aa r X i v : . [ qu a n t - ph ] M a y Exact Abelian and Non-Abelian Geometric Phases Chopin Soo, Huei-Chen Lin
Department of Physics, National Cheng Kung University, Tainan 701, Taiwan email: [email protected], [email protected] Abstract
The existence of Hopf fibrations S N + / S = CP N and S K + / S = HP K allows us to treat the Hilbert spaceof generic finite-dimensional quantum systems as the total bundle space with respectively U ( ) and SU ( ) fibersand complex and quaternionic projective spaces as base manifolds. This alternative method of studying quan-tum states and their evolution reveals the intimate connection between generic quantum mechanical systems andgeometrical objects. The exact Abelian and non-Abelian geometric phases, and more generally the geometricalfactors for open paths, and their precise correspondence with geometric Kahler and hyper-Kahler connectionswill be discussed. Explicit physical examples are used to verify and exemplify the formalism. Introduction
The study of geometric phases in quantum mechanics is a fruitful and active endeavor (see, for instance, Ref. [1] for a review,and references therein for a selected sample of the literature and history of the subject). It reveals that fundamental geomet-rical structures are present in generic quantum systems; and the rich and exact interplay that can exist between geometricalmathematical structures (e.g. Hopf fibrations), physical solitons (monopoles and instantons, just to cite lowest dimensionalexamples) and generic quantum systems is both fascinating and of great pedagogical value.The existence of Hopf fibrations S N + / S = CP N and S K + / S = HP K allows us to treat the Hilbert space of genericfinite-dimensional quantum systems as the total bundle space with respectively U ( ) and SU ( ) fibers and complex andquaternionic projective spaces as base manifolds. (In the latter case of quaternionic projective space, only even dimensionalsystems are permitted). This pedagogical review relies heavily on two recent in-depth studies of Abelian and non-Abeliangeometric phases in quantum mechanics[2]. Previously Aharonov-Anandan presented the Abelian geometric phase as thedifference between the total and dynamical phase[3], and Page formulated the result in terms of complex projective Hopffibrations[4]. Non-Abelian geometric phases were also discussed in terms of Hopf fibrations by Adler and Anandan[5]. Ageneral discussion on Hopf fibrations can be found in [6]. The method of using Hopf fibrations to study quantum statesand their evolution reveals the intimate connection between generic quantum mechanical systems and geometrical objects.The exact Abelian and non-Abelian geometric phases, and more generally the geometrical factors for open paths, and theirprecise correspondence with geometric Kahler and hyper-Kahler connections will be discussed. The emphasis here is on theapplicability of the formulation to generic finite-dimensional systems and on the exactness of the resultant geometric phases.Explicit physical examples are used to verify and exemplify the formalism. Hilbert space of finite-dimensional quantum systems and Hopf fibrations
