Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)
aa r X i v : . [ qu a n t - ph ] J a n Geometric phases and Bloch sphere constructions for SU(N), with a completedescription of SU(4)
D. Uskov and A. R. P. Rau ∗ Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001, USA
A two-sphere (“Bloch” or “Poincare”) is familiar for describing the dynamics of a spin-1/2 particleor light polarization. Analogous objects are derived for unitary groups larger than SU(2) through aniterative procedure that constructs evolution operators for higher-dimensional SU in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possiblelogic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters,and for Hamiltonians of its various sub-groups so that fewer parameters suffice, we derive Bloch-likerotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitaryevolution of a quantal spin pair is thereby expressed as rotations of real vectors. Correspondingly,the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) sub-group and an eight-dimensional Grassmannian manifold for the generalSU(4). This latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectorsmoving on a five-sphere with an additional phase constraint.
PACS numbers: 03.67.-a, 02.20.Qs, 03.65.Vf, 03.65.Fd, 02.40.Yy
I. INTRODUCTION: THE BLOCH SPHEREAND ITS EXTENSION
In the study of the dynamics of a spin-1/2 particle, avisual metaphor that has played a powerful role is thatof the “Bloch sphere” [1]. Pure states of the system arerepresented by the tip of a vector from the origin to thesurface of such a unit sphere S . In the field of nuclearmagnetic resonance (nmr) [2] and elsewhere, transfor-mations between states are then viewed as rotations ofthat vector, described by the Bloch equation of motion,˙ ~m = − ~B × ~m , for a magnetic moment in a magneticfield ~B . Thus, various sequences of nmr manipulationscan be pictured in a nice geometrical way as successive ro-tations, and this has now become central to our intuitionof spin dynamics. The relevant group of unitary trans-formations is SU(2), a rank-one, three-parameter groupthat is the double covering group of the three-dimensionalrotation group SO(3) [3]. The three operators of angu-lar momentum, ( J x , J y , J z ), are the generators of thesegroups. A canonical set of parameters of SO(3) are theEuler angles. Integer values j = 0 , , . . . provide various(2 j +1)-dimensional representations, while for SU(2), thehalf-odd integers occur as well. The two coordinates onS , together with a phase, provide the three parametersdescribing the full state.This latter phase is often not accessible as, for instance,when dealing with the density matrix ρ . Mixed statesalso are naturally accommodated in this picture. Theyare represented by points inside the sphere so that thevector is of length less than unity. Correspondingly, Tr ρ < Tr ρ , which constitutes a definition of a mixed state[4]. States of light polarization, also a two-valued object,map onto the same mathematics and geometry throughthe “Poincare” sphere [5].It would be of interest to have analogous geometri-cal pictures for multiple spins, especially in today’s fields of quantum computation, cryptography, and teleporta-tion, because the fundamental elements of these subjectsare built up of a few qubits [6]. Thus, all logic gatesfor quantum computation can be built up from qubitpairs, while teleporting one qubit state requires an en-tangled pair held by the sender and receiver, for a totalof three qubits. With SU(2 p ) being the relevant group for p qubits, this calls for a similar geometrical description ofhigher SU( N ). In this paper, we develop such a picture,through an easily accessible procedure which iterativelydescends from N to N − n , with n < N , in a mannerthat closely follows the description of SU(2).Our procedure also applies when N is odd, a situationthat does not arise with qubits but elsewhere widely inphysics (for example, qutrits [7], neutrino oscillations [8],the quark model and quantum chromodynamics (QCD),etc.) The N × N matrices of Hamiltonians and evolutionoperators are viewed as built up of 2 × N = ( N − n ) + n decomposition, the blockmatrices then described in terms of the Pauli spinors ofSU(2). Each step of this iterative reduction introducesan analog of the Bloch sphere, albeit of higher dimensionand more complex structure, and constructs the effectiveHamiltonians of dimension ( N − n ) and n for the nextstep. In this manner, using no more than the operationsfamiliar from the SU(2) case, the full construction forSU( N ) is achieved.The philosophy behind such a construction may beseen as generalizing Schwinger’s philosophy for