Rotationally invariant bipartite states and bound entanglement

0.22 MB·English

by  Remigiusz Augusiak

Rotationally invariant bipartite states and bound entanglement
Remigiusz Augusiak and Julia Stasin´ska
∗
Faculty of Applied Physics and Mathematics, Gdan´sk University of Technology, Gdan´sk, Poland
(Dated: February 1, 2008)
We consider rotationally invariant states in CN1 ⊗CN2 Hilbert space with even N1 ≥ 4 and
arbitrary N ≥N , and show that in such case therealways exist states which are inseparable and
2 1
remain positive after partial transposition, and thus the PPT criterion does not suffice to prove
separability of such systems. Wedemonstrate it applyinga map developed recently byBreuer [H.-
P. Breuer, Phys. Rev. Lett 97, 080501 (2006)] to states that remain invariant after partial time
reversal.
PACSnumbers: 03.67.Mn
Keywords: Rotationallyinvariantstates;Boundentanglement; Partialtransposition;Separabilitycriteria
7
0
0
2

n I. INTRODUCTION
a
J
One of the most important problems of rapidly developing branch of science Quantum Information Theory [1] is
9
to determine whether a given quantum state is separable or entangled. We say that a given state ̺ acting on a finite
2
dimensionalproductHilbertspace isseparableorclassicallycorrelatedifitcanbe writtenasaconvexlinear
1 2
H ⊗H
1 combination of product density operators [2], i.e.,
v
2 ̺= p ̺(1) ̺(2), (1)
2 n n ⊗ n
n
2 X
1 where̺(1(2)) ( )forallnandp arenonnegativecoefficientsfulfillingthecondition p =1. Otherwisethe
0 n ∈B H1(2) n n n
7 stateiscalledentangledorinseparable. Animportantnecessaryandsufficientcriterionusedtocheckthe separability
P
0 of a state was developed by the Horodecki’s [3]. It applies the positive but not completely positive maps, precisely,
/ it states that a density matrix ̺ is separable if and only if the operator (I Λ)(̺) is positive for all positive but not
h ⊗
completely positive maps Λ: ( ) ( ). Even though making use of the criterion is not an easy task, together
p 2 1
B H →B H
- with the result of Peres [4] it provides a strong condition for separability, based on partial transposition. It says that
t ̺ is entangled if it has a nonpositive partial transposition. For low dimensional systems such as 2 2 and 2 3
n
⊗ ⊗
a this is also a sufficient criterion for separability [3], however, in general there exist states that have positive partial
u transposition (PPT) and simultaneously are entangled [5]. Such operators are certainly nondistillable [6] and belong
q to the class of bound entangled (BE) states.
:
v Aninterestingquestionisthuswhetherthereexistclassesofstates(exceptfor2 2and2 3systems)wherepartial
⊗ ⊗
i transpositionprovidesanecessaryandsufficienttestforseparability. Onecouldexpectthatinvarianceofstatesunder
X
certain group of symmetry would lead to some interesting results. Such states have a relatively simple structure and
r therefore have been studied extensively in the literature [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In particular, it was
a
shown recently that positive partial transposition is a sufficient criterion in the case of 2 N rotationally invariant
⊗
states [11, 12] and 3 N rotationally invariant states with integer total angular momentum [13, 14]. On the other
⊗
hand in Ref. [18] it was shown that for N N rotationally invariant systems with even N 4 there always exist
⊗ ≥
bound entangled states detected by certain map.
It is the purpose of the present paper to consider the separability of more general SO(3) invariant states. We
concentrate on N N systems with even N 4 and arbitrary N N and show that there is always a region
1 2 1 2 1
⊗ ≥ ≥
in the PPT set where states are bound entangled proving,thus, that in such systems positive partial transpositionis
only a necessarycondition for separability. We achieve this using a recently introduced positive indecomposable map
Φ given by Eq.(2) [18]. The map belongs to the class of indecomposable positive maps arising from the reduction
criterion[19, 20] which was studied in detail by Hall [21]. The actionof Φ on a givenoperatorB from (CN) is such
B
that
Φ(B)=(TrB) B ϑ(B), (2)
N
1 − −
∗Electronicaddress: jul˙[email protected]
2
where ϑ denotes the time reversal operation and is a N N identity matrix. The map Φ is positive if N is an
N
even number and therefore leads to the following1necessary×condition for separability on CN1 CN2 Hilbert space
⊗
with even N :
1
Φ (̺) (Φ I)(̺) 0. (3)
1
≡ ⊗ ≥
We show that when applying the Breuer’s map to rotationally invariant states one can restrict to a family of
operatorsinvariantunder partialtime reversal. Moreoverthe resultsobtainedfor ϑ -invariantstatescanbe extended
1
to the PPT rotationally invariant states. We also show that in our case the set of states invariantunder partial time
reversal can be easily found and therefore it is possible to prove the existence of BE states in higher dimensional
rotationally invariant systems.
