Important Notice to Authors Attached is a PDF proof of your forthcoming article in PRA. Your article has 4 pages and the Accession Code is

(1)

Important Notice to Authors

Attached is a PDF proof of your forthcoming article in PRA. Your article has 4 pages and the Accession Code isLQ14022A.

Please note that as part of the production process, APS converts all articles, regardless of their original source, into standardized XML that in turn is used to create the PDF and online versions of the article as well as to populate third-party systems such as Portico, CrossRef, and Web of Science. We share our authors’ high expectations for the fidelity of the conversion into XML and for the accuracy and appearance of the final, formatted PDF. This process works exceptionally well for the vast majority of articles; however, please check carefully all key elements of your PDF proof, particularly any equations or tables.

Figures submitted electronically as separate PostScript files containing color usually appear in color in the online journal.

However, all figures will appear as grayscale images in the print journal unless the color figure charges have been paid in advance, in accordance with our policy for color in print (http://journals.aps.org/authors/color-figures-print) and the relevant figure captions read “Color”. For figures that will be color online but grayscale in print, please ensure that the text and captions clearly describe the figures to readers who view the article only in grayscale.

No further publication processing will occur until we receive your response to this proof.

Specific Questions and Comments to Address for This Paper 1 Please check added multiplication sign in Eq. (2). As meant?

Q: This reference could not be uniquely identified due to incomplete information or improper format. Please check all information and amend if applicable.

FundRef:Information about an article’s funding sources is now submitted to CrossRef to help you comply with current or future funding agency mandates. Please ensure that your acknowledgments include all sources of funding for your article following any requirements of your funding sources. CrossRef’s FundRef registry (http://www.crossref.org/fundref/) is the definitive registry of funding agencies. Please carefully check the following funder information we have already extracted from your article and ensure its accuracy and completeness:

NSFC (China), 10875061 Other Items to Check

r

Please note that the original manuscript has been converted to XML prior to the creation of the PDF proof, as described above.

Please carefully check all key elements of the paper, particularly the equations and tabular data.

r

Please check PACS numbers. More information on PACS numbers is available online athttp://journals.aps.org/PACS/.

r

Title: Please check; be mindful that the title may have been changed during the peer review process.

r

Author list: Please make sure all authors are presented, in the appropriate order, and that all names are spelled correctly.

r

Please make sure you have inserted a byline footnote containing the email address for the corresponding author, if desired.

Please note that this is not inserted automatically by this journal.

r

Affiliations: Please check to be sure the institution names are spelled correctly and attributed to the appropriate author(s).

r

Receipt date: Please confirm accuracy.

r

Acknowledgments: Please be sure to appropriately acknowledge all funding sources.

r

Hyphenation: Please note hyphens may have been inserted in word pairs that function as adjectives when they occur before a noun, as in “x-ray diffraction,” “4-mm-long gas cell,” and “R-matrix theory.” However, hyphens are deleted from word pairs when they are not used as adjectives before nouns, as in “emission by x rays,” “was 4 mm in length,” and “theRmatrix is tested.”

Note also that Physical Review follows U.S. English guidelines in that hyphens are not used after prefixes or before suffixes:

superresolution, quasiequilibrium, nanoprecipitates, resonancelike, clockwise.

r

Please check that your figures are accurate and sized properly. Make sure all labeling is sufficiently legible. Figure quality in this proof is representative of the quality to be used in the online journal. To achieve manageable file size for online delivery, some compression and downsampling of figures may have occurred. Fine details may have become somewhat fuzzy, especially in color figures. The print journal uses files of higher resolution and therefore details may be sharper in print. Figures to be published in color online will appear in color on these proofs if viewed on a color monitor or printed on a color printer.

r

Overall, please proofread the entire article very carefully.