represen-tations of SU(2) or SO(3), where higher j -representationsare constructed from those of the fundamental, j = 1 / N ). In particular, for theimportant case of SU(4) for two qubits, we give a com-plete description of the manifolds and phases involvedand analytical expressions for them. Note again, as withlight polarization and spin-1/2, that the mathematics of N -level systems in quantum optics, atomic and molecularphysics, and elsewhere, is the same as that we describein the language of multiple qubits. This provides an evenwider context for our results.The arrangement of this paper is as follows. Section IIdescribes the basic iterative decomposition of the evolu-tion operator for SU( N ), mimicking the familiar proce-dure for spin-1/2. With N = 4, and n = 2, Section IIIspecializes the results to SU(4), the case of two qubits,when all the manipulations involved are in terms of Paulispinors. It also applies these results to Hamiltonians in-volving a restricted set of operators of the full group. Aninteresting one is SO(5), which can be described by a5 × × ~B in the Blochequation. Section IV then considers Hamiltonians re-quiring the full SU(4) group for their description. Linearequations, analogous to the Bloch equation, are derivedin terms of vectors ~m , five- and six-dimensional vectors,respectively, for the SO(5) and full SU(4) cases. Thelatter also correspond to so-called “Pl¨ucker coordinates”[10] which are also presented. Appendix A deals with thegeneralization to non-Hermitian Hamiltonians, and Ap-pendix B presents the isomorphism between SU(4) andthe groups Spin(6) and SO(6) which we exploit. II. ITERATIVE CONSTRUCTION OFEVOLUTION OPERATOR IN N DIMENSIONS
We wish to obtain the evolution operator U ( N ) ( t ) forthe N -dimensional time-dependent Hamiltonian H ( N ) : H ( N ) ( t ) = (cid:18) H ( N − n ) ( t ) V ( t ) V † ( t ) H ( n ) ( t ) (cid:19) . (1)We have blocked the Hamiltonian into ( N − n )- and n -dimensional blocks, the diagonal blocks being square ma-trices while the off-diagonal V is ( N − n ) × n and V † is n × ( N − n ). Although our discussion is for Hermitian H ( N ) , the procedure can also apply more generally, inwhich case the off-diagonal blocks will not be simply re-lated as adjoints (see Appendix A). We will also assume H ( N ) to be traceless, again a restriction that can be eas-ily relaxed, the time integral of the trace becoming anoverall phase of U ( N ) .To solve the evolution equation, with an over-dot de-noting derivative with respect to time, i ˙ U ( N ) ( t ) = H ( N ) ( t ) U ( N ) ( t ) , U ( N ) (0) = I , (2)we similarly block the unitary matrix, writing it also asa product of three factors, the first two further groupedas ˜ U and the second, ˜ U , block-diagonal in form: U ( N ) ( t ) = ˜ U ˜ U , ˜ U = e z ( t ) A + e w † ( t ) A − , ˜ U = (cid:18) I ( N − n ) z ( t ) † I ( n ) (cid:19) (cid:18) I ( N − n ) † ( t ) I ( n ) (cid:19) , ˜ U = (cid:18) ˜ U ( N − n ) ( t ) † ˜ U ( n ) ( t ) (cid:19) , (3)where A ± are matrix generalizations of the Pauli spinstep-up/down σ ± , and z and w † are rectangular matricesof complex parameters.The above structure, with ˜ U having blocks of zeroin the lower and upper off-diagonal blocks of its matrixfactors, is crucial in our method. For the case of spin-1/2 and SU(2), the form of a product of three factors,each an exponentiation of one of the Pauli spinors, iswell known [3]. Their Cartesian form, with Euler anglesin the exponents, is the familiar choice but we choose in-stead the triplet, ( σ ± , σ z ), when the first two factors havezero off-diagonal entries. This introduces complex z and w † in place of the Euler angles, and makes the individ-ual factors in Eq. (3) not separately unitary although ourconstruction ensures unitarity of the full U ( N ) ( t ). Fur-ther, for non-Hermitian H when U is non-unitary, ourconstruction still applies. The specific structure of anupper and lower triangular matrix and a diagonal oneproves fruitful, giving simpler equations for z and w † ,which will have at most quadratic nonlinearity in theseparameters and not more complicated trigonometric de-pendences as with the Euler angle decomposition [11, 12].They also yield more naturally to a geometrical pictureof the manifolds they describe.A remark about notation. We will use the symbol tildewhen the corresponding Hamiltonians or evolution oper-ators may not be Hermitian or unitary, respectively. Uni-tarity of the full U ( N ) ( t ) leads to relations between z and w † which would otherwise be independent for evolutionunder a non-Hermitian Hamiltonian (see Appendix A), z = − w γ = − γ w , γ ≡ ˜ U ( N − n ) ˜ U ( N − n ) † = I ( N − n ) + zz † , γ − ≡ ˜ U ( n ) ˜ U ( n ) † = ( I ( n ) + z † z ) − . (4)With U = ˜ U ˜ U , Eq. (2) formally reduces to the evo-lution of ˜ U alone with an effective Hamiltonian [12, 13], i ˙˜ U = ˜H eff ˜U , ˜H eff = ˜U − H ˜U − i ˜U − ˙˜U . (5)A key element of our construction lies in this effectiveHamiltonian and corresponding evolution for the reducedproblem. Since ˜ U and this equation are block diag-onal, the off-diagonal blocks in H eff on the right-handside must vanish. This condition leads to the definingequation for z , i ˙ z = H ( N − n ) z + V − z ( V † z + H ( n ) ) . (6)For SU(2), when N = 2 , n = 1, all the matrices abovereduce to single numbers and Eq. (6) is a Riccati equa-tion for the complex z . More generally, it is a matrix Ric-cati equation [14], and its solutions are involved in thesubsequent construction. With the off-diagonal blocksof Eq. (5) accounted for, the diagonal ones defining theHamiltonians for the ( N − n ) and n problems remain,and are given by ( H ( N − n ) − zV † ) and ( H ( n ) + V † z ), re-spectively. Although the overall trace is preserved in ourconstruction and remains zero, these individual Hamilto-nians are neither traceless nor Hermitian. The equationsfor z need to be solved numerically in general but form asmaller set than the N elements in the original Eq. (2).To set up the process for iteration, the above individualHamiltonians in ( N − n )- and n -dimensional subspacesmust be rendered Hermitian and traceless. The latter iseasily achieved, by subtracting Tr ( H ( N − n ) − zV † ) andTr ( H ( n ) + V † z ) from them. These traces being equaland opposite, this translates into the introduction of aphase, the integral of the trace, in U ( N ) , representing arelative phase between the two subspaces.There are alternative methods for rendering the Hamil-tonians Hermitian, the most accessible one being through˜ U † ˜ U = (cid:18) γ − † γ (cid:19) ≡ (cid:18) g g † † g g † (cid:19) − . (7)The first part of this equation is the observation that˜ U † ˜ U is block diagonal. This suggests the second part ofthe equation, namely, the definition of an inverse throughtwo “Hermitian square-root” matrices g i . Together, theyserve as a gauge factor to unitarize according to U = ˜ U (cid:18) g † g (cid:19) . (8)With that, the second factor, ˜ U , in Eq. (3) is also uni-tarized, U = (cid:18) g − † g − (cid:19) ˜ U . (9)After some algebra, the explicitly Hermitian forms ofthe two diagonal block Hamiltonians of dimension ( N − n )and n are H ( N − n ) = i ddt g − , g ]+12 (cid:16) g − ( H ( N − n ) − zV † ) g +hc (cid:17) , H ( n ) = i ddt g − , g ]+12 (cid:16) g − ( H ( n ) + z † V ) g +hc (cid:17) , (10)with commutator brackets in the first term, and hc in thesecond term denoting the Hermitian conjugate of the pre-ceding expression. Again, the trace of each Hamiltonianin Eq. (10) can be subtracted to render them traceless;as clear by inspection, this is the same trace discussed just above. These Hamiltonians in Eq. (10) can now betreated further as SU( N − n ) and SU( n ) problems.The γ matrices in Eq. (4) are Hermitian with non-negative eigenvalues because of their origin from ˜ U † ˜ U .This permits their decomposition into g as shown inEq. (7). The g matrices and their inverses in Eq. (7)-Eq. (10), are square roots of them, and because anypower, including fractional ones, are Hermitian term byterm in a formal power-series expansion, we can choose g also as Hermitian. The use of identities such as z † γ p = γ p z † , γ p z = z γ p , (11)serves to express all g in terms of the linearly indepen-dent set of matrices of dimension ( N − n ) or n , whicheveris smaller. With n = 2, this means that all the algebraof calculating such square-root matrices and the subse-quent evaluation of the effective Hamiltonian in Eq. (10)reduces to manipulation of Pauli matrices.A count of the parameters is instructive. The originalSU( N ) evolution involves ( N −
1) elements and, there-fore, grows quadratically with N . These are divided inthe above construction into the 2 n ( N − n ) elements in z , which for small n grows only linearly with N . Therest are contained in the elements of the SU( N − n ) andSU( n ) and the single phase between those two subspaces.Our construction of higher SU( N ) evolution in terms ofsmaller ones, with the template in Eq. (3) of three factorsas in SU(2), resembles the Schwinger scheme of generat-ing higher j representations of SU(2) or SO(3) from thefundamental one of j = 1 / N ).In mathematical language of base manifolds and fiberbundles [15], the SU(2) and its Bloch sphere are seenas the bundle [SU(2)/U(1)] × U(1), the former the two-sphere S base and the latter U(1) phase the fiber. Like-wise, our construction is in terms of the base manifold[SU( N )/(SU( N − n ) × SU( n ) × U(1))] and the fiber(SU( N − n ) × SU( n ) × U(1)). For SU(2), there is asingle complex z that defines the base manifold. TheBloch sphere of a unit three-dimensional vector ~m cor-responding to z is then constructed by inverse stereo-graphic projection from R to S . Similar structures of a ~m associated with the larger z will be considered in thenext sections. III. THE CASE OF SU(4), WITHAPPLICATION TO ITS SUB-GROUPS
