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A Natural Entanglement Test

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A Natural Entanglement Test
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    a  r   X   i  v  :   1   2   1   1 .   2   3   0   6  v   3   [  q  u  a  n   t  -  p   h   ]   1   5   N  o  v   2   0   1   3 Constructive entanglement test from triangle inequality Łukasz Rudnicki, 1,2,  ∗ Zbigniew Puchała, 3,4 Paweł Horodecki, 5,6 and Karol Życzkowski 1,4 1 Center for Theoretical Physics, Polish Academy of Sciences,Aleja Lotników 32/46, PL-02-668 Warsaw, Poland  2 Freiburg Institute for Advanced Studies, Albert-Ludwigs University of Freiburg, Albertstrasse 19, 79104 Freiburg, Germany  3  Institute of Theoretical and Applied Informatics,Polish Academy of Sciences, Bałtycka 5, PL-44-100 Gliwice, Poland  4 Smoluchowski Institute of Physics, Jagiellonian University, ul. Reymonta 4, PL-30-059 Kraków, Poland  5  Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, PL-80-952 Gdańsk, Poland  6  National Quantum Information Centre of Gdańsk, PL-81-824 Sopot, Poland  We derive a simple lower bound on the geometric measure of entanglement for mixed quantumstates in the case of a general multipartite system. The main ingredient of the presented derivationis the triangle inequality applied to the root infidelity distance in the space of density matrices.The obtained bound leads to entanglement criteria with a straightforward interpretation. Proposedcriteria provide an experimentally accessible, powerful entanglement test. PACS numbers: 03.67.Mn, 03.67.Lx, 42.50.Dv Quantum entanglement characterizes non–classicalcorrelations in a quantum system consisting of severalsubsystems [1–5]. In the case of a pure quantum state,any correlations between subsystems, that can be de-tected in coincidence experiments, confirm entanglement.However, in any realistic experiment one has to cope withmixed quantum states, for which the problem becomesmore involved, as quantum and classical correlations mayexist. To detect reliably quantum entanglement for amixed quantum state one needs to rule out the morecommon case of classical correlations.While efficient detection of quantum entanglement isnot an easy task in quantum information theory, it ismore difficult to characterize this phenomenon quanti-tatively basing on results of partial measurements thatare not sufficient for full state reconstruction. Knownschemes of such experimental procedure require interac-tions between many copies of the state investigated [6].With an interaction between two copies of the state onecan estimate a lower bound on an entanglement measure[7] in terms of a two–copy entanglement witness that re-produces the difference between global and local entropy[8]. The things become more complicated in the caseof the restriction to single-copy measurements. Despitesome recent progress (see [9]) still there is no generalsatisfactory answer to the question how well one can es-timate given entanglement measure on the basis of non-complete (ie. non–tomographic) experimental data.In this work we build on a pragmatic approach ad-vocated in [5, 10], in which one attempts to constructentanglement measures accessible in an experiment. Wederive a lower bound for the  geometric measure of entan-glement   (GME) [11, 12] capable to describe entanglementof an arbitrary mixed quantum state. We shall empha-size that we are unaware of any results concerning lowerbounds for GME (an upper bound given in terms of thegeneralized robustness of entanglement can be found in[13]). We demonstrate that our quantity can be used tocompare the amount of entanglement between differentmixed states of a  d × d  system and provides a separabilitytest