aa r X i v : . [ qu a n t - ph ] O c t Fractional topological phase for entangled qudits
L. E. Oxman and A. Z. Khoury
Instituto de F´ısica, Universidade Federal Fluminense, 24210-346 Niter´oi - RJ, Brasil.
We investigate the topological structure of entangled qudits under unitary local operations. Dif-ferent sectors are identified in the evolution, and their geometrical and topological aspects areanalyzed. The geometric phase is explicitly calculated in terms of the concurrence. As a mainresult, we predict a fractional topological phase for cyclic evolutions in the multiply connected spaceof maximally entangled states.
PACS numbers: PACS: 03.65.Vf, 03.67.Mn, 07.60.Ly, 42.50.Dv
In a seminal work, M. Berry [1] showed the impor-tant role played by geometric phases in quantum the-ory. Since then, the interest for geometric phases wasrenewed by potential applications to quantum computa-tion. The experimental demonstration of a conditionalphase gate was provided both in Nuclear Magnetic Reso-nance (NMR) [2] and trapped ions [3]. Optical geometricphases have already been discussed both for polarization[4] and vortex mode transformations [5, 6]. The role ofentanglement in the phase evolution of qubits was inves-tigated in refs.[7, 8]. Recently, P. Milman and R. Mosseri[9, 10] investigated the geometric phase and the topolog-ical structure associated with cyclic evolutions of arbi-trary two-qubit pure states. This structure has been ex-perimentally evidenced in the context of spin-orbit modetransformations of a laser beam [11] and in NMR [12].Although the topological nature of the phase acquired bymaximally entangled states is well settled, the distinctionbetween geometrical and topological phases has not beenestablished clearly for partially entangled states. In thiswork we present a group theoretical approach which al-lows for a clear distinction between the two aspects. As abonus, this approach is easily extended to higher dimen-sions, bringing an interesting prediction of a fractionaltopological phase.Let | ψ i = P di,j =1 α ij | ij i be the most general two-qudit pure state. We shall represent this state by the d × d matrix α whose elements are the coefficients α ij .With this notation the norm of the state vector becomes h ψ | ψ i = T r ( α † α ) = 1 and the scalar product betweentwo states is h φ | ψ i = T r ( β † α ), where β is the d × d matrix containing the coefficients of state | φ i in the cho-sen basis. We are interested in the phase evolution ofthe state | ψ i under local unitary operations. So let ustake two unitary matrices U A and U B belonging to U ( d )and representing the operations performed in each sub-system separately. Under these unitary operations thestate matrix will evolve as α ( t ) = U A α (0) U ⊺ B , where U j ( t ) = e iφ j ( t ) ¯ U j ( t ) ( j = A, B ) and ¯ U j ∈ SU ( d ). Onecan identify the following invariants under local unitaryevolutions: T r [ ρ pj ], p = 1 , . . . , d , where ρ j is the reduceddensity matrix with respect to qudit j ( ρ A = α ⊺ α ∗ and ρ B = αα † ). In fact, the invariants are j -independent. The first one ( p = 1) is simply the norm of the statevector. One can readily relate the second invariant tothe I-concurrence of a two-qudit pure quantum state [13] C = p − T r [ ρ ]), so that its invariance expresses thewell known fact that entanglement is not affected by localunitary operations. The p = d invariant can be rewrittenin terms of the former and D = | det[ α ] | . In particular,for qubits we have C = 2 D .In the case of a cyclic evolution, ¯ U A ( τ ) α (0) ¯ U ⊺ B ( τ ) = e i ∆ φ α (0) . By taking the determinant of both sides weget: e i d ∆ φ = 1 as long as D 6 = 0. This implies that thepossible acquired phases due to the SU ( d ) part of a cyclicevolution are ∆ φ = 2 πn/d , with n = 0 , , , ..., d − d = 2) one recovers the well known result∆ φ = 0 , π . However, for d > π/d . Now, we are interestedin discussing in what sense this fractional phase can beconsidered as topological. For this aim, we will analyzethe topology of the space of two-qudit states and howthe total phase is built. In this regard, we would like tounderline that according to ref. [14], the geometric phaseacquired by a time evolving quantum state α ( t ) is alwaysdefined as φ g = arg h ψ (0) | ψ ( t ) i + i Z dt h ψ ( t ) | ˙ ψ ( t ) i , (1)that corresponds to the total phase minus the dynamicalphase. Therefore, a topological phase, that is, an objectthat only depends on a given class of paths, can only findroom as a part of the geometric phase, an object that isinvariant under reparametrizations and gauge transfor-mations. Gauge invariance corresponds to the fact thatthe phase factors φ j ( t ) do not contribute to φ g , whichis completely determined by ¯ U j ( t ), the sector where thefractional values occur.In order to characterize the space of states, we notethat any invertible matrix admits a polar decomposition α = Q S , where Q = d √D e M is a positive definite Her-mitian matrix, M is a traceless Hermitian matrix, and S = e iφ ¯ S , ¯ S ∈ SU ( d ). Since det[ e M ] = e T r [ M ] = 1, oneeasily finds det( α ) = D e i d φ . We can identify the timeevolution as occurring in different sectors α ( t ) = d √D e iφ ( t ) e M ( t ) ¯ S ( t ) , (2)where we have denoted, φ ( t ) = φ (0) + φ A ( t ) + φ B ( t ), M ( t ) = ¯ U A ( t ) M (0) ¯ U A ( t ) † , and ¯ S ( t ) = ¯ U A ( t ) ¯ S (0) ¯ U ⊺ B ( t ).Therefore, we identify the evolution in three sectors ofthe matrix structure: an explicit phase evolution φ ( t ),an evolution closed in the space of traceless Hermitianmatrices M ( t ), and the evolution ¯ S ( t ) closed in SU ( d ).Now we are able to discuss the topological aspectsof the entangled state evolution in terms of these sec-tors. The space of positive definite Hermitian matrices Q has trivial topology. This is a noncompact manifoldisomorphous to R d − , as it can be parametrized in theform Q = e β a T a , where β a are real numbers and T a ( a = 1 , , ..., d −
1) is a basis in the space of Hermi-tian traceless matrices. These T a ’s are the generators of SU ( d ), so that ¯ S = e i ω a T a , with ω a real. They can benormalized in the form tr ( T a T b ) = δ ab and obey theLie algebra (cid:2) T a , T b (cid:3) = if abc T c , where f abc are the struc-ture constants of SU ( d ). The first homotopy group of SU ( d ) is also trivial, however, the physical equivalenceof α matrices differing by a global phase corresponds toconsidering the identification in SU ( d ), e i πn/d ¯ S ≡ ¯ S .This can be naturally implemented by associating the SU ( d ) sector of the matrix α with a corresponding sectorfor the quantum states, represented by transformations R ( ¯ S ) in the adjoint representation ¯ ST a ¯ S − = ˆ n a · ~T ,ˆ n a = R ( ¯ S )ˆ e a . In this manner, the matrices e i πn/d ¯ S aremapped to the same point R ( ¯ S ). In other words, a partof the evolution can be parametrized as R ( t ) ∈ Adj( d ), orequivalently, in terms of a time dependent frame ˆ n a ( t ).Note that for qubits the adjoint representation corre-sponds to SO (3), the manifold used in ref. [9, 10] todescribe maximally entangled states. An evolution ¯ S ( t )starting at ¯ S (0) and ending at e i π/d ¯ S (0) defines an openpath in SU ( d ) and a topologically nontrivial closed path R ( t ) ∈ Adj( d ). If this cyclic evolution were composed d times, we would get a trivial path in Adj( d ), so that thenumber of nonequivalent classes is given by d .The total phase can be written as φ tot = arg { T r [ α † (0) α ( t )] } = φ A + φ B + arg { T r [ α † (0) ¯ U A ( t ) α (0) ¯ U ⊺ B ( t )] } , (3)while the dynamical phase is, φ dyn = − i Z t dt ′ T r [ α † ( t ′ ) ˙ α ( t ′ )] = φ A + φ B − i Z t dt ′ T r [ ρ B (0) ¯ U † A ˙¯ U A + ρ ⊺ A (0) ˙¯ U ⊺ B ¯ U ∗ B ] , (4)where ρ A = ( S † Q S ) ∗ , ρ B = Q . For cyclic evolutions wehave ¯ U j ( τ ) = e i πn j /d ¯ U j (0). Then, the total generatedphase is φ tot = φ A + φ B + 2 πn/d , n = n A + n B , wherethe values n = 0 , d, d, . . . , correspond to topologicallynontrivial paths. As already discussed, the total phase isalways written as a dynamical plus a geometric part. Inorder to consider a fractional phase as topological, it mustbe built only as a part of the geometric phase, receiving no relevant contribution from the dynamical part. Thismeans that at any time t , 0 ≤ t ≤ τ , we must have, Z t dt ′ T r [ ρ B (0) ¯ U † A ˙¯ U A + ρ ⊺ A (0) ˙¯ U ⊺ B ¯ U ∗ B ] = 0 . (5)This is satisfied by the maximally entangled states, forevery possible local evolution ¯ U j . In this regard, theinvariant quantities in the