Classification of Tripartite Entanglement with one Qubit. Marcio F. Cornelio and A. F. R. de Toledo Piza

Transcription

1 Classification of Tripartite Entanglement with one Qubit Marcio F. Cornelio and A. F. R. de Toledo Piza Universidade de São Paulo, Instituto de Física, CP 66318, São Paulo, S.P., Brazil July 05, quant-ph/

2 Introduction Local operation on a shared entangled states CC CC Source Alice n level ψ qubit Carol n level CC Bob CC Stochastical local operations and classical communication (SLOCC). Bipartite case ψ = n i λi λ i λ i Schmidt Rank C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, and A. V. Thapliyal, arxiv:quant-ph/ (2000). 2

3 Three qubits case two classes: W and GHZ W = 1 3 ( ) GHZ = 1 2 ( ) W Dür, G. Vidal and J. I. Cirac, Phys. Rev. A63, (2000). 2-a

4 Other works concerning SLOCC classification: 1. Four qubits 2. 2 qubits and one n-level system 3. Aspects of SLOCC classification F. Verstraete, J. Dehaene, B. De Moor and H. Verschelde, Phys. Rev. A65, (2002). A. Miyake and F. Verstraete, Phys. Rev. A69, (2004). A. Miyake, Phys. Rev. A67, (2003). 2-b

5 Our work 1. the number of products in the smallest decomposition is a SLOCC invariant. 2. Describe how to find these decompositions for entangled states with local supports (n, n, 2). 3. Use these decompositions to get the SLOCC classification. W Dür, G. Vidal and J. I. Cirac, Phys. Rev. A63, (2000). 2-c

6 Tripartite states with one qubit 1. Let ψ an entangled state with local supports (n, n, 2). 2. The local support of ψ on s ab = s a + s b is a 2D plane P C n a Cn b. 3. P C 2 a C2 b generated by entangled states has either: one product state W class two product states GHZ class A. Sanpera, R. Tarrach and G. Vidal, Phys. Rev. A58, 826 (1998). W Dür, G. Vidal and J. I. Cirac, Phys. Rev. A63, (2000). 3

7 4. ψ = k=0,1 c k r k k φ = α 0 r 0 + α 1 r 1 where r k P C n a Cn b r k span P. 5. φ can be seen as the linear mapping φ : Ca n Cb n u a u a φ The rank of this linear mapping is the Schmidt rank of φ. 3-a

8 6. We are looking for α 0 and α 1 such that the equation u a (α 0 r 0 + α 1 r 1 ) = 0 (1) has a at least one non-trivial solution u a C n a. 7. Interpretation of u a : If we found s a in state u a ψ reduces to a product state. Three qubits: A. Acín, A. Andrianov, L. Costa, E. Jané, J. I. Latorre and R. Tarrach, Phys. Rev. Lett. 85, 1560 (2000). 4

9 8. Let { i } and { j } being basis in C n a and Cn b (α 0 R 0 + α 1 R 1 ) u a = 0 (R 1 1 R 0 λ)u a = 0 where [R k ] ij = ji r k, u a i = u a i and λ = α 1 /α Choosing another base { φ k } for P φ = β 0 φ 0 + β 1 φ 1 (Φ 1 1 Φ 0 µ)u a = 0 where µ = β 1 /β 0 and [Φ k ] ij = ji φ k. 10. What aspects are common to matrices R1 1 R 0 and Φ 1 and how are their respective eigenvalues λ l and µ l related? 1 Φ 0 5

10 11. Definition 2: Jordan family two matrices, A and B, are at the same Jordan family iff λ l of A µ l of B rank(a λ l ) k = rank(b µ l ) k 5-a

11 12. Theorem 1: Let R 0 and R 1 be two n by n matrices, R 1 invertible Φ 0 = ar 0 + br 1 Φ 1 = cr 0 + dr 1 with (ad bc) = 1 and Φ 1 invertible R 1 1 R 0 and Φ 1 1 Φ 0 are at the same Jordan family and µ l = aλ l + b cλ l + d. 13. Interchanging the subsystems s a and s b, the result is equivalent. R k goes to R T k and R 1 1 R 0 goes to (R 0 R1 1 )T which is similar to R1 1 R 0. 5-b

12 14. Let φ 1 and φ 2 to states in P with Schmidt rank smaller than n ψ = φ 1 c 1 + φ 2 c 2, (2) where c 1 and c 2 are appropriate non-normalized states in C 2 c. 15. R 1 1 R 0 may have only one eigenvalue. 16. In general, R 1 1 R 0 has m solutions, there are combinations. ( m 2 ) 17. For each Jordan family of R 1 1 R 0 we can associate a family of entangled states ψ. States which belong to distinct families belong also to distinct SLOCC classes. 5-c

