arXiv:quant-ph/0506043v1 6 Jun 2005 n igeqbtadtoqbtsae sdsusdb .Mseiand Mosseri R. that by shown discussed have is They states H [6]. two-qubit e between doloff and relation Segre d The qubit The geometrical another single fibration. point. also Hopf and and is called Hug There P. state qubits, pure L. of [5]. describe and in pair to discussed Brody a been C. for also D. map has see ding this embedding, illustrate Segre They called concurrence map [4]. between a connection in The found [3]. is concurrence of terms in u hoy nqatmmcais h pc fpr ttsi sa is in states pure of feature space interesting the most mechanics, quantum the In of one theory. is tum 2] entanglement[1, Quantum Introduction 1 iesoa ibr pc a edsrbdb h ope projectiv complex the by described be can CP space Hilbert dimensional hc setnlmn estv,with sensitive, entanglement is which 7,weeteHletsaeo h he-ui tt stefifteen- the is state three-qubit the of pre space been sphere Hilbert has state the three-qubit where to [7], fibration Hopf of generalization ui vrl hs ereo reo.Frtoqbtsae,th states, two-qubit For sphere rep fibres seven-dimensional freedom. circular a of the is degree while phase sphere, overall Bloch the qubit but nothing is fibration motn nsrn hoy .. ntepoeso opcicto of compactification of process Conifold the in manifold. i.e., five-dimensional theory, a string usually con in a is important like base c look The conifold neighborhood a base. whose manifolds, certain points Unlike i.e., singularities, manifold. conical a tain geometry, of algebraic notion relation two In the establish of pure fibration. will generalization a Hopf we and of Moreover, conifold variety, concurrence variety. qu Segre Segre the concurren a the the an of is with with complete vanishing variety which coincides a The Segre variety, giving the Segre by compare states. 12], the will two-qubit 11, We describe 10, it. will 9, for [8, we formula geometry paper algebraic this in space In fiber. as N er ait,cnfl,Hp bain and fibration, Hopf conifold, variety, Segre o iatt,pr tts h nageeto omto a b can formation of entanglement the states, pure bipartite, For . h emtyadtplg fsprbest fpr multi-qu highe pure and in variety of we Segre sets multi-projective Moreover, fibration. separable complex a of states. on topology two-qubit based and pure geometry of the sets separable and S 15 eetbihrltosbtenSgevrey oiod Ho conifold, variety, Segre between relations establish We hc losfrtetidHp bainwith fibration Hopf third the for allows which , -,KnaSrgdi hyd-u oy 0-80,Japan 8308, 101- Tokyo Chiyoda-ku, Kanda-Surugadai, 1-8, eaal ut-ui states multi-qubit separable nttt fQatmSine io University, Nihon Science, Quantum of Institute ohn Heydari Hoshang S 7 hc loalw o eodHp fibration Hopf second a for allows also which , S Abstract 2 aesaeo utbyoriented suitably a of space base S 3 be n a and fibres S 4 ae oevr a Moreover, base. S 8 re Hopf order r ffibration, pf sbs and base as ibr space Hilbert e i states bit vestigate etdi Ref. in sented oiodi a is conifold a n geometry and p fibration opf dimensional qbtstate -qubit Calabi-Yau fquan- of s eo pure, of ce eetthe resent between s escription explicit d n written e S .Dan- R. r very are iha with e space e ncon- an N 3 mbed- hston adric Hopf 1- + S 7 manifolds. A Calabi-Yau manifold is a compact K¨ahler manifold with a vanish- ing first Chern class. A Calabi-Yau manifold can also be defined as a compact Ricci-flat K¨ahler manifold. Finally, we will discuss the geometry and topology of pure multi-qubit states based on some mathematical tools from algebraic ge- ometry and algebraic topology, namely the multi-projective Segre variety and higher-order Hopf fibration. Let us start by denoting a general, pure, composite p quantum system with m subsystems = m(N1,N2,...,Nm)= 1 2 m, N1 Q NQ2 Nm Q Q ···Q consisting of a pure state Ψ = αi ,i ,...,i i1,i2,...,im | i i1=1 i2=1 ··· im=1 1 2 m | i and corresponding Hilbert space as Q = Q Q Q , where the P HP H 1 ⊗P H 2 ⊗···⊗H m dimension of the jth Hilbert space is given by Nj = dim( Qj ). We are going to use this notation throughout this paper, i.e., we denoteH a pure two-qubit p states by (2, 2). Next, let ρQ denotes a density operator acting on Q. Q2 H The density operator ρQ is said to be fully separable, which we will denote by sep ρQ , with respect to the Hilbert space decomposition, if it can be written as sep N m k N ρQ = k=1 pk j=1 ρQj , k=1 pk = 1 for some positive integer N, where pk k are positive real numbers and ρQj denotes a density operator on Hilbert space Pp N P Qj . If ρQ represents a pure state, then the quantum system is fully separable H p sep m if ρQ can be written as ρQ = j=1 ρQj , where ρQj is a density operator on Q . If a state is not separable, then it is said to be entangled state. H j N 2 Complex projective variety