Ways to Respond

r

_Web:If you accessed this proof online, follow the instructions on the web page to submit corrections.

r

_Email:Send corrections to praproofs@aptaracorp.com Subject:LQ14022Aproof corrections

(2)

r

_Fax:Return this proof with corrections to +1.703.791.1217. WriteAttention:PRA Project Manager and the Article ID,LQ14022A, on the proof copy unless it is already printed on your proof printout.

r

_Mail:Return this proof with corrections toAttention:PRA Project Manager, Physical Review A, c/o Aptara, 3110 Fairview Park Drive, Suite #900, Falls Church, VA 22042-4534, USA.

(3)

PHYSICAL REVIEW A00, 002300 (2015)

1

Entanglement classification and invariant-based entanglement measures

2

Xiangrong Li¹and Dafa Li^2,3

3

1Department of Mathematics, University of California Irvine, Irvine, California 92697, USA

4

2Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

5

3Center for Quantum Information Science and Technology, Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing, 100084, China

6 7

(Received 2 March 2014; published xxxxxx)

8

We propose a method of classifyingn-qubit states into stochastic local operations and classical communication inequivalent families in terms of the rank of the square matrixC(iσy)^⊗^kC^T, whereCis the rectangular coefficient matrix of the state and σy is the Pauli operator. The rank of the square matrix C(iσy)^⊗kC^T is capable of distinguishing betweenn-qubit Greenberger-Horne-Zeilinger andWstates. The determinant of the matrix gives rise to a family of polynomial invariants fornqubits which include as special cases well-known polynomial invariants in the literature. In addition, explicit expressions can be given for these polynomial invariants and this allows us to investigate the properties of entanglement measures built upon the absolute values of polynomial invariants for product states.

9 10 11 12 13 14 15 16

DOI:10.1103/PhysRevA.00.002300 PACS number(s): 03.67.Mn,03.65.Ud

17

I. INTRODUCTION

18

Quantum entanglement is an essential resource in quantum

19

teleportation, quantum cryptography, and quantum compu-

20

tation [1]. A crucial task in entanglement theory is the

21

classification of entanglement. To classify entangled states,

22

some equivalence relation has to be introduced. Of particular

23

importance is the equivalence under stochastic local operations

24

and classical communication (SLOCC). If two states are

25

SLOCC equivalent then they can perform the same tasks

26

in quantum information theory [2]. As equivalence under

27

SLOCC induces a natural partition of quantum states, the key

28

point of SLOCC classification is to classify quantum states

29

according to a criterion that is invariant under SLOCC. While

30

entanglement classification of two and three qubits is well

31

understood, the task of classifying entanglement beyond three

32

qubits becomes increasingly difficult. For four or more qubits,

33

there exists an infinite number of inequivalent SLOCC classes.

34

It is highly desirable to partition the infinite SLOCC classes

35

into a finite number of families such that states belonging to

36

the same family possess similar properties, according to some

37

criteria for determining to which family a given state belongs.

38

Considerable efforts have been undertaken over the last decade

39

for SLOCC entanglement classification of four-qubit states,

40

resulting in a finite number of families or classes [3–10]. For

41

more than four qubits, a few attempts have been made for

42

SLOCC classification [11–16]. Despite these efforts, a SLOCC

43

classification for generaln-qubit states which results in a finite

44

number of families with Greenberger-Horne-Zeilinger (GHZ)

45

andW states belonging to different families is still beyond

46

reach.

47

This paper is organized as follows. We first construct a

48

matrix for an n -qubit state and we show that the rank of

49

the matrix is preserved under SLOCC. The rank provides

50

a simple way of classifying n-qubit states into a number of

51

SLOCC-inequivalent families. We then exemplify the use of

52

the rank in distinguishingn-qubit GHZ andW states as well

53

as some four-, five-, and six-qubit states. The determinant

54

of the matrix gives rise to a polynomial invariant of degree

55

2^k (kn/2+1) for n qubits and this construction allows

56

one to derive the expressions for these polynomial invariants ⁵⁷ explicitly. Intriguingly, the even n-qubit concurrence, the 58

even n-tangle, the odd n-tangle, and polynomial invariants ⁵⁹ of degree 2^n/2for evennqubits all turn out to be special cases ⁶⁰ of polynomial invariants of degree 2^k. We also discuss the ⁶¹ properties of entanglement measures built upon the polynomial ⁶² invariants of degree 2^kfor product states. ⁶³ II. THE INVARIANCE OF THE RANK 64