The paperis organizedasfollows. InSec. II we giveabriefdescriptionofrepresentationsofSO(3)-invariantstates
andactionofcertainpositivebutnotcompletely positivemapsonthis classofstates. InSec. III we presenta special
case of 4 N and on the basis of this example analyze more general case N N with even N 4 and arbitrary
1 2 1
⊗ ⊗ ≥
N N . In particularwe showthat to determine the set ofBE states detected by the Breuer’smap one canrestrict
2 1
≥
to the subset of states invariant under partial time reversal and then extend the result to the set of PPT states.
The results also suggest that the map (2) could provide both necessary and sufficient separability criterion for states
invariant under partial time reversal.
II. ROTATIONALLY INVARIANT STATES
A. Representations
Assume that we are given a bipartite quantum state represented by a density matrix ̺ acting on a finite product
Hilbert space 1 2 =CN1 CN2 such that N1 N2. The angular momenta of the particles are j1 =(N1 1)/2
H ⊗H ⊗ ≤ −
and j = (N 1)/2, respectively. The characterization of entanglement of such states is in general a difficult
2 2
−
task. However, it simplifies when one impose some constraints on considered density matrices. Hereafter we shall
be assuming that ̺ is invariant under the action of SO(3) group. More rigorously this assumption means that the
following relation holds
(j1)(R) (j2)(R),̺ =0 (4)
D ⊗D
h i
for all proper rotations R from SO(3) group. Here (j1(2))(R) denote the unitary irreducible representation of the
D
SO(3) groupon the respective state spaces. In view of the Shur’s lemma, the states which obey the above condition,
can be written in the following form
1 j1+j2 α J
J
̺= P , P = JM JM , (5)
J J
√N N √2J +1 | ih |
1 2
J=X|j1−j2| MX=−J
where JM arethe commoneigenvectorsof the squareof totalangularmomentum operatorandof its z-component.
Thus t|he siet of rotationally invariant states is isomorphic to a proper subset of vectors α from RN1, which are
nonnegative, i.e., α 0 and fulfil the following normalization condition
J
≥
j1+j2 2J +1
α =1. (6)
J
N N
J=X|j1−j2|r 1 2
On the other hand, as proposed by Breuer in Refs. [13, 14], each rotationally invariant state can be written as a
combination of Hermitian operators
K
QK = TK(1,)q⊗TK(2,)q†, K =0,1,...,2j1, (7)
q= K
X−
where T(i) are the components ofan irreducible tensor operator[22]. Subscript K is the rank of this tensor operator
K,q
andq takeson2K+1values,q = K,...,K. TheoperatorsQ arerotationallyinvariantandformacompletesetin
K
−
3
the space of rotationally invariantstates. Thus any SO(3) invariant state canbe written as their linear combination,
i.e.,
1 2j1 β
K
̺= Q . (8)
K
√N N √2K+1
1 2
K=0
X
Like in the case of PJ representation the state is uniquely characterized by a parameter vector β RN1. Due to the
∈
fact that tensor operators Q are traceless for K = 0 the normalization condition gives β = 1. As it was shown in
K 0
6
Ref. [14], the N -dimensionalvectors α and β are relatedby a linear transformationβ =Lα with matrix elements of
1
the orthogonalmatrix L given by
j j J
L = (2K+1)(2J +1)( 1)j1+j2+J 1 2 . (9)
KJ − j2 j1 K
(cid:26) (cid:27)
p
Thereasonforintroducingtwodifferentrepresentationsistheirconvenienceforcertainpurposes. P -representation
J
is suitable to determine the state space, whereas the operation of partial time reversal, unitarily equivalent to the
partial transposition, can be performed much more easily in the Q basis. Thus we apply the latter to identify the
K
set of rotationally invariant states with positive partial time reversal which is the same as set of states that remain
positive after partial transposition.
B. Separability
Todetermine the setofrotationallyinvariantPPTstates (fromnowondenotedby )weapply amapunitarily
ppt
R
equivalenttopartialtransposition,i.e.,thepartialtimereversalmap[14]. Itismoreusefulinourcaseduetothe fact
that, unlike the partial transposition, it preserves the SO(3) invariance of the state. The operation of partial time
reversalacts as follows
ϑ (B)=(ϑ I)(B), (10)
1
⊗
where ϑ is the time reversal map acting on a given operator B as ϑ(B)=V BT V . Here V is a unitary rotation by
†
the angle π about the y-axis. As mentioned before, the action of partialtime reversaloperator is especially simple in
the Q representation
K
ϑ :β ( 1)Kβ , (11)
1 K K
→ −
which follows directly from the relationϑ (Q )=( 1)KQ and equation(8). Let us recallone more property of ϑ
1 K K 1
−
map, i.e., preservationof separability. It means that if ̺ is separable then ϑ (̺) is also separable. We apply this fact
1
in the next section to prove the separability of certain PPT states.