In this section we will review basic definition of complex projective variety. Let n f1,f2,...,fq be continuous functions K K, where K is field of real {R or complex} number C. Then we define real−→ (complex) space as the set of simultaneous zeroes of the functions n K(f1,f2,...,fq)= (z1,z2,...,zn) K : fi(z1,z2,...,zn)=0 1 i q . V { ∈ ∀ ≤ ≤ (1)} These real (complex) spaces become a topological spaces by giving them the n induced topology from K . Now, if all fi are polynomial functions in coordinate functions, then the real (complex) space is called a real (complex) affine variety. A complex projective space CPn which is defined to be the set of lines through the origin in Cn+1, that is, CPn = (Cn+1 0)/ , where is an equivalence relation define by (x ,...,x ) (y ,...,y− ∼) λ ∼ C 0 such that 1 n+1 ∼ 1 n+1 ⇔ ∃ ∈ − λxi = yi 0 i n. For n = 1 we have a one dimensional complex manifold CP1 which∀ is≤ very≤ important one, since as a real manifold it is homeomorphic to the 2-sphere S2. Moreover every complex compact manifold can be embedded in some CPn. In particular, we can embed a product of two projective spaces into a third one. Let f1,f2,...,fq be a set of homogeneous polynomials in the { } n+1 coordinates α1, α2,...,αn+1 of C . Then the projective variety is defined to be the subset{ } n (f1,f2,...,fq)= [α1,...,αn+1] CP : fi(α1,...,αn+1)=0 1 i q . V { ∈ ∀ ≤ ≤ (2)} n+1 We can view the complex affine variety C(f1,f2,...,fq) C as complex V ⊂ n cone over projective variety (f1,f2,...,fq). We can also view CP as a quo- tient of the unit 2n + 1 sphereV in Cn+1 under the action of U(1) = S1, that is CPn = S2n+1/U(1) = S2n+1/S1, since every line in Cn+1 intersects the unit sphere in a circle. 3 Hopf fibration and two- and three-qubit states

p For a pure one-qubit state 1(2) with Ψ = α1 1 + α2 2 , where α1, α2 C, and α 2 + α 2 = 1, we canQ parameterize| i this state| i as | i ∈ | 1| | 2| ϑ ϕ χ α1 cos( 2 )exp(i( 2 + 2 )) = ϑ ϕ χ (3) α2 cos( )exp(i( ))    2 2 − 2 

where ϑ [0, π], ϕ [0, 2π] and χ [0, 2π]. The Hilbert space Q of a single qubit is∈ the unit 3-dimensional∈ sphere∈ S3 R4 = C2. But sinceH quantum mechanics is U(1) projective, the projective⊂ Hilbert space is defined up to a phase exp (iϕ), so we have CP1 = S3/U(1) = S3/S1 = S2. Now, the first Hopf S1 map, S3 / S2 as an S1 fibration over a base space S2. For a pure two- qubit state p(2, 2) with Ψ = α 1, 1 +α 1, 2 +α 2, 1 +α 2, 2 , where Q2 | i 1,1| i 1,2| i 2,1| i 2,2| i α , α , α , α C and 2 α 2 = 1. The normalization condition iden- 1,1 1,2 2,1 2,2 ∈ k,l | k,l| tifies the Hilbert space to be the seven dimensional sphere S7 R8 = C4 Q P and the projective HilbertH space to be CP3 = S7/U(1). Thus we can⊂ parame- S3 terized the sphere S7 as a S3 fiber over S4, that is S7 / S4 which is called the Hopf second fibration. This Hopf map is entanglement sensitive and the separable states satisfy α1,1α2,2 = α1,2α2,1, see Ref. [6].