We follow the notation of [14]. Let|ψ =₂ⁿ₋₁

k=0 ak|kbe 65

any pure state of any n qubits. We let C_1,2,...,i⁽ⁿ⁾ (|ψ) be the ⁶⁶ 2ⁱ×2ⁿ⁻ⁱ coefficient matrix of the state |ψ, whose entries 67

are the coefficientsa₀,a₁, . . . ,a₂ⁿ₋₁of the state|ψarranged ⁶⁸ in ascending lexicographical order. Here the bits 1 toi and ⁶⁹ i+1 to n are referred to as the row bits and column bits, ⁷⁰ respectively. In particular, wheni=0,C_∅⁽ⁿ⁾(|ψ) reduces to ⁷¹ the row vector (a0, . . . ,a₂ⁿ₋₁) and, wheni=n,C_1,...,n⁽ⁿ⁾ (|ψ) ⁷² reduces to the column vector (a0, . . . ,a₂ⁿ₋₁)^T. Note that qubits ⁷³ q₁,q₂, . . . ,qican also be chosen as row bits. ⁷⁴ Recall that two n-qubit states |ψ and|ψ are SLOCC ⁷⁵ equivalent if and only if there are local invertible operators ⁷⁶ A1,A2, . . ., andAnsuch that [2] ⁷⁷

|ψ =A1⊗A2⊗ · · · ⊗An|ψ. (1) When qubitsq₁,q₂, . . . ,q_i are chosen as row bits, the coeffi- ⁷⁸ cient 2ⁱ×2ⁿ⁻ⁱ matrixC⁽ⁿ⁾_q₁_,q₂_,...,q_i(|ψ) satisfies the following 79

matrix equation [13,14]: ⁸⁰

C_q⁽ⁿ⁾₁_,q₂_,...,q_i(|ψ)

=(Aq₁⊗ · · · ⊗Aq_i)C_q⁽ⁿ⁾₁_,q₂_,...,q_i(|ψ)(Aq_i+1⊗ · · · ⊗A_qn)^T. (2)

1 Taking the transpose of both sides of Eq. (2) and after some ⁸¹

algebra, we obtain ⁸²

C⁽ⁿ⁾_q

1,q₂,...,q_i(|ψ)υ^⊗(n⁻ⁱ⁾ C_q⁽ⁿ⁾

1,q₂,...,q_i(|ψ)T

=(Aq1⊗ · · · ⊗Aqi)C_q⁽ⁿ⁾₁_,q₂_,...,q_i(|ψ)

ACC. CODE LQ14022A AUTHOR Li

(4)

XIANGRONG LI AND DAFA LI PHYSICAL REVIEW A00, 002300 (2015)

×

A^Tq_i+1υAq_i+1⊗ · · · ⊗A^T_qnυA_qn

× C_q⁽ⁿ⁾

1,q₂,...,q_i(|ψ)T

(Aq₁⊗ · · · ⊗Aq_i)^T, (3) whereυ=√

−1σyandσyis the Pauli operator.

83

We let

84

⁽ⁿ⁾_q₁_,q₂_,...,q_i(|ψ

=C⁽ⁿ⁾_q

1,q2,...,qi(|ψ)υ^⊗⁽ⁿ⁻ⁱ⁾ C_q⁽ⁿ⁾

1,q2,...,qi(|ψ)T

. (4) It is clear that⁽ⁿ⁾_q

1,q₂,...,q_i(|ψis a square matrix of order 2ⁱfor

85

nqubits. Invoking the fact thatA^TkυAk=(detAk)υ, we may

86

rewrite Eq. (3) as

87

⁽ⁿ⁾_q

1,q2,...,qi(|ψ)

=

ⁿ_k₌_i₊₁detAqk

(Aq1⊗ · · · ⊗Aqi)

×⁽ⁿ⁾_q₁_,q₂_,...,q_i(|ψ)(Aq₁⊗ · · · ⊗Aq_i)^T. (5) In the following, we will omit the state |ψ and simply

88

write⁽ⁿ⁾_q₁_,q₂_,...,q_i whenever the state is clear from the context.