SincethesetofseparablestatesisasubsetofPPTstates,oneshouldhavetheconditionunambiguouslydetermining
theseparabilityofanalyzedstates. Amethodusefultoidentifytheseparableinvariantstateswasdevelopedin[7]and
isbasedonthe actionofaprojectionsuper-operatorΠ. The super-operatorforSO(3)groupofsymmetry,considered
in this paper, projects each state onto a rotationally invariant state space as follows
j1+j2 Tr(P ̺) 2j1 Tr(Q ̺)
J K
Π(̺)= P = Q , (12)
J K
2J +1 2K+1
J=X|j1−j2| KX=0
preserving the separability of a state. The last property allows us to identify the set of separable states with a
sep
R
convex linear combination of Π-projections of pure normalized product states φ(1)φ(2) CN1 CN2:
| i∈ ⊗
=conv ̺:̺=Π P P , (13)
Rsep φ(1) ⊗ φ(2)
where Pφ(i) is the projection onto the state |φ(i)i.(cid:8) Consequ(cid:0)ently to find(cid:1)s(cid:9)eparable states among PPT states it is
enough to show that for the extreme points of (denoted by ̺ ) there exist pure normalized product states
sep ext
R
satisfying relation Π(P P ) = ̺ . However to find the ”suspected” extreme points ̺ one should have a
φ(1) ⊗ φ(2) ext ext
criterion detecting at least some bound entanglement, so that one could look for separable states in a smaller set.
In our case the map developed recently in [18] proved to be especially useful. It leads to the necessary criterion
(3). Using this criterion we show that the set of PPT rotationally invariant states for even N N always contains
1 2
≤
entangled states. To prove this it is enough to check the criterion on the subset of ϑ -invariant states
1
1
̺ = [̺+ϑ (̺)], ̺ , (14)
inv 1 ppt
2 ∈R
4
since the action of Breuer’s map is the same for all states ̺ satisfying above equation and equivalent to
1
Φ (̺)= Tr (̺) ̺ ϑ (̺)= 2̺ . (15)
1 1N1 ⊗ 1 − − 1 N 1N1 ⊗1N2 − inv
2
In the second equality appearing in Eq. (15) we make use of the fact that Tr (̺) = (1/N ) for all rotationally
1 2 1N2
invariant states. As a result of the above relations we can simplify the criterion (3) to
1
2̺ 0. (16)
N 1N1 ⊗1N2 − inv ≥
2
It automatically follows that if ̺ fulfils the criterion (16) then all ̺’s from equation (14) satisfy (3).
inv
III. ROTATIONALLY INVARIANT STATES AND BOUND ENTANGLEMENT
In this section we shall consider rotationally invariant states in the context of bound entanglement. We show that
in CN1 CN2 Hilbert space with even N1 and arbitraryN2 N1 there always exist PPT entangled states.
⊗ ≥
At the very beginning to provide some insight into the state space structure we focus on the case of the 4 N
system, since it may be easily visualized in R3. ⊗
A. 4⊗N system
Let us now restrict our attention to the case of N = 4 (j = 3/2) and arbitrary N = 2j +1 N from now on
1 1 2 2 1
≥
denoted by N. In this paragraphwe will also denote j by j. The total angular momentum of the system J takes on
2
the values J =j 3/2,...,j+3/2 and thus every rotationally invariant state is represented by 4 coordinates which
−
satisfy the conditions α 0 and
J
≥
j+3/2
2J +1
α =1. (17)
J
4N
r
J=Xj−3/2
By the normalization condition the number of independent parameters α describing a state can be reduced to
J
3. In order to characterize this set of rotationally invariant states we need to find its extreme points in α space.
Straightforwardcalculations lead to
4N 4N 4N 4N
A = ,0,0,0 , B = 0, ,0,0 , C = 0,0, ,0 , D = 0,0,0, .
α α α α
N 3 N 1 N +1 N +3
r − ! r − ! r ! r !
