The twistor geometry of three-qubit entanglement

a r X i v :q u a n t -p h /0403060v 1 8 M a r 2004

The twistor geometry of three-qubit entanglement

P′e ter L′e vay

Department of Theoretical Physics,Institute of Physics,

Budapest University of Technology and Economics

(Dated:February 1,2008)

A geometrical description of three qubit entanglement is given.A part of the transformations corresponding to stochastic local operations and classical communication on the qubits is regarded as a gauge degree of freedom.Entangled states can be represented by the points of the Klein quadric Q a space known from twistor theory.It is shown that three-qubit invariants are vanishing on special subspaces of Q .An invariant vanishing for the GHZ class is proposed.A geometric interpretation of the canonical decomposition and the inequality for distributed entanglement is also given.

PACS numbers:03.65.Ud,02.40.-k,02.20-a

The basic role played by geometry in such areas of modern physics as general relativity,quantum ?eld the-ory,and string theory is well-known.Further examples for the usefulness of geometric ideas for a deeper un-derstanding of quantum theory are provided by the rich research ?eld of geometric phases,fractional spin and statistics [1]and holonomic quantum computation [2].Recently some authors [3,4,5,6,7,8]have also tried to understand quantum entanglement in geometric terms.In particular it was observed[5,6]that two and a spe-cial class of three-qubit entangled states can be described by certain maps that are entanglement sensitive.These maps enable a geometric description of entanglement in terms of ?ber bundles.Fiber bundles are spaces that are locally look like the product of two spaces the base and the ?ber,globally however,they can exhibit a nontriv-ial twisted structure.In this picture this twisting of the bundle accounts for some portion of quantum entangle-ment.For two qubits using the correspondence between ?bre bundles and the language of gauge ?elds these ideas were elaborated[8].The essence of this approach was to provide a description of entanglement by regarding a part of the LOCC transformations of local operations and classical communication corresponding to a ?xed subsys-tem as a gauge degree of freedom.For example for two qubits our Hilbert space is C 2?C 2?C 4hence the space of normalized states is the seven sphere S 7that can be regarded as a ?ber bundle over the four sphere S 4.The ?ber is SU (2)?S 3which is precisely the group of LOCC (gauge)transformations corresponding to one of the subsytems.Then two-qubit entanglement as seen from the viewpoint of this subsystem can be described by characterizing submanifolds of the four sphere S 4.In this letter we generalize this gauge theory mo-tivated approach to understand three-qubit entangle-ment of three parties A,B and C in geometric terms.In order to make our presentation more transparent it is convenient to enlarge our gauge group correspond-ing to party A from LOCC to SLOCC (i.e.stochastic LOCC)transformations.As it is well-known two entan-gled states are SLOCC equivalent if they can be con-

verted back and forth with (maybe di?erent)nonvanish-ing probabilities[9,10].In this case the states in question can preform the same tasks in quantum information pro-cessing although with di?erent probabilities of success.Group theoretically this means that if we wish to account for SLOCC equivalence we have to enlarge our group of gauge transformations from SU (2)to SL (2,C ).More-over,it is more comfortable to work with unnormalized states hence we do not ?x the determinant of the invert-ible operator corresponding to SLOOC transformations of subsystem A to unity.This introduces an extra com-plex constant,hence in the following we use as the group of gauge transformations the full group of invertible 2×2complex matrices i.e.GL (2,C ).

Consider now a three-qubit state |ψ ∈C 2?C 2?C 2of the form

|ψ =

a,b,c =0,1

C abc |abc ,|abc ≡|a A ?|b B ?|c C .(1)

Let us de?ne two four component vectors Z μand W νμ,ν=1,2,3,4as

C 0bc ≡

12

Z μ

E μbc ,C 1bc ≡

1

2

W μE μbc ,(2)

where E k =?iσk ,k =1,2,3and E 4=I with I the 2×2unit matrix and σk are the Pauli matrices.Notice that the vectors Z and W are the components of C 0and C 1in a basis equivalent to the magic base of Hill and Wootters[11].If these vectors are linearly independent they de?ne a complex two-plane in C 4.We de?ne two scalar products on C 4as Z |W ≡

2

of the manifold of two-planes in C4,i.e.the four complex dimensional Grassmannian Gr(2,4).