89

It immediately follows from Eq. (5) that the rank of the square

90

matrix⁽ⁿ⁾_q

1,q₂,...,q_iof ann-qubit state is invariant under SLOCC.

91

This leads to the following theorem.

92

Theorem.If twon-qubit states are SLOCC equivalent then

93

their square matrices⁽ⁿ⁾_q₁_,q₂_,...,q_i given above have the same

94

rank.

95

Restated in the contrapositive the theorem reads: If two

96

square matrices⁽ⁿ⁾_q

1,q₂,...,q_i associated with twon-qubit states

97

differ in their ranks, then the two states belong necessarily to

98

different SLOCC classes.

99

III. APPLICATIONS TO SLOCC CLASSIFICATION

100

A. Classification of four-qubit states

101

Verstraete et al. partitioned four-qubit states into nine

102

SLOCC-inequivalent families, two of which are Labc₂ and

103

L_ab₃ [3]. Later, it was pointed out that L_ab₃ is equivalent

104

to the subfamily L_abc₂(a=c) of L_abc₂ obtained by setting

105

a=c [6]. In [13], we showed that Lab₃(a=b=0) is

106

inequivalent to Labc₂(a =c) using the rank of coefficient

107

matrices. Alternatively, Sharmaet al.proved thatLabc₂(a =c)

108

andL_ab₃are not SLOCC equivalent using negativity fonts [17].

109

Here by elaborating further on the relationship betweenL_ab₃

110

andLabc2(a=c), we show by using the rank of the square

111

matrix that, whena=0,Lab₃is contained inLabc₂. In TableI

112

we list the rank of⁽⁴⁾_1,2forLab3andLabc2(a =c).

113

It follows from TableIthat, whena=0,Lab₃andLabc₂(a =

114

c) are inequivalent to each other. Furthermore,Lab₃(a=0)

115

and Lab₃(a=0) are inequivalent to each other. Likewise,

116

L_abc₂(a=c=0) and L_abc₂(a=c=0) are inequivalent to

117

TABLE I. The rank of⁽⁴⁾_1,2forLab₃andLabc₂(a=c).

a=0 a=0 a=0 a=0 b=0 b=0 b=0 b=0

L_ab₃ 1 2 3 4

Labc₂(a=c) 0 1 3 4

TABLE II. SLOCC classification of some five-qubit and six-qubit states.

Five-qubit Rank of Rank of Six-qubit Rank of Rank of state ⁽⁵⁾₁ ⁽⁵⁾₃ state ⁽⁶⁾_1,2 ⁽⁶⁾_1,2,3,4

|2 2 2 |2 2 2

|4 0 0 |4 0 2

|5 0 1 |5 0 3

|₆ 1 1 |₆ 1 3

|7 2 3

each other. It turns out that, whena =0,L_ab₃andL_abc₂(a=c) ¹¹⁸ are equivalent to each other. This can be seen as follows. A ¹¹⁹ tedious calculation shows that, whena=0,Labc₂(a =c) and ¹²⁰ Lab₃satisfy the following equation: ¹²¹

Labc₂(a=c)=A1⊗A2⊗A3⊗A4Lab₃, (6) whereA1,A2,A3, andA4are invertible local operators given 122

by ¹²³

A1= 1

2a^3/2 0

0 ¹

2√ 2a²

, A2= 0 1

−√ 2a 0

,

A3= −√

2a 0

−i√ 2 2a

, A4= −i √

−√ 2a

a 0

.