To determine the set of PPT states we transform the vectors α to β using the linear transformation L which matrix
elements are given by Eq. (9). In the considered case of 4 N the transformation matrix is of the form
⊗
N 3 N 1 N+1 N+3
N− N− N N
L= 12 −(3Nq−q3()N(5N(−N+3)−1()N1()N+N+1)2) −(N√q−55NN)√(+NN7++12) √(Nq5+NN5(+)N√7−N1−)2 −(3Nq−q1()N(5N(+N−3)+2()N1()N−N+1)3) . (18)
−q (NN5(+(NN1−)−(2N1))(+(NN2)−−(N12))+N3) 3√(N((NN−−22−2)N)9N)((N(NN+++121))) −−3√(N((NN−−21−1)N)9N)((N(NN+−+222))) q(N5N−N(3(N)N(+N+1−1))(2(N)N(+N+22−))1) 
 q q q q 
Atthesametimeweobserve,thatthe4-dimensionalvectorsβ areunambiguouslycharacterizedbythreecoordinates
since β =1 for all rotationally invariant density matrices. This allows us to restrict our considerations to the three
0
parameters β ,β ,β and visualize all considered sets in R3. The L transformation carried on the extreme points
1 2 3
gives us the vertices of the tetrahedron in the space of parameters β ,β ,β as follows
1 2 3
N +1 (N +1)(N +2) (N +1)(N +2)(N +3)
A = 3 , , ,
− rN −1 s(N −1)(N −2) s5(N −1)(N −2)(N −3)!
N +7 N +1 N 5 (N +1)(N +2) (N +2)(N +3)(N 3)
B = , − ,3 − ,
−N +1s5(N 1) −N +1s(N 1)(N 2) s5(N 2)(N 1)(N +1)!
− − − − −
5
N 7 N +1 N +5 (N 1)(N 2) (N 2)(N 3)(N +3)
C = − , − − , 3 − − ,
N +1s5(N 1) −N 1s(N +1)(N +2) − s5(N +2)(N 1)(N +1)!
− − −
N 1 (N 2)(N 1) (N 1)(N 2)(N 3)
D = 3 − , − − , − − − .
s5(N +1) s(N +2)(N +1) s5(N +1)(N +2)(N +3)!
-1 β
1
11
0 β --11 00
2 2 -1
1
C’ A2’ B 0 β2
B 1 C’ 1
β 0 E F’
3 E’
C D E F’ A’
β 0
D’ G 3
F G’ E’ D
-2
B’ -1 D’ G C G’
F
B’
A -1 0 1 A
β
1
β1 11 β1
--11 00 --11 00 1
-1 -1
B 0 β2 B 0 β2
1 C’ 1 1 C’ 1
E
E F’ F’
β 0 A’ A’
3 β 0
3
D D
E’ G’ G’
-1 D’ G C -1 D’ G
E’
F C
F
B’ A
B’
A
FIG. 1: The set of rotationally invariant states in C4 ⊗ CN (tetrahedron ABCD), its image under partial time reversal
(A′B′C′D′),andintersectionofthesets(DD′EE′FF′GG′)forvariousvaluesofj,namelyj =3/2(leftupper),j =5/2(right
upper),j =7/2(leftlower),j =11/2(rightlower). ThePPTregionbecomeslargerwiththegrowthofdimensionofthesecond
subsystemand thePPT criterion fails todetect manyentangled stateswith thegrowth ofasymmetrybetween thedimensions
of subsystems.
Now we can find the image of the tetrahedron ABCD under the action of ϑ . The extreme points of this set
1
transform according to the relation (11), consequently only the sign of β and β coordinates change. We denote
1 3
the points corresponding to A, B, C, D by A, B , C and D , respectively. To identify the set of PPT states we
′ ′ ′ ′
need to find the intersection of the tetrahedrons ABCD and AB C D . Straightforward calculations lead to the set
′ ′ ′ ′
DD EE FF GG (see Fig. 1). One may easily verify that the pairs of points (E,E ),(F,F ),(G,G) are the images
′ ′ ′ ′ ′ ′ ′
ofeachotherunderϑ (e.g. E =ϑ (E))andthus itis sufficientto findcoordinatesofthe pointsE, F,andG,which
1 ′ 1
are as follows
6
N 1 (N 1)(N 2) (N 1)(N 2)(N 3)
E = − , − − ,3 − − − ,
−s5(N +1) −s(N +1)(N +2) s5(N +1)(N +2)(N +3)!
9(N 4) N 1 N +4 (N 1)(N 2) 13N +28 (N 1)(N 2)(N 3)
F = − − , − − , − − − ,
−7N 20s5(N +1) 7N 20s(N +1)(N +2) − 7N 20 s5(N +1)(N +2)(N +3)!
− − −
3(N +1) N 1 N +2 (N 1)(N +1) N 7 (N 1)(N +1)(N +2)
G = − , − , − − .