which is experimentally accessible.Consider an arbitrary  K  –partite quantum system de-scribed in the Hilbert space  H  =   K I  =1 H I  with noassumption about the dimensionality of the particularsubspace  H I  representing the  I  -th subsystem. We de-note by  S  m ,  m  = 1 ,...,K   the set of   m –separable purestates  | φ  m sep  =  ⊗ mI  =1 | φ I   . We have the following chain S  K   ⊂ S  K  − 1  ⊂···⊂ S  1  = H .In our considerations we shall use the root infidelitydistance between two mixed states  ρ 1  and  ρ 2  [14]: C  F   ( ρ 1 ,ρ 2 ) =   1 − F   ( ρ 1 ,ρ 2 ) ,  (1)defined with the help of the fidelity  F   ( ρ 1 ,ρ 2 ) . To deriveour result we use the fidelity involving at least one purestate, thus we need only the restricted, simpler formulafor the fidelity F   ( ρ, | Ψ  Ψ | ) ≡ F   ( ρ, | Ψ  ) =  Ψ | ρ | Ψ  .  (2)Finally we need to introduce the hierarchy of GME[11–13], which in the case of pure states is defined as: E  m  ( | Ψ  ) = 1 −  max | φ ∈S  m | φ | Ψ | 2 (3) ≡  min | φ ∈S  m C  2 F   ( | φ  , | Ψ  ) , m  = 2 ,...,K. The second, equivalent definition follows directly fromEqs. (1, 2). The operational interpretation of the mea-sure  E  m  ( | Ψ  )  is straightforward. If the state  | Ψ   is  m –separable it belongs to the set  S  m , thus the minimal infi-delity distance is  0 , since one can always chose  | φ ∈S  m to be equal  | Ψ  .The geometric measure of entanglement for mixedstates is defined [12] with the help of the convex roof   2construction: E  m  ( ρ ) = min E   i  p i  min | φ ∈S  m C  2 F   ( | φ  , | Ψ i  ) ,  (4)where the ensemble  E   =  {  p i , | Ψ i }  represents the mixedstate  ρ , i.e.  ρ  =   i  p i | Ψ i  Ψ i | . Surprisingly, it wasshown [15] that  E  m  ( ρ )  is simultaneously a distance mea-sure  E  m  ( ρ ) = min σ  C  2 F   ( σ,ρ )  with  σ  being a  m –separablemixed state, for  m  = 2 ,...,K  . The lower bound on   E  m  ( ρ ) .—   Any density matrixrepresenting a multipartite system can be characterizedby its  product numerical radii   L m ( ρ ) , often used in thetheory of quantum information [16]. These quantities canbe defined as the maximal expectation value of   ρ  amongnormalized pure product states, L m  ( ρ ) = max | φ ∈S  m  φ | ρ | φ  , m  = 2 ,...K.  (5)Note that  E  m  ( | Ψ  ) ≡ 1 − L m  ( | Ψ  ) .The main result of this paper is the following lowerbound for the square root of the geometric measures of entanglement   E  m  ( ρ ) ≥R m  ( ρ ) =   1 − L m  ( ρ ) −   1 − Tr ρ 2 .  (6)We start the derivation of (6) with an arbitrary expan-sion  ρ  =   i  p i | Ψ i  Ψ i |  of the mixed state  ρ . For somefixed index  i  we chose a pure state  | Ψ i  , and anotherpure state  | φ   to be specified. Since the root infidelity(1) is a legitimate metric we can write down the triangleinequality for  C  F   ( | φ  ,ρ )  with  | Ψ i  Ψ i |  as a third state: C  F   ( | φ  ,ρ ) ≤ C  F   ( | φ  , | Ψ i  ) + C  F   ( | Ψ i  ,ρ ) .  (7)If we next take the minimum with respect to  | φ  ∈ S  m and use the definitions (1, 3, 5) we obtain   1 − L m  ( ρ ) ≤   E  m  ( | Ψ i  ) + C  F   ( | Ψ i  ,ρ ) .  (8)In the next step we shall multiply the resulting inequalityby  p i  and sum over  i . The term    1 − L m  ( ρ )  is indepen-dent of   i , while for the two terms on the right hand sidewe shall apply the following estimates srcinating fromthe concavity of the  √ ·  function:  i  p i   E  m  ( | Ψ i  ) ≤   i  p i E  m  ( | Ψ i  ) ,  (9)  i  p i C  F   ( | Ψ i  ,ρ ) ≤   1 −  i  p i  Ψ i | ρ | Ψ i  .  (10)In the final step we shall recognize that the sum over i  on the right hand side of (10) is equal to Tr ρ 2 , sothat is independent of the given ensemble  E   = {  p i , | Ψ i } .This implies that we can immediately minimize with re-spect to  E   =  {  p i , | Ψ i }  producing the quantity    E  m  ( ρ ) FIG. 1: (color online). Parameter space for the generalizedWerner states of a  3 × 3  system. Red volume corresponds tothe states  ρ p ′ , λ  the entanglement of which is shown by thebound (6) to be larger than this of reference state  ρ p, ¯ λ . Here d  = 3  so that  p cr  = 3 / 4  and  Λ min  = 1 / 3 . on the right hand side of (9). Applying the above es-timates to Eq. (8) we obtain the desired lower bound(6) after a one–step rearrangement. From (6) one canobviously find the lower bound for  E  m  ( ρ ) , which reads (max[ R m  ( ρ );0]) 2 . It is important to take the maxi-mum first, in order to avoid cases when negative valuesof   R m  ( ρ )  can give a positive, unphysical contribution R 2 m  ( ρ )  to the lower bound of   E  m  ( ρ ) .We shall further observe that the quantity  1 − L m  ( ρ ) provides a natural (but typically rough) upper bound for E  m  ( ρ ) , To prove this statement it is sufficient to re-strict the minimization in  E  m  ( ρ ) = min σ  C  2 F   ( σ,ρ )  topure states  σ  =  | φ  φ | . This upper bound, as well asour lower bound are in the case of pure states equal to E  m  ( ρ ) . In the case of   pseudo–pure   states characterizedby Tr ρ 2  1  we thus get a sharp estimate of the value of the entanglement measure in question.Let us now focus on the family of generalized  d  ×  d Werner states ( 0 ≤  p ≤ 1 ): ρ  p, λ  = (1 −  p ) | Ψ λ  Ψ λ | +  pI  d ⊗ I  d d 2  ,  (11)where  | Ψ λ   =   di =1 √    λ i | ii  . One can straightforwardlycalculate (as  ρ  p, λ  is bipartite we can skip the index  m ):Tr ρ 2  p, λ  = 1+   p 2 − 2  p  d 2 − 1  d 2  , L ( ρ  p, λ ) =  pd 2 +(1 −  p )Λ , (12)where  Λ = max i λ i  and  1 /d  ≤  Λ  ≤  1 . The abovefamily possesses a distinguished member given by  ¯ λ  =(1 /d,..., 1 /d ) , which for  p  = 0  represents the maximallyentangled state. Comparing  E   ρ  p, ¯ λ   with the lowerbound based on  R ( ρ  p ′ , λ )  for  p ′ < p  and vector  λ  ma- jorizing [17]  ¯ λ  we can look for the states  ρ  p ′ , λ  which aremore entangled than  ρ  p, ¯ λ  (see Fig. 1 and [18]).  3 Entanglement criteria.—   Eq. (6) provides the entan-glement criteria  L m  ( ρ )  <  Tr ρ 2   ⇒  ( ρ  is not m–separable ) ,  (13)which for  m  =  K   have been recently recognized in[19, 20]. In [19] they appear in the form of a nonlin-ear entanglement witness  L K   ( ρ )1l  −  ρ , while in [20] amore general object (see Eq. (12) from [20]) that con-tains  L K   ( ρ )  as a special case has been introduced. Theabove situation is similar to the case of   purity–entropy  entanglement criteria [21] given in terms of the Rényientropy  H  α , which for  α  = 2  was shown [7] to establishthe lower bound for the concurrence [22]. Let us empha-size in passing that the criteria (13) are strong enough todetect bound entanglement [18] of a concrete family [23].Let us now study the entanglement criteria (13) in thecase of a bipartite  M  × N   system. A general mixed stateof such system can be written as ρ  = 1 MN   I  M   ⊗ I  N   + k M M  2 − 1  i =1 q  i σ i ⊗ I  N   (14) +  k N N  2 − 1  j =1  p j I  M   ⊗ ˜ σ j  + k M  k N M  2 − 1  i =1 N  2 − 1  j =1 B ij σ i ⊗ ˜ σ j  , where  σ i ,  i  = 1 ,...,M  2 − 1  and  ˜ σ j ,  j  = 1 ,...,N  2 − 1  aretraceless, hermitian generators of   SU   ( M  )  and  SU   ( N  ) groups respectively, normalized such that Tr σ i σ i ′  = 2 δ  i ′ i and Tr ˜ σ j ˜ σ j ′  = 2 δ  j ′ j . For further convenience we set k M   =   MM  − / 2  where  M  −  =  M   − 1  and similarly for k N  . In the above representation the state is describedby two Bloch vectors  p ,  q  of the partially reduced statesand the   M  2 − 1  ×  N  2 − 1   correlation tensor  B  [3].Let us recall that the Bloch vector  q  belongs to thespace  B  ( M  )  defined by the constraints  q  ·  q  = 1  and 2( M   − 2) q  =  k M  Tr  ( q · σ ) 2 σ   [24, 25], and an equiv-alent definition holds for  p ∈B  ( N  ) .The pure separable state  | φ  φ |  present in (5) can becompletely characterized by a couple of Bloch vectors v  ∈ B  ( M  )  and  w  ∈ B  ( N  ) . In that representation theproduct numerical radius  L ( ρ )  reads 1 + max v , w  ( M  − v · q  + N  − w ·  p + M  − N  − v · B w ) MN  .  (15)The above maximization over  ( v , w )  ∈ B  ( M  ) × B  ( N  ) can be efficiently performed numerically even for largersystems, eg.  M,N   = 10 . To get however a deeper in-sight we provide in the Supplemental Material [18] thefollowing upper bound for (15): L ( ρ ) ≤  1 + N  −   p  + M  −  q  + M  − N  −   ξ  1  ( C ) MN  ,  (16)where  ξ  1  ( C )  denotes the largest eigenvalue of  C = B T  B or  C  =  BB T  (both quadratic matrices  B T  B  and  BB T  possess the same trace and nonzero eigenvalues). Thepurity of   ρ  can also be easily computed:Tr ρ 2 = 1 + N  −   p  2 + M  −  q  2 + M  − N  − Tr C MN  .  (17)In fact, if any upper bound on  L m  ( ρ )  is smaller thanTr ρ 2 then the condition (13) is satisfied. In the abovecase our entanglement test can thus be rewritten as: If  M  −  q  (1 − q  )+ N  −   p  (1 −  p  )+ M  − N  − f   ( C )  <  0 , (18)where  f   ( C ) =   ξ  1  ( C ) − Tr C , then the mixed state  ρ is entangled. Note that the above criterion is invariantunder local unitary operations U  A ⊗ U  B  what follows fromthe fact that a unitary rotation  UXU  † of any matrix  X  is an isometry in the Hilbert-Schmidt space. This meansthat for a given state  ρ AB  entanglement of its all  U  A ⊗ U  B transformations is detected with the same efficiency.In order to investigate the performance of the new cri-teria (18) we shall use the state  ρ  p,λ  ≡ ρ  p, λ  given by Eq.(11) with  d  = 2 , so that  λ  = ( λ, 1 − λ ) . This state isdescribed by  p  =  q  = (0 , 0 ,z ) , with  z  = (1 −  p )(2 λ − 1) ,and  B = (1 −  p ) diag ( η,η, 1)  with  η  = 2   λ (1 − λ ) . Bya direct substitution and comparison with (12) one cancheck that the bound (16) becomes tight.From the PPT criteria we know that  ρ  p,λ  is separa-ble when  p  ≥  1 − (1 + 2 η ) − 1 . In the maximally entan-gled case  λ  = 1 / 2  the threshold for separability is thus  p sep  = 2 / 3 . According to the criteria (18) the state  ρ  p,λ  isentangled for  p ≤ 4(1 − max[ λ ;1 − λ ]) / 3 . Note that for λ  = 1 / 2  we obtain the separability threshold  p  = 2 / 3 , sothat a full range of entangled Werner states is detected.This conclusion remains valid for an arbitrary dimension d , where [26]  p  = 0 =  q  and  C  is the identity matrixmultiplied by  (1 −  p ) 2 / ( d − 1) 2 .In Fig. 2 we compare the criteria (18) (dashed greencurve) and the purity–entropy test [21] (dotted bluecurve) given by the condition Tr ρ 2 A/B  ≥  Tr ρ 2 satisfiedwhen  ρ  is separable. Here  ρ A/B  denote density operatorsof single subsystem  A  and  B  respectively. In the neigh-borhood of the maximally entangled state ( λ  = 1 / 2 ) thecriteria (13) outperform the purity test.The above criteria has somehow built in the purity re-quirement but, as we have seen above, its power does notnecessarily depend on how pure the state is. It mighthowever be sensitive to the degree of purity of the ele-ment in the mixture that is responsible for entanglement.To study this possibility we check the family of   U   ⊗ U  invariant  d ⊗ d  srcinal Werner states [27]. This familyis defined by  ρ ( α,d ) = ( I  d ⊗ I  d  + αV  ) /  d 2 + αd   withthe swap operator  V  , a real parameter  α  ∈  [ − 1 , 1]  andsharp entanglement condition  α  ∈  [ − 1 , − 1 /d )  