Therefore, whena=0,L_ab₃andL_abc₂(a =c) are inequivalent ¹²⁴ to each other, whereas whena=0,L_ab₃andL_abc₂(a=c) are ¹²⁵ equivalent to each other. In particular, whena=0, we have 126

SLOCC equivalence betweenLab₃ andLabc₂(a=c) for the ¹²⁷ following cases: (i)a=b; (ii)a= −b; (iii)b=3a; (iv)b= ¹²⁸

−3a; (v)b=0; (vi)ab=0 anda= ±bandb= ± −3a. ¹²⁹

B. Classification of some five-qubit and six-qubit states 130

We list in TableIIthe rank of⁽ⁿ⁾_q

1,q₂,...,q_i for the five-qubit ¹³¹ states and six-qubit states introduced in [18]. Consulting ¹³² the table, the five-qubit states |₂, |₄, |₅, and |₆ ¹³³ are inequivalent to one another under SLOCC and they can 134

be distinguished via the rank of ⁽⁵⁾₁ and ⁽⁵⁾₃ . Similarly, ¹³⁵ the six-qubit states |₂, |₄, |₅, |₆, and |₇ are ¹³⁶ inequivalent to one another under SLOCC and they can be ¹³⁷ distinguished via the rank of⁽⁶⁾_1,2and⁽⁶⁾_1,2,3,4. 138

C. Classification ofn-qubit states 139

We exemplify the classification withn-qubit GHZ andW 140

states. We find that⁽ⁿ⁾₁ has rank 2 forn-qubit GHZ states, ¹⁴¹ rank 1 for 3-qubitW states, and rank 0 forn-qubitW states ¹⁴² for n4. Hence n-qubit GHZ states can be distinguished ¹⁴³ fromn-qubitW states under SLOCC via the rank of⁽ⁿ⁾₁ . In ¹⁴⁴ addition, one may also distinguish cluster states from GHZ ¹⁴⁵ (W) states using the ranks of ⁽ⁿ⁾_q

1,q2,...,qi. For example, the ¹⁴⁶ cluster state of four qubits can be readily distinguished from ¹⁴⁷ a four-qubit GHZ (four-qubitW) state using the ranks of⁽⁴⁾₁ 148

and⁽⁴⁾_1,2. ¹⁴⁹

More generally, let σ denote the sequence q₁,q₂, . . . ,q_i 150

andF_r^σbe the set ofn-qubit states with the rank of⁽ⁿ⁾_q

1,q2,...,qi 151

being equal tor. Thus,n-qubit states are partitioned into 2ⁱ+1 ¹⁵² 002300-2

(5)

ENTANGLEMENT CLASSIFICATION AND INVARIANT- . . . PHYSICAL REVIEW A00, 002300 (2015)

TABLE III. The polynomial invariant|det⁽ⁿ⁾_∅ |of degree 2.

Qubits Expressions Polynomial invariants

n=2 |det⁽²⁾_∅ | Concurrence [29]

evenn |det⁽ⁿ⁾_∅ | n-qubit concurrence [28,30]

oddn |det⁽ⁿ⁾_∅ | =0 ^a

aNote that for odd n, |det⁽ⁿ⁾_∅ | =0. This reveals that no such nontrivial polynomial invariant of degree 2 exists for odd-nqubits.

SLOCC-inequivalent families by the theorem above, i.e.,F₀^σ,

153

F₁^σ,. . ., andF₂^σi.

154

IV. POLYNOMIAL INVARIANTS OF DEGREE 2^k

155

Several approaches have been proposed to construct poly-

156

nomial invariants [19–23]. However, the computational com-

157

plexity grows very rapidly as the number of qubits increases

158

(e.g., some methods are not readily generalized to more

159

complicated Hilbert spaces). Accordingly, the expressions of

160

polynomial invariants have thus far been given only up to five

161

qubits. Recently, a few attempts have been made to construct

162

polynomial invariants with explicit expressions [24–28].

163

Corollary 1. Let|ψand|ψbe two SLOCC-equivalent

164

states ofnqubits, then the following equation holds for 0

165

in:

166

det⁽ⁿ⁾_q

1,q₂,...,q_i(|ψ)

=det⁽ⁿ⁾_q₁_,q₂_,...,q_i(|ψ)

ⁿ_k₌₁detAk

2ⁱ

. (7) Proof.Taking the determinant of both sides of Eq. (5) yields

167

the desired result.