A group element g∈GL(2,C)with matrix elements α,β,γ,δacts on the pair of vectors Z and W as Zμ→αZμ+βWμ,Wμ→γZμ+δWμ.It is easy to show that the action of GL(2,C)on the pair of linearly independent vectors(Z,W)is free and transitive hence we can regard the space of such pairs(Z,W)as a?ber bundle over Gr(2,4)with gauge group GL(2,C).Since we would like to understand the geometry of three-qubit entanglement by examining the structure of subspaces of Gr(2,4)it is convenient to introduce coordinates for this space.Let us de?ne these coordinates(Pl¨u cker coordinates)as

Pμν≡ZμWν?ZνWμ(3) i.e.Pμνis a separable bivector.These6complex co-ordinates transform under the gauge group GL(2,C)as Pμν→(αδ?βγ)Pμν,hence they are de?ned only up to a complex scalar factor.Notice,however that P is in-variant with respect to the SL(2,C)subgroup of SLOCC transformations.Moreover,since the Pl¨u cker relation

P12P34?P13P24+P23P14=0(4) holds,we can remove all the redundancies accounting for the4complex dimensions of Gr(2,4).Note that(4)de-?nes a quadric Q(the Klein-quadric)embedded in C P5. It can be shown[12]that(4)is a su?cient and necessary condition for an arbitrary bivector P to be separable i.e. of the(3)form.Hence we can also look at Gr(2,4)as the Klein-quadric Q in C P5which is the space encoding the information we need on the geometry of three-qubit entanglement.

In order to extract this information we will study spe-cial values of both LOCC and SLOCC invariants.Let us?rst consider the SLOCC invariant three-tangle[13]τABC≡4|D(C)|where

D(C)≡C2000C2111+C2001C2110+C2010C2101+C2011C2100?2(C000C001C110C111+C000C010C101C111

+C000C011C100C111+C001C010C101C110

+C001C011C110C100+C010C011C101C100)

4(C000C011C101C110+C001C010C100C111).(5)

is the Cayley hyperdeterminant[14].Due to the method of Schl¨a?i[14]we can express D(C)as the discrimi-nant of the quadratic form Det(xC0+yC1)in the vari-ables x and y as D(C)=(Tr C0Tr C1?Tr(C0C1))2?4Det(C0)Det(C1).For a complex2×2matrix M de?n-ing its”adjoint”as M′≡Det(M)M?1this expression can be written as D(C)=[Tr(C′0C1)]2?4Det(C′0C1). Noting that E′4=E4,E′k=?E k and using(2)we ob-tain the following nice expression for the three tangle τABC=2|PμνPμν|=4|(Z·Z)(W·W)?(Z·W)2|.(6)Using(6)the SL(2,C)×SL(2,C)×SL(2,C)?SL(2,C)×SO(4,C)invariance ofτABC can be imme-diately established.Moreover,we have also learnt that the residual SLOCC transformations on parties B and C are represented by the adjoint action of SO(4,C)on the separable bivector P∈Q in the form P→SP S T,S∈SO(4,C).

In order to gain further insight into the geometry of three-qubit entanglement we write out4DetρA which is called[13]the”concurrence between qubit A and the pair BC”in terms of P.Denoting this quantity byτA(BC)we get

τA(BC)=2Pμν

Pμν,(8) where?P is the dual of the bivector P i.e.?Pμν≡1

Zν+Wμ

ρ(σ2?σ2)which in the magic base amounts to merely complex conjuga-tion,we obtain the formula

Tr(ρBC?ρBC)=|Z·Z|2+|W·W|2+2|Z·W|2.(9) From Eqs.(8)and(9)we see that Tr(ρBC?ρBC)= 2(DetρB+DetρC?DetρA)as it has to be[13].After a cyclic permutation of this relation we see that

Tr(ρ±?ρ±)=(Pμν±?Pμν)

3

that the squared di?erence of the square-root of the two nonzero eigenvalues of ρ±?ρ±de?ne the two-tangles τAB and τAC subject to the Co?man-Kundu-Wootters relation[13]τA (BC )=τABC +τAB +τAC .All the quan-tities in this relation are gauge invariant up to a com-plex factor hence this relation is characterizing relation-ships between special submanifolds of the Klein quadric Q .What are these special submanifolds?