An important case is of N = 4. Four-level systems arecommonly considered in quantum optics and molecularsystems and, of course, in today’s quantum computationwhere they describe two qubits [6]. Since all logic gatescan be built up from such qubit pairs, the study of theevolution operator for such N = 4 problems is of currentinterest. As a combined description of spin and isospin,SU(4) also has central importance in the study of nucleiand particles [16]. The group also occurs in the descrip-tion of unusual magnetic phases of f electron states inCeB [17]. Both choices n = 1 , N = 4 , n = 2, all the matrices involved in theprevious section can be rendered in terms of Pauli spinorsand the unit 2 × × z comprises four complex quantities, ( z , z i ), and the ma-trix Riccati equation reduces to coupled first-order equa-tions in them with quadratic nonlinearity. Deferring thisgeneral case to the next section, we consider first thesmaller sets of operators of various sub-groups of SU(4). su(2) × su(2) sub-algebra: Consider first a Hamilto-nian consisting of only six of the 15 operators. Since ourconstruction is representation independent, in a suitablerepresentation, the six may be viewed as two indepen-dent, mutually commuting, triplets that obey su(2) al-gebra. Clearly, each then may be expected to have itsown geometrical description in terms of a Bloch sphereand phase. In our above, general formulation, this re-sult is realized as follows. Thus, consider two indepen-dent magnetic moments, characterized by the standardPauli matrices σ , in time-varying magnetic fields A ( t )and B ( t ) which may also be independent, with Hamil-tonian H = ~σ (1) · ~A + ~σ (2) · ~B . Using a standard set of4 × V = ( A x − iA y ) I and H (1 , = ~σ · ~B ± A z I .The z in Eq. (3) also reduces, as with V , to a unit opera-tor with a single complex coefficient z obeying a Riccatiequation in Eq. (6). The gamma matrices in Eq. (4) arealso proportional to the unit operator, thus simplifyingEq. (10), the g dropping out. As a result, the Hermitianmatrices in the block-diagonal effective Hamiltonian takethe form of the same ~σ · ~B plus/minus a term proportionalto a unit matrix. The first term is viewed as for a singlespin with a Bloch sphere and a phase, the second repre-sents a phase between the two 2 × z can again be inverse stereographically projected into another two-sphere as in the Bloch construction. We ar-rive, therefore, at the same initial expectation, that asimultaneous viewing in terms of two Bloch vectors inindividual two-spheres, along with their fibers, providesthe geometrical picture for all such qubit-pair systems.A specific physical example occurs in the construction ofoptimal quantum NOT operations [20]. su(2) × su(2) × u(1) sub-algebra: Another sub-algebra, involving seven of the 15 operators, has beenconsidered before [18, 21]. It has the symmetry of SU(2) × SU(2) × U(1). In a suitable representation, such aHamiltonian can be cast as a diagonal form in Eq. (1)plus a term which is proportional to the unit operator inboth diagonal blocks but with equal and opposite sign.Such an operator commutes with all the other six, them-selves comprised of two mutually commuting triplets of4 × V = 0, z also vanishes andwe reduce trivially to the two independent SU(2) and aphase between the two spaces, together accounting forthe 7 parameters of this problem. An example is pro-vided by the CNOT gate constructed with two Josephsonjunctions [22]. Many such sets of seven operators, one ofwhich commutes with all the remaining six, have beenidentified through a general procedure in footnote 11 of[21]. so(5) sub-algebra: Proceeding further to other sub-groups, a non-trivial example is provided by a H thatinvolves ten operators satisfying an so(5) sub-algebra ofsu(4). Again, there are many such sets of ten opera-tors/matrices which close under commutation within thefull set of 15 as noted in footnote 11 of [21]. As a physicalexample, a four-level system of two symmetric pairs, asnaturally so with two identical qubits, has only two realparameters along the diagonal in its H . Selection rulesoften restrict the off-diagonal