168

As an immediate consequence of Corollary 1, det⁽ⁿ⁾_q₁_,q₂_,...,q_i

169

is a polynomial invariant of degree 2ⁱ⁺¹, where 0in/2.

170

Otherwise,⁽ⁿ⁾_q

1,q₂,...,q_i is not full rank.

171

Polynomial invariants constructed above include as special

172

cases several well-known polynomial invariants in the litera-

173

ture. Below are some examples.

174

Example 1. Set i=0. Then |det⁽ⁿ⁾_∅ | is a polynomial

175

invariant of degree 2. See TableIII.

176

Example 2. We set i=1. Then det⁽ⁿ⁾₁ is a polynomial

177

invariant of degree 4. In particular, 4|det⁽³⁾₁ |is equal to the 3-

178

tangle [25,31]. Furthermore, 4|det⁽ⁿ⁾₁ |is a natural extension

179

of the 3-tangle to generalnqubits. See TableIV.

180

We remark that forneven, det⁽ⁿ⁾₁ (i.e., evenn-tangle) is

181

invariant under permutations [33]. Fornodd, we may choose

182

qubitj,j =1, . . . ,n, as the row bit for the coefficient matrix

183

TABLE IV. The polynomial invariant det⁽ⁿ⁾₁ of degree 4.

Qubits Expressions Polynomial invariants

n=3 4det⁽³⁾₁ 3-tangle [25,31]

evenn 4det⁽ⁿ⁾₁ evenn-tangle [32]

oddn 4det⁽ⁿ⁾₁ oddn-tangle [25]

C_j⁽ⁿ⁾. This yieldsnpolynomial invariants det⁽ⁿ⁾_j of degree 4 ¹⁸⁴ for oddn(5) qubits (for five qubits, see [23]) [25,33,34]. ¹⁸⁵ We emphasize that these npolynomial invariants det⁽ⁿ⁾_j 186

of degree 4 for any oddn5 qubits are linearly independent ¹⁸⁷ (this is particularly true for five qubits [23]). This can be proved ¹⁸⁸ by resorting to the following properties of det⁽ⁿ⁾_j forn-odd ¹⁸⁹

qubits [34]: 190

(1) (i,j) det⁽ⁿ⁾_j =det⁽ⁿ⁾_i , where (i,j) is the transposition ¹⁹¹

of qubitsiandj. ¹⁹²

(2) det⁽ⁿ⁾_j is invariant under any permutation of qubits not ¹⁹³

involving qubitj. ¹⁹⁴

Example 3.Letnbe even andi=n/2. ThenC_1···(n/2)⁽ⁿ⁾ is a ¹⁹⁵ square matrix. In view of Eqs. (4) and (7), we have 196

detC⁽ⁿ⁾_1···(n/2)(|ψ)

=detC₁⁽ⁿ⁾_···_(n/2)(|ψ)

ⁿ_k₌₁detAk

2^(n−2)/2

. (8) As an immediate consequence, detC₁⁽ⁿ⁾_···_(n/2) is a determinant ¹⁹⁷ invariant of degree 2^n/2 and we recover the result in [26] (in ¹⁹⁸ particular we recover the polynomial invariants of degree 4 for ¹⁹⁹

four qubits given in [19]). ²⁰⁰

In light of Eq. (7), we may determine whether twon-qubit ²⁰¹ states are inequivalent to each other under SLOCC via the ²⁰² vanishing or not of their associated polynomial invariants ²⁰³ det⁽ⁿ⁾_q

1,q₂,...,q_i. More precisely, we have the following result. ²⁰⁴ Corollary 2. For any two SLOCC-equivalent pure states 205

|ψand|ψofnqubits, either both det⁽ⁿ⁾_q₁_,q₂_,...,q_i(|ψ) and ²⁰⁶ det⁽ⁿ⁾_q

1,q₂,...,q_i(|ψ) vanish or neither vanishes. In other words, ²⁰⁷ if one of det⁽ⁿ⁾_q