4 Segre variety for a general bipartite state and concurrence

p N1−1 For given general pure bipartite state 2(N1,N2) we want make CP CPN2−1 into a projective variety by itsQ Segre embedding which we construct×

as follows. Let (α1, α2,...,αN1 ) and (α1, α2,...,αN2 ) be two points defined on CPN1−1 and CPN2−1, respectively, then the Segre map

: CPN1−1 CPN2−1 CPN1N2−1 (4) SN1,N2 × −→

((α ,...,α ), (α ,...,α )) (α ,...,α ,...,α ,...,α ) 1 N1 1 N2 7−→ 1,1 1,N1 N1,1 N1,N2

is well defined. Next, let αi,j be the homogeneous coordinate function on CPN1N2−1. Then the image of the Segre embedding is an intersection of a family of quadric hypersurfaces in CPN1N2−1, that is

Im( ) = < α α α α >= (α α α α ) . (5) SN1,N2 i,k j,l − i,l j,k V i,k j,l − i,l j,k This quadric space is the space of separable states and it coincides with the p definition of general concurrence ( 2(N1,N2)) of a pure bipartite state [13, 14] because C Q

1 2 N1 N2 ( p(N ,N )) = α α α α 2 , (6) C Q2 1 2 N | i,k j,l − i,l j,k|  j,i=1 X l,kX=1   where is a somewhat arbitrary normalization constant. The separable set is N defined by αi,kαj,l = αi,lαj,k for all i, j and k,l. I.e., for a two qubit state we have : CP1 CP1 CP3 and S2,2 × −→ Im( )= (α α α α ) α α = α α (7) S2,2 V 1,1 2,2 − 1,2 2,1 ⇐⇒ 1,1 2,2 1,2 2,1 is a quadric surface in CP3 which coincides with the space of separable set of pairs of qubits. In following section comeback to this result.

5 Conifold

In this section we will give a short review of conifold. An example of real (complex) affine variety is conifold which is defined by

4 4 2 4 2 C( z )= (z ,z ,z ,z ) C : z =0 . (8) V i { 1 2 3 4 ∈ i } i=1 i=1 X X and conifold as a real affine variety is define by

4 4 4 8 2 2 R(f ,f )= (x ,...,x ,y ,...,y ) R : x = y , x y =0 . (9) V 1 2 { 1 4 1 4 ∈ i j i i } i=1 j=1 i=1 X X X 4 2 2 4 where f1 = i=1(xi yi ) and f2 = i=1 xiyi. This can be seen by defining − 4 2 z = x + iy and identifying imaginary and real part of equation i=1 zi = P P n 0. As a real topological space R(f1,...,fn) R , x R(f1,...,fn) is a V ⊂ ∈ V P smooth point of R(f1,...,fn) if there is a neighborhood V of x such that V is homeomorphic toV Rd for some d which is usually called the local dimension of R(f ,...,f ) in x. If there is no such neighborhood V , then x is said to be a V 1 n singular point of R(f1,...,fn). Now, we can call R(f1,...,fn) a topological V V n manifold if all points x R(f1,...,fn) are smooth. S is compact, since it is a closed and bounded subset∈V of Rn+1. Now, let us define a cone as a real space n R(f1,...,fn) R with a specified point s such that for all x R(f1,...,fn) V ⊂ ∈Vn we have that the line sx R(f1,...,fn). But every line s R intersect any n−1 ∈V ∈ sphere S with center s, the cone R(f1,...,fn) can be determined by a Vn−1 compact space = R(f1,...,fn) S called the base space of the cone. As a real space,B the conifoldV is cone∩ in R8 with top the origin and base space the compact manifold S2 S3. One can reformulate this relation in term of a × 4 2 theorem. The conifold C( i=1 zi ) is the complex cone over the Segre variety CP1 CP1 S2 SV2. To see this let us make a complex linear change of × ′ ≃ × P′ ′ ′ coordinate α1,1 = z1 +iz2, α1,2 = z4 +iz3, α2,1 = z4 +iz3, and α2,2 = z1 iz2. Thus after this linear coordinate transformation− we have −

