Entanglement in SU(2)-invariant quantum systems: The positive partial transpose criterion and others
John Schliemann
Department of Physics and Astronomy, University of Basel, CH-4056 Basel, Switzerland 共Received 14 March 2005; published 11 July 2005兲
We study entanglement in mixed bipartite quantum states which are invariant under simultaneous SU共2兲 transformations in both subsystems. Previous results on the behavior of such states under partial transposition are substantially extended. The spectrum of the partial transpose of a given SU共2兲-invariant density matrixis entirely determined by the diagonal elements ofin a basis of tensor-product states of both spins with respect to a common quantization axis. We construct a set of operators which act as entanglement witnesses on SU共2兲-invariant states. A sufficient criterion forhaving a negative partial transpose is derived in terms of a simple spin correlator. The same condition is a necessary criterion for the partial transpose to have the maximum number of negative eigenvalues. Moreover, we derive a series of sum rules which uniquely deter- mine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we compare our findings with other entanglement criteria including the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
DOI:10.1103/PhysRevA.72.012307 PACS number共s兲: 03.67.Mn, 03.65.Ud
I. INTRODUCTION
As it was recognized already in the 1930s by some of the founding fathers of modern physics, the notion of entangle- ment is one of the most intriguing properties of quantum mechanics, distinguishing the quantum world form the clas- sical one关1,2兴. Moreover, quantum entanglement is the key ingredient to many if not almost all concepts and proposal in the field of quantum information theory and processing关3兴;
recent reviews on progress in the theoretical description and analysis of entanglement are listed in Refs.关4–6兴.
As far as pure states of a quantum system are concerned, the situation is, from a theory point of view, very clear: there are simple and efficient methods to detect and quantify en- tanglement in a given pure state. One of the most widely used entanglement measures for this case is certainly the von Neumann entropy of partial density matrices constructed from the full pure-state density matrix 关7兴. However, the problem of entanglement in mixed states is in general an open one. A mixed state is said to be nonentangled共or sepa- rable兲if it can be represented as a convex sum of projectors onto nonentangled pure states. In the following we shall con- centrate on bipartite systems. As it was noticed by Peres关8兴, a necessary criterion for a mixed state of a bipartite system to be separable is that its partial transpose with respect to one of the subsystems is positive关9兴. Subsequently it was shown by the Horodecki family that this condition is also sufficient if the Hilbert space of the bipartite system has dimension 2
⫻2 or 2⫻3 关10兴. For larger dimensions, inseparable states with positive partial transpose共PPT兲exist关11兴, i.e., the PPT 共or Peres-Horodecki兲 criterion is in general a necessary but not a sufficient one.
More recently, mixed states being invariant under certain joint symmetry operations of the bipartite system have been studied关12兴. Probably the oldest example known to the lit- erature of this kind of objects are the Werner states 关13兴.
Here both parties have local Hilbert spaces of the same di-
mension, and the Werner states are defined by being invari- ant under all simultaneous unitary transformations U丢U.
Another important example from this class of states but with an in general much smaller symmetry group are the so-called SU共2兲-invariant states关14,15兴. Here we regard the two sub- systems as spinsSជ1,Sជ2, where 2S1+ 1, 2S2+ 1 are the dimen- sions of the corresponding Hilbert spaces. SU共2兲-invariant states are defined to be invariant under all uniform rotations U1丢U2of both spinsSជ1andSជ2, whereU1/2= exp共iជSជ1/2兲are transformations corresponding to the same set of real param- etersជ in the representation of SU共2兲appropriate for the spin lengthsS1 andS2 共ប= 1兲. Werner states and SU共2兲invariant states are identical for S1=S2= 1 / 2, but for larger spin lengths the SU共2兲-invariant states have a clearly smaller symmetry group. By construction, SU共2兲-invariant states commute with all components of the total spinJជ=Sជ1+Sជ2. In particular, for SU共2兲-invariant states acting on bipartite Hil- bert spaces with dimension 2⫻N, the Peres-Horodecki cri- terion can be shown to be necessaryand sufficient关14兴, i.e., there are no entangled states of this kind with a positive partial transpose.
SU共2兲-invariant density matrices arise from thermal equi- librium states of low-dimensional spin systems with a rota- tionally invariant Hamiltonian by tracing out all degrees of freedom but those two spins. In fact, in the recent years, entanglement in generic quantum spin models has developed to a major direction of research, see, e.g., Refs. 关16–20兴.
Most recently, SU共2兲-invariant states were also studied as a model for entangled multiphoton states produced by para- metric down conversion关21兴.
Most recently, and while the present work was being com- pleted, a preprint by Breuer appeared 关22兴 where SU共2兲- invariant states with common spin length,S1=S2are studied.
The approach there is so far restricted to small spin lengths 共S1=S2艋3 / 2兲, but has the merit to allow for an analysis on the sufficiency of the Peres-Horodecki criterion.
1050-2947/2005/72共1兲/012307共6兲/$23.00 012307-1 ©2005 The American Physical Society
In the present work we extend previous results on en- tanglement properties of SU共2兲-invariant states 关14兴 and compare the PPT criterion with other entanglement criteria including the reduction criterion 关23,24兴, the majorization criterion关25兴, and the local uncertainty relations studied very recently关26,27兴. The latter criteria are very readily applied to SU共2兲-invariant states, and these considerations provide in- structive illustrations of the logical hierarchy of those en- tanglement criteria.
This paper is organized as follows. In Sec. II we summa- rize important properties of SU共2兲-invariant states under par- tial transposition and derive a series of additional results which allow to extend previous findings关14兴to the case of larger spin lengths. In the following section we apply the above-mentioned other entanglement criteria to SU共2兲- invariant density matrices and compare the results with each other. We close with conclusions in the last section.
II. SU(2)-INVARIANT STATES UNDER PARTIAL TRANSPOSITION
An SU共2兲-invariant state of a bipartite system of two spinsSជ1,Sជ2 has the general form关14兴
=
兺
J=兩S1−S2兩 S1+S2
A共J兲 2J+ 1
兺
Jz=−J J
兩J,Jz典00具J,Jz兩, 共1兲 where the constants A共J兲 fulfill A共J兲艌0, ⌺JA共J兲= 1. Here 兩J,Jz典0 denotes a state of total spinJandz-componentJz. In particular,commutes with all components of the total spin Jជ=Sជ1+Sជ2. Obviously the SU共2兲-invariant density matrices from a convex set, i.e., with two given SU共2兲-invariant states
1,2any convex combination1+共1 −兲2,苸关0 , 1兴, has the same property. Let us now consider the partial transpose of an SU共2兲-invariant state,T2, where we take, without loss of generality, the partial transpositions to be performed of the second subsystem describing the spin Sជ2. Moreover, let us assume that the partial transposition is performed in the stan- dard basis of joint tensor-product eigenstates ofS1z andS2z. As shown earlier 关14兴, under these conditions T2 commutes with all components of the vectorKជ defined by Kx=S
1 x−S2x, Ky=S1y+S2y, Kz=S1z−S2z, and these operators also furnish a representation of su共2兲, 关K␣,K兴=i␣␥K␥ 共using standard notation兲. We note that the above result relies on the trans- formation properties ofT2关14兴. The form of the operatorsKជ depends on the basis with respect to the partial transposition performed. For any choice of basis one finds a set of opera- torsKជ commuting withT2and fulfilling the angular momen- tum algebra, but the form of the operators will in general be different from the above one obtained in the standard basis.
From the above observations it follows that the eigensystem of T2 has the same multiplet structure as 关14兴 and can therefore be written in the general form
T2=
兺
K=兩S1−S2兩 S1+S2
B共K兲 2K+ 1
兺
Kz=−K K
兩K,Kz典00具K,Kz兩, 共2兲
where the multiplets are labeled by the value of Kជ2=K共K + 1兲 with兩S1−S2兩艋K艋S1+S2 and have degeneracy 2K+ 1.
Again, the real coefficients B共K兲 fulfill ⌺KB共K兲= 1 共since tr= trT2兲but are not necessarily positive. As pointed out by Peres 关8兴, negative B共K兲 indicate entanglement in the original state . The coefficient of the largest multiplet, K
=S1+S2, is given by关14兴 B共S1+S2兲
2共S1+S2兲+ 1=具±S1,⫿S2兩兩±S1,⫿S2典艌0, 共3兲 where兩S1z,S2z典are tensor-product eigenstates ofS1z andS2z. In particular,B共S1+S2兲is always non-negative and can alterna- tively be expressed as
B共S1+S2兲 2共S1+S2兲+ 1= tr关P˜
nជ共S1+S2兲T2兴, 共4兲 where P˜nជ共L兲 is the projector onto the subspace with nជ·Kជ
=L, and nជ is an arbitrary unit vector. As it follows from the above multiplet structure, each eigenvalue ofT2 in the sub- space with nជ·Kជ=L+ 1⬎0 occurs also exactly once in the subspace with nជ·Kជ=L. Thus, for 兩S1−S2兩艋K⬍S1+S2 the above relation can be generalized to
B共K兲 2K+ 1= tr关P˜
nជ共K兲T2兴−关P˜
nជ共K+ 1兲T2兴, 共5兲 where the right-hand side can be rewritten as
tr兵关P˜
nជ共K兲−P˜
nជ共K+ 1兲兴T2其= tr兵关P˜
nជ共K兲−P˜
nជ共K+ 1兲兴T2其 共6兲
=tr兵关Pnជ共K兲−Pnជ共K+ 1兲兴其. 共7兲 Here Pnជ共L兲 is the projector onto the subspace with nជ共Sជ1
−Sជ2兲=L. In the last equation we have used the fact that the projectorsP˜
nជ共L兲are polynomials in the operator nជ·Kជ which turns, in the standard basis, intonជ共Sជ1−Sជ2兲. However, the ex- pression共7兲 contains only the spin operatorsSជ1,Sជ2 and the density matrix itself; therefore this expression is indepen- dent of any choice of basis,
B共K兲
2K+ 1= tr兵关Pnជ共K兲−Pnជ共K+ 1兲兴其. 共8兲 Hence any separable SU共2兲-invariant density matrix fulfills
tr兵关Pnជ共K兲−Pnជ共K+ 1兲兴其艌0 共9兲 for兩S1−S2兩艋K⬍S1+S2, while
tr兵关Pnជ共K兲−Pnជ共K+ 1兲兴其⬍0, 共10兲 indicates the presence of entanglement in the state . Thus, when restricting the full space of density operators to the convex submanifold of SU共2兲-invariant states, the operators 关Pnជ共K兲−Pnជ共K+ 1兲兴,兩S1−S2兩艋K⬍S1+S2, have the properties of entanglement witnesses关10,28兴. It is an interesting ques- tion whether and, if so, to what extent, one can relax the restriction to SU共2兲-invariant states with this property of the operators关Pnជ共K兲−Pnជ共K+ 1兲兴 being unaltered. Note also that the above operators can, by construction, only detect en-
tanglement in SU共2兲-invariant states with negative partial transpose, although these operators do not fulfill the con- struction recipe of decomposable entanglement witnesses 关5兴.
Moreover, the contributions to the right-hand side of Eq.
共8兲can be expressed as tr关Pnជ共K兲兴=
兺
S1z−S2z=K
具S1 z,S2z兩兩S1
z,S2z典. 共11兲 Thus, the eigenvalues of the partial transposeT2are entirely determined by the diagonal elements of in a basis of tensor-product states of both spins with respect to a common quantization axis. In particular, the relation
B共K兲 2K+ 1=
兺
S1z−S2z=K
具S1z,S2z兩兩S1z,S2z典−
兺
S1z−S2z=K+1
具S1z,S2z兩兩S1z,S2z典 共12兲 provides a convenient way to compute the eigenvalues ofT2 without explicitly solving for the zeros of a characteristic polynomial. Below we shall encounter yet another method to determine the spectrum of T2 based on sum rules for its eigenvalues.
To gain further insight into the properties ofT2 consider tr共Kជ2T2兲= tr关共Kជ2兲T2兴 共13兲
=tr兵关共Sជ1−Sជ2兲2兴其 共14兲 for 0艋n艋2 min兵S1,S2其. In the last equation we have used the fact that the operatorS2y, when expressed in the standard basis, changes sign under partial transposition while S2x and S2z remain unaltered. Alternatively, the left-hand side of Eq.
共13兲can also be evaluated using Eq.共2兲leading to 具共Sជ1−Sជ2兲2典=
兺
K=兩S1−S2兩 S1+S2
K共K+ 1兲B共K兲, 共15兲 where具·典 denotes an expectation value with respect to.
It is instructive to investigate the condition
具共Sជ1−Sជ2兲2典艌共S1+S2兲共S1+S2+ 1兲 共16兲 which is equivalent to
具Sជ1·Sជ2典⬍−S1S2 共17兲 and implies that T2 has at least one negative eigenvalue, since otherwise we had
具共Sជ1−Sជ2兲2典=
兺
K=兩S1−S2兩 S1+S2
K共K+ 1兲B共K兲艋共S1+S2兲共S1+S2
+ 1兲
兺
K=兩S1−S2兩 S1+S2
B共K兲=共S1+S2兲共S1+S2+ 1兲.
共18兲 Thus, the inequalities共16兲and共17兲are asufficient condition forT2 having at least one negative eigenvalue, and, in turn, for being entangled. The latter statement follows also di-
rectly from共17兲, because the right-hand side of this inequal- ity represents the minimum value the correlator具Sជ1·Sជ2典can attain in a separable state. Therefore, if共17兲is fulfilled, the underlying state must be entangled. Note that for general spinsSជ1,Sជ2 the above correlator is bounded by −共S1+ 1兲S2
艋具Sជ1·Sជ2典艋S1S2共assumingS1艌S2兲.
Moreover, the conditions共16兲and共17兲are also a neces- sary criterion forT2 having the maximum possible number of negative eigenvalues. Here all B共K兲 with 兩S1−S2兩艋K
⬍S1+S2 are negative, while B共S1+S2兲¬B¯⬎1 because of the normalization condition ⌺KB共K兲= 1. The assertion is proved as follows:
具共Sជ1−Sជ2兲2典艌共S1+S2− 1兲共S1+S2兲共1 −B¯兲+共S1+S2兲共S1 +S2+ 1兲B¯=共S1+S2− 1兲共S1+S2兲+ 2B¯共S1
+S2兲共S1+S2兲共S1+S2+ 1兲. 共19兲 The above considerations can obviously be extended to higher powers ofKជ2, i.e., 共Kជ2兲n withn⬎1. However, when performing the partial transposition, more complicated op- erator products occur which give rise to additional contribu- tions. For example, for the next higher powers one finds
关共Kជ2兲2兴T2=关共Sជ1−Sជ2兲2兴2+ 4Sជ1·Sជ2 共20兲 and
关共Kជ2兲3兴T2=关共Sជ1−Sជ2兲2兴3− 32共Sជ1·Sជ2兲2+ 4兵3关S1共S1+ 1兲 +S2共S2+ 1兲兴− 4其Sជ1·Sជ2+ 8S1共S1+ 1兲S2共S2+ 1兲
共21兲 leading to the additional sum rule
具
关共Sជ1−Sជ2兲2兴2+ 4Sជ1·Sជ2典
=兺
K=兩S1−S2兩 S1+S2
关K共K+ 1兲兴2B共K兲 共22兲 and an analogous relation forn= 3 following from Eq.共21兲.
Equations 共15兲 and 共22兲, together with the normalization condition⌺KB共K兲= 1, form a series of sum rules being linear in the coefficients B共K兲. This series can obviously be ex- tended to arbitrary high powers of the spin operators. The number of independent sum rules, however, is in general given by 2 min兵S1,S2其+ 1. Thus, for given S1,S2, the rela- tions arising from n= 0 ,…, 2 min兵S1,S2其 constitute a linear system of equations which uniquely determines the spectrum ofT2. Note that the coefficients in this system of equations are of the form共K共K+ 1兲兲n, i.e., the corresponding matrix is of the Vandermonde type with its determinant given by
K,L=兩
兿
S1−S2兩 K⬎L S1+S2关K共K+ 1兲−L共L+ 1兲兴, 共23兲
which is always positive. Such a system of linear equations for the coefficientsB共K兲provides an alternative way to com- pute the eigenvalues ofT2in terms of spin correlators.
Moreover, using the relation 具共Sជ1+Sជ2兲2典=
兺
J=兩S1−S2兩 S1+S2
J共J+ 1兲A共J兲 共24兲 and Eq.共15兲one derives the following sum rule:
2关S1共S1+ 1兲+S2共S2+ 1兲兴=
兺
L=兩S1−S2兩 S1+S2
L共L+ 1兲关A共L兲+B共L兲兴, 共25兲 where the left-hand side is independent of the given state. Let us illustrate the above findings on some examples.
The simple case when one of the spins, say Sជ2, has length 1 / 2 was already fully discussed in Ref.关14兴. Here one finds B
冉
S−12冊
=2S1+ 1共S+ 2具Sជ1·Sជ2典兲, 共26兲 B冉
S+12冊
=2S1+ 1共S+ 1 − 2具Sជ1·Sជ2典兲, 共27兲 where SªS1. Clearly, B共S+ 1 / 2兲 is always non-negative 共since具Sជ1·Sជ2典艋S/ 2兲, whileB共S− 1 / 2兲 becomes negative if 具Sជ1·Sជ2典⬍−S/ 2, in accordance with the above results for gen- eral spin lengths. Moreover, as shown in Ref. 关14兴, in the caseS2= 1 / 2, there are no entangled states with positive par- tial transpose, i.e., the Peres-Horodecki criterion for separa- bility is necessary and sufficient.Next let us considerS2= 1,S1=S艌1. Here we can use the relations共15兲and 共22兲along with the normalization condi- tion to obtain the coefficientsB共K兲as
B共S− 1兲= 1
2S+ 1
冉
− 1 +具Sជ1·Sជ2典+1S具共Sជ1·Sជ2兲2典冊
, 共28兲B共S兲= 1 − 1
S共S+ 1兲具共Sជ1·Sជ2兲2典, 共29兲
B共S+ 1兲= 1
2S+ 1
冉
1 −具Sជ1·Sជ2典+S+ 11 具共Sជ1·Sជ2兲2典冊
.共30兲 Again the the coefficient of the largest multiplet is of course always non-negative, B共S+ 1兲艌0, while the conditions for B共S− 1兲⬍0 andB共S兲⬍0 read
1⬎具Sជ1·Sជ2典+1
S具共Sជ1·Sជ2兲2典, 共31兲 具共Sជ1·Sជ2兲2典⬎S共S+ 1兲, 共32兲 respectively. These inequalities generalize the conditions given in Ref.关14兴forS= 1 to the case of general spin length S. Besides, demanding that bothB共S− 1兲andB共S兲should be negative leads to the necessary condition
具Sជ1·Sជ2典⬍−S, 共33兲
and it is also easy to explicitly show from the above relations that at least one eigenvalue ofT2must be negative if共33兲is fulfilled, both in accordance with our earlier general findings.
Alternatively, the coefficientsB共K兲characterizingT2can be expressed in terms of the quantitiesA共J兲 describing,
B共S− 1兲=2S− 1 2S+ 1−S− 1
S A共S− 1兲−2S− 1 2S+ 1
S+ 1 S A共S兲,
共34兲
B共S兲= 1
S+ 1− 2S+ 1
S共S+ 1兲A共S− 1兲+ S− 1
S A共S兲, 共35兲 B共S+ 1兲= 1
共2S+ 1兲共S+ 1兲+S+ 2
S+ 1A共S− 1兲+ 2 2S+ 1A共S兲.
