γl i∈γl ⊗ i/∈γl P Q N N N N It shows that inequality (6) is right for pure state containing k unentangled particles. Now, we consider the case of mixed state. Suppose ρ = pj ρj is a mixed state with pure states ρj containing j at least k unentangled particles, then P

N A B h i iiρ i=1 N N p A B ≤ j h i iiρj j i=1 k+1 1 P N † † † † 2k+2 pj ( AiA ) ( B Bi) ( B Bi) ( AiA ) i i ρj i i ρj , ≤ j γ l=1 i∈γl ⊗ i/∈γl i∈γl ⊗ i/∈γl 1 k+1 1 1 k+1 P P Q N † N † 2 N † N † 2 pj ( AiA ) ( B Bi) ( B Bi) ( AiA ) i i ρj i i ρj ≤ γ l=1 j i∈γl ⊗ i/∈γl i∈γl ⊗ i/∈γl k+1 1 P Q P N N N N 2k+2 A A† B†B B†B A A† ( i i ) ( i i) ρ ( i i) ( i i ) ρ ≤ γ l=1 i∈γl ⊗ i/∈γl i∈γl ⊗ i/∈γl P Q N N N N 4 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. If an N-partite quantum states ρ is fully separable state (that is, it is the quantum state containing at least N 1 unentangled particles), then we have −

N N N A B A A† B†B . (7) h i iiρ ≤ i i ρ i i ρ i=1 s i=1 i=1 N N N

Of course, ρ is entangled if it violates the inequality (7). N Proof. Suppose that ψ = ψi is fully separable pure state, we have | i i=1 | i N N 2 AiBi |ψi hi=1 i N N N † † = AiBi |ψi Bi Ai |ψi hi=1 i h i=1 i N N N N † † = AiBi |ψii Bi Ai |ψii i=1h i l=1h i N N Q † Q † † † AiAi |ψii Bi Bi |ψii Bi Bi |ψii AiAi |ψii ≤ i=1 h i h i i=1 h i h i N q q Q † † Q = AiAi |ψii Bi Bi |ψii i=1h i h i N N Q † † A A B B , = i i ρ i i ρ i=1 i=1 N N where we have used the Cauchy-Schwarz inequality (3). Hence, inequality (7) holds for fully separable pure state. Similar to Theorem 1, we can ﬁnd that (7) is guaranteed for fully separable mixed states. Here we need to point out that Theorem 2 is inequality (44) of Ref.[26]. Theorem 3. For any N-partite density matrix acting on Hilbert space ρ containing at ∈ H1 ⊗ H2 ⊗···⊗HN least k unentangled particles, where 1 k N 2, we have ≤ ≤ −

N N N N † † † † † † † † U ( A A )U A A U U ( A A )UnUm + (N k 1) U ( A A )U h m i i niρ ≤ h i i iρh m n i i iρ − − h m i i miρ m6=n i=1 m6=n s i=1 i=1 m i=1 P N P N N P N (8) where A are any operator of the subsystem , and U = 1 1 u 1 1 with u being i Hi m 1 ⊗···⊗ m−1 ⊗ m ⊗ m+1 ⊗···⊗ N m any operator of the subsystem and 1 being identity matrix of the subsystem . If ρ violates (8), then it Hm j Hj contains at most k 1 unentangled particles. − Proof. We begin with pure state. Suppose that the pure state ψ contains k unentangled particles, then there | i is a partition γ1 γk+1 with γl containing one particle for 1 l k, and γk+1 containing N k particles, it can | · · · |k+1 ≤ ≤ − be written as ψ = ψl γl . When m,n in γk+1, we can obtain | i l=1 | i N N U ( A A†)U † h m i i niρ i=1 NN N U A A† U † U A A† U † m( i i ) m ρ n( i i ) n ρ (9) ≤ sh i=1 i h i=1 i NN NN U ( A A†)U † + U ( A A†)U † h m i i miρ h n i i niρ i=1 i=1 , ≤ N 2 N where ﬁrst inequality holds because of Cauchy-Schwarz inequality (3) and second inequality follows from mean inequality. When m γ , n γ ′ and l = l′, we have ∈ l ∈ l 6

N N N † † † † † † Um( AiA )U ρ ( A A ) U U ( A A )UnUm (10) h i ni ≤ h i i iρh m n i i iρ i=1 s i=1 i=1

N N N

5 by Cauchy-Schwarz inequality (3). Combining (9) and (10) leads to

N U ( A A†)U † h m i i niρ m6=n i=1 P N N N † † † † = Um( A iAi )Un ρ + Um( AiAi )Un ρ ′ ′ m∈γl,n∈γl ,l6=l h i=1 i | m,n∈γk+1,m6=n |h i=1 i

P NN N P N † † † † ( AiAi ) ρ UmUn( AiAi )UnUm ρ ′ ′ ≤ m∈γl,n∈γl ,l6=l sh i=1 i h i=1 i N N 1 P N † † N † † + ( Um( AiAi )Um ρ + Un( AiAi )Un ρ) 2 m,n∈γl,m6=n h i=1 i h i=1 i PN N n N N † † † † † † AiAi ρ UmUn( AiAi )UnUm ρ + (N k 1) Um( AiAi )Um ρ. ≤ m6=n shi=1 i h i=1 i − − m h i=1 i P N N P N Hence, inequality (8) holds for any pure state containing k unentangled particles. It is easy to prove that it is also right for any mixed state containing at least k unentangled particles by utilizing absolute value inequality, inequality (8) for pure states, and the Cauchy-Schwarz inequality (4). Theorem 4. For any N-partite fully separable state ρ, one has