Consider ?rst an arbitrary separable bivector P giv-ing rise to the plane λZ μ+κW μ.Then we can ?nd

the principal null directions of this plane by solving the quadratic equation λ2(Z ·Z )2+2λκ(Z ·W )+κ2(W ·W )2.According to (6)the discriminant of this equation is just the Cayley hyperdeterminant so we have two prin-cipal directions for τABC =0and one for τABC =0.One can show that these directions are given by the

formula U μ

±=Z νP νμ±1τABC e i?/2Z μ,where ?≡arg[(Z ·W )2?(Z ·Z )(W ·W )].It is obvious that for normalized states ?nding the canonical form of a three-qubit state in the method of Acin et.al.[16]is equivalent to rotating one of the vectors Z or W to one of these principal directions using the LOCC subgroup of our gauge group then applying further LOCC transforma-tions.This observation also accounts for the two possi-bilities for this decomposition (apart from the case when τABC =0when this decomposition is unique).Moreover,the complex phase phactor appearing in the canonical form[16]can be related to our ?as de?ned above.

Let us now characterize geometrically submanifolds of Q correspondingto B (AC )and C (AB )separable states.For such states we have τ?or τ+equals zero.Let us chose a subset of two-planes characterized by the con-ditions ?P =?P and Z ·W =0,i.e.the bivector P is anti-self-dual and is de?ned by two orthogonal vec-tors.Notice that such vectors are automatically null i.e.Z ·Z =W ·W =0as can be seen by contracting the equation of anti-self-duality with Z and W .Subspaces of this form are called β-planes in twistor theory[15].Hence from (8)we see that for β-planes τ?=0.However,we can even state more after noticing that the four sum-mands in (8)are all nonnegative.Indeed taking into ac-count the nonegativity of the left hand side of (10)a mo-ment thought shows that τ?=0precisely for β-planes.A similar line of reasoning shows that τ+=0precisely for α-planes characterized by the analogous conditions and P =?P i.e.self duality.Let us illustrate this on the B (AC )separable state |ψ? =α|100 +γ|001 .In this case Z μ=γ2(i,?1,0,0)T and W ν=α2(0,0,i,1)T .These vectors are null,orthogonal and a calcualtion shows ?P =?P .Similarly for the C (AB )separable state |ψ+ =α|100 +β|010 we have Z μ=β2(i,1,0,0)T and W νis just the same as above.Here the change of sign in Z results in the condition ?P =P .Clearly all αand βplanes through the origin of C 4can be obtained as SO (4,C )orbits of a ”canonical”P corresponding to

either of the states |ψ± .Since two-planes in C 4are represented by points on Q ,SO (4,C )orbits of the form SP S T then give rise to submanifolds of Q after factoring out with those transformations which ?x the canonical P .One can show[15]that the set of αplanes in Q can be parametrized by the three-dimensional complex projec-tive space C P 3.Similarly the set of β-planes is another

C P 3?which is the projective dual of the previous one[15].

Consider next the Werner state |W =α|100 +β|010 +γ|001 .For this state we have Z μ=12((β+γ)i,β?γ,0,0)T ,W ν=α2(0,0,i,1)T .We see that Z is not null but orthogonal to the null vector W .More-over,the null vector W lies in the intersection of the αand βplanes of the previous paragraph.The sepa-rable bivector P corresponding to |W has the impor-tant property τABC =0as one can quickly check from (6).Separable bivectors satisfying τABC =0are called null separable bivectors or null-twistors in twistor the-ory.Let us show that the aforementioned properties characterize precisely the Werner-class!For this de?ne