4 ′ ′ ′ ′ 2 4 C(α α α α )= C( z ) C . (10) V 1,1 2,2 − 1,2 2,1 V i ⊂ i=1 X We will comeback to this result in section 6 where we establish a relation be- tween these varieties, Hopf fibration and two-qubit state. Moreover, removal of singularity of a conifold leads to a Segre variety which also describes the separable two-qubit states. We will investigate this connection in the following 2 2 2 2 section. We can also define a metric on conifold as dS6 = dr + r dST 1,1 , where

2 2 2 2 1 1 2 2 2 2 dS 1,1 = dψ + cos θ dφ + dφ + sin θ dφ , (11) T 9 i i 6 i i i i=1 ! i=1 X X
1,1 SU(2)×SU(2) is the metric on the Einstein manifold T = U(1) , with U(1) being a diagonal subgroup of the maximal torus of SU(2) SU(2). Moreover, T 1,1 is a 2 2 × U(1) bundle over S S , where 0 ψ 4 is an angular coordinate and (θi, φi) for all i = 1, 2 parameterize× the two≤ S2≤, see Ref. [15, 16]. One can even relate ′ these angular coordinate to the αk,l for all k,l =1, 2 as follows

′ i ′ i 3/2 2 (ψ−φ1−φ2) θ1 θ2 3/2 2 (ψ+φ1−φ2) θ1 θ2 α1,1 = r e sin 2 sin 2 α1,2 = r e cos 2 sin 2 ′ i ′ i . 3/2 2 (ψ−φ1+φ2) θ1 θ2 3/2 2 (ψ+φ1+φ2) θ1 θ2 α2,1 = r e sin 2 cos 2 α2,2 = r e cos 2 cos 2

4 2 Moreover, if we define the conifold as C( i=1 zi ), then we identify the Einstein 1,1 V 4 2 3 manifold T as the intersection of conifold with the variety C( i=1 zi r ) 1,1 P V | |− and T is invariant under rotations SO(4) = SU(2) SU(2) of zi coordinate and under an overall phase rotation. × P

6 Conifold, Segre variety, and a pure two-qubit state

In this section we will investigate relations between pure two-qubit states, Segre variety, and conifold. For a pure two-qubit state the Segre variety is given by : CP1 CP1 CP3 and S2,2 × −→

Im( 2,3) = (α1,1α2,2 α1,2α2,1) (12) S V ′ ′− ′ ′ = (α 2 + α 2 + α 2 + α 2 ) V 1,1 2,2 1,2 2,1 = CP1 CP1 S2 S2 × ≃ × S3 S1 S4 o S7 / S7/U(1) = CP3 . ⊂ where we have performed a coordinate transformation on ideal of Segre variety Im( ). Moreover, we have the following commutative diagram S2,2 id S7 / S7

S1 S3   S2 CP3 = S7/U(1) / S4 = S7/SU(2) = HP1 where HP1 denotes projective space over quaternion number field and we have S3 the second Hopf fibration S7 / S4 . Thus we have established a direct relation between two-qubit state, Segre variety, conic variety and Hopf fibration. Thus the result from algebraic geometry and algebraic topology give a unified picture of two-qubit state. Now, let us investigate what happens to our state, when we do the coordinate transformation to establish relation between conic ′ variety and Segre variety. By the coordinate transformation α1,1 = α1,1 + iα1,2, ′ ′ ′ α1,2 = α2,2 + iα2,1, α2,1 = α2,2 + iα2,1, and α2,2 = α1,1 iα1,2 we perform the − − ′ following map Ψ = α1,1 1, 1 + α1,2 1, 2 + α2,1 2, 1 + α2,2 2, 2 Ψ which is given by | i | i | i | i | i −→ | i ′ ′ ′ ′ ′ Ψ = α 1, 1 + α 1, 2 + α 2, 1 + α 2, 2 (13) | i 1,1| i 1,2| i 2,1| i 2,2| i = α ( 1, 1 + 2, 2 )+ iα ( 1, 1 2, 2 ) 1,1 | i | i 1,2 | i − | i +iα ( 1, 2 + 2, 1 ) α ( 1, 2 2, 1 ) 2,1 | i | i − 2,2 | i − | i = √2 α Ψ+ + iα Ψ− + iα Φ+ α Φ− . 1,1| i 1,2| i 2,1| i− 2,2| i Thus the equality between Segre variety, conic variety means that we
rewrite a pure two-qubit state in terms of Bell’s basis. For higher dimensional space we have Segre variety but we couldn’t find any relation between these two variety.