coupling between the levelsfrom six to four, thus introducing four complex, or eightreal, parameters. The net result of such symmetric four-level systems is a ten-parameter problem [23]. Such H fall into this so(5) sub-algebra. The corresponding groupis the so-called spin group Spin(5) which is the double-covering group of SO(5), the group of five-dimensionalrotations, much as Spin(3), isomorphic to SU(2), is thecovering group of SO(3) [24]. All such Spin(5) or SO(5)will themselves have a Spin(4) or SO(4) sub-group, whichin turn has the two mutually commuting SU(2) or SO(3)discussed above so that the ten matrices can be conve-niently viewed as two sets of commuting triplets plus fourmore which transform like a four-dimensional vector un-der SO(4). For completeness here in this paper, we brieflysummarize results on this so(5) sub-algebra that werepublished elsewhere [12]; see also [25].In a convenient representation that uses Pauli matricesfor two spins [18], we have H ( t ) = F σ (2) z − F σ (2) y + F σ (2) x − F i σ (1) z σ (2) i + F i σ (1) x σ (2) i − F σ (1) y , where the tenarbitrarily time-dependent coefficients F µν ( t ) form a 5 × µ, ν = 1 − i, j, k = 1 − H (1 , = ( ∓ F k − ǫ ijk F ij ) σ k , V = iF I (2) + F i σ i . (12)With the matrix Riccati equation in Eq. (6) cast interms of Pauli spinors together with coefficients z µ = z , z i : z = z I (2) − iz i σ i , it takes the form˙ z µ = F µ (1 − z ν ) + 2 F µν z ν + 2 F ν z ν z µ . (13)(As an alternative, V and z can also be rendered in termsof quaternions (1 , − iσ i ).) γ and γ in Eq. (4) becomeequal and proportional to a unit matrix, (1 + z µ z µ ) I (2) .The structure of Eq. (13) admits to the four quantities z being real. The effective Hamiltonian in Eq. (10) interms of these z becomes H (1 , = H (1 , − ǫ ijk z i F j σ k ∓ F j z σ j ± F z i σ i . (14)We can now construct a five-dimensional unit vector ~m out of the four z , m µ = − z µ (1 + z ν ) , m = (1 − z ν )(1 + z ν ) , µ, ν = 1 − . (15)The nonlinear Eq. (6), or Eq. (13) in z , becomes of simple,linear Bloch-like form,˙ m µ = 2 F µν m ν , µ, ν = 1 − . (16)As in the single spin case, this represents an inversestereographic projection, now from the four-dimensionalplane z ∈ R to the four-sphere S . It provides a higher-dimensional polarization vector for describing such twospin problems. With z so described, the two effectiveSU(2) Hamiltonians in Eq. (14), when solved in turn,give the complete solution. In all, such Hamiltonianspossessing Spin(5) symmetry are, therefore, described bythe geometrical picture of one S and two S spheresalong with two phases. su(3) sub-algebra: Four-level systems with only twoindependent energy parameters along the Hamiltonian’sdiagonal and three complex off-diagonal couplings con-stitute a su(3) sub-algebra with 8 parameters. A generalthree-level system, embedded into four with the fourthlevel completely uncoupled, constitutes a trivial exampleof such an su(3) sub-algebra but less trivial examples canalso occur. The z now has two non-zero complex z fora total of four parameters. The description of this four-dimensional manifold, as well as the remaining SU(2) and a U(1) phase, parallel the discussion of the general SU(4)in the next section, and will be presented elsewhere [26].Therefore, we omit details except to note that setting z = − iz and z = − iz in Section IV reduces to such aSU(3) symmetry. IV. THE GENERAL SU(4) HAMILTONIANINVOLVING ALL FIFTEEN OPERATORS
Instead of the Hamiltonians considered in Section IIIwhich involve sub-algebras of the full two-qubit system,consider an arbitrary 4 × H is ob-tained by adding to the previous Spin(5) Hamiltonianconsidered above the five additional terms, F σ (1) z + F σ (1) x + F i σ (1) y σ (2) i . Correspondingly, Eq. (11) gets anadditional term ± F I (2) in the diagonal H (1 , while in V , the F µ are replaced by F µ − iF µ . Thus, the fullSU(4) amounts to a simple modification of the previouslyconsidered Spin(5) by adding a term proportional to theunit operator to the diagonal blocks and making the four F µ complex, with F µ absorbed as their imaginary parts.The Riccati Eq. (13), now for complex z , becomes˙ z µ = F µ (1 − z ν ) − iF µ (1 + z ν ) + 2 F µν z ν + 2( F ν + iF ν ) z ν z µ − iF z µ , µ, ν = 1 − . (17)The two gammas in Eq. (4) are given by γ , = (1 + z µ ) I (2) + i ( z ∗ i z − z ∗ z i ) σ i ± iǫ ijk ( z i z ∗ j − z j z ∗ i ) σ k . (18)Their square-root matrices g , can also be evaluated interms of the Pauli matrices and the two SU(2) effectiveHamiltonians then constructed in explicitly traceless andHermitian form.Just as the very structure of Eq. (13) suggests that z µ and (1 − z ν ) with suitable