− N +5 s5(N +1) N +5s(N 2)(N +2) −N +5s5(N 3)(N +3)(N 2)!
− − −
ThedescribedsetsarepresentedinFig. 1forN =4, 6, 8, 12. OnecouldseethattheoverlapoftetrahedronsABCD
andAB C D growswith the increaseofN. This immediately leads to the conclusionthatthe NPT setshrinks with
′ ′ ′ ′
increasing N. Now applying the methods described in the previous section we characterize the separability of PPT
states.
Firstly we apply the Breuer’s criterion to the subset of ϑ -invariant states, namely states represented by points
1
lying on the line E G (see Fig. 2)
′′ ′′
̺ (t)=(1 t)E +tG , t [0,1]. (19)
inv ′′ ′′
− ∈
The criterion (15) applied to ̺ (t) leads to the operator with α given by:
inv j 3/2
−
N 3 (N 1)(N +4)
α (t)= − 1 − t . (20)
j−3/2 r N (cid:18) − (N −2)(N +5) (cid:19)
Asitmaybeeasilyverifiedthatremainingα ’sarenonnegativeforallvaluesoftandthusthenonpositivitycondition
J
reduces to
(N 1)(N +4)
1 − t<0. (21)
− (N 2)(N +5)
−
States ̺(t) for t satisfying the inequality are entangled. The parameter t for which the LHS of (21) equals zero
represents the point D lying in the middle of the line DD . At the same time the inequality is satisfied by t = 1
′′ ′
which implies that points between D and G , including G represent entangled states.
′′ ′′ ′′
β3 β1 β3 β1
-101 11 0 --11 1 -101 11 0 --11 1
G’ G’’ G G’ G’’ G
0.5 ΓD D’’ D’0.5
Γ D F’D’’ D’
F F’
0 β2 F 0 β2
0 0
-0.5 -0.5
E
E’ E’’ E
-1 -1
E’’
E’
FIG.2: Thesetof rotationally invariantPPTstatesfor j =3/2 and3,respectively. The γ plain labelled inthepicturesisthe
boundary of the region in which entanglement is detected by Breuer’s map. BE states which can be detected by the Breuer’s
map lie above theγ plain in theset of PPT operators.
Now we denote by γ the plain perpendicular to the line E G and intersecting point D (Fig. 2). From relation
′′ ′′ ′′
(16) and a remark below it we can immediately conclude that all points lying in the PPT set above the γ plain are
entangled.
NowwemovetothepointsforwhichΦ (̺) 0(suchpointslieonandbelowγ)andusetheΠprojectionargument
1
≥
introduced in Sec II.B to show the separability of D and E (the separability of D , E results immediately from the
′ ′
properties of ϑ map, namely preservation of separability).
1
7
Let us first recall the special symmetric case 4 4 solved in [10, 13]. In this case points D, D , F, F lie on the
′ ′
⊗
γ plain and are all separable. The points E, E are also separable and thus the γ plain is a boundary between the
′
separable and bound entangled region.
β
3
-101 1
G’ G’’ G
0.5
D D’’ D’
F’
0 β
F 2
-0.5
E
E’ E’’ -1
1 0 -1
β
1
′ ′
FIG. 3: The set of rotationally invariant PPT states for j = 2. The borders of the minimal separable region DDEE are
marked with thick lines.
In the general case 4 N, the full characterization of the separable set is more difficult. Following the method
developed by Breuer [14]⊗we show that certain separable pure product states φ(1)φ(2) are projected by Π (see (12))
| i
onto points D and E. For this purpose we take the functionals
K
4N
β˜K[φ(1),φ(2)]= 2K+1 hφ(1)|TK(1,)q|φ(1)ihφ(2)|TK(2,)q†|φ(2)i, (22)
r q= K
X−
which, applying relation (12), map pure product states into the β space. Namely the β˜ functional is the β
K K
coordinate of a pure product state φ(1)φ(2) after action of Π projection. The matrix elements of T used in the
K,q
| i
above equation are related to Wigner 3-j symbol by the formula [22]:
j j K
j,mT j,m =√2K+1( 1)j m (23)
h | K,q| ′i − − m m′ q
(cid:18) − (cid:19)
anTdobpyrtohveertehlaattiEoniTsK†se,qpa=ra(b−le1)iqtTsKu,ffi−qceosnteoccaonnsdiedteerrmthienesttahteesmatrix elements of the conjugate.