7 Segre variety, Hopf fibration, and multi-qubit states

In this section, we will generalize the Segre variety to a multi-projective space and then we will establish connections between Segre variety for multi-qubit state and Hopf fibration. As in the previous section, we can make CPN1−1 N2−1 N −1 × CP CP m into a projective variety by its Segre embedding fol- ×···× lowing almost the same procedure. Let (α1, α2,...,αNj ) be points defined on CPNj −1. Then the Segre map

N1−1 N2−1 Nm−1 N1N2···Nm−1 N1,...,Nm : CP CP CP CP S ((α , α ,...,α ×),..., (α , α×···×,...,α )) −→ (...,α ,...). 1 2 N1 1 2 Nm 7−→ i1,i2,...,im (14) is well defined for α ,1 i N , 1 i N ,..., 1 i N as a i1,i2,...,im ≤ 1 ≤ 1 ≤ 2 ≤ 2 ≤ m ≤ m homogeneous coordinate-function on CPN1N2···Nm−1. Now, let us consider the composite quantum system p (N ,N ,...,N ) and let the coefficients of Ψ , Qm 1 2 m | i namely αi1,i2,...,im , make an array as follows

= (αi ,i ,...,i ) , (15) A 1 2 m 1≤ij ≤Nj for all j = 1, 2,...,m. can be realized as the following set (i1,i2,...,im): 1 i N , j , inA which each point (i ,i ,...,i ) is assigned{ the value ≤ j ≤ j ∀ } 1 2 m αi1,i2,...,im . Then and it’s realization is called an m-dimensional box-shape matrix of size N AN N , where we associate to each such matrix a sub- 1 × 2 ×···× m ring SA = C[ ] S, where S is a commutative ring over the complex number field. For eachAj ⊂=1, 2,...,m, a two-by-two minor about the j-th coordinate of is given by A = α α (16) Ck1,l1;k2,l2;...;km,lm k1,k2,...,km l1,l2,...,lm α − α − SA. − k1,k2,...,kj 1,lj ,kj+1,...,km l1,l2,...,lj 1,kj ,lj+1,...,lm ∈ m Then the ideal of SA is generated by and describes the sep- IA Ck1,l1;k2,l2;...;km,lm arable states in CPN1N2···Nm−1. The image of the Segre embedding Im( ) SN1,N2,...,Nm which again is an intersection of families of quadric hypersurfaces in CPN1N2···Nm−1 is given by Im( ) = < > (17) SN1,N2,...,Nm Ck1,l1;k2,l2;...;km,lm = ( ) . V Ck1,l1;k2,l2;...;km,lm In our paper [17], we showed that the Segre variety defines the completely separable states of a general multipartite state. Furthermore, based on this sub- determinant, we define an entanglement measure for general pure bipartite and three-partite states which coincide with generalized concurrence. Let us consider p a general multi-qubit state m(2,..., 2). For this state the Segre variety is given by equation (17) and Q

Im( ) = ( ) (18) S2,...,2 V C1,2;1,2;...;1,2 m times = CP1 CP1 S2 S2 ×···× ≃ ×···× S2m−1 S1 z m o }| m+1{ / m+1− 2m−1 S2 S2 −1 S2 1/U(1) = CP . ⊂

m+1 m m We can parameterized the sphere S2 −1 as a S2 −1 fiber over S2 , that is S2m−1 m+1 / m S2 −1 S2 which are higher order Hopf fibration. Moreover, we have the following commutative diagram

m+1 id / m+1 S2 −1 S2 −1

S1 S2m−1

 2m−2  m m+1 S m CP2 −1 = S2 −1/U(1) / S2

Thus we have established relations between Segre variety and higher order Hopf fibration and separable set of a multi-qubit state. As an example, let us look at a pure three-qubit state. For such state we have