Consider the Hilbert space of an arbitrary “ ( N + ) -state” pure system: Let {| a i} , a = , , ··· , N be a time-independentorthonormal basis. An arbitrary normalized state may be expressed as: | Y i = z a p ¯ z b z b | a i ≡ c a | a i ; (1)wherein ~ z ( t ) = ( z ( t ) , z ( t ) , ··· , z N ( t )) ∈ C N + − { } . Writing the complex coefficients c a = x a + iy a as real numbers x a ∈ R and y a ∈ R , and noting the normalization condition,1 = N (cid:229) a = | c a | = N (cid:229) a = h ( x a ) + ( y a ) i , (2)lead to the conclusion of the correspondence { c a } ⇔ { x a , y a } ∈ S N + . Thus the Hilbert spaces of respectively 2-state,3-state, 4-state, 5-state, ··· systems are associated with S , S , S , S , ··· , i.e. odd-dimensional spheres which have veryspecial properties! Besides the simple Hopf fibration S M / { + , − } = RP M over M-dimensional real projective space andthe octonionic Hopf fibration S L + / S = OP = S , the other two series of Hopf fibrations over complex and quaternionicprojective spaces are of great interest. These are the complex Hopf fibrations: S N + / S = CP N over N-dimensional complexprojective spaces (e.g. S / S = [ CP = S ] (Dirac monopole)), and the quaternionic Hopf fibrations: S K + / S = HP K overK-dimensional quaternionic projective spaces (e.g. S / [ S = SU ( )] = [ HP = S ] (BPST instanton)). omplex Hopf fibration over CP N and exact Abelian geometric phase Complex projective spaces CP N are defined as spaces of ~ z modulo the equivalence relation ~ z ( t ) ∼ l ( t ) ~ z ( t ) ; l ∈ C − { } i.e.with ( z , z , ··· , z N ) and ( l z , l z , ··· , l z N ) identified. In any local patch or chart U ( h ) wherein z h =
0, the inhomogeneouscoordinates z a ( h ) ( t ) = z a ( t ) / z h ( t ) are well-defined, and we can pass from homogeneous coordinates z a to z a which is explic-itly invariant under the complex l scaling. The Hopf projections for C N + − { } → S N + → CP N can be explicitly realizedby z a → c a ≡ z a p ¯ z b z b → c a / c = z a / z = z a ( h = ) ; (3)with each projection specified in local chart U ( h = ) and extended to the atlas of all charts, ∪ U ( h ) . This constitutes an explicitHopf map of the S N + bundle over CP N base manifold with U ( ) fiber.The exact formula of geometric factor in quantum mechanics can be obtained from the following considerations: Locally, S N + ∼ (cid:8) partof CP N (cid:9) × S and we may express | Y i in local coordinates. In the local patch U ( ) wherein z a = z a / z , and e i f z ≡ z / | z | lead to | Y i = z a p ¯ z b z b | a i = c a | a i = e i f z z a q ¯ z b z b | a i . (4)Substituting into the Schrodinger equation, i ¯ h ddt | Y i = H ( t ) | Y i yields d f z dt + ¯ z a ( d z a / dt ) − z a ( d ¯ z a / dt ) i ¯ z b z b = − ¯ z a H ab z b ¯ h ( ¯ z h z h ) . (5)Identifying A ≡ ¯ z a ( d z a / dt ) − z a ( d ¯ z a / dt ) i ¯ z b z b dt implies the overall phase can be solved as f z ( t ) = f z ( ) − ( Z z ( t ) z ( ) A ) − h Z t h Y | H | Y i dt . (6)It follows that the generic state is expressible, in terms of z -coordinates of CP N and the phase f z , as | Y ( t ) i = z a ( t ) q ¯ z b ( t ) z b ( t ) | a i = e i f z ( t ) z a ( t ) q ¯ z b ( t ) z b ( t ) | a i . (7)Moreover, in the overlap U ( h ) ∩ U ( x ) , we have z a ( x ) = z a / z x = ( z h / z x ) z a ( h ) ∀ a , and the transition function ( z x / z h ) ≡ Re i f ∈ C .The geometric