1,q₂,...,q_i(|ψ) and det⁽ⁿ⁾_q

1,q₂,...,q_i(|ψ) vanishes ²⁰⁸ while the other does not, then the two states|ψand|ψare ²⁰⁹

SLOCC inequivalent. ²¹⁰

For example, the n-qubit GHZ andW states can also be ²¹¹ distinguished under SLOCC as det⁽ⁿ⁾_∅ =0 for theW state ²¹² and det⁽ⁿ⁾_∅ =0 for the GHZ state. ²¹³

V. INVARIANT-BASED ENTANGLEMENT MEASURES 214

The explicit expressions of these polynomial invariants ²¹⁵ det⁽ⁿ⁾_q

1,q₂,...,q_i make it possible for us to investigate the ²¹⁶ properties of det⁽ⁿ⁾_q

1,q2,...,qi. We next explore the properties ²¹⁷ of |det⁽ⁿ⁾_q₁_,q₂_,...,q_i| by use of the product state |ψ1···n= ²¹⁸

|φj₁···j⊗ |ϕj₊₁···j_n, where |φj₁···j is a state of qubits, ²¹⁹ j₁, . . . ,j, and |ϕj₊₁···j_n is a state of the remaining (n−) ²²⁰ qubits,j+1, . . . ,jn. We letC_q⁽ⁿ⁾

1,...,q_i(|ψ1···n) be the coefficient ²²¹ matrix associated with the state|ψ_1···n, whereq₁, . . .andq_iare ²²² chosen as row bits. We letC_q⁽⁾∗

1,...,q^∗_s(|φj1···j) be the 2^s×2⁻^s ²²³ coefficient matrix associated with the-qubit state|φj1···j. ²²⁴ Here {q₁^∗, . . . ,q_s^∗} = {q₁, . . . ,qi} ∩ {j₁, . . . ,j} are the row ²²⁵ bits. We letC⁽ⁿ_q⁻⁾

1,...,q_t(|ϕj₊₁···j_n) be the 2^t×2ⁿ⁻⁻^t coefficient ²²⁶ matrix associated with the (n−)-qubit state|ϕj₊₁···j_n. Here 227

{q₁, . . . ,q_t} = {q₁, . . . ,qi} ∩ {j+1, . . . ,jn} are the row bits. ²²⁸ Note thats+t=i. From [14], we have ²²⁹

C_q⁽ⁿ⁾₁_,...,q_i(|φj₁···j ⊗ |ϕj₊₁···j_n)

=C_q⁽⁾∗

1,...,q^∗_s(|φj1···j)⊗C_q⁽ⁿ⁻⁾

1,...,q_t(|ϕj₊₁···jn). (9) 002300-3

(6)

XIANGRONG LI AND DAFA LI PHYSICAL REVIEW A00, 002300 (2015) Using the notation of Eq. (4), a simple calculation yields

230 det⁽ⁿ⁾_q₁_,...,q_i(|φj₁···j⊗ |ϕj₊₁···j_n)

=det⁽⁾_q∗

1,...,q_s^∗(|φj1···j)²^tdet⁽ⁿ_q⁻⁾

1,...,q_t(|ϕj₊₁···jn)²^s. (10) Clearly, the absolute value of the SLOCC polyno-

231

mial invariant det⁽ⁿ⁾_q

1,q₂,...,q_i is not additive for product

232

states. Note that it vanishes for product states with s >

233

/2 or t >(n−)/2. Consider, for example, a prod-

234

uct state of four qubits |ψ1234= |φ13⊗ |ϕ24. Then a

235

straightforward calculation yields that det⁽⁴⁾₁₂(|ψ1234)=

236

[det⁽²⁾₁ (|φ13)]²[det⁽²⁾₂ (|ϕ24)]².

237

Recently, it was shown that a positive homogeneous

238

SLOCC polynomial invariant defines ann-qubit entanglement

239

monotone if and only if the homogeneous degree is less

240

than or equal to 4 [35]. Accordingly, the absolute value of

241

the polynomial invariant det⁽ⁿ⁾_q₁_,q₂_,...,q

i with degree 4 is

242

an entanglement monotone and it can be considered as an

243

entanglement measure fornqubits.

244

VI. CONCLUSION 245

We have constructed a matrix whose rank is preserved under ²⁴⁶ SLOCC and given examples of classifyingn-qubit states via ²⁴⁷ the rank for n up to 6. Polynomial invariants in the form ²⁴⁸ of determinants of the square matrix not only have explicit ²⁴⁹ expressions but also, as special cases, recover several existing ²⁵⁰ polynomial invariants in the literature. We have also studied ²⁵¹ the properties of the entanglement measures built from the ²⁵² absolute values of polynomial invariants on product states. ²⁵³ We expect that the proposed approach for classifyingn-qubit ²⁵⁴ states and constructing polynomial invariants may find further ²⁵⁵

applications. 256

ACKNOWLEDGMENTS 257

This work was supported by the NSFC (Grant No. ²⁵⁸ 10875061) and Tsinghua National Laboratory for Information ²⁵⁹

Science and Technology. ²⁶⁰

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, 2000).

[2] W. D¨ur, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000).

[3] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde,Phys.

Rev. A65,052112(2002).

[4] A. Miyake,Phys. Rev. A67,012108(2003).

[5] D. Li, X. Li, H. Huang, and X. Li,Phys. Rev. A76,052311 (2007).

[6] O. Chterental and D. Z. Djokovi´c, inLinear Algebra Research Advances, edited by (Nova Science, Hauppauge, NY, 2007), Chap. 4, p. 133.

[7] L. Lamata, J. Le´on, D. Salgado, and E. Solano,Phys. Rev. A75, 022318(2007).

[8] D. Li, X. Li, H. Huang, and X. Li, Quantum Inf. Comput.9, 0778 (2009).

Q

[9] L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, and W. Rubens,Phys. Rev. Lett.105,100507(2010).

[10] O. Viehmann, C. Eltschka, and J. Siewert,Phys. Rev. A83, 052330(2011).

[11] T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano,Phys. Rev. Lett.103,070503(2009).

[12] P. Ribeiro and R. Mosseri,Phys. Rev. Lett.106,180502(2011).

[13] X. Li and D. Li,Phys. Rev. Lett.108,180502(2012).

[14] X. Li and D. Li,Phys. Rev. A86,042332(2012).

[15] S. S. Sharma and N. K. Sharma,Phys. Rev. A85,042315(2012).

[16] G. Gour and N. R. Wallach, Phys. Rev. Lett. 111, 060502 (2013).

[17] S. S. Sharma and N. K. Sharma,Phys. Rev. A87,022335(2013).

[18] A. Osterloh and J. Siewert,Int. J. Quantum. Inf.4,531(2006).

[19] J.-G. Luque and J.-Y. Thibon,Phys. Rev. A67,042303(2003).

[20] M. S. Leifer, N. Linden, and A. Winter,Phys. Rev. A69,052304 (2004).

[21] P. Levay,J. Phys. A38,9075(2005).

[22] A. Osterloh and J. Siewert,Phys. Rev. A72,012337(2005).

[23] J.-G. Luque and J.-Y. Thibon,J. Phys. A39,371(2006).

[25] D. Li,Quantum Inf. Process.11,481(2012).

[26] X. Li and D. Li,J. Phys. A45,075308(2012).

[27] X. Li and D. Li,J. Phys. A46,135301(2013).

[28] X. Li and D. Li,Phys. Rev. A88,022306(2013).

[29] W. K. Wootters,Phys. Rev. Lett.80,2245(1998).

[30] X. Li and D. Li, Quantum Inf. Comput.10, 1018 (2010). Q [31] V. Coffman, J. Kundu, and W. K. Wootters,Phys. Rev. A61,

052306(2000).

[32] A. Wong and N. Christensen,Phys. Rev. A63,044301(2001).

[33] D. Li, X. Li, H. Huang, and X. Li,J. Math. Phys.50,012104 (2009).

[35] C. Eltschka, T. Bastin, A. Osterloh, and J. Siewert,Phys. Rev.

A85,022301(2012).

002300-4