13 Example 1: Three qubits. two Jordan families (a): (b): ( λ1 1 0 λ 1 ( λ1 0 0 λ 2 ) ) W = 1 3 ( ) GHZ = 1 2 ( ). where λ 1 λ 2. 6

14 Example 2: ψ has local supports 3, 3 and 2 - Five Jordan families: (a): λ λ 1 1 ψa = 1 [( ) 0 + ( ) 1 ]. 0 0 λ 1 5 (b): λ λ 1 1 ψb = 1 [ ( ) 1 ]. 0 0 λ 2 1 (c): (d): (e): λ λ λ 2 λ λ λ 2 λ λ λ 3 ψc = 1 [( ) 0 + ( ) 1 ]. 2 ψd = 1 3 [ ( ) 1 ]. where λ l λ l for l l. ψ e = 1 [( ) 0 + ( ) 1 ]. 2 7

15 Example 3: ψ has local supports 4, 4 and 2-13 Jordan families λ GHZ φ + = 0 λ A = λ 2 0 2[ ( 00, , 01 ) λ +( 10, , 11 ) 1 ] 2 B = λ λ λ λ 1 W φ + = 1 6 [ ( 00, , , , 01 ) 0 +( 00, , 01 ) 1 ] Note that the Jordan family corresponding to B differs from that corresponding to λ λ C = λ 1 1 ψ [ c = 1 6 ( 00, , , , 11 ) 0 +( 10, , 10 ) 1 ] λ 1 only in that the ranks of (B λ 1 ) k and (C λ 1 ) k differ for k = 2. 8

16 W φ + = 1 6 [ ( 00, , , , 01 ) 0 +( 00, , 01 ) 1 ] φ 0 φ 1 φ 0. φ 0 φ 1 φ ψ c = 1 6 [ ( 00, , , , 11 ) 0 + ( 10, , 10 ) 1 ] φ 0. φ 0 φ 1 φ 0 φ 1 φ a

17 Another interesting family is (d) λ λ λ λ 4 ψ d = a 2 [ ( 11 + a ) 0 +( 00 + a ) 1 ] which is the only one at this entanglement dimensionality that needs to be subdivided into an infinity of SLOCC classes and where 0 a 1. 8-b

18 ψ SLOCC Classification SLOCC ψ iff ψ = A B C ψ where A, B and C are linear operators in Ca n, Cn b and C2 c respectively. ψ = A B r k C(c k k ) = φ k c k k=0,1 where c k = C(c k k ) and φ k = A B r k. k=0,1 There exist C which maps c k k into any two distinct c k. Φ k = BR ka T Φ 1 1 Φ 0 interchanging the subsystems s a and s b, 1 = AT R1 1 R 0A T (Φ 0 Φ 1 1 )T = B T 1 (R 0 R1 1 )T B T. + Theorem 1 ψ SLOCC ψ only if they are in the same Jordan family. W Dür, G. Vidal and J. I. Cirac, Phys. Rev. A63, (2000). 9

19 ψ ψ R1 1 R 0 in the same Jordan family of R 1 1 R 0 {λ l,r } {λ l,r } rank(r1 1 R 0 λ l,r ) k = rank(r 1 1 R 0 λ l,r )k look for two matrices, Φ 1 and Φ 2 that are superpositions of R 1 and R 0 and such that Φ 1 1 Φ 0 are similar to R 1 1 R 0. λ l,r = µ l,φ = aλ l,r + b cλ l,r + d λ l,rλ l,r c + λ l,r d λ l,r a b = 0 with the additional condition that (ad bc) = 1. Any non-trivial solution of linear system intersects (ad bc) = 1. There always exist at least one solution if that L 3. 9-a

20 Discussion: More General Tripartite Entangled States When neither one of the subsystems is a qubit, we get the equation ( k α k R k )u a = 0 When the entanglement has local supports n,m and 2, with m n, there is no invertible matrix. 10

21 Conclusion We have described a constructive method to find decompositions of tripartite entangled pure states which involve a number of terms smaller than one obtains using two successive Schmidt decompositions for entangled states with local supports on each part n, n and 2. We use these decompositions to classify these states according their inter-convertibility through SLOCC. We show how to find the SLOCC operation which transform one state in another when they are in the same SLOCC class. 11

22 Acknowledgments D. Tausk for crucial helping on the proof of Theorem 1. MFC acknowledges financial support of FAPESP (Fundação de Amparo a Pesquisa do Estado de São Paulo). 12