Im( ) = ( ) (19) S2,2,2 V C1,2;1,2;1,2 = α α α α , α α α α h 1,1,1 2,1,2 − 1,1,2 2,1,1 1,1,1 2,2,1 − 1,2,1 2,1,1 , α α α α , α α α α 1,1,1 2,2,2 − 1,2,2 2,1,1 1,1,2 2,2,1 − 1,2,1 2,1,2 , α α α α , α α α α 1,1,2 2,2,2 − 1,2,2 2,1,2 1,2,1 2,2,2 − 1,2,2 2,2,1 , α α α α , α α α α 1,1,1 1,2,2 − 1,1,2 1,2,1 1,1,1 2,2,2 − 1,2,1 2,1,2 , α α α α , , α α α α 1,1,2 2,2,1 − 1,2,2 2,1,1 2,1,1 2,2,2 − 2,1,2 2,2,1 , α α α α , α α α α 1,1,1 2,2,2 − 1,1,2 2,2,1 1,2,1 2,1,2 − 1,2,2 2,1,1i = CP1 CP1 CP1 S2 S2 S2 × × ≃ × × S7 S1 S8 o S15 / S15/U(1) = CP7 . ⊂ This is what we have expected to see. Moreover, we have the following commu- tative diagram id S15 / S15

S1 S7  S6  CP7 = S15/U(1) / S8

S7 where we have the third Hopf fibration S15 / S8 for three-qubit state which has been discussed in Ref. [7]. 8 Conclusion

In this paper, we have discussed a geometric picture of the separable pure two- qubit states based on Segre variety, conifold, and Hopf fibration. We have shown that these varieties and mappings give a unified picture of two-qubit states. Moreover, we have discussed the geometry and topology of pure multi-qubit states based on multi-projective Segre variety and higher-order Hopf fibration. Thus we have established relations between algebraic geometry, algebraic topol- ogy and fundamental quantum theory of entanglement. Perhaps, these geomet- rical and topological visualization puts entanglement in a broader perspective and hopefully gives some hint about how we can solve the problem of quantify entanglement.

Acknowledgments: This work was supported by the Wenner-Gren Foundation

References

[1] E. Schr¨odinger, Naturwissenschaften 23, 807-812, 823-828, 844-849 (1935). Translation: Proc. of APS, 124, 323 (1980).

[2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).

[3] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

[4] D. C. Brody and L. P. Hughston, Journal of Geometry and Physics 38, 19 (2001).

[5] A. Miyake, Phys. Rev. A 67, 012108 (2003).

[6] R. Mosseri, R. Dandoloff, e-print quant-ph/0108137.

[7] B. A. Bernevig, H. D. Chen, e-print quant-ph/0302081.

[8] H. Li and F. Van Oystaeyen, A primer of Algebraic geometry, Marcel Dekker, Inc., New York, 2000.

[9] C. Musili, Algebraic geometry, Hindustan book agency, India, 2001.

[10] K. Ueno, An introduction to algebraic geometry, American Mathematical Society, Providence, Rhode Island, 1997.

[11] P. Griffiths and J. Harris, Principle of algebraic geometry, Wiley and Sons, New York, 1978.

[12] D. Mumford, Algebraic Geometry I, Complex Projective Varieties, Springer-Verlag, Berlin, 1976.

[13] S. Albeverio and S. M. Fei, J. Opt. B: Quantum Semiclass. Opt. 3, 223 (2001).

[14] E. Gerjuoy, Phys. Rev. A 67, 052308 (2003).

[15] I. R. Klebanov, E. Witten, e-print hep-th/9807080. [16] D. R. Morrison, M. R. Plesser , e-print hep-th/9810201. [17] H. Heydari and G. Bj¨ork, J. Phys. A: Math. Gen. 38, 3203-3211 (2005).