connection is thus revealed to be A ≡ − i ¯ z a d z a − z a d ¯ z a z b z b = − i ¯ z i d z i − z i d ¯ z i ( + ¯ z j z j ) , j = , ,... N ; (8)which is an Abelian connection whose curvature is F = dA = K , wherein K is the Kahler 2-form (which is real and closed( dK = CP N which is a Kahler-Einstein manifold with the Fubini-Study metric.The preceding formulas straightforwardly imply that the overlap function at different times is given by h Y ( T ) | Y ( o ) i = ¯ z a ( T ) z a ( o )[ ¯ z b ( T ) z b ( T )] [ ¯ z k ( o ) z k ( o )] e − i ( f z ( T ) − f z ( o )) = ¯ z a ( T ) z a ( o )[ ¯ z b ( T ) z b ( T )] [ ¯ z k ( o ) z k ( o )] exp (cid:18) i Z z ( T ) z ( o ) A + i ¯ h Z To h Y ( t ) | H ( t ) | Y ( t ) i dt (cid:19) . (9)By subtracting R To h Y ( t ) | H ( t ) | Y ( t ) i dt which is referred to as the ”dynamical phase”, the geometric phase factor is the residualentity in the overlap function. In the special case of a closed path c = ¶ S bounding a two-surface S , z a ( T ) = z a ( o ) ( closedpath means the wave function at o and T differs by only a total phase ⇐⇒ z a ( T ) = z a ( o ) ∀ a ), the geometric phase factorwith z a ( h = ) = z a / z ; z ( h = ) = [ e i H c = ¶ S A ] = arg h exp (cid:16) I c ¯ z i d z i − z i d ¯ z i ( + ¯ z j z j ) (cid:17)i = Z S F . (10) A note on the gauge symmetry of the geometric phase:
There are two connections: h Y ( t ) | ddt | Y ( t ) i dt , and the Kahler connection A ≡ − i ¯ z a d z a − z a d ¯ z a z b z b . They are related by h Y ( t ) | d | Y ( t ) i = A ( h ) + d f ( h ) ( t ) ; (11)herein f ( h ) = z h / | z h | . The L.H.S begets additional term d c ( t ) under | Y ( t ) i 7→ e i c ( t ) | Y ( t ) i . Consistently, on the R.H.S. f ( h ) ( t ) f ( h ) ( t ) + c ( t ) . But A ( z ) remains explicitly unchanged (an overall scaling for all z a does not change z a ≡ z a / z h )!In other words, despite the similarities with the approach by Aharonov-Aharonov [3], the Kahler potential A does not gaugethe symmetry | Y ( t ) i 7→ e i c ( t ) | Y ( t ) i (which Aharonov-Anandan advocated [3]). Rather, the Kahler connection A transformsas an Abelian U ( ) gauge potential under local coordinate transformations between patches and gauges this symmetry. Inthe overlap U ( h ) ∩ U ( x ) , the coordinates are related by z a ( x ) = z a / z x = ( z a / z x ) z a ( h ) ∀ a ; thus the transition function is just ( z x / z h ) ≡ Re i c ∈ C . Under this change of coordinates, the connection A = − i ¯ z a d z a − z a d ¯ z a z b z b transforms as A A ′ = A + d c .The geometric phase/factor and the state remain invariant under such coordinate transformations between different patches. Explicit examples
Generic qubit systems, S / S = CP , and the Dirac monopole: A generic qubit (or 2-state) system corresponds to the Hopf fibration S / [ U ( ) = S ] = [ CP = S ] . For the qubit system, thestate is | Y i = (cid:229) a = c a | a i = (cid:229) a , b = e i f z z a q ¯ z b z b | a i . (12)The explicit parametrization c = e i ( c − f ) cos ( q ) and c = e i ( c + f ) sin ( q ) with | c | + | c | =
