Classification of four-qubit states by means of a stochastic local

Construct SLOCC invariants via square matrix Xin-Wei Zha School of Science, Xi’an Institute of Posts and Telecommunications, Xi’an, 710121, P R China Abstract

We define a square matrices, by which some stochastic local operations and classical communication (SLOCC) invariants can be obtained. The relation of SLOCC invariants and character polynomial of square matrix are given for three and four qubit states. Keywords: Square matrice; SLOCC invariant; four qubit

PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w 1. Introduction Quantification of quantum entanglement is one of the most important problems in quantum information theory. Therefore, it is important to find ways of classifying and quantifying the entanglement properties of quantum states. Invariants under stochastic local operations and classical communication(SLOCC), have been extensively studied in this context[1–13].Recently, Li et al. [14]proposed a method of classifying n-qubit states into stochastic local operations and classical communication inequivalent families in terms of the rank of the square matrix. According to this inspiration, a improved square matrix definition is put forward, by which some SLOCC invariants can be obtained. Furthermore , the relation of SLOCC invariants and character polynomial of square matrix is given for three and four-qubit states.

2.

SLOCC invariant and square matrix

In a recent paper [Phys. Rev. A 91, 012302, (2015)], Li et al. proposed a definition of square matrices, which can be defined as[14] Qq1 ,q2 ,,qi =Cq1 ,q2 ,,qi i y  n

 n  i 

Cq1 ,q2 ,,qi 

T

(1)

Where Cq1 ,q2 ,,qi is the coefficient matrix and  y is the Pauli operator. In the follows, we will give the revise definition of the square matrices. It is known that two states 

and 

are equivalent under SLOCC if and only if there exist invertible local

operators M A , M B , M C  , and such that



ABC

 AA  AB  AC   

ABC

(2)

where Ak  SL  2  are SLOCC operations with det  Ak   1 . In an analogous way, we define a square matrices, which can be expressed as n F1,2, i     v 

 i 

n C1,2, i 

 n  i 

v 

n C1,2,  i  

T

(3)

where v  i y and  y is the Pauli operator. Invoking the fact that AkT vAk  det Ak v , for det Ak  1 AkT vAk  v we may have , n  F1,2, i     v 

 i 

 n  i 

n  C1,2, i   v 

n   C1,2, i  

T

(4)

n n  using C1,2, i     A1   Ai  C1,2,i   Ai 1   An 

We can obtain

n 1 n  F1,2, i    [  A1   Ai  ] F1,2,i  T

T

(5)

 A1   Ai 

T

(6)

Therefore, the square matrices F   and F    is similar transformation’ Let the character polynomial of F   is given by det F     I   am m    a1  a 0

(7)

This means that f     det F     I  is a SLOCC invariant polynomial and every λ coefficient of ai is a SLOCC invariant. 3.

The relation of SLOCC invariant with square matrix for three-qubit states An arbitrary pure state  of the three qubits is expressed in the form of



123

 a0 000  a1 001  a2 010  a3 011  a4 100  a5 101  a6 110  a7 111 (8)

It is well known that the SLOCC invariant is[4,15]

F13   * ˆ Ayˆ Byˆ Cx 

2

  * ˆ Ayˆ Byˆ Cz 

2

  * ˆ Ayˆ By 

2

 4[ a0 a7  a1a6  a2 a5  a3 a4   4  a0 a3  a1a2  a5 a6  a4 a7 ] 2

(9)

A convenient, and physically significant, choice is the 3-tangle identified by Coffman, Kundu and Wootters[16] :  2123 ，

 123  F1 3

(10)

For three qubit pure state，the coefficient matrix can be defined a C1 23   0  a4

a1

a2

a5

a6

a3   a0  , C213   a7   a2

a1

a4

a3

a6

a5   a0  , C3 21   a7   a1

a2

a4

a3

a5

a6   a7 

(11)

From [3,11], we have  0 1   a0 F1 23        1 0   a4

a1

a2

a5

a6

a a a a a a a a  0 7 1 6 2 5 3 4 2  a0 a3  a1a2  

0 0 0 a3   0 0 1  a7   0 1 0  1 0 0 2 a4 a7  a5 a6

1   a0 a4    0   a1 a5  0  a2 a6    0   a3 a7 



   a0 a7  a1a6  a2 a5  a3a4 

(12)

The character polynomial of F1 23   is given by





det F1 23     I  a2 2  a1  a0

(13)

Using [12,13], we have the SLOCC invariant a2  1 a1  0 a0    a0 a7  a1a6  a2 a5  a3a4   4 a0 a3  a1a2 a5 a6  a4 a7 2



(14)

1 3   F1  4

Therefore, the determinant of the square matrix F1 23   is the SLOCC invariant. 4. The relation of SLOCC invariant with square matrix for four-qubit states. The states of a four-qubit system can be generally expressed as  1234  a0 0000  a1 0001  a2 0010  a3 0011  a4 0100  a5 0101  a6 0110  a7 0111  a8 1000  a9 1001  a10 1010  a11 1011  a12 1100  a13 1101  a14 1110  a15 1111

(15) We know there are four independent algebraic SLOCC invariants for a 4-qubit system[6,8], that is  H , L, M , Dxt  . Where H  2  a0a15  a1a14  a2a13  a3a12  a4a11  a5a10  a6a9  a7 a8