normalization define a five-dimensional unit vector ~m in Eq. (15), the occurrence of z µ , (1 − z ν ) , (1 + z ν ) in Eq. (17) suggests now the intro-duction of six quantities according to m µ = − z µ De iφ , m = (1 − z ν ) De iφ , m = − i (1 + z ν ) De iφ , (19)with D ≡ (1 + 2 | z ν | + z µ z ∗ ν ) / , ˙ φ = − F + iF µ ( z ∗ µ − z µ ) + F µ ( z ∗ µ + z µ ) . (20)As with the so(5) case in Section III, with such a set ofsix complex quantities ~m , the nonlinear Riccati equationfor the four complex z µ in Eq. (17) becomes a linearBloch-like equation as before,˙ m µ = 2 F µν m ν , µ, ν = 1 − . (21)Once again, the m µ obey a first-order equation with anantisymmetric matrix which describes rotations. Sincethe 15 F µν are real, the real and imaginary parts ofthe six m µ each obey such a rotational transformation.These six-dimensional rotations reflect the isomorphismbetween the groups SU(4) and SO(6) (more accurately,its covering group Spin(6)) and suggest a mapping be-tween their generators (see Appendix B).To get a geometrical picture of the manifold m , wenote first the relations, m µ = 0 , | m µ | = 2 , (22)which amount to three constraints. In addition, only thederivative, not the value, of φ is determined in Eq. (20).Thereby, the number of independent parameters in m µ is eight just as in the complex z µ , themselves built from z . The description of such an eight-dimensional manifoldwill be taken up in the next sub-section but we note herethe reduction to the previous so(5) example. This followsupon setting F = 0 , F µ = 0 which makes φ = 0 and D = (1 + z ν ) in Eq. (20), and reduces m µ and m tothe values in Eq. (15) whereas m = − i . This, of course,makes ~m a five-dimensional unit vector and its manifoldthe four-sphere S . The first relation in Eq. (22), of thevanishing of a square, hints at Grassmannian elements,to be discussed further below. A. Nature of the manifold describing ( z, m ) forgeneral SU(4) Our construction of the evolution operator for ( N =4 , n = 2) in Eq. (3) is in terms of the eight-dimensionalbase manifold z and a fiber consisting of two residualSU(2) along its diagonal blocks and a U(1) phase betweenthem: SU(4) → [SU(4)/SU(2) × SU(2) × U(1)] × [SU(2) × SU(2) × U(1)]. To describe the former base manifold,consider first [SU(4)/SU(2) × SU(2)], which is a nine-dimensional manifold. It can also be described in termsof spin-groups as Spin(6)/Spin(4). The six complex m µ in Eq. (18) with the three constraints in Eq. (22) consti-tute such a manifold called a Stiefel manifold St (6, 2, R ) ∼ = ℜ , this name being given to manifolds consisting of n orthogonal vectors from an N -dimensional space ℜ N [27].Geometrically, the second relation in Eq. (22) states thatthe real and imaginary parts of m are six-dimensionalunit vectors while the first relation expresses their mutualorthogonality. Therefore, one can view the manifold asa five-sphere S with another four-sphere S attached ateach point on it. The absolute value of the phase param-eter φ in Eq. (19) and Eq. (20) being undefined, reduces such a manifold by one dimension to [SU(4)/SU(2) × SU(2) × U(1)], which is equivalent to the reduction fromthe Stiefel to a Grassmannian manifold G (4, 2, C ) ac-cording to St (6, 2, R ) ∼ = G (4 , , C ) × U(1). Such a Grass-mannian manifold, which has eight dimensions, therebydescribes the z in Eq. (3) or its equivalent z µ in Eq. (17)or m µ in Eq. (19).A more accessible geometrical picture is to consider asingle five-sphere S embedded in six-dimensional spaceand two six-dimensional unit vectors from the origin tothe surface to represent the real and imaginary parts of m µ . The two vectors are always taken as orthogonal, sothat one views such an orthogonally-coupled pair rotat-ing within the sphere [28]. This nine-dimensional object,combined with the zero reference of φ being undefined,is our eight-dimensional manifold of interest. B. Description in Pl¨ucker coordinates
An alternative view of these manifolds is pro-vided in terms of what are termed Pl¨ucker coordi-nates, defined as a set of six complex parameters( P , P , P , P , P , P ) formed as minors of the 2 × U = u u u u u u u u u u u u u u u u . (23)They obey the relations P P − P P + P P = 0 , X | P ij | = 1 . (24)They are combinations of the m µ according to P P P P P P = 12 im − m im + m − im + m − im − m − im + m im + m . (25)The linear equations for m µ in Eq. (20) translate intoa similar linear equation i ˙ P = HP , P ≡ ( P , − P , P , P , P , P ) , (26)with H P = H , H H − H H H H , − H − H − H H − H H , H H − H − H H , H − H H H H H , − H − H H − H − H H , , (27)where we have adopted the notation for the diagonal en-tries: H ii,jj = H