1, leads to the Hopf map pro-jection c a ( c , q , f ) → z a ∈ CP : z = c / c = , z = c / c = e i f tan ( q ) . (13)The geometric Kahler connection computed in accordance with our previous discussion is A = ¯ z d z − z d ¯ z i ( + ¯ z z ) = ( − cos q ) d f , e i f z = z / | z | = c / | c | = e i ( c − f ) , (14)which is precisely the gauge potential for a Dirac monopole connection with Chern number12 p Z F = p Z dA = p Z pq = Z p
12 sin q d q ∧ d f = . (15)Note also that the local chart fails at the south pole q = p where c vanishes, and we need more than one patch for the atlas.A chart which fails only at the north pole ( q =
0) is z a ( h = ) = c a / c . In the overlap U ( ) T U ( ) , we have z a ( ) = ( c / c ) z a ( ) with transition function ( c / c ) = e i f tan ( q ) . Moreover, the phase of the coordinate transition function e i f tan ( q ) is precisely f ; hence following our discussions in section VII, A ( ) = A ( ) + d f = A ( ) + e i f ide − i f . The monopole charge can alsobe deduced, via the Wu-Yang formulation, from the P ( U ( )) homotopy map of the transition function, e i f : f ∈ S → e i f ∈ U ( ) = S , which has winding number 1. Note the distinction between the Wu-Yang transition function relating themonopole potentials A ( ) and A ( ) (which are connected by gauge transformation e i f ∈ U ( ) ) and the transition function e i f tan ( q ) between coordinate patches which is a complex scaling. Remarkably the setup in the previous sections yield theseresults self-consistently. Furthermore, according to the rules of the formalism, the general state is | Y ( t ) i = z a ( t ) q ¯ z b ( t ) z b ( t ) | a i = e i f z ( t ) z a ( t ) q ¯ z b ( t ) z b ( t ) | a i = e i f z ( t ) p ( + tan ( q / ) h | i + e i f tan ( q / ) | i i = e i f z ( t ) [ cos ( q ( t ) / ) | i + e i f ( t ) sin ( q ( t ) / ) | i ] . (16)It should be noted that in addition to the ( q , j ) Bloch sphere characterization of the usual 2-state density matrix, the quantum state depends additionally on e i f z which contains the geometric phase and connection. A 2-state subsystem of the harmonic oscillator:
A very simple example is the time-independent harmonic oscillator with Hamiltonian and eigenvalues, H = p / m + m w x ; E n = (cid:0) n + (cid:1) ¯ h w . If we are restricted to a normalized 2-state basis | n = i and | n = i , it follows that | Y ( ) i = cos ( q / ) | i + sin ( q / ) | i ; and | Y ( t ) i = e − i ¯ h Ht | Y ( ) i = cos ( q / ) e − i w t | i + sin ( q / ) e − i w t | i = e − i w t h cos ( q / ) | i + sin ( q / ) e − i w t | i i . (17)hus we read off f z = − w t ; f = − w t ; A = − d f z + d c − cos q d f . At time t = T = p / w , we then have | Y ( t = p / w ) i = e − i p | Y ( ) i . Furthermore, R t = p / w A = R p / w w dt − w dt + ( cos q ) w dt = p ( cos q − ) and h R t = p / w h Y ( t ′ ) | H | Y ( t ′ ) i dt ′ =( − cos q ) p . It can also be verified explicitly that these results confirm the general formula h Y ( T ) | Y ( o ) i = ¯ z a ( T ) z a ( o )[ ¯ z b ( T ) z b ( T )] [ ¯ z k ( o ) z k ( o )] e − i ( f z ( T ) − f z ( o )) = ¯ z a ( T ) z a ( o )[ ¯ z b ( T ) z b ( T )] [ ¯ z k ( o ) z k ( o )] exp (cid:18) i Z z ( T ) z ( o ) A + i ¯ h Z To h Y ( t ) | H ( t ) | Y ( t ) i dt (cid:19) . (18) Arbitrary spin J system in a rotating magnetic field:
Consider, as shown in the figure below, a particle of angular momentum J in a rotating magnetic field ~ B = B ( sin a cos w t , sin a sin w t , cos a ) inclined at angle a with respect to the z-axis. B Figure 1: The rotating magnetic field.The time-dependent Hamiltonian of the system has the form H ( t ) = − µB ( sin a cos w tJ + sin a sin w tJ + cos a J ) . with H ( ) = − µB ( sin a J + cos a J ) . Furthermore, the Hamiltonian H ( t ) = V † H ( ) V = e − i w tJ [ − µB ( sin a J + cos a J )] e i w tJ ; V = e i w tJ (19)does not commute at different times; and the evolution of the state is governed by | Y ( t ) i = U ( t ) | Y ( ) i , i ¯ h ddt | Y ( t ) i = H ( t ) | Y ( t ) i with time-ordered evolution operator, U ( t ) = T [ exp ( − i ¯ h R T H ( t ′ ) dt ′ )] . This implies, H ( ) = V H ( t ) V † = V (cid:20) i ¯ h ( ddt U ) U † (cid:21) V † = i ¯ h ( − i w J ) + i ¯ h ( ddt VU )( VU ) † ; (20)with U ( t ) = V † exp (cid:26) i (cid:20) µB ¯ h sin a J + ( µB ¯ h cos a + w ) J (cid:21) t (cid:27) = V † exp { i [ W tan b J + W J ] t } . It can be worked out that the unitaryevolution operator (for arbitrary J) takes the form ( U ) MM ′ = (cid:28) J , M | V † exp (cid:26) i (cid:20) µB ¯ h sin a J + ( µB ¯ h cos a + w ) J (cid:21)(cid:27) | J , M ′ (cid:29) = ( e − i w tJz ¯ h e − iJz g ¯ h e − iJy b ′ ¯ h e − iJz a ′ ¯ h ) MM ′ = e − i [ M w t +( M + M ′ ) g ] ( cos b ′ ) M + M ′ ( sin b ′ ) M − M ′ ( − ) M − M ′ e iM ′ p (cid:20) ( J − M ) ! ( J − M ′ ) ! ( J + M ) ! ( J + M ′ ) ! (cid:21) · J (cid:229) n = ( − ) n ( J + M + n ) ! ( J − M − n ) ! ( M − M ′ + n ) ! n ! ( sin b ′ ) n ; (21)wherein sin b ′ = sin b sin J t , sin g = cos J t q cos J t + cos b sin J t , cos b ′ = r cos J t + cos b sin J t , cos g = cos b sin J t q cos J t + cos b sin J t , J = W cos b , a ′ = g − p . (22) Qubit spin 1/2 system in rotating magnetic field:
Specializing to a spin 1/2 or 2-state system in a rotating magnetic field, the evolution operator is then U ( t ) = e − i w t (cid:16) cos J t + i cos b sin J t (cid:17) e − i w t (cid:16) i sin b sin J t (cid:17) e i w t (cid:16) i sin b sin J t (cid:17) e i w t (cid:16) cos J t − i cos b sin J t (cid:17) . (23)s an example, the choice of the initial state | Y ( ) i = (cid:18) (cid:19) , and | Y ( t ) i = U ( t ) | Y ( ) i yield the following results: A = A ( ) = (cid:16) J cos b − w ( + cos ( b ) + ( J t ) sin b ) (cid:17) dt , (24)1¯ h h Y ( t ) | H ( t ) | Y ( t ) i = ( − J cos b + w cos b + w cos J t sin b ) , t = , (25) A + h h Y ( t ) | H ( t ) | Y ( t ) i dt = − w dt ; (26) h Y ( T ) | Y ( t ) i = e − i ( t + T ) w (cid:26) e it w sin b sin J t sin J T + e iT w (cid:18) cos J t + i cos b sin J t (cid:19) · (cid:18) cos J T − i cos b sin J T (cid:19)(cid:27) (27)¯ z a ( T ) z a ( t )[ ¯ z b ( T ) z b ( T )] [ ¯ z k ( t ) z k ( t )] = sin b · sin J t · sin J T + e − i ( t − T ) w (cid:18) cos b sin J t − i cos J t (cid:19) (cid:18) cos b sin J T + i cos J T (cid:19) . (28)It can again be checked from these that the formula h Y ( T ) | Y ( t ) i = ¯ z a ( T ) z a ( t )[ ¯ z b ( T ) z b ( T )] [ ¯ z k ( t ) z k ( t )] e i R Tt ( A + h h Y ( t ) | H ( t ) | Y ( t ) i dt ) is satis-fied. Qutrit 3-state system:
A 3-state J = U ( t ) = − cos b ′ e − i ( g + w t ) − √ sin b ′ e − i ( g + w t ) − sin b ′ e − i w t − √ sin b ′ e − i g cos b ′ √ sin b ′ e i g − sin b ′ e i w t √ sin b ′ e i ( g + w t ) − cos b ′ e i ( g + w t ) . (29)For arbitrary initial state to return to original value modulo an overall phase (i.e. closed path in CP ) after time T, theconditions w T = p ; J T = m p m = , , , ··· (30)must be satisfied. We consider some specific cases as illustrations: Case (1):