(16)

and H is a degree-2 invariant whose each term involves only two coefficients. L and M are degree-4 invariant ,are given by the determinants of matrices: a0 a4 a8 a12 L

a1

a5

a9

a13

a2

a6

a10

a14

a3

a7

a11

a15

M

a0

a8

a2

, a10

a1

a9

a3

a11

a4

a12

a6

a14

a5

a13

a7

a15

(17)

(18) ,

Dxt is a degree-6 invariants[8],and can be expressed as the determinants of three 3  3 matrices:

a0a6 a2a4,

Dxt 

a0a14 a6a8

a0a7 aa 1 6 a2a5 a3a4,

a0a15 a6a9 aa 1 14 a7a8

aa 1 7 a3a5,

aa 1 15 a7a9

a2a12 a4a10, a2a13 a3a12 a4a11 a5a10, a3a13 a5a11, a8a14 a10a12, a8a15 a9a14 a10a13 a11a12, a9a15 a11a13,

The coefficient matrix of four-qubit system can be expressed as

(19) .

a C1 234    0  a8 a C2134    0  a4 a C3124    0  a2

C1234 

C14 23

 a0 a  4  a8   a12  a0 a  1  a8   a9

a1

a2

a9

a10

a1

a2

a5

a6

a1

a4

a3

a6

a1

a2

a5

a6

a9

a10

a13 a14 a2

a4

a3

a5

a10

a12

a11

a13

a3

a4

a5

a6

a7  a11 a12 a13 a14 a15  , a3 a8 a9 a10 a11  a7 a12 a13 a14 a15  , a5 a8 a9 a12 a13  a7 a10 a11 a14 a15  , a3   a0 a1  a a7 a3 , C  2 13 24   a8 a9 a11    a15   a10 a11 a6  a7   a14   a15 

a4

a5  a7  , a13   a15 

a6 a12 a14

(20)

From [3,20], we have  1  2 H   F1 234     v1C1234   v2  v3  v4  C1 234  =     0   T

 0   1   H 2 

(21)

Therefore, The character polynomial of F1 234    is given by 1  2  H  H 2 4 Obviously, this character polynomial determine the degree-2 invariant H.

(22)

Similarly, we have F1234      v1  v2  C12 34  v3  v4  C12 34      a0 a15  a1a14  a4 a11  a5 a10   a0 a11  a1a10  a2 a9  a3a8    a0 a7  a1a6  a2 a5  a3a4  2  a0 a3  a1a2  

T

a4 a15  a5 a14  a6 a13  a7 a12  a4 a11  a5 a10  a6 a9  a7 a8  2 a4 a7  a5 a6



a0 a7  a1a6  a2 a5  a3a4

a8 a15  a9 a14  a10 a13  a11a12  2a 8 a11  a9 a10



 a4 a11  a5 a10  a6 a9  a7 a8 a0 a11  a1a10  a2 a9  a3a8

2a12 a 15 a a13



   a8 a15  a9 a14  a10 a13  a11a1 2  a4 a15  a5 a14  a6 a13  a7 a12   a0 a15  a1a14  a2 a13  a3a12  14

(23) It is easy to show that character polynomial of F1234    is given by

 4  a3 3  a2 2  a1  a0 where

a3   H  , a1  4 D xt  6HM  HL ,

1 2 H  2  L  2M  , 4 a0  L2 . a2 

Therefore, the character polynomial of F1234    SLOCC invariants 5. Conclusion

 H , L, M , Dxt  .

(24)

determine four independent algebraic

In summary, using revise definition of the square matrices F   , we can know if two n-qubit states are SLOCC equivalent then their square matrices F   given above have the same character polynomial. Further, we find that just use one square matrices F1234    , can we determine if two n-qubit states are SLOCC equivalent or not for four-qubit system. We expect this method will be useful in the determine SLOCC invariants for n-qubit states. Acknowledgments

This work was supported by the foundation of Shannxi provincial Educational Department under Contract No. 15JK1668 and National Science Foundation of Shannxi Province Grant Nos 2017JQ6024. References [1] Dur W, Vidal G and Cirac J I 2000 Phys. Rev. A 62 062314 [2] Verstraete F, Dehaene J, De Moor B and Verschelde H 2002 Phys. Rev. A 65 052112 [3] Miyake A 2003 Phys. Rev. A 67 012108 [4] Osterloh A and Siewer J 2005 Phys. Rev. A 72 012337 [5] Osterloh A and Siewert J 2006 Int.J. Quantum Inf. 4 531 [6] Luque J G and Thibon J Y 2003 Phys. Rev. A 67 042303 [7] Li D F 2007 Phys. Rev. A 76, 052311 . [8] Ren X J, Jiang W, Zhou X X, Zhou Z W and Guo G C 2008 Phys. Rev. A 78 012343 [9] Gour G and Wallach N R 2013 Phys. Rev. Lett. 111 060502 [10] Wang S H, Lu Y, Gao M, Cui J L and Li J L 2013 J. Phys. A: Math. Theor. 46 105303 [11] Jing N, Li M, Li-Jost X, Zhang T and Fei S M 2014 J. Phys. A: Math. Theor. 47 215303 [12] Sun L L, Li J L and Qiao C F 2015 Quantum Inf. Process. 14 229 [13] Zhang T G , Zhao M J and Huang X F 2016 J. Phys. A: Math. Theor. 49 405301 [14] Li X R and Li D F 2015 Phys. Rev. A 91 012301 [15] Li D, Li X, Huang H and Li X 2007 Phys. Rev. A 76 032304 [16] Coffman V, Kundu J and Wootters W K 2000 Phys. Rev. A 61 052306