ii + H jj .Actually, the above equations for P can be arrived atdirectly from the evolution equation i ˙ U = HU becausethe elements of P are quadratic in the elements of U inEq. (23): P ij = iε ijkl u (3) k u (4) l , and i ˙ u (3) k = H kj u (3) j . Also, z can be defined in terms of the two minors on the rightin Eq. (23): z = (cid:18) u u u u (cid:19) / (cid:18) u u u u (cid:19) , (28)the matrix in the denominator assumed to be non-singular. Writing U in Eq. (23) in the form in Eq. (3),the first factor ˜ U involving z is a map of the Grass-mannian manifold G (4, 2, C ) onto C , and providesa partial coordinization of that manifold. Elements of G (4, 2, C ) are two-dimensional complex hyperplanesspanned by vectors u = ( u , u , u , u ) T and u =( u , u , u , u ) T . The Pl¨ucker coordinates provide aunique identification of such planes. They are an analogof the coordinization of the n -dimensional sphere S n byan ( n + 1)-dimensional unit vector ~m as in Section III.The matrix H P in Eq. (27) being Hermitian, P † P = constant = 1. This can be verified by the relationbetween P ’s and m ’s in Eq. (25) which involves a uni-tary matrix so that P † P = m † m , and combining withEq. (22). Further, a symplectic structure can be intro-duced. Defining a 6 × Ω ≡ δ i, − j with non-zeroentries of 1 only along the anti-diagonal, the first rela-tion in Eq. (24) can be rendered as P T ΩP = 0, and thematrix H P , a generator of the symplectic group Sp (6, C), H P Ω + ΩH TP = Tr( H P ) Ω = 0 . (29) Any two vectors P i , evolving according to Eq. (26), satisfy P T ( t ) ΩP ( t ) =constant. If P ΩP = 0 , then the twohyperplanes defined by P i intersect, and if | P ΩP | = 1 ,they do not.Geometrically, the set P † P = 1 is a sphere S , the alge-braic relation PΩP = 0 determining a 9-dimensional sub-manifold, an intersection between S and the affine varietyof roots of the polynomial equation PΩP = 0 . This mani-fold may be denoted ℜ . Multiplication by a phase acts asa transformation group on this manifold, that is, if P ∈ ℜ ,then P e iφ ∈ ℜ . Therefore, G (4, 2, C) is a quotientspace ℜ / U(1) and has eight dimensions. The connection to SU(4) is, as noted before, ℜ ∼ = SU(4)/(SU(2) × SU(2)) ∼ = Spin(6)/ Spin(4). The stability sub-group of a vector P ∈ ℜ is SU(2) × SU(2) while the stability sub-group of ℜ /U(1) is SU(2) × SU(2) × U(1). Since Spin(6)/Spin(5) ∼ = S and Spin(5)/Spin(4) ∼ = S , we can identify the fibra-tion of ℜ with S × S . V. SUMMARY
We have presented a complete analysis of the evolutionoperator for SU( N ), setting up its construction in a hier-archical way in terms of those for smaller SU( N − n ) andSU( n ), with n < N and arbitrary. The evolution operatoris written as a product of two N × N matrices, the secondof which is block diagonal in ( N − n ) × ( N − n ) and n × n of the smaller groups. The first factor is obtained througha z , which is an ( N − n ) × n complex matrix obeying a ma-trix Riccati equation. Its solutions determine both the firstfactor as well as the Hermitian matrices for the subsequent N − n and n evolution problems.This general constructive method is applied especially toa four-level system with special emphasis on two qubits.The general symmetry is of SU(4), a 15-parameter group.Our procedure expresses the evolution operator as a prod-uct of two 4 × z is also a × matrix with complex entries in general and obeys amatrix Riccati equation. Alternatively, we transform z intoa six-dimensional complex vector ~m , whose real and imag-inary parts both separately undergo linear, six-dimensionalrotational transformations. This is exactly analogous to thelinear Bloch equation for real three-dimensional rotations ofa vector to represent the evolution operator for a single spinin a magnetic field.Just as a Bloch sphere describes the three-dimensionalvector ~m for a single spin (and, together with a phase, thecomplete SU(2)), we also present the geometrical manifolddescribing z or its equivalent six-dimensional complex vector ~m . Together with two residual SU(2) problems and a phase,this provides a complete description of the quantum evolu-tion operator for SU(4). For certain sub-algebras of SU(4),the manifold is an analogous higher-dimensional sphere; afour-sphere, for example, for an so(5) sub-algebra. For themost general SU(4), we have an eight-dimensional Grass-mannian manifold. We provide a picture of it as two five-spheres with an orthogonality and phase constraint. Thesegeometrical objects may serve for all possible four-level andtwo qubit systems the useful purpose that the Bloch spherehas for two-level and single qubit problems in physics. APPENDIX A: EXTENSION TO NON-UNITARYEVOLUTION FOR A NON-HERMITIANHAMILTONIAN