the quadratic form Q (P 1,P 2)≡εμν?σP μν1P ?σ

2.Clearly relation (4)is equivalent to Q (P,P )=0,moreover one has τABC =2|Q (P,?P )|.Then one can easily prove the lemma that the intersection of two planes P 1and P 2is nonzero if and only if Q (P 1,P 2)=0.From this lemma it follows that τABC =0precisely when the plane P intersects with its dual plane ?P .Moreover for αand β-planes by virtue of ?P =±P ,τABC =0.Now the Werner class is characterized by the conditions[10]τA (BC )=0,τ±=0and τABC =0.The ?rst three of this condi-tions excludes the possibilities of collinear Z and W and αand β-planes.The planes P and ?P corresponding to these cases are either degenerate or beeing identical up to sign hence intersect trivially.Then we see that the Werner class is characterized precisely by the condition that P and ?P intersect along a line .This was what we illustrated for the state |W .Notice that equations Q (P,P )=Q (?P,?P )hold trivially,showing that P and ?P are points lying on the Klein quadric Q .Condition Q (P,?P )=0characterizing the Werner class shows that the tangent vector ?P of the tangent plane of Q at P lies entirely in Q .Moreover,the line L ≡tP +(1?t )?P through P and ?P also satisfying Q (L,L )=0lies entirely in the Klein quadric.Such lines are called in twistor the-ory null lines [12].Now Q can be regarded as the com-pacti?cation and complexi?cation of Minkowski space-time.From twistor theory it is well-known that the null lines of Q represent the light-cone (conformal)structure of Minkowski spacetime.Hence we found an interesting connection between three-qubit states belonging to the Werner class and special null lines in Minkowski space time.Is there any deeper physical reason for this con-nection?

How to characterize geometrically the GHZ class,which is known to be the other class from the two inequiv-

4 alent ones[10]characterizing genuine three-qubit entan-

glement?In order to attack this problem we recall that

there is still one more independent invariant we have

not discussed.This is the Kempe invariant character-

izing hidden nonlocalities(i.e.ones that cannot be re-

vealed by inspection of local density matrices)[17].This

invariant[17,18]rewritten in a form convenient for our

purposes is

ξABC=Tr(A3+B3+3C?CA+3CC?B),(11)

where A≡C0C?0,B≡C1C?1and C≡C0C?1.After a

straightforward but tedious calcuation we getξABC=

N3+3

P P).

Now we claim that the new permutation and LOCC in-

variant characterizing the GHZ class is

σABC≡NτABC?4Tr(ρBC

2(||U+||2+

||U?||2+||V+||2+||V?||2)≡||U||2+||V||2.Here we ob-serve that the principal null directions Vμ±≡PμνWν±

1τ

ABC

e i?/2Wμare up to a complex number the same as the corresponding ones Uμ?de?ned previously.σABC is LOCC and permutation invariant and nonegative.It is zero i?U=V=0i.e.when Z and W are precisely the two di?erent principal null directions.Such vectors bee-ing the eigenvectors o

f Pμνwith the nonzero eigenvalues ±1τABC e i?/2are clearly null hence Z·Z=W·W=0. Moreover,the eigenvalues are nonzero hence Z·W=0. The conditions characterizin

g the GHZ class[10]are τA(BC)=0,τ±=0,andτABC=0.These conditions can be shown to follow fromσABC≡0,hence the van-ishing of this invariant characterizes the GHZ class.In order to check these statements,take the standard GHZ state|GHZ =α|000 +β|111 .For this state we have

Zμ=α

2(0,0,i,1)T and Wν=β

2

(0,0,?i,1)T.These

vectors are null,Z·W=0andτABC=0,they are the null directions of P with the only nonvanishing compo-nent P34=iαβ,and an explicit calculation shows that σABC=0.It is well-known[16,19]that for normal-ized statesτA(BC),τ±andτABC are entanglement mono-tones satisfying the inequalities0≤τA(BC),τ±,τABC≤https://www.sodocs.net/doc/0d11646715.html,ing|Tr(A?B)|≤(Tr(A?A))1/2(Tr(B?B)1/2and Tr(