Let ¯ h w µB = √ − √ , and ¯ h J µB = √ − √ . In this case, tan b is chosen to be 1, and J = w . Thus m = n = h Y ( t ) |−→ J | Y ( t ) i with initial state h m = , , − | Y ( ) i =( , √ , √ ) is plotted in Fig.(2). Since in this case the final state differs from the initial state by an overall phase, h Y ( t ) |−→ J | Y ( t ) i must be closed. And it is. Cases (2) and (3):
We next choose ¯ h w µB = q −√ and ¯ h J µB = r ( −√ ) . The irrational ratio for Jw implies that states with thisconfiguration do not obey the periodicity conditions. For the same initial state as in Case (1) , the evolution for two differentfinal times T are plotted in Fig.(3) and Fig.(4). As expected, h Y ( t ) | ~ J | Y ( t ) i does not describe closed paths in either instance,implying that the state cannot differ by just an overall phase after time T when the periodicity conditions are not satisfied. Qutrit 3-state system and S / S = CP fibration: It should be mentiond that a generic 3-state system corresponds tothe Hopf fibration S / S = CP . The state can be expressed as | Y i = c | i + c | i + c | i , and explicit parametrization of S by c = e i ( c + f ) cos ( q / ) , c = e i ( c − f ) sin ( q / ) cos ( q / ) , c = e i ( f ) sin ( q / ) sin ( q / ) , lead to z ( ) = , z ( ) = e i f tan ( q / ) cos ( q / ) , z ( ) = e i g tan ( q / ) sin ( q / ) , g ≡ f − c + f . (31)From these, it can be computed that A = ( − cos q )[ d ( f + g ) + cos q d ( f − g )] , and its curvature 2-form F = p sin q d q ∧ h ( q / ) d f + sin ( q / ) d g i − ( − cos q ) sin q d q ∧ d ( f − g ) = ∗ F . (32)Moreover, it can be checked that integrated over the entire 4-dimensional CP manifold the self-dual curvature F yields p R CP F ∧ F = + - - Figure 2: h Y ( t ) | ~ J | Y ( t ) i from t = t = pw with J = w . - - - Figure 3: h Y ( t ) | ~ J | Y ( t ) i from t = t = √ pw . - - - Figure 4: h Y ( t ) | ~ J | Y ( t ) i from t = t = · pw . Quaternionic Hopf fibration and non-Abelian geometric phase factor
In a manner analogous to the construction of Hopf fibration over complex projective space, the formalism for quaternionicprojective space can be obtained by studying the geometry of S K + / [ S = SU ( )] = HP K , but with the caveat that it isapplicable only to finite-dimensional systems with even number of states. The reason is that each quaternion has to beassociated with a pair of complex state coefficients. Starting with a generic state as | Y i = N = K + (cid:229) a = C a | a i ; C a = N (cid:229) b = a z a √ z b ¯ z b ; (33)we may defined the associated quaternions through q a = Re ( z a ) I + Im ( z a ) s i + Re ( z a ) s i + Im ( z a ) s i ; a = , , ··· , K , a ≡ a + K + s , , = Paulimatrices . (34)It follows that | Y i = K (cid:229) a = Tr (cid:0) P − Q a (cid:1) | a i + Tr (cid:16) P + ( i s ) Q a (cid:17) , Q a ≡ q a q Tr (cid:0) q b q † b (cid:1) ; (35)with the projector P ± = ( I ± s ) . For the bundle S K + / S = HP K , the quaternionic Hopf fibration can be realized throughthe projection H K + − { } → S K + → HP K which is q a → Q a = q a p q b q † b → