Constructing Entanglement Witness Via Real Skew-Symmetric Operators
aa r X i v : . [ qu a n t - ph ] A ug Constructing Entanglement Witness Via RealSkew-Symmetric Operators
M. A. Jafarizadeh a,b,c ∗ , N. Behzadi a,b † a Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz 51664, Iran. b Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Iran. c Research Institute for Fundamental Sciences, Tabriz 51664, Iran.
October 30, 2018 ∗ E-mail:[email protected] † E-mail:[email protected] ntanglement Witnesses Abstract
In this work, new types of EWs are introduced. They are constructed by using realskew-symmetric operators defined on a single party subsystem of a bipartite d ⊗ d systemand a maximal entangled state in that system. A canonical form for these witnesses isproposed which is called canonical EW in corresponding to canonical real skew-symmetricoperator. Also for each possible partition of the canonical real skew-symmetric operatorcorresponding EW is obtained. The method used for d ⊗ d case is extended to d ⊗ d systems. It is shown that there exist C d d distinct possibilities to construct EWs for agiven d ⊗ d Hilbert space. The optimality and nd-optimality problem is studied foreach type of EWs. In each step, a large class of quantum PPT states is introduced. Itis shown that among them there exist entangled PPT states which are detected by theconstructed witnesses. Also the idea of canonical EWs is extended to obtain other EWswith greater PPT entanglement detection power.
Keywords: Canonical Entanglement Witness, Skew-Symmetric Matrices, En-tangled PPT states.
PACS: 03.67.Mn, 03.65.Ud ntanglement Witnesses Entanglement is one of the essential features of quantum physics which has no analogousin classical one. Entanglement lies in the heart of the quantum information and quantumcomputation. It is used as a physical resource which allows to realize various quantum infor-mation and quantum computation tasks such as quantum cryptography, teleportation, densecoding, and key distribution [1, 2, 3]. The most important problem in quantum entanglementis determining boundary of the separable states and entangled ones, which is still not wellcharacterized. Although the famous Peres-Horodecki criterion based on positive partial trans-pose (PPT) explicitly determines this boundary for low dimensional bipartite systems suchas 2 ⊗ ⊗ ρ sep ,Tr(W ρ sep ) ≥ ρ ent , Tr(W ρ ent ) < ρ ent is detected by W). Clearly, the construction of EWs is a hard task, although it is easy toconstruct an operator W which has negative expectation value with an entangled state, but itis very difficult to check that Tr(W ρ ) ≥ ρ [10, 11, 12, 13]. There aretwo types of EWs: decomposable EW (d-EW) that can not detect any entangled PPT statesand non-decomposable one (nd-EW) which can detect at least one entangled PPT state. It ntanglement Witnesses h α ∗ | U | α i = 0 (for every state vector | α i thatlives in the single party subsystem). It is shown that, by using this property of skew-symmetricmatrices, the related EWs can be constructed analytically for any d ⊗ d systems. From thetheory of matrices [22], the rank of any skew-symmetric matrix is equal to an even number(2 n ). Also every skew-symmetric matrix U can be written as U = QJQ t in which J is thecanonical form of U (or canonical skew-symmetric matrix) and Q is an orthogonal matrix(QQ t = Q t Q = I d ). A witness which is established by J is called canonical EW. It is shownthat a large number of witnesses are obtained by doing orthogonal transformations specifiedby Q, on the characteristic elements of canonical EW. Thus one is interested to study andanalyze the canonical EWs in details.To develop the idea of the canonical EW for various ranks that J can achieve, the relatedcanonical EW is derived. It is proved that for all possible values of rank ( J ), ( rank ( J ) =0 , ... n = d ), the corresponding canonical EWs is optimal. It is also shown that for full-rankJ, the related EW is optimal nd-EW and, when the rank ( J ) <
4, it is optimal d-EW. Onthe other hand, we introduce a new class of PPT states among which there exist entangled ntanglement Witnesses d = 2 n ). In this case, J is partitionedas a direct sum of block-diagonal matrices, which corresponds to a partition of n. As illustrated, all of witnesses obtained according to possible partition of n are optimal EWs. Also in thisstep for every partition of n, we construct entangled PPT state detected by the nd-EW whichis given in the same partition. It is shown that our method can be extended to the bipartite d ⊗ d systems. The shape of canonical EWs are similar to the those ones obtained for d ⊗ d systems. In this case, PPT states are also introduced and by the constructed canonical EWs,it is shown that some of them are entangled.The paper is organized as follows: In section 2, we investigate our approach to constructEWs for d ⊗ d systems. The canonical EWs along with the PPT states are introduced. Op-timality and nd-optimality for canonical EWs are discussed and also the positive maps corre-sponding to canonical EWs are obtained. In section 3, the canonical EWs corresponding topossible partition of J and also the related PPT states are discussed. Section 4 is devoted todescribe the canonical EWs and corresponding PPT states for d ⊗ d systems. Also in thissection, the idea of canonical EW is extended to obtain other EWs. In the end, we summarizeour result and present our conclusions. d ⊗ d Systems
In this section, we are going to describe a method for constructing EWs. Let us consider H as a d-dimensional Hilbert space devoted to a single particle subsystem. U is defined as a realskew-symmetric operator (U T = − U) which acts on the H . U in matrix form isU = d − X i ≤ j a ij ( | i ih j | − | j ih i | ) i, j = 0 , ..., d − . (2.1) ntanglement Witnesses | α i ∈ H and for every real skew-symmetric matrix U thefollowing relation is satisfied h α ∗ | U | α i = 0 , (2.2)i.e., | α ∗ i and U | α i are orthogonal to each other. Now let us introduce the following Hermitianoperator W = I d ⊗ I d − d | ψ ih ψ | − d (U T ⊗ I d ) | ψ ih ψ | T A (U ⊗ I d ) , (2.3)where I d ⊗ I d is the identity operator, T is transposition and | ψ i , as defined below, is the d ⊗ d maximal entangled Bell-state | ψ i = 1 √ d d − X i =0 | ii i , i = 0 , ..., d − . (2.4)W has the following expectation value with an arbitrary product state | η i ⊗ | ζ ih η | ⊗ h ζ | W | η i ⊗ | ζ i = 1 − |h ζ | η ∗ i| − |h ζ | U | η i| , (2.5)in which | η ∗ i and U | η i are orthogonal to each other. It is assumed h η | U T U | η i ≤
1. If U | η i is normalized then | ζ i , in general, can be written as | ζ i = α | η ∗ i + β U | η i with | α | + | β | = 1otherwise it can not. Consequently, in both cases, the expectation values of the W with respectto all separable states are positive, so the Hermitian operator W can be considered as an EW.We will show that this type of EWs have ability to detect entangled PPT states, so they arend-EWs. It should be noted that, if we omit the last part of the EW in (2.3), the remainderpart is the reduction EW [10]; in other words,W = W red − d (U T ⊗ I d ) | ψ ih ψ | T A (U ⊗ I d ) , (2.6)where W red = I d ⊗ I d − d | ψ ih ψ | . (2.7) ntanglement Witnesses In this subsection, we introduce canonical EW and investigate how other EWs are obtainedfrom it. To illustrate, we know that, from the theory of matrices [22], every real skew-symmetricmatrix on the d-dimensional Hilbert space H can be decomposed asU = QJQ T , (2.8)where Q is an orthogonal matrix, i.e. QQ T = Q T Q = I d , and J is a block-diagonal one whichis called the canonical form of U J = j ⊕ j ⊕ j ⊕ ... ⊕ j n − , (2.9)where j i = λ i − λ i . (2.10)The scalars λ , λ , · · · , λ n − , which appear in the J, are invariant factors of U (they are invariantunder orthogonal transformations). It is clear that the rank of every real skew-symmetricmatrix U is always an even number (2 n ); therefore, if the matrix U is full-rank, then 2 n = d and if d is an odd number, U can not be full-rank anywise. Also it is clear that the eigenvaluesof U are complete imaginary or zero, but, if U is full-rank, then all of its eigenvalues will beimaginary. In addition, the condition h η | U T U | η i ≤ T = J T J ≤ I d which leads to acondition on λ i s, i.e. λ i ≤ i = 0 , · · · , n −
1. Now we are ready to introduce canonicalEW. Consider the following Hermitian operatorW C = I d ⊗ I d − d | ψ ih ψ | − d (J T ⊗ I d ) | ψ ih ψ | T A (J ⊗ I d ) (2.11)By the same prescription proposed for W in equation (2.3) to be as an EW, W C can also be anEW. It is based on two characteristic elements: J and | ψ ih ψ | . If we do some transformationson these operators, we obtain other EWs. To further illustrate, let | ψ ih ψ | be transformed to ntanglement Witnesses T ⊗ I d ) | ψ ih ψ | (Q ⊗ I d ) then W C becomes asW ψ = I d ⊗ I d − d (Q T ⊗ I d ) | ψ ih ψ | (Q ⊗ I d ) − d (J T Q T ⊗ I d ) | ψ ih ψ | T A (QJ ⊗ I d ) (2.12)where Q can be any orthogonal matrix corresponding to any orthogonal transformation in thed-dimensional Hilbert space. The subscript | ψ i denotes the class of operators obtained by anylocal orthogonal transformation Q ⊗ I d which is preformed on | ψ ih ψ | . Now the calculation ofthe expectation values of W ψ over all separable states yields the following equation h η | ⊗ h ζ | W ψ | η i ⊗ | ζ i = 1 − |h ζ | Q | η ∗ i| − |h ζ | QJ | η i| . (2.13)By the same argument sketched in equation (2.5), the above mentioned expectation values arealways positive. Therefore, the class of operators, W ψ , can be considered as EWs. The otherclass is obtained by transforming J in to QJQ T asW J = I d ⊗ I d − d | ψ ih ψ | − d (QJ T Q T ⊗ I d ) | ψ ih ψ | T A (QJQ T ⊗ I d ) , (2.14)where the subscript J denotes the class of all operators obtained by any orthogonal transfor-mation Q which is done on J. By the equation (2.8), the operator in (2.14) is the EW denotedin (2.3). On the other hand, we can see that the class of EWs W ψ and W J are related to eachother by local orthogonal transformation (Q ⊗ I d ), i.e.W ψ = (Q T ⊗ I d )W J (Q ⊗ I d ) , (2.15)therefore they are locally equivalent. These results strongly motivate us to go to study canon-ical EW, W C , in detail. Hence the rest of the paper will be devoted to describe the propertiesof the canonical EW. To this end, we see that the action of J on the basis states of thed-dimensional single party Hilbert space H is asJ | i i = −| i + 1 i , J | i + 1 i = | i i , i = 0 , ..., n − , (2.16)and J | i i = 0 , i = 2 n, ..., d − . (2.17) ntanglement Witnesses C is obtained inexpanded form asW C = n − X i =0 (1 − λ i )( | i, i + 1 ih i, i + 1 | + | i + 1 , i ih i + 1 , i |−| i, i ih i + 1 , i + 1 | − | i + 1 , i + 1 ih i, i | )+ n − X i = j =0 ( | i, j ih i, j | + | i, j + 1 ih i, j + 1 | + | i + 1 , j ih i + 1 , j | + | i + 1 , j + 1 ih i + 1 , j + 1 | − | i, i ih j, j | − | i, i ih j + 1 , j + 1 |−| i + 1 , i + 1 ih j, j | − | i + 1 , i + 1 ih j + 1 , j + 1 |− λ i λ j ( | i + 1 , j ih j + 1 , i | − | i + 1 , j + 1 ih j, i |−| i, j ih j + 1 , i + 1 | + | i, j + 1 ih j, i + 1 | ))+ n − X i =0 d − X j =2 n ( | i, j ih i, j | + | i + 1 , j ih i + 1 , j | + | j, i ih j, i | + | j, i + 1 ih j, i + 1 |−| i, i ih j, j | − | i + 1 , i + 1 ih j, j | − | j, j ih i, i | − | j, j ih i + 1 , i + 1 | )+ d − X i,j =2 n ( | i, j ih i, j | − | i, i ih j, j | ) . (2.18)Consider the following operatorO T A = n − X i =0 (1 − λ i )( | i, i + 1 ih i, i + 1 | + | i + 1 , i ih i + 1 , i |− | i, i ih i + 1 , i + 1 | − | i + 1 , i + 1 ih i, i | ) . (2.19)The W C is composed of the O t A and the remainder one which is called W C ( λ , λ , ..., λ n − ),i.e. W C = O T A + W C ( λ , λ , ..., λ n − ) . (2.20)We see that the operator O is positive; therefore, from [24] for every PPT state ρ , Tr( O t A ρ ) ≥ C ρ ) ≥ Tr(W C ( λ , λ , ..., λ n − ) ρ ) . (2.21) ntanglement Witnesses λ i = 1 for i = 0 , ..., n − T A becomes zero and W C =W C (1 , , ..., C for λ i = 1 ( i = 0 , ..., n −
1) andrewriting W C asW C = n − X i = j =0 ( | i, j ih i, j | + | i, j + 1 ih i, j + 1 | + | i + 1 , j ih i + 1 , j | + | i + 1 , j + 1 ih i + 1 , j + 1 | − | i, i ih j, j | − | i, i ih j + 1 , j + 1 |−| i + 1 , i + 1 ih j, j | − | i + 1 , i + 1 ih j + 1 , j + 1 |−| i + 1 , j ih j + 1 , i | + | i + 1 , j + 1 ih j, i | + | i, j ih j + 1 , i + 1 | − | i, j + 1 ih j, i + 1 | )+ n − X i =0 d − X j =2 n ( | i, j ih i, j | + | i + 1 , j ih i + 1 , j | + | j, i ih j, i | + | j, i + 1 ih j, i + 1 |−| i, i ih j, j | − | i + 1 , i + 1 ih j, j | − | j, j ih i, i | − | j, j ih i + 1 , i + 1 | )+ d − X i,j =2 n ( | i, j ih i, j | − | i, i ih j, j | ) . (2.22)To disambiguate, W C can be briefly written asW C = O T A + O T A + W OP C , (2.23)whereO T A = n − X i =0 d − X j =2 n ( | i, j ih i, j | + | i + 1 , j ih i + 1 , j | + | j, i ih j, i | + | j, i + 1 ih j, i + 1 |− | i, i ih j, j | − | i + 1 , i + 1 ih j, j | − | j, j ih i, i | − | j, j ih i + 1 , i + 1 | ) , (2.24)O T A = d − X i,j =2 n ( | i, j ih i, j | − | i, i ih j, j | ) (2.25)and W OP C = n − X i = j =0 ( | i, j ih i, j | + | i, j + 1 ih i, j + 1 | + | i + 1 , j ih i + 1 , j | + | i + 1 , j + 1 ih i + 1 , j + 1 | − | i, i ih j, j | − | i, i ih j + 1 , j + 1 | ntanglement Witnesses −| i + 1 , i + 1 ih j, j | − | i + 1 , i + 1 ih j + 1 , j + 1 |−| i + 1 , j ih j + 1 , i | + | i + 1 , j + 1 ih j, i | + | i, j ih j + 1 , i + 1 | − | i, j + 1 ih j, i + 1 | ) . (2.26)Note that if the rank of J (2 n ) is two then W OP C will be zero and therefore by consideringthe equation (2.23), it is concluded that W C is a d-EW. Since we are interested in dealingwith nd-EWs then those d ⊗ d quantum systems for which rank ( J ) ≥ d ≥
4. Clearly if J is full-rank (2 n = d ) then in equation (2.23), O T A and O T A will be zero.Now we claim that the W C type witnesses are able to detect entangled PPT states so they arend-EWs. This subsection is devoted to construct PPT states and determine a subset of them as a set ofentangled PPT states whose entanglement are detected by the EWs introduced in the previoussubsection. Let us write the following operator ρ = 1 N ( a | ψ ih ψ | + a n − X i =0 ( | i, i ih i, i | + | i + 1 , i + 1 ih i + 1 , i + 1 |−| i, i ih i + 1 , i + 1 | − | i + 1 , i + 1 ih i, i | )+ n − X i =0 ( a i +2 , i | i + 2 , i ih i + 2 , i | + a i +1 , i +3 | i + 1 , i + 3 ih i + 1 , i + 3 |− C i ( | i + 2 , i ih i + 1 , i + 3 | + | i + 1 , i + 3 ih i + 2 , i | ))+ n − X i = j =0 ,i − j =1 ,j − i = n − a i, j | i, j ih i, j | + n − X i = j =0 ,j − i =1 ,i − j = n − a i +1 , j +1 | i +1 , j +1 ih i +1 , j +1 | + n − X i,j =0 ( a i, j +1 | i, j + 1 ih i, j + 1 | + a i +1 , j | i + 1 , j ih i + 1 , j | )+ d − X i =2 n n − X j =0 ( a j,i | j, i ih j, i | + a j +1 ,i | j +1 , i ih j +1 , i | + a i, j | i, j ih i, j | + a i, j +1 | i, j +1 ih i, j +1 | ) ntanglement Witnesses d − X i = j =2 n a ij | i, j ih i, j | ) , (2.27)where N is the normalization factor and equals to N = ( d + 2 n ) a + n − X i = j =0 ( a i, j + a i +1 , j +1 ) + n − X i,j =0 ( a i, j +1 + a i +1 , j )+ d − X i =2 n n − X j =0 ( a j,i + a j +1 ,i + a i, j + a i, j +1 ) + d − X i = j =2 n a i,j . (2.28)The positivity conditions impose that all of the multipliers which appear in the ρ are positivesemi definite and in addition the following inequality must be satisfied a i +1 , i +3 a i +2 , i ≥ C i , i = 0 , ..., n − , (2.29)where addition in the subscripts is done by module (2n). Also the PPT conditions are asfollows: a i, j a j, i ≥ a , i, j = 0 , ..., n − , i = j,a i +1 , j +1 a j +1 , i +1 ≥ a , i, j = 0 , ..., n − , i = j,a i, j +1 a j +1 , i ≥ a , i, j = 0 , ..., n − , i = j,a i +1 , i a i +2 , i +3 ≥ C i , i = 0 , ..., n − ,a j,i a i, j ≥ a , j = 0 , ..., n − , i = 2 n, ..., d − ,a j +1 ,i a i, j +1 ≥ a , j = 0 , ..., n − , i = 2 n, ..., d − ,a i,j a j,i ≥ a , i, j = 2 n, ..., d − , i = j. (2.30)Therefore, by these two groups of inequalities, the operator ρ becomes as a density operatorwith positive partial transpose. In the next step, it is shown that the expectation value ofthe witness W C with respect to the ρ , under positivity and PPT conditions, really fulfills thefollowing inequality (the proving of the following Lower bound is given in the appendix A.)Tr(W C ρ ) ≥ − nd + 4 n + ( d − n )( d + 2 n − , n = 2 , , , · · · . (2.31) ntanglement Witnesses ρ satisfies the following inequality − nd + 4 n + ( d − n )( d + 2 n − ≤ Tr(W C ρ ) < , n = 2 , , , · · · , (2.32)then it will be an entangled PPT state whose entanglement is detected by W C . On the otherhand, the witness W C which can detect the entangled PPT state ρ , becomes as a nd-EW whichproves our claim. It is also seen that the lower bound of the inequality in (2.27) depends onthe rank of the matrix J and the dimension of the Hilbert space of the single party subsystem.It is also obvious that if J is full-rank, then the lower bound becomes smaller. Finally; whenthe state ρ violate the PPT conditions, the expectation value of the entanglement witness W C with respect to the density operator ρ satisfies the following inequality − n ( n −
1) + ( d − n )( d + 2 n − d + 2 n ≤ Tr(W C ρ ) < − nd + 4 n + ( d − n )( d + 2 n − ,n = 2 , , , · · · . (2.33).Clearly the lower bound becomes greater when the matrix J is full-rank. Now we discuss at first the optimality of canonical EWs by investigating optimality of canoni-cal EW when J is full-rank (2n=d). The proving of optimality for the others (2 n < d ) is similarto this one. Secondly, we describe nd-optimality of canonical EWs. There exist different defi-nitions of optimal entanglement witness. Our description is based on the definition introducedby Lewenstein et.al., [24]. One has two EWs W and W , W is finer than W if they differ bya positive operator P. We say that W is optimal iff for all P and ǫ >
0, W ′ = (1 + ǫ )W − ǫ P isnot an EW. P is positive operator and PP W = 0 where P W = {| γ i , Tr(W | γ ih γ | = 0) } in which | γ i is separable state. Since any positive operator can be written as a convex combination ofpure product states so let us assume that P = | ψ ih ψ | in which | ψ i = d − X i,j =0 a i,j | i, j i . (2.34) ntanglement Witnesses | γ i ∈ P W C has the following form | γ i = | η i ⊗ ( α | η ∗ i + βJ | η i ) , (2.35)where | α | + | β | = 1. We define | η i = P n − i =0 ( η i | i i + η i +1 | i + 1 i ) such that P n − i =0 ( | η i | + | η i +1 | ) = 1. For simplicity we choose α = 1 and β = 0 ( | γ i = | η i ⊗ | η ∗ i ). Therefore, theseparable state | γ i in expanded form is written as | γ i = n − X i,j =0 η i η ∗ j | i, j i + η i η ∗ j +1 | i, j + 1 i + η i +1 η ∗ j | i + 1 , j i + η i +1 η ∗ j +1 | i + 1 , j + 1 i . (2.36)It is proven P=0 as: . η i = δ ik which gives the following separable state | γ i = | k, k i −→ h γ | ψ i = a k, k = 0 . (2.37). . η i +1 = δ ik | γ i = | k + 1 , k + 1 i −→ h γ | ψ i = a k +1 , k +1 = 0 . (2.38) . η k = α ′ , η k +1 = β ′ , | α ′ | + | β ′ | = 1, | γ i = | α ′ | | k, k i + α ′ β ′∗ | k, k + 1 i + β ′ α ′∗ | k + 1 , k i + | β ′ | | k + 1 , k + 1 i , , h γ | ψ i = β ′ α ′∗ a k +1 , k + α ′ β ′∗ a k, k +1 = 0 −→ a k, k +1 = 0 , a k +1 , k = 0 . (2.39) . η k = α ′ , η l = β ′ , k < l , | α ′ | + | β ′ | = 1 , | γ i = | α ′ | | k, k i + α ′ β ′∗ | k, l i + β ′ α ′∗ | l, k i + | β ′ | | l, l i , ntanglement Witnesses h γ | ψ i = α ′ β ′∗ a k, l + β ′ α ∗ a l, k = 0 −→ a k, l = 0 , a l, k = 0 . (2.40) . η k +1 = α ′ , η l +1 = β ′ , k < l , | α ′ | + | β ′ | = 1 , | γ i = | α ′ | | k + 1 , k + 1 i + α ′ β ′∗ | k + 1 , l + 1 i + β ′ α ′∗ | l + 1 , k + 1 i + | β ′ | | l + 1 , l + 1 i , h γ | ψ i = α ′ β ′∗ a k +1 , l +1 + β ′ α ′∗ a l +1 , k +1 = 0 −→ a k +1 , l +1 = 0 , a l +1 , k +1 = 0 . (2.41) . η k = α ′ , η l +1 = β ′ , k < l , | α ′ | + | β ′ | = 1 | γ i = | α ′ | | k, k i + α ′ β ′∗ | k, l + 1 i + β ′ α ′∗ | l + 1 , k i + | β ′ | | l + 1 , l + 1 i , h γ | ψ i = α ′ β ′∗ a k, l +1 + β ′ α ′∗ a l +1 , k = 0 −→ a k, l +1 = 0 , a l +1 , k = 0 . (2.42) . η k +1 = α ′ , η l = β ′ , k < l , | α ′ | + | β ′ | = 1 , | γ i = | β ′ | | l, l i + β ′ α ′∗ | l, k + 1 i + α ′ β ′∗ | k + 1 , l i + | α ′ | | k + 1 , k + 1 i , h γ | ψ i = α ′ β ′∗ a k +1 , l + β ′ α ′∗ a l, k +1 = 0 −→ a l, k +1 = 0 , a k +1 , l = 0 . (2.43)These equations explicitly show that P=0 and therefore the canonical EW is optimal. Theproving of optimality for W C , when J is not full-rank (and specially is zero), is similar tothe previous one except by noting that the typical separable state | γ i ∈ P C has the form | γ i = | η i ⊗ | η ∗ i where | η i = P n − i =0 ( η i | i i + η i +1 | i + 1 i ) + P d − i =2 n η i | i i . Therefore, it isconcluded that the canonical EW (2.11) is optimal for all ranks of J ( rank (J) = 0 , ..., n = d ),in other words, its optimality is independent from the rank of J. On the other hand, to discussnd-optimality, we define d W C = { ρ ≥ | ρ T A ≥ , Tr(W c ρ ) < } , i.e., the set of entangled ntanglement Witnesses C . Given two nd-EWs W C and W C , we say that W C is nd-finerthan W C , if d W C ⊆ d W C , i.e., if all of the entangled PPT states detected by W C are alsodetected by W C . We say that W C is optimal nd-EW, if there exists no other nd-EW which isnd-finer than it. So by keeping this in mind, we determine optimal nd-EW among canonicalEWs. By referring to equation (2.23), we see that the operators O and O are positive, soO T A and O T A are optimal d-EWs [24] and W OP C is a canonical EW on the 2 n ⊗ n Hilbertspace which is a subspace of d ⊗ d one (the proving that W OP C is an EW, is given in appendixB). We know that for any PPT state the following inequality is satisfiedTr(W C ρ ) ≥ Tr(W
OP C ρ ) . (2.44)From the Lewenstein definition of an optimal nd-EW, it is clear that each entangled PPT statedetected by W C , is also detected by W OP C . It is proven that W
OP C is an optimal nd-EW.To this aim we say that a nd-EW, W, is optimal nd-EW iff for all decomposable operator D(D = P + Q t A where P and Q are positive operators) and ǫ >
0, W ′ = (1 + ǫ )W − ǫ D is not anEW. In the proving of optimality for canonical EWs, It was shown that P = 0. To illustratethat W
OP C is an optimal nd-EW, it must be shown that for Q T A P W OPC = 0 then Q T A = 0. Letus assume that Q = | ϕ ih ϕ | in which | ϕ i = P d − i,j =0 a i,j | i, j i . As previously, since J is full-rank on2 n -dimensional subspace of the d-dimensional one party Hilbert space H , a typical separablestate | γ i ∈ P W OPC which lies in the 2 n ⊗ n subspace (see appendix B), is written as | ϑ i = | η i ⊗ ( α | η ∗ i + β J | η i ). On the other hand, since Tr(Q t A | α ih α | ⊗ | β ih β | ) = Tr(Q | α ∗ ih α ∗ | ⊗ | β ih β | )then we calculate the expectation values of Q with the products | ϑ i = | η ∗ i ⊗ ( α | η ∗ i + β J | η i ). | ϑ i in expanded form is | ϑ i = n − X i,j =0 η ∗ i ( αη ∗ j + βη j +1 ) | i, j i + η ∗ i ( αη ∗ j +1 − βη j ) | i, j + 1 i + η ∗ i +1 ( αη ∗ j + βη j +1 ) | i + 1 , j i + η ∗ i +1 ( αη ∗ j +1 − βη j ) | i + 1 , j + 1 i . (2.45) ntanglement Witnesses n ⊗ n as below . η i = δ ik , α = 1 , β = 0 which gives the following separable state | ϑ i = | k, k i −→ h ϑ | ϕ i = a k, k = 0 . (2.46) . η i = δ ik , α = 0 , β = 1 | ϑ i = | k, k + 1 i −→ h ϑ | ϕ i = a k, k +1 = 0 . (2.47) . η i +1 = δ ik , α = 0 , β = 1 | ϑ i = | k + 1 , k i −→ h ϑ | ϕ i = a k +1 , k = 0 . (2.48) . η i +1 = δ ik , α = 1 , β = 0 | ϑ i = | k + 1 , k + 1 i −→ h ϑ | ϕ i = a k +1 , k +1 = 0 . (2.49) . η k = α ′ , η l = β ′ , | α ′ | + | β ′ | = 1 , k < l , α = 0 , β = 1 | ϑ i = | α ′ | | k, k + 1 i + α ′ β ′∗ | l, k + 1 i + β ′ α ′∗ | k, l + 1 i + | β ′ | | l, l + 1 i , h ϑ | ϕ i = α ′ β ′∗ a l, k +1 + β ′ α ′∗ a k, l +1 = 0 −→ a l, k +1 = 0 , a k, l +1 = 0 . (2.50) . η k +1 = α ′ , η l +1 = β ′ , | α ′ | + | β ′ | = 1 , k < l , α = 0 , β = 1 | ϑ i = | α ′ | | k + 1 , k i + α ′ β ′∗ | l + 1 , k i + β ′ α ′∗ | k + 1 , l i + | β ′ | | l + 1 , l i , h ϑ | ϕ i = α ′ β ′∗ a l +1 , k + β ′ α ′∗ a k +1 , l = 0 −→ a l +1 , k = 0 , a k +1 , l = 0 . (2.51) . η k = α ′ , η l +1 = β ′ , | α ′ | + | β ′ | = 1 , k < l , α = 0 , β = 1 | ϑ i = −| α ′ | | k, k + 1 i − α ′ β ′∗ | l + 1 , k + 1 i + β ′ α ′∗ | k, l i + | β ′ | | l + 1 , l i , ntanglement Witnesses h ϑ | ϕ i = − α ′ β ′∗ a l +1 , k + β ′ α ′∗ a k, l = 0 −→ a l +1 , k +1 = 0 . a k, l = 0 . (2.52) . η k +1 = α ′ , η l = β ′ , | α ′ | + | β ′ | = 1 , k < l , α = 0 , β = 1 | ϑ i = | α ′ | | k + 1 , k i + α ′ β ′∗ | l, k i − β ′ α ′∗ | k + 1 , l + 1 i − | β ′ | | l, l + 1 i , h ϑ | ϕ i = α ′ β ′∗ a l, k − β ′ α ′∗ a k +1 , l +1 = 0 −→ a k +1 , l +1 = 0 , a l, k = 0 . (2.53)therefore, Q = 0 on the 2 n ⊗ n subspace. From the other side, since all of the separable stateswhich lie on the complement subspace of the 2 n ⊗ n one belong to the P W OPC (see appendixB), then it is obvious that Q is also zero on that subspace. Consequently Q = 0 on d ⊗ d Hilbert space hence W
OP C is optimal nd-EW. Clearly, if J is full-rank, then O T A and O T A willbe zero so W C = W OP C and W
OP C is an optimal nd-EW on the d ⊗ d Hilbert space. It isconcluded that when J is full-rank, W C is optimal nd-EW and when J is not full-rank W C is not optimal nd-EW; therefore, despite the optimality, the nd-optimality of canonical EWsdepends on the rank of J. Since the nd-EWs have an essential role in the studying of separability problem in quan-tum theory, by using Jamiolkowski isomorphism [23] between operators and maps, the non-decomposable positive maps (or nd-positive maps) have the same role as nd-EWs. By this iso-morphism, one can obtain the corresponding positive map of the canonical EW W C ∈ H d ⊗ H d (2.11) as discussed in subsection (2.1). Consider the following equation φ ( ρ ) = Tr B (W C (I d ⊗ ρ T )) . (2.54)Where ρ is a density operator on the d-dimensional Hilbert space. This equation shows how toconstruct the map φ from a given operator W C . After some calculations, the following result ntanglement Witnesses φ ( ρ ) = I d Tr( ρ ) − ρ − J T ρ T J . (2.55)From the properties of W C discussed earlier, we expect that, if the rank ( J ) <
4, then the φ ( ρ )is decomposable positive map (or d-positive map), especially when rank ( J ) = 0, φ ( ρ ) is thewell-known reduction map [25]. Therefore, for rank ( J ) ≥ φ ( ρ ) is nd-positive map. In this section, by referring to each possible partition of J, we are going to construct a newset of canonical EWs. Therefore, for a given J, we have a set of canonical EWs correspondingto the set of possible partitions of J. It is shown that, for a given partition of J, a PPTstate is constructed for that partition. The entanglement of this PPT state in some range ofparameters is detected by the corresponding canonical EW established in the same partition.Suppose that J is full-rank, i.e., d = 2 n . Consider a partition of d-dimensional single partyHilbert space H to its 2 µ i -dimensional subspaces, H µ i s, through the following direct sum H = H µ ⊕ H µ ⊕ H µ ⊕ · · · ⊕ H µ ν . (3.56)Also consider the following Hermitian operatorW C = I d ⊗ I d − d | ψ ih ψ | − d (U T ⊗ I d ) | ψ ih ψ | T A (U ⊗ I d ) − d (U T ⊗ I d ) | ψ ih ψ | T A (U ⊗ I d ) − d (U T ⊗ I d ) | ψ ih ψ | T A (U ⊗ I d ) − · · · − d (U Tν ⊗ I d ) | ψ ih ψ | T A (U ν ⊗ I d ) , (3.57)in which every U i is block-diagonal full-rank matrix on H µ i ( k = 1 , · · · , ν ) such thatJ = U + U + U + · · · + U ν , (3.58) ntanglement Witnesses i U j = 0 ( i = j = 1 , ..., ν ). Each U i is a full-rank canonical skew-symmetric matrix in H µ i . Therefore the rank of J is the sum of the ranks of U i s that is2 n = 2 µ + 2 µ + 2 µ + · · · + 2 µ ν (3.59)hence we obtain the next result n = µ + µ + µ + · · · + µ ν . (3.60)It is well-known that the numbers ( µ , µ , µ , · · · , µ ν ) are a partition of n . Generally from [26],for a given number n there are p(n) number of partitions ( µ , µ , µ , · · · , µ ν ) with µ ≥ µ ≥ µ ≥ , · · · , ≥ µ ν (for example consider 5, the number of partitions for it, is p (5) = 7). Hencewe say that the U k s in (3.58) are a partition of J. Therefore for the other possible partitionsof n , we have corresponding partitions for J and corresponding Hermitian operators such asW C which is renamed as W C ( µ , µ , µ , · · · , µ ν ).In the same way, as described in section 2, the expectation values of Hermitian operatorW C with all product states are given as h η | ⊗ h ζ | W C ( µ , µ , · · · , µ ν − , µ ν ) | η i ⊗ | ζ i = 1 − |h ζ | η ∗ i| − |h ζ | U | η i| − |h ζ | U | η i| − |h ζ | U | η i| − · · · − |h ζ | U ν | η i| , (3.61)in which the states {| η ∗ i , U | η i , U | η i , U | η i , · · · , U ν | η i} are orthogonal to each other.If U i | η i , for each i, is normalized then | ζ i can be written as | ζ i = α | η ∗ i + P νi =0 β i U i | η i with | α | + P νi =1 | β i | = 1 otherwise it can not. Therefore W C ( µ , µ , µ , · · · , µ ν ) has positiveexpectation values with all separable states so it can be considered as a canonical EW for thepartition ( µ , µ , µ , · · · , µ ν ). par The action of U i ( i = 1 , ..., ν ) on the basis states of H µ i ( i = 1 , ..., ν ) is as U i | k i = −| k + 1 i , U i | k + 1 i = | k i ,i = 1 , ..., ν, k = 0 , ..., µ i − , (3.62) ntanglement Witnesses C ( µ , µ , µ , · · · , µ ν ) in equation (3.57) for a given partition ( µ , µ , µ , · · · , µ ν ) canbe obtained as W C ( µ , µ , µ , · · · , µ ν ) = W C ( µ , µ , µ , · · · , µ ν )+ O t A ( µ ) + O t A ( µ ) + O t A ( µ ) + · · · + O t A ( µ ν − ) , (3.63)where W C ( µ , µ , µ , · · · , µ ν ) = W C ( µ ) ⊕ W C ( µ ) ⊕ W C ( µ ) ⊕ · · · ⊕ W C ( µ ν ) (3.64)The operators W C ( µ i ) ( i = 1 , · · · , ν ) and O t A ( µ i ) ( i = 1 , · · · , ν −
1) have been given in theappendix C. By the same prescription shown in section (2.3), the W C ( µ , µ , µ , · · · , µ ν ) isoptimal EW for each partition of n. Each W C ( µ i ) ( i = 1 , · · · , ν ) is an EW in the 2 µ i ⊗ µ i subspace of the d ⊗ d Hilbert space. Also for every µ i > C ( µ i ) is anoptimal nd-EW and hence W c ( µ , µ , µ , · · · , µ ν ) which is the direct some of the W C ( µ i )s, isalso an optimal nd-EW. The operators O( µ i ) ( i = 1 , ..., ν −
1) are positive operators then theoperators O t A ( µ i ) ( i = 1 , · · · , ν −
1) are optimal d-EWs.As mentioned in section (2.1), if a given µ i be equal to one then the corresponding W C ( µ i )will be zero. Therefore if all of µ i s become one, i.e. for the partition (1 , , , · · · , C (1 , , , · · · ,
1) will be an optimal d-EW. The witness corresponding to the partition ( n = µ ), i.e. W C ( µ ), is an optimal nd-EW which was discussed earlier. In the end, the witnesseswhich correspond to the other partitions between these two partitions , as the equation (3.63),are a mixture of optimal d-EWs and the optimal nd-EWs. Therefore, by considering theoptimality of the nd-EWs discussed in subsection (2.3), these witnesses are not optimal nd-EWs.In the next step, for a given partition ( µ , µ , µ , · · · µ ν ), a PPT state is introduced. Thisstate can be entangled and the signature of entanglement for it is shown by the witness whichcorresponds to the same partition discussed above. This state is the following one ρ ( µ , µ , µ , · · · , µ ν ) ntanglement Witnesses
22= 1 N ( µ , µ , µ , · · · , µ ν ) ( ̺ ( µ , µ , µ , · · · , µ ν ) + σ ( µ ) + σ ( µ ) + σ ( µ ) + · · · + σ ( µ ν − )) (3.65)such that ̺ ( µ , µ , µ , · · · , µ ν ) = ρ ( µ ) ⊕ ρ ( µ ) ⊕ ρ ( µ ) ⊕ · · · ⊕ ρ ( µ ν ) , (3.66)where N ( µ , µ , µ , · · · , µ ν ) is the norm of the ρ ( µ , µ , µ , · · · , µ ν ). The operators ρ ( µ i )s( i = 1 , · · · , ν ) are also (unnormalized) PPT states in the subspace 2 µ i ⊗ µ i and by thewitnesses W C ( µ i )s, they are entangled PPT states. Also the operator ̺ ( µ , µ , µ , · · · , µ ν )which is the direct sum of the ρ ( µ i )s, is PPT state so its entanglement is detected by thewitness W c ( µ , µ , µ , · · · , µ ν ) and finally σ ( µ i )s ( i = 1 , · · · , ν −
1) are Hermitian operators.All these operators together with the positivity and PPT conditions for ρ ( µ , µ , µ , · · · , µ ν )have been given in appendix D.Now at the end of this section the expectation value of W C ( µ , µ , µ , · · · , µ ν ) with respectto the PPT state ρ ( µ , µ , µ , · · · , µ ν ), by considering both positivity and PPT conditions,is calculated. In the same way as in subsection (2.2), after some calculations we obtain thefollowing lower bound Tr[W C ( µ , µ , µ , · · · , µ ν ) ρ ( µ , µ , µ , · · · , µ ν )] ≥ − d P νǫ =1 (2 µ ǫ (2 µ ǫ + 1) + 4 µ ǫ ( d − P ǫθ =1 µ θ )) . (3.67)Therefore we say that if a PPT state ρ ( µ , µ , µ , · · · , µ ν ) satisfy the following inequality − d P νǫ =1 (2 µ ǫ (2 µ ǫ + 1) + 4 µ ǫ ( d − P ǫθ =1 µ θ )) ≤ Tr[W C ( µ , µ , µ , · · · , µ ν ) ρ ( µ , µ , µ , · · · , µ ν )] < , (3.68)then it is an entangled PPT state detected by the witness W C ( µ , µ , µ , · · · , µ ν ). Finally if ρ ( µ , µ , µ , · · · , µ ν ) violate the PPT condition, it always satisfies the next inequality − P νǫ =1 (2 µ ǫ ( µ ǫ −
1) + 2 µ ǫ ( d − P ǫθ =1 µ θ )) d ≤ Tr[W C ( µ , µ , µ , · · · , µ ν ) ρ ( µ , µ , µ , · · · , µ ν )] < − d P νǫ =1 (2 µ ǫ (2 µ ǫ + 1) + 4 µ ǫ ( d − P ǫθ =1 µ θ )) . (3.69) ntanglement Witnesses In this section, at first, we extend our approach used for d ⊗ d cases in the previous sections toobtain EWs in d ⊗ d quantum systems ( d < d ). Secondly, we extend the idea of canonicalEW introduced in subsection (2.1) to construct other EWs. Let us introduce a projectionoperator P c on the d -dimensional party in which Im(P c ) is a d -dimensional Hilbert space.Since in the d -dimensional vector space one can construct d -dimensional subspace by C d d (the number of combinations of d distinct objects taken d at a time without repetitions)distinct ways; therefore, the subscript c ( c = 1 , · · · , C d d ) dentes a projection operator amongthe set of C d d ones . From the linear algebra, we know that H d = KerP c ⊕ ImP c . Thereforefor every | α i ∈ H d , we have | α i = | α c i + | α c ′ i where | α c i ∈ ImP c and | α c ′ i ∈ KerP c withdim(ImP c )= d and dim(KerP c )= d − d . Now we introduce the following Hermitian operatorW = I d ⊗ I c − d | ψ c ih ψ c | − d (U Td ⊗ I c ) | ψ c ih ψ c | T A (U d ⊗ I c ) (4.70)where I c = P d − i =0 | i c ih i c | is the identity operator in the projected subspace of H d correspondingto a combination denoted by c . So the identity operator in the corresponding d ⊗ d subspaceof d ⊗ d system is given as I d ⊗ I c = d − X i,j =0 | i, j c ih i, j c | (4.71)and | ψ c i is the the maximal entangled Bell-state in that subspace | ψ c i = 1 √ d d − X i =0 | i, i c i , (4.72)in which | i c i ∈ ImP c for i = 0 , · · · , d −
1. It should be noted that the skew-symmetricoperator U d is defined on the d -dimensional party of the d ⊗ d system. To clarify, we givean example in 4 ⊗
5. Sine C = 5 then we have five 4-dimensional projected subspace forH . The first one spanned by {| i = | i , | i = | i , | i = | i , | i = | i} , the second one by {| i = | i , | i = | i , | i = | i , | i = | i} , the third one by {| i = | i , | i = | i , | i = | i , | i = | i} , the fourth one by {| i = | i , | i = | i , | i = | i , | i = | i} and the fifth ntanglement Witnesses {| i = | i , | i = | i , | i = | i , | i = | i} . Therefore one has five ways to constructthe operator (4.70). It is clear that the operator (4.70) is the same operator (2.3) in d ⊗ d bipartite quantum system which has been embedded in d ⊗ d Hilbert space. Consequently,the number of such operators obtained by embedding in this way is, in fact, C d d .By the same arguments proposed in section 2, the operator (4.70) can be considered as anentanglement witness which is defined on the d ⊗ d Hilbert space. Using J d instead of U d in (4.70), we obtain a canonical form for W asW C = I d ⊗ I c − d | ψ c ih ψ c | − d (J Td ⊗ I c ) | ψ c ih ψ c | T A (J d ⊗ I c ) , (4.73)where Q d is an orthogonal matrix (Q d Q T d1 = Q T d1 Q d = I d ) and J d is the canonical formof U d i.e., U d = Q d J d Q T d . The action of J d (whose rank is 2 n ) on the basis of the d -dimensional single party Hilbert space is similar to (2.16) and (2.17). Consequently, after somecalculations, the expanded form of W C becomesW C = n − X i = j =0 ( | i, (2 j ) c ih i, (2 j ) c | + | i, (2 j + 1) c ih i, (2 j + 1) c | + | i + 1 , (2 j ) c ih i + 1 , (2 j ) c | + | i + 1 , (2 j + 1) c ih i + 1 , (2 j + 1) c | − | i, (2 i ) c ih j, (2 j ) c | − | i, (2 i ) c ih j + 1 , (2 j + 1) c |−| i + 1 , (2 i + 1) c ih j, (2 j ) c | − | i + 1 , (2 i + 1) c ih j + 1 , (2 j + 1) c |−| i + 1 , (2 j ) c ih j + 1 , (2 i ) c | + | i + 1 , (2 j + 1) c ih j, (2 i ) c | + | i, (2 j ) c ih j + 1 , (2 i + 1) c | − | i, (2 j + 1) c ih j, (2 i + 1) c | )+ n − X i =0 d − X j =2 n ( | i, j c ih i, j c | + | i + 1 , j c ih i + 1 , j c | + | j, (2 i ) c ih j, (2 i ) c | + | j, (2 i + 1) c ih j, (2 i + 1) c |−| i, (2 i ) c ih j, j c | − | i + 1 , (2 i + 1) c ih j, j c | − | j, j c ih i, (2 i ) c | − | j, j c ih i + 1 , (2 i + 1) c | )+ d − X i,j =2 n ( | i, j c ih i, j c | − | i, i c ih j, j c | ) . (4.74)These witnesses are optimal d-EWs for d <
4; otherwise, they are nd-EWs, so they candetect the entanglement of some PPT states. Optimality and nd-optimality problem for these ntanglement Witnesses ρ = 1 N ( a | ψ c ih ψ c | + a n − X i =0 ( | i, (2 i ) c ih i, (2 i ) c | + | i + 1 , (2 i + 1) c ih i + 1 , (2 i + 1) c |−| i, (2 i ) c ih i + 1 , (2 i + 1) c | − | i + 1 , (2 i + 1) c ih i, (2 i ) c | )+ n − X i =0 ( a i +2 , (2 i ) c | i + 2 , (2 i ) c ih i + 2 , (2 i ) c | + a i +1 , (2 i +3) c | i + 1 , (2 i + 3) c ih i + 1 , (2 i + 3) c |− C i ( | i + 2 , (2 i ) c ih i + 1 , (2 i + 3) c | + | i + 1 , (2 i + 3) c ih i + 2 , (2 i ) c | ))+ n − X i = j =0 ,i − j =1 ,j − i = n − a i, (2 j ) c | i, (2 j ) c ih i, (2 j ) c | + n − X i = j =0 ,j − i =1 ,i − j = n − a i +1 , (2 j +1) c | i + 1 , (2 j + 1) c ih i + 1 , (2 j + 1) c | + n − X i,j =0 ( a i, (2 j +1) c | i, (2 j + 1) c ih i, (2 j + 1) c | + a i +1 , (2 j ) c | i + 1 , (2 j ) c ih i + 1 , (2 j ) c | )+ d − X i =2 n n − X j =0 a j,i c ( | j, i c ih j, i c | + a j +1 ,i c | j + 1 , i c ih j + 1 , i c | + a i, (2 j ) c | i, (2 j ) c ih i, (2 j ) c | + a i, (2 j +1) c | i, (2 j + 1) c ih i, (2 j + 1) c | )+ d − X i = j =2 n a i,j c | i, j c ih i, j c | + d − X i =0 d − d − X j =0 a i,j c ′ | i, j c ′ ih i, j c ′ | , (4.75)it is clear that the number of such states is C d d . The positivity and PPT conditions are thesame as for d ⊗ d cases except that we have an additional condition for positivity through theinequalities a i,j c ′ ≥ i = 0 , · · · , d − j = 0 , · · · , d − d − c ′ denotesthose states which lie in the KerP c ( i = 0 , · · · , d − d − ρ gives out the following lower boundTr(W C ρ ) ≥ − n d + 4 n + ( d − n )( d + 2 n − , n = 2 , , , · · · . (4.76) ntanglement Witnesses − n d + 4 n + ( d − n )( d + 2 n − ≤ Tr(W C ρ ) < , n = 2 , , , · · · , (4.77)then they are entangled. And, as before, if the states ρ violate the PPT conditions, then theyalways fulfill the following inequality − n ( n −
1) + ( d − n )( d + 2 n − d + 2 n ≤ Tr(W C ρ ) < − n d + 4 n + ( d − n )( d + 2 n − ,n = 2 , , , · · · . (4.78)In the end of this paper, the idea of canonical EW introduced in equation (2.11) can beextended to construct other EWs. Let us , at first, assume that the single party Hilbert spaces H and H and the tensor product Hilbert space H ⊗ H are defined on real field. As anillustration to this restriction, any entangled state which lie in a real tensor product Hilbertspace H ⊗ H can be generated, by interactions or any entanglement generating process, fromsingle party states which lie in real single party Hilbert spaces H and H . It should be notedthat any LOCC (local operation and classical communication) must be restricted on the realfield. Therefore by these considerations, when we deal to detect the entanglement of a statewhich lies in a real tensor product Hilbert space H ⊗ H by constructing an EW, it is sufficientthat our EW should have positive expectation value with respect to all real separable states.Now consider the following operatorW = I d ⊗ I d − d | ψ ih ψ | − d (J T ⊗ I d ) | ψ ih ψ | T A (J ⊗ I d ) − d (J ′ T ⊗ I d ) | ψ ih ψ | T A (J ′ ⊗ I d ) − d (J ′′ T ⊗ I d ) | ψ ih ψ | T A (J ′′ ⊗ I d ) , (4.79)where J = j ⊕ j ⊕ j ⊕ ... J ′ = j ′ ⊕ j ′ ⊕ j ′ ⊕ ... J ′′ = j ′′ ⊕ j ′′ ⊕ j ′′ ⊕ ..., (4.80) ntanglement Witnesses
27, in which the J, J ′ and J ′′ are real skew-symmetric matrices in d-dimensional Hilbert space H with j = − − , j ′ = − − , j ′′ = − − . (4.81)Clearly the ranks of J, J ′ and J ′′ are 4 n , n = 1 , , , · · · with d = 4 n + m where 0 ≤ m ≤ ′ and J ′′ on the basis states of the d-dimensional single party subsystem areas J | k i = −| k + 1 i J | k + 1 i = | k i J | k + 2 i = −| k + 3 i J | k + 3 i = | k + 2 i , J ′ | k i = −| k + 3 i J ′ | k + 1 i = −| k + 2 i J ′ | k + 2 i = | k + 1 i J ′ | k + 3 i = | k i , J ′′ | k i = −| k + 2 i J ′′ | k + 1 i = | k + 3 i J ′′ | k + 2 i = | k i J ′′ | k + 3 i = −| k + 1 i . (4.82). The expectation values of the operator W with respect to the product states are as h η | ⊗ h ζ | W | η i ⊗ | ζ i = 1 − |h ζ | η ∗ i| − |h ζ | J | η i| − |h ζ | J ′ | η i| − |h ζ | J ′′ | η i| . (4.83)It is easy to see that the states | η ∗ i , J | η i , J ′ | η i and J ′′ | η i are orthogonal to each other when theyare belong to a real single particle Hilbert space. Therefore, the expectation values becomepositive with respect to all real separable states. The relation between the operator (2.11) and(4.79) is W C = W + D T A + D T A , (4.84)where D T A = d (J ′ T ⊗ I d ) | ψ ih ψ | T A (J ′ ⊗ I d ) and D T A = d (J ′′ T ⊗ I d ) | ψ ih ψ | T A (J ′′ ⊗ I d ). The operatorsD and D are positive operators so the PPT entanglement detection power of W is greaterthan W C . Consider, for example, the following state with d = 4 n which is defined in a realtensor product Hilbert space H ⊗ H (and keeping in mind that any LOCC must be restricted ntanglement Witnesses ρ = a d | ψ ih ψ | + a n − X i =0 [3( | i, i ih i, i | + | i + 1 , i + 1 ih i + 1 , i + 1 | + | i + 2 , i + 2 ih i + 2 , i + 2 | + | i + 3 , i + 3 ih i + 3 , i + 3 | ) −| i, i ih i + 1 , i + 1 | − | i, i ih i + 2 , i + 2 | − | i, i ih i + 3 , i + 3 |−| i + 1 , i + 1 ih i, i | − | i + 1 , i + 1 ih i + 2 , i + 2 | − | i + 1 , i + 1 ih i + 3 , i + 3 |−| i + 2 , i + 2 ih i, i | − | i + 2 , i + 2 ih i + 1 , i + 1 | − | i + 2 , i + 2 ih i + 3 , i + 3 |−| i + 3 , i + 3 ih i, i | − | i + 3 , i + 3 ih i + 1 , i + 1 | − | i + 3 , i + 3 ih i + 2 , i + 2 | ]+ n − X i = j =0 ( a i, j | i, j ih i, j | + a i, j +1 | i, j + 1 ih i, j + 1 | + a i, j +2 | i, j + 2 ih i, j + 2 | + a i, j +3 | i, j +3 ih i, j +3 | + a i +1 , j | i +1 , j ih i +1 , j | + a i +1 , j +1 | i +1 , j +1 ih i +1 , j +1 | + a i +1 , j +2 | i + 1 , j + 2 ih i + 1 , j + 2 | + a i +1 , j +3 | i + 1 , j + 3 ih i + 1 , j + 3 | + a i +2 , j | i + 2 , j ih i + 2 , j | + a i +2 , j +1 | i + 2 , j + 1 ih i + 2 , j + 1 | + a i +2 , j +2 | i + 2 , j + 2 ih i + 2 , j + 2 | + a i +2 , j +3 | i + 2 , j + 3 ih i + 2 , j + 3 | + a i +3 , j | i + 3 , j ih i + 3 , j | + a i +3 , j +1 | i + 3 , j + 1 ih i + 3 , j + 1 | + a i +3 , j +2 | i + 3 , j + 2 ih i + 3 , j + 2 | + a i +3 , j +3 | i + 3 , j + 3 ih i + 3 , j + 3 | ) n − X i =0 ( a i, i +1 | i, i + 1 ih i, i + 1 | + a i, i +2 | i, i + 2 ih i, i + 2 | + a i, i +3 | i, i + 3 ih i, i + 3 | + a i +1 , i | i + 1 , i ih i + 1 , i | + a i +1 , i +2 | i + 1 , i + 2 ih i + 1 , i + 2 | + a i +1 , i +3 | i + 1 , i + 3 ih i + 1 , i + 3 | + a i +2 , i | i + 2 , i ih i + 2 , i | + a i +2 , i +1 | i + 2 , i + 1 ih i + 2 , i + 1 | + a i +2 , i +3 | i + 2 , i + 3 ih i + 2 , i + 3 | + a i +3 , i | i + 3 , i ih i + 3 , i | + a i +3 , i +1 | i + 3 , i + 1 ih i + 3 , i + 1 | + a i +3 , i +2 | i + 3 , i + 2 ih i + 3 , i + 2 | ) ntanglement Witnesses a n − X i =0 ( | i + 1 , i + 2 ih i + 3 , i | + | i, i + 3 ih i + 2 , i + 1 | + | i + 3 , i ih i + 1 , i + 2 | + | i + 2 , i + 1 ih i, i + 3 | ) (4.85)Its positivity and PPT conditions are as a ≥ , a i +1 , i +2 a i +3 , i ≥ a , a i, i +3 a i +2 , i +1 ≥ a ,i = 0 , ..., n − a i, j a j, i ≥ a , a i, j +1 a j +1 , i ≥ a , a i, j +2 a j +2 , i ≥ a ,a i, j +3 a j +3 , i ≥ a , a i +1 , j +1 a j +1 , i +1 ≥ a , a i +1 , j +2 a j +2 , i +1 ≥ a ,a i +1 , j +3 a j +3 , i +1 ≥ a , a i +2 , j +2 a j +2 , i +2 ≥ a , a i +2 , j +3 a j +3 , i +2 ≥ a ,a i +3 , j +3 a j +3 , i +3 ≥ a ,i, j = 0 , ..., n − , i = j,a i +1 , i a i +3 , i +2 ≥ a , a i, i +1 a i +2 , i +3 ≥ a , i = 0 , ..., n − , (4.87)respectively. The expectation value of W with respect to ρ , along with the positivity and PPTconditions, satisfies the following inequalityTr(W ρ ) ≥ − a n. (4.88)Clearly, by the relation (4.84), the PPT entanglement detection power of W C for this state isweaker than W. In this paper, we have constructed new types of EWs, by using real skew-symmetric operatorsand maximal entangled state for bipartite d ⊗ d quantum systems analytically. The proving ntanglement Witnesses n < d ) the corresponding canonicalEW is not optimal nd-EW. In these cases, optimal nd-EW lies in the 2 n ⊗ n subspace of d ⊗ d Hilbert space. We have also constructed positive maps corresponding to canonical EWsby Jamiolkowski isomorphism. On the other hand, we have constructed the other types ofwitnesses in d ⊗ d Hilbert space corresponding to the possible partitions of full-rank J. It hasbeen shown that for a full-rank J there exist p(n) (the number of partitions of n) number ofoptimal EWs. Among these witnesses, there are one optimal d-EW and one optimal nd-EW.The other ones are composed of optimal nd-EWs and optimal d-EWs. We have also generalizedour approach to the d ⊗ d ( d ≤ d ) quantum systems. We have shown that there exist C d d distinct possibilities to construct EWs for a given d ⊗ d Hilbert space. In all of the caseswhere we have discussed, we have emphasized that the rank (J) ≥ rank (J ) ≥ rank (J) < rank (J ) <
4) then we can not find out any nd-EWs. Also the idea ofcanonical EW has been extended to produce other EWs allowed to use only for detecting theentanglement of states which lie in a real tensor product Hilbert space.In each step, we have constructed a class of PPT states. The expectation values ofconstructed witnesses with respect to the corresponding PPT states give out a negative lowerbound. This lower bound lies at the boundary of the entangled PPT states and entangledstates that violate the PPT conditions. Finally we must mention that the other interestingissues remain unsolved such as for positive expectation values; we can not conclude that thecorresponding states are separable or not. On the other hand, by replacing the W red by thegeneralized reduction EW introduced in [27], the generalized canonical EWs may be obtained.The work on these important points is under investigation by these authors. ntanglement Witnesses Appendix A:
Proving the lower bound (2.31) :The expectation value of the EW (2.22) with density matrix ρ (without normalization) in(2.27) is given asTr(W C ρ ) = − (4 n ( n −
1) + ( d − n )( d + 2 n − a + n − X i = j =0 ( a i, j + a i +1 , j +1 + a i +1 , j + a i, j +1 ) − n − X i =0 C i + d − X i =2 n n − X j =0 ( a i,j + a j, i + a i +1 ,j + a j, i +1 ) + d − X i = j =2 n a i,j (A-1)Let us rewrite the the PPT condition in (2.30) as a i, j a j, i = δ i, j a ; δ i, j ≥ i, j = 0 , ..., n − i = ja i +1 , j +1 a j +1 , i +1 = δ i +1 , j +1 a ; δ i +1 , j +1 ≥ i, j = 0 , ..., n − i = ja i, j +1 a j +1 , i = δ i, j +1 a ; δ i, j +1 ≥ i, j = 0 , ..., n − i = ja i +1 , i a i +2 , i +3 = δ i, i +2 a , δ i, i +2 a ≥ C i ; i = 0 , ..., n − a j,i a i, j = δ j,i a ; δ j,i ≥ j = 0 , ..., n − i = 2 n, ..., d − a j +1 ,i a i, j +1 = δ j +1 ,i a ; δ j +1 ,i ≥ j = 0 , ..., n − i = 2 n, ..., d − a i,j a j,i = δ i,j a ; δ i,j ≥ i, j = 2 n, ..., d − i = j (A-2)It is clear that δ is invariant under permutation of its subscripts. It should be noted thatfor two variables with constant product, their summation becomes minimum when they areequal. The corresponding summation of each product in (A-2) appears in (A-1). Hence theminimum value for (A-1) is obtained when a i, j = a j, i = δ i, j a ; δ i, j = δ j, i ; i, j = 0 , ..., n − i = ja i +1 , j +1 = a j +1 , i +1 = δ i +1 , j +1 a ; δ i +1 , j +1 = δ j +1 , i +1 ; i, j = 0 , ..., n − i = j ntanglement Witnesses a i, j +1 = a j +1 , i = δ i, j +1 a ; δ i, j +1 = δ j +1 , i ; i, j = 0 , ..., n − i = ja j,i = a i, j = δ i, j a ; δ i, j = δ j,i ; j = 0 , ..., n − i = 2 n, ..., d − a j +1 ,i = a i, j +1 = δ i, j +1 a ; δ i, j +1 = δ j +1 ,i ; j = 0 , ..., n − i = 2 n, ..., d − a i,j = a j,i = δ i,j a ; δ i,j = δ j,i ; i, j = 2 n, ..., d − i = jC i = δ i, i +2 a ; i = 0 , ..., n − C i ≤ δ i +1 , i +3 a ( i = 0 , ..., n −
1) givesthe positivity conditions for ρ . To check this matter, one can write the first and second line of(A-3) for j = i + 1 and therefore a i +2 , i = δ i, i +2 a and a i +1 , i +3 = δ i +1 , i +3 a ( i = 0 , ..., n − a i +2 , i a i +1 , i +3 = δ i, i +2 δ i +1 , i +3 a ≥ C i ( i = 0 , ..., n −
1) which is the same as the positivitycondition for ρ in (2.29). The equation (A-1) becomesTr(W C ρ ) = − (4 n ( n −
1) + ( d − n )( d + 2 n − a + a n − X i = j =0 ( δ i, j + δ i +1 , j +1 + δ i +1 , j + δ i, j +1 ) − a ( n − X i =0 δ i, i +2 + δ i +2 , i ) + a d − X i =2 n n − X j =0 ( δ i,j + δ j, i + δ i +1 ,j + δ j, i +1 ) + a d − X i = j =2 n δ i,j (A-4)It is clear that we have used the invariancy of δ with respect to permutation of its subscripts.Since Tr(W C ρ ) is a linear strictly increasing function of various δ s then its minimum takesplace in lower bounds of various δ s. Thus, when all of δ s are equal to one the lower bound of(A-4) is ( − na ). Consequently, by considering the normalization factor N (2.28), which by(A-3) gets its minimum value, we obtainTr(W C ρ ) ≥ − nd + 4 n + ( d − n )( d + 2 n −
1) ; n = 2 , , , · · · (A-5)So we conclude that, the lower bound is obtained in the boundary of the entangled PPT statesand entangled states that violate the PPT conditions. Appendix B: ntanglement Witnesses n ⊗ H n is a subspace of H d ⊗ H d Hilbert space. W
OP C in (2.26), is aHermitian operator in H n ⊗ H n which is embedded in the H d ⊗ H d and then its compact formis as W OP C = I n ⊗ I n − n | ψ ih ψ | − n (J T ⊗ I n ) | ψ ih ψ | T A (J ⊗ I n ) (B-1)where | ψ i = 1 √ n n − X i =0 | i, i i (B-2)Now we define | η ′ i ⊗ | ζ ′ i ∈ H n ⊗ H n as the projection of products | η i ⊗ | ζ i ∈ H d ⊗ H d so theexpectation value of W OP C with normalized products | η i ⊗ | ζ i is h η | ⊗ h ζ | W OP C | η i ⊗ | ζ i = h η ′ | η ′ ih ζ ′ | ζ ′ i − |h ζ ′ | η ′∗ i| − |h ζ ′ | J | η ′ i| (B-3)By normalizing the states | η ′ i and | ζ ′ i , we obtain the next equation h η | ⊗ h ζ | W OP C | η i ⊗ | ζ i = h η ′ | η ′ ih ζ ′ | ζ ′ i (1 − |h ζ ′′ | η ′′∗ i| − |h ζ ′′ | J | η ′′ i| ) (B-4)where | η ′′ i = 1 p h η ′ | η ′ i | η ′ i , | ζ ′′ i = 1 p h ζ ′ | ζ ′ i | ζ ′ i (B-5)The right hand side of the equation (B-4), apart from the multiplier h η ′ | η ′ ih ζ ′ | ζ ′ i , is similar tothe right hand side of the equation (2.5). Since J is full-rank in 2n-dimensional subspace then,by the same argument sketched in section 2, the above mentioned expectation value is zero bythe products such as | η ′′ i ⊗ ( α | η ′′∗ i + βJ | η ′′ i ) and positive with other separable states so W OP C is an EW. To further illustrate, if | η i ⊗ | ζ i ∈ H n ⊗ H n then the equation (B-4) becomes asthe following equation h η | ⊗ h ζ | W OP C | η i ⊗ | ζ i = 1 − |h ζ | η ∗ i| − |h ζ | J | η i| (B-6)which was discussed in section 2. On the other hand, if | η i ⊗ | ζ i ∈ H d − n ⊗ H d − n , whereH d − n ⊗ H d − n is the complement subspace of H n ⊗ H n , then the expectation value of the ntanglement Witnesses n ⊗ H n .Consequently, the expectation value of W OP C with the following products is always zero,P W OPC = {| η i ⊗ | ζ i ∈ H n ⊗ H n ; | ζ i = α | η ∗ i + β J | η i}∪ {| η i ⊗ | ζ i ∈ H d ⊗ H d ; | ζ ′′ i = α | η ′′∗ i + βJ | η ′′ i} ∪ {| η i ⊗ | ζ i ∈ H d − n ⊗ H d − n } (B-7).where | η ′′ i and | ζ ′′ i have been defined in equation (B-5) and | η ′ i ⊗ | ζ ′ i is the projection of | η i ⊗ | ζ i in to subspace H n ⊗ H n and with other products is positive. Appendix C:
As we considered in the paper, for a given partition ( µ , µ , µ , · · · , µ ν ) of n , the corre-sponding entanglement witness is given byW C ( µ , µ , µ , · · · , µ ν ) = W C ( µ , µ , µ , · · · , µ ν )+O t A ( µ )+O t A ( µ )+O t A ( µ )+ · · · +O t A ( µ ν − )in which W C ( µ , µ , µ , · · · , µ ν ) = W C ( µ ) ⊕ W C ( µ ) ⊕ W C ( µ ) ⊕ · · · ⊕ W C ( µ ν )Each of W C ( µ ǫ ) ( ǫ = 1 , , · · · , ν ) is canonical EW on the 2 µ ǫ ⊗ µ ǫ subspace of d ⊗ d Hilbertspace such asW C ( µ ǫ ) = µ + µ + ··· + µ ǫ − X i = j = µ µ ··· + µǫ − ( | i, j ih i, j | + | i, j + 1 ih i, j + 1 | + | i + 1 , j ih i + 1 , j | + | i + 1 , j + 1 ih i + 1 , j + 1 | − | i, i ih j, j | − | i, i ih j + 1 , j + 1 |−| i + 1 , i + 1 ih j, j | − | i + 1 , i + 1 ih j + 1 , j + 1 |−| i + 1 , j ih j + 1 , i | + | i + 1 , j + 1 ih j, i | + | i, j ih j + 1 , i + 1 | − | i, j + 1 ih j, i + 1 | ) ntanglement Witnesses µ ǫ = 1 are zero and for the others are optimalnd-EWs. Consequently the W C ( µ , µ , µ , · · · , µ ν ) is zero or optimal nd-EW and the operatorO t A ( µ ǫ ) for ( ǫ = 1 , , , · · · ν −
1) is written as belowO t A ( µ ǫ ) = µ + µ + ··· + µ ǫ − X i = µ µ ··· + µǫ − d − X j = µ µ ··· + µǫ ( | i, j ih i, j | + | i, j +1 ih i, j +1 | + | i +1 , j ih i +1 , j | + | i + 1 , j + 1 ih i + 1 , j + 1 | − | i, i ih j, j | − | i, i ih j + 1 , j + 1 |−| i + 1 , i + 1 ih j, j | − | i + 1 , i + 1 ih j + 1 , j + 1 || j, i ih j, i | + | j, i + 1 ih j, i + 1 | + | j + 1 , i ih j + 1 , i | + | j + 1 , i + 1 ih j + 1 , i + 1 | − | j, j ih i, i | − | j, j ih i + 1 , i + 1 |−| j + 1 , j + 1 ih i, i | − | j + 1 , j + 1 ih i + 1 , i + 1 | )It is clear that the O( µ ǫ ) is positive operator so the O t A ( µ ǫ ) is optimal d-EW. Appendix D:
In this appendix we give the positivity and PPT conditions for the following PPT statewhich corresponds to the partition ( µ , µ , µ , · · · , µ ν ) of n = d . ρ ( µ , µ , µ , · · · , µ ν )= 1 N ( µ , µ , µ , · · · , µ ν ) ( ̺ ( µ , µ , µ , · · · , µ ν ) + σ ( µ ) + σ ( µ ) + σ ( µ ) + · · · + σ ( µ ν − ))where ̺ ( µ , µ , µ , · · · , µ ν ) = ρ ( µ ) ⊕ ρ ( µ ) ⊕ ρ ( µ ) ⊕ · · · ⊕ ρ ( µ ν )Each of the ρ ( µ ǫ ) with ( ǫ = 1 , , , · · · , ν ) and σ ( µ ǫ ) with ( ǫ = 1 , , , · · · , ν −
1) is given asbelow ρ ( µ ǫ ) = a | ψ ( µ ǫ ) ih ψ ( µ ǫ ) | ntanglement Witnesses a µ + µ + µ + ··· + µ ǫ − X i = µ + µ + µ + ··· + µ ǫ − ( | i, i ih i, i | + | i + 1 , i + 1 ih i + 1 , i + 1 |−| i, i ih i + 1 , i + 1 | − | i + 1 , i + 1 ih i, i | )+ µ + µ + µ + ··· + µ ǫ − X i = µ + µ + µ + ··· + µ ǫ − ( a i +2 , i | i + 2 , i ih i + 2 , i | + a i +1 , i +3 | i + 1 , i + 3 ih i + 1 , i + 3 |− C i ( | i + 2 , i ih i + 1 , i + 3 | + | i + 1 , i + 3 ih i + 2 , i | ))+ µ + µ + µ + ··· + µ ǫ − X i = j = µ + µ + µ + ··· + µ ǫ − ,i − j =1 ,j − i = µ ǫ − a i, j | i, j ih i, j | + µ + µ + µ + ··· + µ ǫ − X i = j = µ + µ + µ + ··· + µ ǫ − ,j − i =1 ,i − j = µ ǫ − a i +1 , j +1 | i + 1 , j + 1 ih i + 1 , j + 1 | + µ + µ + µ + ··· + µ ǫ − X i,j = µ + µ + µ + ··· + µ ǫ − ( a i, j +1 | i, j + 1 ih i, j + 1 | + a i +1 , j | i + 1 , j ih i + 1 , j | )in which | ψ ( µ ǫ ) i = 1 √ µ ǫ µ ǫ − X i =1 | i, i i is the maximal entangled state in 2 µ ǫ ⊗ µ ǫ subspace. and σ ( µ ǫ ) = µ + µ + µ + ··· + µ ǫ − X i = µ µ µ ··· + µǫ − d − X j = µ µ µ ··· + µǫ ( a i, j | i, j ih i, j | + a i, j +1 | i, j + 1 ih i, j + 1 | + a i +1 , j | i + 1 , j ih i + 1 , j | + a i +1 , j +1 | i + 1 , j + 1 ih i + 1 , j + 1 | + a ( | i, i ih j, j | + | i, i ih j + 1 , j + 1 | + | i + 1 , i + 1 ih j, j | + | i + 1 , i + 1 ih j + 1 , j + 1 | )+ a j, i | j, i ih j, i | + a j, i +1 | j, i + 1 ih j, i + 1 | + a j +1 , i | j + 1 , i ih j + 1 , i | + a j +1 , i +1 | j + 1 , i + 1 ih j + 1 , i + 1 | + a ( | j, j ih i, i | + | j, j ih i + 1 , i + 1 | + | j + 1 , j + 1 ih i, i | + | j + 1 , j + 1 ih i + 1 , i + 1 | )) ntanglement Witnesses N ( µ , µ , µ , · · · , µ ν ) = ν X ǫ =1 N ( µ ǫ ) + ν − X ǫ =1 n ( µ ǫ )with N ( µ ǫ ) = 4 a µ ǫ + µ + µ + µ + ··· + µ ǫ − X i = j = µ + µ + µ + ··· + µ ǫ − ( a i, j + a i +1 , j +1 ) + µ + µ + µ + ··· + µ ǫ − X i,j = µ + µ + µ + ··· + µ ǫ − ( a i, j +1 + a i +1 , j )and n ( µ ǫ ) = µ + µ + µ + ··· + µ ǫ − X i = µ + µ + µ + ··· + µ ǫ − d − X j = µ + µ + µ + ··· + µ ǫ ( a i, j + a i, j +1 + a i +1 , j + a i +1 , j +1 + a j, i + a j, i +1 + a j +1 , i + a j +1 , i +1 ) The Positivity Conditions:
The following conditions must be satisfied for each ǫ = 1 , , , · · · , ν a ≥ a i, j ≥ f or i, j = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − with i = j ; i − j = 1 ; j − i = µ ǫ − a i +1 , j +1 ≥ f or i, j = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − with i = j ; j − i = 1 ; i − j = µ ǫ − a i, j +1 ≥ a i +1 , j ≥ f or i, j = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − a i +2 , i a i +1 , i +3 ≥ C i f or i = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − ntanglement Witnesses a i, j ≥ a i, j +1 ≥ a i +1 , j ≥ a i +1 , j +1 ≥ a j, i ≥ a j, i +1 ≥ a j +1 , i ≥ a j +1 , i +1 ≥ f or i = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − and j = µ + µ + · · · + µ ǫ , · · · , d − ρ ( µ , µ , µ , · · · , µ ν ) and ρ ( µ ǫ ) ( ǫ = 1 , ..., ν ) are positiveoperators. The PPT Conditions:1) a i, j a j, i ≥ a ; a i +1 , j +1 a j +1 , i +1 ≥ a ; a i, j +1 a j +1 , i ≥ a f or i, j = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − with i = j a i +1 , i a i +2 , i +3 ≥ C i f or i = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − a i, j a j, i ≥ a ; a i, j +1 a j +1 , i ≥ a ; a i +1 , j a j, i +1 ≥ a ; a i +1 , j +1 a j +1 , i +1 ≥ a f or i = µ + µ + · · · + µ ǫ − , · · · , µ + µ + · · · + µ ǫ − and j = µ + µ + · · · + µ ǫ , · · · , d − ρ ( µ , µ , µ , · · · , µ ν ) and ρ ( µ ǫ ) ( ǫ = 1 , ..., ν ). References [1] M. A. Nielsen and I. L. Chuang,
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