evolution can be written as T r [( Q ) p ]. In terms of the concurrence we can write Q = (1 /d ) I + p C m − C ˆ q · ~T , (6)where C m = p d − /d . The C = 0 value correspondsto separable states. For maximally entangled states C = C m , giving Q = (1 /d ) I , and ρ A = ρ B = (1 /d ) I .In addition, for any ¯ U j ∈ SU ( d ), the matrices ¯ U † j ˙¯ U j arecombinations of the generators T a . Therefore, using thisinformation, the trace in the integrand of eq. (5) van-ishes.Now, let us consider an evolution on the first qudit A . In this case, h ψ (0) | ψ ( t ) i = T r [ Q (0) ¯ U A ( t )], while thedynamical phase is, φ dyn = φ A − i Z t dt ′ T r [ Q (0) ¯ U † A ( t ′ ) ˙¯ U A ( t ′ )] . (7)These phases do not depend on ¯ S (0) so that for simplicitywe can consider ¯ S (0) = I , that is, ¯ U A ( t ) = ¯ S ( t ). Forqubits T a = σ a / a = 1 , , σ a are the Paulimatrices. We shall assume that the basis is chosen sothat ˆ q (0) = ˆ e , that is, Q = I/ √ − C σ /
2. Theunitary sector of the state evolution can be put in termsof Euler angles, in the form ¯ U A ( t ) = U m ( t ) V ( t ), where U m = e − iϕT e iθT e iϕT , V = e iχT . (8)Note that for cyclic evolutions, U m (0) = U m ( τ ), while V (0) = ± V ( τ ). In addition, U m can be expanded interms of I , T and T , as the term proportional to T isobtained from T r [ T e − iϕT e iθT e iϕT ] = T r [ T e iθT ] =0. Here, we have used that the latter exponential is acombination of I and T . Using a similar expansion for e iϕT , we arrive to the conclusion that the terms in U m proportional to T , T do not contribute to h ψ (0) | ψ ( t ) i .With the ingredients above we can work out the ex-pression for the time evolving overlap h ψ (0) | ψ ( t ) i = e iφ A cos θ h cos χ i p − C sin χ i . (9)In terms of Q , U m , and V , the dynamical phase is φ dyn = φ A − i Z t dt ′ T r [( I + p − C σ ) × ( U † m ˙ U m + V † ˙ V )] . (10) -1 -0.5 0 0.5 1 Re[< ψ(0)|ψ( t)>] -1-0.50.51 I m [ < ψ ( ) | ψ ( t ) > ] C=0.8C=0.99C=0 PP ` FIG. 1. Time evolution of the quantum state overlap for apair of qubits with different concurrences.
Using ˙ V = i ( ˙ χ/ σ V and defining the unit vectors ˆ m a so that U m σ a U † m = ˆ m a · ~σ, we get φ g = arctan hp − C tan( χ/ i − p − C ( χ/ p − C (Φ / , (11)with Φ ≡ R t dt ′ ˆ m · ˙ˆ m . In the last term, the frame ˆ m a depends on θ ∈ [0 , π ) and ϕ ∈ [0 , π ] defining a pointon S , the surface of a sphere with unit radius. Then,ˆ m a ( θ, ϕ ) is a mapping S → ˆ m a , and the evolution onthis sector is given by a curve, defined by θ ( t ) , ϕ ( t ), con-tained on S . In this regard, for a cyclic evolution, oneeasily shows that Φ = Ω, where Ω is the solid angle sub-tended by the closed path [15, 16]. This term can beassociated to the usual Berry phase for a single qubit.For a general evolution, we see that for product states( C = 0), the first two terms in eq. (11) cancel each otherwhile the last term coincides with the one given by theusual picture of the Bloch sphere evolution of a singlequbit. On the other hand, for maximally entangled states( C = 1), the last two terms vanish while the first termcan assume only two discrete values 0 or π . In fig.1 thisevolution is represented as paths in the complex plane,where the overlap h ψ (0) | ψ ( t ) i is plotted for different val-ues of the concurrence. This path degenerates to a circlefor product states and to a straight line on the real axisas the concurrence approaches its maximum value C = 1.It gives a graphical picture of the phase jump between 0and π discussed in ref. [10]. This jump occurs when theevolving state crosses the subspace orthogonal to the ini-tial one. Note that the solid lines in fig.1 correspond toclosed paths since points P and P ′ represent physicallyequivalent quantum states. Dashed lines correspond toadditional closed paths.Now, let us study a simple nontrivial path that gener-alizes the V -sector for qubits (cf. eq. (8)) to the case of qudits. Consider an evolution of the form ¯ U A ( t ) = V N ( t ), V N ( t ) = e iχ ( t ) E , χ (0) = 0 , (12)where E is a diagonal traceless matrix with components, E αα = (cid:26) (1 /d ) , α = 1 , . . . , d − /d ) − , α = d . (13)This matrix can be written in terms of the N ’th generatorof SU ( d ), N = d − E = C m T N . When χ ( τ ) = 2 π , itis simple to see that V N ( τ ) = e i π/d I . In the case whereˆ q (0) = ˆ e N , we have, Q (0) = (1 /d ) I + p − ( C / C m ) E . (14)By expanding the exponential in eq. (12) and usingeq.