N N N † † † † † † U ( A A )U ( A A ) U U ( A A )UnUm (11) h m i i niρ ≤ h i i iρh m n i i iρ i=1 s i=1 i=1

N N N for any m = n. If ρ does not satisfy the above inequality (11), then it is entangled. 6 Proof. The proof of this result is quite similar to Theorem 3. Note that there is only one case that m,n belong to diﬀerent γ if ρ = ψ ψ is fully separable pure state, which ensures that inequality (11) is true for fully separable l | ih | pure state. Hence, inequality (11) also holds for fully separable mixed states. The following examples shows that the power of our results by comparison with Observation 5 in Ref.[25]. Example 1. For the family of quantum states 1 p ρ(p)= p Ψ Ψ + − 1, | 5ih 5| 55 1 4 where Ψ5 = iiiii . | i √5 i=0 | i Applying TheoremP 1 by choosing Ai = 1 0 ,Bi = 0 0 , we can get that, if p > 0.0016, ρ contains at most 3 | ih | | ih | unentangled particles; if p> 0.0173, ρ contains at most 2 unentangled particles; if p> 0.0325, ρ contains at most 1 unentangled particles; if p > 0.0399, ρ contains at most 0 unentangled particles. But Observation 5 in Ref.[25] cannot detect any quantum states containing at most k unentangled particles for 0 k 3. ≤ ≤ Example 2. Consider the N-qubit mixed states, 1 p ρ(p)= p G G + − 1, | ih | 2N 1 where G = ( 0 ⊗N + 1 ⊗N ). | i √2 | i | i Let A = 1 0 ,B = 0 0 , then by using Theorem 1, we know that ρ(p) contains at most N 3 unentangled i | ih | i | ih | − particles when p

1 − p ′ TABLE I: For ρ(p) = p|GihG| + 1, the thresholds of pN−2, pN 2 for the quantum states containing at most N − 3 2N − unentangled particles detected by Theorem 1 and Observation 5 in Ref.[25] for 9 ≤ N ≤ 15, respectively, are illustrated. ′ When pN−2

N 9 10 11 12 13 14 15 pN−2 0.1263 0.0824 0.0519 0.0317 0.0189 0.0111 0.0064

′ pN−2 0.1547 0.1350 0.1197 0.1076 0.0977 0.0894 0.0824

k=2 p 1.0 a : N = 6 b : N = 7 0.8 c : N = 8 d : N = 9 0.6

0.4

0.2 a d c b q 0.0 0.2 0.4 0.6 0.8 1.0

⊗N ⊗N 1 − p − q FIG. 1: Detection quality of Theorem 3 for the state ρ(p, q) = p|WN ihWN | + qσx |WN ihWN |σx + 1 for k = 2 2N when N = 6, 7, 8, 9. The area enclosed 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.

When p = 0, the quantum state ρ(p, q) is 1 q ρ(q)= qσ⊗N W W σ⊗N + − 1. x | N ih N | x 2N By choosing u = σ , A = 1 0 , our Theorem 3 can identify more quantum states containing at most k 1 m x i | ih | − unentangled particles than Observation 5 in Ref.[25] when N =8. For1 k 6, when q < q q′ , these quantum ≤ ≤ k ≤ k states containing at most k 1 unentangled particles which only can be detected by our Theorem 3, but not by − Observation 5 in Ref.[25]. The exact value of q and q′ for 1 k 6 are shown in the Table II. k k ≤ ≤

IV. CONCLUSION

In this paper, we have investigated the problem of detection of quantum states containing at most k 1 unentan- − gled particles. Several criteria for detecting states containing at most k 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. 7

⊗N ⊗N 1 − q ′ TABLE II: For ρ(q)= qσx |WN ihWN |σx + 1 when N = 8, the thresholds of qk, qk for the quantum states containing 2N at most k − 1 unentangled particles detected by Theorem 3 and Observation 5 in Ref.[25] for 1 ≤ k ≤ 6, respectively, are ′ illustrated. When qk < q ≤ 1 and qk < q ≤ 1, ρ(q) contains at most k − 1 unentangled particles detected by our Theorem 3 and Observation 5 in Ref.[25], respectively. The symbol \ means that Observation 5 in Ref.[25] cannot identify any quantum states containing at most 0 unentangled particles and at most 1 unentangled particles. k 1 2 3 4 5 6

qk 0.2889 0.1579 0.1028 0.0725 0.0533 0.0400

′ qk \ \ 0.8647 0.6392 0.4587 0.3234

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos: 12071110, 11701135 and 11947073; the Hebei Natural Science Foundation of China under Grant Nos: A2020205014, A2018205125 and A2017403025, and the Education Department of Hebei Province Natural Science Foundation under Grant No: ZD2020167, and the Foundation of Hebei GEO University under Grant No. BQ201615.

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