3 1 N 1 N 1
φ˜(1) = , , φ˜(2) = − , − . (24)
| i 2 −2 | i 2 2
(cid:12) (cid:29) (cid:12) (cid:29)
(cid:12) (cid:12)
Due to the selection rules for 3-j symbol t(cid:12)he only nonvanishing(cid:12) elements of the sum in (22) are those with q = 0,
(cid:12) (cid:12)
hence
4N N 1
β˜1[φ˜(1),φ˜(2)] = r 3 hφ˜(1)|T1(,10)|φ˜(1)ihφ˜(2)|T1(,20)†|φ˜(2)i=−s5(N−+1),
4N (N 1)(N 2)
β˜2[φ˜(1),φ˜(2)] = r 5 hφ˜(1)|T2(,10)|φ˜(1)ihφ˜(2)|T2(,20)†|φ˜(2)i=−s(N −+1)(N −+2),
4N (N 1)(N 2)(N 3)
β˜3[φ˜(1),φ˜(2)] = r 7 hφ˜(1)|T3(,10)|φ˜(1)ihφ˜(2)|T3(,20)†|φ˜(2)i=3s5(N−+1)(N−+2)(N−+3).
These are indeed the coordinates of E which implies that E represents the separable state.
8
To provethe separability of D we follow the argumentsgivenby Breuer in [14]. Since D is anextreme point of the
set of SO(3) invariant states it has a single nonzero coordinate in α-space. The nonzero coordinate corresponds to
the largest J =J =j +j . Consequently the spectral decomposition of the state always contains the projection
max 1 2
on J ,J whichis a separablepure productstate j ,j j ,j . This implies thatforarbitraryj ,j one can
max max 1 1 2 2 1 2
| i | i⊗| i
find a separable state j ,j j ,j which is mapped under Π to the state represented by point D.
1 1 2 2
| i⊗| i
The tetrahedron with vertices DD EE in Fig. (3) represents the minimal separable set (see also [10]). The
′ ′
separabilityofthepointsF andF aswellaspointslyingbetweenγ andthetetrahedronDD EE withinthePPTset
′ ′ ′
is still undetermined. However in the case of 4 5 considered by Hendriks in [10] the set of separable states is given
⊗
not only by points DD EE but extends to the regionnear points F and F . The separable states found numerically
′ ′ ′
by Hendriks lie on lines E F and EF . This result allows us to suspect that for N 5 there also exist separable
′ ′
≥
states outside the tetrahedron DD EE .
′ ′
B. Bound entanglement in higher dimensional systems
Let us now move on to the general case of CN1 CN2 Hilbert space with even N1 4 and arbitrary N2 N1.
⊗ ≥ ≥
For the purpose of proving the existence of BE in this state space we confine ourselves to the subset of ϑ -invariant
1
operators.
A set of rotationally and ϑ -invariant states ( ) can be easily determined in the β-space because the action of
1 inv
R
ϑ in this representation is just a change of sign for β with odd K. As a result the states which remain unchanged
1 K
after the partial time reversal must have coordinates indexed by odd K’s equal zero. Moreover, a normalized state
has always β0 = 1, so each rotationally and ϑ1-invariant state in CN1 CN2 with even N1 is fully characterized by
⊗
the set of (N 2)/2 parameters β (K =1,2,...,(N 2)/2), i.e.,
1 2K 1
− −
̺ =(1,0,β ,0,β ,...,0,β ,0). (25)
inv 2 4 N1−2
Todeterminetherangeoftheβ parametersforwhichvectorβrepresentsadensityoperatorweimposetheconstraint
2K
of positivity. This can be easily done in P representation. Firstly we employ the N N matrix L 1 =LT (9) to
J 1 1 −
×
find the α coordinates of a ϑ -invariant state. This gives
J 1
αJ = (−1)N1+N22−2+J√2J +1 ((NN12−11))//22 ((NN21−11))//22 J0
(cid:16)(cid:26) − − (cid:27)
(N1−2)/2 (N 1)/2 (N 1)/2 J
+ √4K+1 (N1−1)/2 (N2−1)/2 2K β2K . (26)
2 1
KX=1 (cid:26) − − (cid:27) (cid:17)
The positivity condition α 0 leads to the set of inequalities for β parameters. The β parameters describing
J 2K 2K
≥
the SO(3), ϑ -invariant states lie between the hyperplains given by equations α =0, where α ’s are given by (26).
1 J J
Our further reasoning follows from the structure of 4 N space, where Breuer’s map detects the entanglement of
⊗
ϑ -invariant states above the D point.