The iterative method of Section II for the evolution op-erator in Eq. (2) through writing it as in Eq. (3) applies also when H in Eq. (1) is not Hermitian and, therefore, theevolution not unitary. However, z and w in Eq. (3) are nolonger simply related as in Eq. (4) but obey independentequations, the former still in Riccati form but the lattergiven in terms of z . Thus, instead of Eq. (1), consider ˜ H ( N ) ( t ) = (cid:18) ˜ H ( N − n ) ( t ) V ( t ) Y † ( t ) ˜ H ( n ) ( t ) (cid:19) , (30) where we have again indicated by tildes non-Hermiticity,and V and Y are not equal but independent.Writing ˜ U ( N ) ( t ) again as in Eq. (3), Eq. (6) now becomes i ˙ z = ( ˜ H ( N − z − z ˜ H ( n ) − zY † z + V ,i ˙ w † = w † ( zY † − ˜ H ( N − )+ (˜ H ( n ) − Y † z ) w † + Y † . (31) The residual problems of ( N − n ) and n dimension thenbecome i ˙˜ U ( N − n ) † i ˙˜ U ( n ) ! = (cid:18) ˜ H ( N − − zY † † ˜ H ( n ) + Y † z (cid:19) ˜ U . (32) APPENDIX B: DESCRIPTION OF EVOLUTIONAS SIX-DIMENSIONAL ROTATIONS
For a single spin or qubit, the rewriting of the quantumevolution operator, which is complex, as rotational trans-formations of a real, unit vector in three dimensions givenby the Bloch equation, rests on the isomorphism of thegroup SU(2) to SO(3) (or its double covering Spin(3)). Asimilar isomorphism between the groups SU(4) and SO(6)(or its extension Spin(6)) underlies the construction in Sec-tions III and IV of the complex evolution operator for twoqubits in terms of rotations of a vector in six dimensions.Both groups are described by 15 real parameters throughan antisymmetric F µν , µ, ν = 1 , , . . . . In Sections III andIV, explicit expressions are given for the Hamiltonian witheach of these parameters multiplying one of the 15 complexgenerators of SU(4) in a standard representation of Paulimatrices, ~σ (1) ⊗ I (2) , I (1) ⊗ ~σ (2) , ~σ (1) ⊗ ~σ (2) . An alternativerendering in terms of the 15 generators of SO(6) is usefuland recorded here.The Hamiltonian in Section IV, apart from a factor of ,can be cast in terms of a matrix array σ (2) z − σ (2) y − σ (1) z σ (2) x σ (1) x σ (2) x σ (1) y σ (2) x − σ (2) z σ (2) x − σ (1) z σ (2) y σ (1) x σ (2) y σ (1) y σ (2) y σ (2) y − σ (2) x − σ (1) z σ (2) z σ (1) x σ (2) z σ (1) y σ (2) z σ (1) z σ (2) x σ (1) z σ (2) y σ (1) z σ (2) z − σ (1) y σ (1) x − σ (1) x σ (2) x − σ (1) x σ (2) y − σ (1) x σ (2) z σ (1) y σ (1) z − σ (1) y σ (2) x − σ (1) y σ (2) y − σ (1) y σ (2) z − σ (1) x − σ (1) z , (33) which is explicitly anti-symmetric. We thus have H =2 F µν L νµ . Analogous to the familiar triplet of angular mo-mentum generators, six-dimensional generators of SO(6) aregiven by L µν = − il µν , where the l are 15 real antisymmet-ric × matrices with only two non-zero entries, +1 in the ( µν ) and − in the ( νµ ) position: ( l µν ) ρσ = δ µρ δ νσ − δ µσ δ νρ . (34) Their commutators close: [ l µν , l ρσ ] = δ νρ l µσ + δ µσ l νρ − δ νσ l µρ − δ µρ l νσ , (35) so that L µν form an so(6) algebra.The array in Eq. (33) is also a convenient display of thegenerators of the various sub-groups of SO(6) of lower-dimensional rotations. Either upper left or lower right cor-ner × blocks describe the SO(2) generator of one of thequbits. Adding a third row and column gives the full tripletof SO(3) generators. To this can be added a next row andcolumn of three non-zero entries to give the six generatorsof SO(4). For this purpose, any of the three remainingrow/column can be employed, each giving an SO(4), thethree added entries transforming as a vector under SO(3).This continues. Adding another row and column’s four newentries, which transform as a vector under SO(4) (furthersubdividing into three components that transform as a vec-tor and one as a scalar under the previous SO(3)), gives theten SO(5) generators. The final sixth row/column adds fiveentries, an SO(5) vector, to give the full 15 generators ofSO(6). This hierarchical nesting of SO sub-groups, togetherwith the corresponding Clifford structure with Pauli matri-ces in Eq. (33), accounts for the richness of the structuresin the isomorphic groups SU(4) and SO(6), one we have ex-ploited in Sections III and IV. Note that the linear Bloch-likeequation for ~m in Eq. (21) for a general SU(4) Hamiltonianreduces to the same antisymmetric form for its sub-groupssuch as in Eq. (16), all the way down to the standard Blochequation for a single qubit, whose SO(3) antisymmetric F ij is usually written as a vector product with a magnetic field. [*] Email: [email protected][1] W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001).[2] A. Abragam,
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