共36兲 HereA共S+ 1兲has been eliminated via the normalization con- dition, and the other coefficients can be expressed in terms of spin correlators as follows关14兴:
A共S− 1兲= 1
S共2S+ 1兲关−S−共S− 1兲具Sជ1·Sជ2典+具共Sជ1·Sជ2兲2典兴, 共37兲
A共S兲= 1 − 1
S共S+ 1兲关具Sជ1·Sជ2典+具共Sជ1·Sជ2兲2典兴. 共38兲 Moreover, most recently Breuer has investigated the case S1=S2= 1 using a different approach and concluded that for this case the PPT criterion is necessary and sufficient, i.e., there are no entangled states with positive partial transpose 关22兴. This finding also confirms a conjecture raised recently in Ref.关19兴. The question whether this is also true for gen- eral S1=S⬎1, S2= 1 remains open. The approach of Ref.
关22兴 finds linear expressions for the coefficients B共K兲 in terms of the A共J兲 共in the notation used here兲. Equations 共34兲–共36兲 are an example of such a linear relation for the case ofS2= 1 and generalS1=S艌1, while the results of Ref.
关22兴are restricted to equal spin lengthsS1=S2艋3 / 2.
III. COMPARISON WITH OTHER ENTANGLEMENT CRITERIA
We now compare the above findings from the PPT crite- rion with other entanglement criteria. These criteria are gen- erally weaker than the PPT criterion, but have the merit of being very readily applied to SU共2兲-invariant states.
A. The reduction criterion and the majorization criterion The reduction criterion关23,24兴states that if a given state
is separable, then the operators
1丢1−,
1丢2−,
are also positive, i.e., do not contain any negative eigen- value. Here1/2= tr2/1共兲denotes the reduced density matri- ces of each subsystem. This criterion is in general weaker than the PPT criterion. If one of the subsystems has a Hilbert space of dimension two, however, both criteria are equiva- lent 关24兴. Thus, in particular, the reduction criterion is nec- essary and sufficient for the case of the dimensions 2⫻2 or 2⫻3.
Applying the reduction criterion to SU共2兲-invariant states is technically very easy since, due to the rotational invari- ance of these objects, we have
1/2= 1
2S1/2+ 11. 共39兲
Thus, the criterion is violated if A共J兲
2J+ 1⬎ 1
2S1/2+ 1 共40兲
for some J. If this inequality is fulfilled, the underlying SU共2兲-invariant stateis inseparable. BecauseA共J兲艋1 this is only possible for J⬍S1/2, which strongly restricts the power of this entanglement criterion as applied to SU共2兲- invariant states关22兴.
Let us now compare the reduction criterion with the re- sults obtained from the PPT criterion. For the case S1=S, S2= 1 / 2 the reduction criterion is violated if
A共S− 1/2兲⬎ 2S
2S+ 1, 共41兲
or, usingA共S− 1 / 2兲=共S− 2具Sជ1·Sជ2典兲/共2S+ 1兲 共cf. Ref.关14兴兲, 具Sជ1·Sជ2典⬍−S
2. 共42兲
This condition is of course the same as found from the PPT criterion since both criteria are equivalent for this case.
For the case S1=S艌1, S2= 1 violation of the reduction criterion leads to the condition
A共S− 1兲⬎2S− 1
2S+ 1, 共43兲
or, using Eq.共37兲,
−共S− 1兲
S 具Sជ1·Sជ2典+1
S具共Sជ1·Sជ2兲2典⬎2S. 共44兲 ForS= 1 this inequality is the same as the criterion共32兲. The other inequality共31兲, however, is not reproduced by the re- duction criterion, and forS⬎1 the above inequality共44兲is a weaker criterion for entanglement than共32兲. In fact, demand- ing that T2 is positive, i.e., B共S− 1兲艌0 and B共S兲艌0, one derives from Eqs.共34兲and共35兲the necessary condition
A共S− 1兲艌S2−共S− 1兲 S2
2S− 1
2S+ 1. 共45兲
Thus, whenever the PPT criterion is unable to detect en- tanglement in a given state, the reduction criterion will also
fail. As mentioned above, this is a general property关23,24兴.