followingfrom PPT test. The states are known to have  p  = 0 =  q ,what can be seen immediately by considering their par-tial transpose. With the help of formula Tr ( A 1 ⊗ A 2 V  ) = Tr ( A 1 A 2 )  [27] the matrix  B  =  αdI  d / ( d 2 +  αd )( d  −  1)  4 FIG. 2: (color online). The plane of a family of two–qubitstates  ρ p,λ  (points inside gray circle) in the polar coordinate r  = 1 −  p ,  θ  = 2arccos √  λ  . States in the shaded regionbounded by solid red curve are separable (PPT). Entangledstates outside dashed green curve and outside dotted bluecurve are detected by the criteria (18) and the purity testrespectively. can be easily found. The condition (18) reports entan-glement for  α  ∈  [ − 1 , −  dd +2 )  converging to the single-point extreme  α  =  − 1  rather than the entanglement-corresponding interval  [ − 1 , − 1 /d )  with  d  → ∞ . UsingEq. (15) we see however that ( d − 1) 2 max v , w v · B w  =  α ( d − 1) d + α  max v , w v · w  = −  αd + α, (19)provided that  α <  0 . The last equality appears since if  v , w  ∈ B  ( d )  then  − 1 / ( d − 1)  ≤  v  ·  w  ≤  1  [24]. Withthe above result we recover the entanglement condition α < − 1 /d . Experimental advantages.—   The entanglement test(18) can be successfully used provided that  p ,  q  and  C are known. In fact, in order to determine  ξ  1  ( C )  all ma-trix elements of the positive, symmetric matrix  C  mustbe found, what in principle requires quantum tomogra-phy. In the two–qubit case, our criteria while faithful onthe family of Werner states, will always be less practicalthan the PPT condition. Our aim is thus to reduce thenumber of necessary parameters. To this end we shall up-per bound the function  f   ( C ) , so that the upper bound g ( C ) ≥ f   ( C )  depends on less number of matrix entries.If inequality (18) is satisfied with  f   ( C )  substituted by g ( C )  then the state in question is obviously entangled.In order to achieve this goal we shall distinguish the ma-trix elements of   C  to be measured and maximize  f   ( C ) with respect to the remaining parameters. Performingthe maximization we shall preserve the positivity of  C .Let us explain the above approach using an example of two qubits, so that  C  is a real, symmetric,  3 × 3  matrixgiven by six parameters:  C  11 ,  C  12 ,  C  22 ,  C  13 ,  C  23 ,  C  33 .Assume that we would like to measure  C  11 ,  C  12 ,  C  22  andoptimize  f   ( C )  with respect to  C  13 ,  C  23 ,  C  33 . We obtain: f   ( C ) ≤ g ( C  11 ,C  12 ,C  22 ) ≡  max C  13 ,C  23 ,C  33 f   ( C ) .  (20)For the two–qubit Werner state we have:  C  11  =(1 −  p ) 2 =  C  22 ,  C  12  = 0 , so that the condition for  C to be positive reads: C  213  + C  223  ≤ C  33  (1 −  p ) 2 .  (21)After analytical optimization [18] we find g ( C  11 ,C  12 ,C  22 ) ≡ g (  p )  of the form: g (  p ) =  (1 −  p )(2  p − 1)  for  0 ≤  p ≤  1214  − (1 −  p ) 2 for  12  ≤  p ≤ 1 .  (22)According to the test (18) the Werner state is detectedto be entangled if   g (  p )  <  0 , so that for  p <  1 / 2 . Thisis up to now the best known threshold value for entan-glement verification of the family of the Werner states,obtained without resorting to quantum tomography. Letus remind that the threshold value given by the puritytest is  p  = 1 − 1 / √  3 ≈ 0 . 