1 ′′
The Breuer’s map for rotationally,ϑ -invariantstates has the form presentedin Ineq. (16). The identity matrix in
1
β space is a vector with β = N N and other coordinates equal zero. This follows from Eq. (8) and the fact that
0 1 2
Q operatoris proportionalto identity (Q =(N N ) 1/2 ). Thereforethe β vectorobtainedafter applying
0 0 1 2 − 1N1⊗1N2
the Φ map and normalization is:
1
2 2 N
1
− ,0,β ,0,β ,...,0,β ,0 . (27)
2−N1 (cid:18) 2 2 4 N1−2 (cid:19)
One should notice that the vector is again ϑ -invariant and that the Φ map simply shifts the point with respect to
1 1
point (1,0,...,0).
We define the Γ hyperplain:
1 2 (N1−2)/2 (N 1)/2 (N 1)/2 (N N )/2
Γ: √N1N2 +(−1)N2N1−2 K=1 √4K+1(cid:26)(N21−−1)/2 (N12−−1)/2 2−2K1 (cid:27)β2K =0. (28)
X
Itcrosscutsthe PPTϑ -invariantsetinsuchwaythatpoints lyingononesideofithavealwaysanegativeeigenvalue
1
after the action of map (16). To see this we apply the Φ map to points lying on Γ. The obtained hyperplain is the
1
boundary of given by α = 0. So points lying on one side of Γ are always mapped by Φ onto points
outside the seRtionvf positive ope(Nra2t−oNrs1,)/w2hereas points from the other side remain positive. 1
9
Now we prove that Γ always go through the interior of the . To do it we find the analog of the point
inv
D from the case 4 N. We denote it by D˜ . It correspondsRto the state [̺ +ϑ (̺ )]/2, where ̺ =
′′ ′′ max 1 max max
⊗
[N N (N +N 1)] 1/2P . D˜ is certainly ϑ -invariant since it has the form (14). It has all nonzero
α 1’s,2whi1ch imp2l−ies th−at it b(Nel1o+nNg2s−t2o)/t2he in′t′erior of (r1ecall that the boundaries of the set are given by α =0).
J inv J
R
The even (and thus nonzero) coordinates representing the D˜ state in β space are:
′′
β2(DK˜′′) = N1N2(4K+1)(−1)N2 ((NN12−11))//22 ((NN21−11))//22 (N1+N2K2−2)/2 , (29)
(cid:26) − − (cid:27)
p
with K = 1,2,...,(N 2)/2. The coordinates fulfill the equality (28) (the proof is given in appendix A) so D˜
1 ′′
−
always lies on the Γ plain. This implies that the Γ hyperplain cuts through the set of PPT ϑ -invariant states. The
1
above arguments prove the existence of BE states for all ϑ1-invariantstates from CN1 CN2 Hilbert space with even
⊗
N and arbitrary N N . Moreover, by the relation (14) our arguments immediately extend to the set of PPT
1 2 1
≥
states.
We illustrateourpreviousconsiderationsbyanexamplefrom6 N space(see Fig. 4). Inthiscasethe hyperplains
⊗
defined by (26) become straight lines. One should notice that the BE region (gray area in Fig. 4) detected by the
Breuer’s map (3) shrinks with the growth of N.
β β
4 4 β
α =0G 1.5 α4=0 α6=0 1.5 α5=0 4
5 1 1 α3=0 0.275 G α7=0
α =0 0.25
2
0.225 Ž
0.5 0.5 D’’
Ž Ž 0.2
D’’ β D’’ β 0.175 α8=0
2 2
-1.5 -1 -0.5 0.5 1 1.5 -1α.5=0-1 -0.5 0.5 1 G1.5 0.15
2
-0.5 -0.5 0.125
α =0 α3=0 α =0 α =0 α =0 0.9 0.95 1.05 1.1 1.15 1.2 β2
1 -1 0 4 -1 1
FIG.4: Thesubsetofrotationallyandϑ -invariantstatesfordimensions6⊗6,6⊗8and6⊗14,respectively. Theconstantlines
1
arethebordersoftheϑ1-invariantset,namelyαJ =0(J =(N2−6)/2,...,(N2+4)/2),thethickconstantlineisα1/2(N2−6) =0,
andthethickdashedlineisthelineΓdefinedbyEq. (28). Thegrayareaistheregioninϑ -invariantsetwhereboundentangled
1
states are detected by theBreuer’s map. The point where thedashed lines intersect is thepoint D˜′′.
IV. CONCLUSION
InthepresentpaperwehaveshownthatinCN1 CN2 HilbertspaceofrotationallyinvariantstateswithevenN1 4
⊗ ≥
and arbitrary N N there always exist bound entangled states among PPT states. This means simultaneously
2 1
≥
that partial transposition provides only a necessary criterion for separability in these cases. However,the problem is
still unsolved for rotationally invariant states acting on CN1 CN2 with odd N1 5 and arbitrary N2 N1. Some
⊗ ≥ ≥
preliminary results suggest that bound entanglement do exist in such systems. However, this issue requires further
research.