Another entanglement criterion related to the reduction criterion is the majorization criterion关25,6兴. It states that any separable statefulfills the inequalities
↓Ɱ↓1,
↓Ɱ
2
↓ ,
where the vector↓ consists of the eigenvalues of in de- creasing order. The notationxⱮy means⌺j=1k xj艋⌺j=1k yjfor k苸兵1 ,…,d其, where the equality holds fork=d. Heredis the dimension of the total Hilbert space of the bipartite system, and the vectors
1/2
↓ are extended by zeros in order to make their dimension equal to that of↓. Obviously the majoriza- tion criterion is fulfilled if
A共J兲
2J+ 1⬍ 1
2S1/2+ 1. 共46兲
Thus, when applied to SU共2兲-invariant states, the majoriza- tion criterion is always weaker than the reduction criterion.
B. Local uncertainty relations
Entanglement criteria based on so-called local uncertainty relations were introduced recently by Hofmann and Takeuchi 关26兴, and by Gühne 关27兴. This concept is based on the fol- lowing observation. Let=⌺kpkk,pk艌0,⌺kpk= 1 be a con- vex combination of some stateskand letMibe some set of operators. Then the following inequality holds关26,27兴:
兺
i ␦2共Mi兲艌兺
k pk兺
i ␦2共Mi兲k, 共47兲where
␦2共M兲=具M2典−具M典2 共48兲 and具·典denotes an expectation value with respect to. Con- sider now a bipartite system with operatorsMi共1兲,Mi共2兲acting on one of the subsystems. Then for any separable statekone has
␦2共Mi共1兲+Mi共2兲兲k=␦2共Mi共1兲兲k+␦2共Mi共2兲兲k. 共49兲 Let nowU1/2be the absolute minimum of⌺i␦2共Mi共1/2兲兲with respect to all possible states of each subsystem. Then any separable statemust fulfill the inequality
兺
i ␦2共Mi共1兲+Mi共2兲兲艌U1+U2. 共50兲Violation of this inequality is indicative of entanglement in the underlying state. Note that this observation provides a whole variety of entanglement criteria since the operators Mi共1/2兲 are undetermined so far. In circumstances of SU共2兲- invariant states, however, it is natural to chooseMi共1/2兲=S1/2i , i苸兵x,y,z其 with U1/2=S1/2 关26兴. Then a given SU共2兲- invariant state is entangled if
具Sជ1·Sជ2典⬍−12共S12+S22兲 共51兲
=−S1S2−12共S1−S2兲2. 共52兲 The second version of this inequality suggests that this en- tanglement criterion is strongest if both spins are of the same length,S1=S2. In this case, the criterion again states that the correlator具Sជ1·Sជ2典must be smaller than its minimum value in any separable state, from which it follows that the underlying state has a negative partial transpose.
The local uncertainty relation of the above form is based on a very natural choice of operators, but provides in general only a quite weak entanglement criterion. For instance, the above spin correlator is bounded from below by −共S1+ 1兲S2
艋具Sជ1·Sជ2典 共assuming S1艌S2兲. Thus, the above inequality cannot be fulfilled ifS1 sufficiently exceeds S2. We leave it open whether another choice of operators could lead to stron- ger criteria for inseparability.
IV. CONCLUSIONS
We have investigated entanglement in SU共2兲-invariant bi- partite quantum states and have substantially extended pre- vious results on the behavior of such states under partial
transposition. The spectrum of the partial transpose of a given SU共2兲-invariant density matrixis entirely determined by the diagonal elements of in a basis of tensor-product states of both spins with respect to a common quantization axis. We have constructed a set of operators which act as entanglement witnesses on SU共2兲-invariant states, and we have derived sufficient criterion for T2 having at least one negative eigenvalue in terms of a simple spin correlator. The same condition is a necessary criterion for the partial trans- pose to have the maximum number of negative eigenvalues.
Moreover, we have presented a series of sum rules which uniquely determine the eigenvalues of the partial transpose in terms of a system of linear equations. Finally we have compared our findings with other entanglement criteria in- cluding the reduction criterion, the majorization criterion, and the recently proposed local uncertainty relations.
The key challenge for future investigations of SU共2兲- invariant states 共or states being invariant under other trans- formation groups兲is certain to determine to what extent the PPT criterion is necessary and sufficient. A possible route toward this goal could be given by the methods developed in Ref. 关22兴. This approach, however, is so far limited to the case of equal spin lengthsS1=S2艋3 / 2.
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