4226 .In fact,  10  parameters suffice (see the SupplementalMaterial [18] for explicit relations between the desiredand measured parameters) to determine    p  ,   q   and C  11 ,  C  12 ,  C  22 . It is once again a huge experimental ad-vantage, as in order to measure the global purity Tr ρ 2 of atwo–qubit state one needs  12  parameters. This improve-ment could be obtained because of the interplay betweenthe purity Tr ρ 2 and the product numerical radius  L ( ρ ) .At the end let us analyze the  K  –qubit state ̺ (  p ) = (1 −  p ) | Φ K   Φ K  | +  p 2 K  I  ⊗ K  ,  (23)where  | Φ K    =  | 0  ⊗ K  + | 1  ⊗ K   / √  2 . The high symme-try of the above state provides that  L m  ( ̺ ) = (1 −  p ) / 2+  p/ 2 K  for all  m  ∈ { 2 ,...,K  } , what implies that allbounds  R m  ( ̺ )  capture the  genuine multipartite en-tanglement   of   ̺  associated with  m  = 2 . In fact, R m  ( ̺ ) = 0  leads to the biseparability threshold  p gme  =1 / 2  1 − 2 − K   , which according to [28] is optimal.It is a great pleasure to thank Florian Mintert forhis fruitful comments. This research was supportedby the grant number IP2011 046871 (Ł.R.) of the Pol-ish Ministry of Science and Higher Education, and thegrants number: DEC–2012/04/S/ST6/00400 (Z.P.) and2011/02/A/ST2/00305 (K.Ż.) financed by Polish Na-tional Science Centre. A partial support from ECthroughthe project Q–ESSENCE (P.H.) is gratefully acknowl-edged.  5 ∗ Electronic address: rudnicki@cft.edu.pl[1] E. Schrödinger, Proc. Camb. Phil. Soc.  31 , 555 (1935).[2] A. Einstein, N. Podolsky, B. Rosen, Phys. Rev.  47 , 777(1935).[3] I. Bengtsson and K. 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Olkin,  Inequalities: Theory of Majorization and Its Applications   New York: Academic,1979.[18] See Supplemental Material at ... for the details and ex-amples.[19] G. Sarbicki,  J. Phys.: Conf. Ser.  104 , 012009 (2008).[20] P. Badziąg,  et al. ,  Phys. Rev. Lett.  100 , 140403 (2008).[21] R. Horodecki, P. Horodecki, and M. Horodecki,  Phys.Lett.  A 210 , 377 (1996).[22] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, andW. K. Wootters,  Phys. Rev.  A 54 , 3824 (1996).[23] P. Horodecki, M. Horodecki, and R. Horodecki,  Phys.Rev. Lett  82 , 1056 (1999).[24] M. S. Byrd and N. Khaneja,  Phys. Rev. A  68 , 062322(2003).[25] G. Kimura,  Phys. Lett. A  314 , 339 (2003).[26] J. I. de Vicente,  Quantum Inf. Comput.  7 , 624 (2007).[27] R. F. Werner,  Phys. Rev. A  40 , 4277 (1989).[28] O. Gühne and M. Seevinck,  New J. Phys.  12 , 053002(2010). ENTANGLEMENT ORDERING In [12] it was shown that for the state  ρ  p, ¯ λ  one can findthe exact formula for the geometric measure of entangle-ment (GME): E   ρ  p, ¯ λ   = 1 −  1 d  √ F   +   ( d − 1)(1 −F  )  2 ,  (24)where F   = 1 −  p  d 2 − 1  d 2  .  (25) FIG. 3: Parameter space for the generalized Werner statesof a  10 × 10  system. Red volume corresponds to the states ρ p ′ , λ  the entanglement of which is shown by our bound to belarger than this of reference state  ρ p, ¯ λ . Here  d  = 10  so that  p cr  = 10 / 11  and  Λ min  = 1 / 10 . Using Eq. (12) of our paper we can find that R ( ρ  p ′ , λ ) =   1 −  p ′ d 2  − (1 −  p ′ )Λ −   2  p ′ − (  p ′ ) 2  ( d 2 − 1) d . (26)The relation  R ( ρ  p ′ , λ )  > E   ρ  p, ¯ λ   is a sufficient condi-tion for  ρ  p ′ , λ  to be more entangled than  ρ  p, ¯ λ , which isconsidered to be our reference state.In the main paper we present the first nontrivial caseof   d  = 3 . The volume of states found by our lower boundfor  d  = 10  is shown in Fig. 3. BOUND ENTANGLEMENT Consider now an example of the two–qutrit state [23]: ρ α  = 27 | Ψ +  Ψ + | +  α 21 σ +  + 5 − α 21  σ − ,  (27)where | Ψ +  = ( | 00  + | 11  + | 22  ) / √  3 ,  (28)is now a two–qutrit maximally entangled state and σ +  =  diag (0 , 1 , 0 , 0 , 0 , 1 , 1 , 0 , 0) ,  (29) σ −  =  diag (0 , 0 , 1 , 1 , 0 , 0 , 0 , 1 , 0) .  (30)For  2  ≤  α  ≤  3  the state  ρ α  is separable, for  3  < α  ≤  4 ,it is entangled but PPT (bound entangled), while for 4  < α ≤ 5  the state is entangled and not PPT.A straightforward calculation yieldsTr ρ 2 α  = 37 + 2 α ( α − 5)147  .  (31)
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