(14) we get, h ψ (0) | ψ ( t ) i = A e iχ/d + B e i (1 − d ) χ/d , (15)with A = d − d + p C m − C and B = 1 − A . Using V † N ˙ V N = i ˙ χ E in the dynamical phase, we arrive at φ g = arctan " A sin χd + B sin (1 − d ) χd A cos χd + B cos (1 − d ) χd − p C m − C χ . In the above example, for maximally entangled states,when d ≥ π/d , and the evolving state never becomes orthogonalto the initial state. This is in contrast to what happensin the d = 2 case. The minimum value for |h ψ (0) | ψ ( t ) i| is ( A − B ) = ( d − d ) , attained when χ = π . For d = 3,the minimum overlap is (1 / .It is interesting to look for topologically nontrivial evo-lutions for qudits with similar properties to those dis-played by qubits. In the d = 3 case, this can be realizedas follows. Let us consider the path ¯ U A ( χ ( t )), contin-uously evolving from ¯ U A (0) = I to ¯ U A (2 π ) = e i π/ I ,defined by a diagonal unitary matrix with nontrivial ele-ments e iφ α such that φ = 2 χ/ π − ζ ) / χ − π ), φ = − χ/
3, and φ = − ( φ + φ ); Θ( χ ) is the Heavisidefunction. For maximally entangled states, we have h ψ (0) | ψ ( t ) i = (cid:26) [1 + 2 cos( χ )] , χ ∈ [0 , π ] [1 + 2 cos( χ + π )3 )] e i π , χ ∈ [ π, π ] . Then, we see that the total phase vanishes in the firstpart of the evolution, while it takes the fractional value2 π/ χ = π , when thephase changes discontinuously, the state | ψ ( t ) i becomesorthogonal to the initial state.Both qutrit evolutions are represented in fig.2a, wherethe overlap h ψ (0) | ψ ( t ) i is plotted in the complex planefor maximal concurrence. The first cyclic evolution from P to P ′ is represented by the solid black line clearly show-ing that the overlap between the initial and the evolvingquantum states never vanishes. On the other hand, the -1 -0.5 0 0.5 1Re[< ψ(0)|ψ( t)>]-1-0.500.51 I m [ < ψ ( ) | ψ ( t ) > ] PP ` " P (a) π π π π χ (rad) π /34 π /3 φ g (r a d ) P P P ` "(b) FIG. 2. (a) Complex plane representation of the quantumstate overlap for a pair of qutrits with maximal concurrence.Two different time evolutions are considered. (b) The corre-sponding stepwise evolution of the geometric phase. second evolution (red online) shows a path crossing theorigin of the complex plane, where the evolving quantumstate becomes orthogonal to the initial one. The dashedlines correspond to additional closed paths defining threevertices which evidence the fractional phase values. Infig.2b, we plot the associated geometric phase evolution,showing a stepwise behavior with two jumps between thefractional values 0, 2 π/
3, and 4 π/
3. For the first evolu-tion (black) smooth jumps occur, while for the secondevolution (red online) they are discontinuous.As a conclusion, in this letter we studied unitary localoperations on a pair of qudits, showing that fractionalphases naturally appear when cyclic evolutions are con-sidered. These fractional values are related to differenthomotopy classes of closed paths in the two-qudit Hilbertspace. The geometric phase has been calculated in termsof the I-concurrence introduced in ref.[13]. In the caseof maximally entangled states, the fractional values orig-inate solely from the geometric part of the phase evolu-tion, since the dynamical part vanishes at all times.The fractional phase of maximally entangled states is built in a stepwise evolution, where the phase jumps be-tween discrete values in steps of 2 π/d . For qubits thisjump is strictly discontinuous, while for qutrits, it may bediscontinuous or not, depending on the particular evolu-tion considered. Due to its stepwise evolution, we expectthe fractional phase acquired by maximally entangled qu-dits to be particularly robust against the influence of theenvironment. In order to produce a relevant change, anyexternal noise would have to cause a large fluctuation,driving the two-qudit system through a phase step. Sincethe phase jump for qubits is strictly discontinuous, its ro-bustness should be even more pronounced. These resultscan be important to proposals of quantum gates basedon topological phases.
ACKNOWLEDGEMENTS
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