We have described in more details the cases of N = 4 and N = 6. The provided examples give the geometrical
1 1
description of the action of the map developed recently by Breuer [18] in the space of rotationally invariant states.
MoreovertheyrevealthattheBEPPTregiondetectedbyBreuer’smapshrinkswiththegrowthofasymmetrybetween
the subsystems. Moreover, the example of 4 N shows that in this case the Breuer’s map provides a necessary and
⊗
sufficient criterion for SO(3), ϑ -invariant states. This suggests that for higher dimensional states of the form (25)
1
the region of entanglement might be unambiguously determined with the criterion based on the Φ map.
1
Let us discuss entanglement detection of the rotationally invariant states in the context of a recently proposed
separability criterion involving determinant of the partially transposed density matrix [23]. Namely, it says that a
given two-qubit state ρ is separable if and only if
detρPT 0, (30)
≥
10
where PT denotes standard partial transposition with respect to an arbitrary subsystem. As shown in Ref. [11] and
then in Ref. [14] partial transposition is also a sufficient criterion for separability for rotationally invariant states
acting on C2 CN. Therefore one may expect that the above criterion applies also to this class of states. Using
⊗
simple arguments it may be shown that this, indeed, is the case for even N.
Suchstatesaswellastheirpartialtimereversalcanbewrittenintheform(5)andtheireigenvaluesareproportional
to α ’s. For such systems α of a state after partial time reversal (unitarily equivalent to partial transposition)
J N/2
is positive. This means that if a given rotationally invariant state is entangled its partial time reversal must have
α <0,andtherefore,incase ofevenN, oddnumber ofnegativeeigenvalues(eigenvalues of2 N rotationally
(N 2)/2
inva−riant states with even N have odd degeneracy). of a density matrix after partial time reversal m⊗ust be negative.
Acknowledgments
Authors are grateful to P. Horodecki for fruitful discussions and commenting on the manuscript. Discussions with
M. Czachor are acknowledged. This work was supported by EU-IP Programme SCALA (contract No. 015714)
and Polish Ministry of Scientific Research and Information Technology under the (solicited) grant no. PBZ-MIN-
008/P03/2003.
APPENDIX A
In this appendix we show that point D˜ (29) indeed belongs to the Γ hyperplain (28). To show this we rewrite
′′
Eq. (28) in the following form (here for simplicity we use j instead of (N 1)/2 whenever it leads to shorter
1(2) 1(2)
−
formulas)
(−1)N2(NK1−=20)/2√4K+1(cid:26)jj12 jj21 j22−Kj1 (cid:27)β2K + √2NN1N2 2 =0. (A1)
X
Now inserting coordinates of D˜ given by Eq. (29) we obtain:
′′
(N1−2)/2 j j j j j j j +j 1
(4K+1) 1 2 2− 1 1 2 1 2 + =0. (A2)
K=0 (cid:26)j2 j1 2K (cid:27)(cid:26)j2 j1 2K (cid:27) 2N2
X
We complete the sums on the left hand side with odd elements and receive:
N1−1 j j j j j j j +j N1−1 j j j j j j j +j
(2K+1) 1 2 2− 1 1 2 1 2 + ( 1)K(2K+1) 1 2 2− 1 1 2 1 2
j2 j1 K j2 j1 K − j2 j1 K j2 j1 K
K=0 (cid:26) (cid:27)(cid:26) (cid:27) K=0 (cid:26) (cid:27)(cid:26) (cid:27)
X X
= 1/N . (A3)
2
−
Both sums on the left-hand side of the above equation can be calculated with the help of the following relations [22]
a b J a b J
(2J +1)(2K+1) c d K c d K′ =δJJ′ (A4)
K (cid:26) (cid:27)(cid:26) (cid:27)
X
and
( 1)K(2K+1) a b J a c J′ =( 1)J+J′ a b J . (A5)
− c d K b d K − d c J′
K (cid:26) (cid:27)(cid:26) (cid:27) (cid:26) (cid:27)
X
The first sum in Eq. (A3) equals 0 by the orthogonality relation (A4), and the second sum reduces to:
N1−1 j j j j j j j +j j j j j 1 1
( 1)K(2K+1) 1 2 2− 1 1 2 1 2 =( 1)2j2 1 2 2− 1 = = , (A6)
K=0 − (cid:26)j2 j1 K (cid:27)(cid:26)j2 j1 K (cid:27) − (cid:26)j1 j2 j1+j2 (cid:27) −2j2+1 −N2
X
which is the right hand side of Eq. (A3).(cid:4)
[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cam-
bridge, U.K. 2000).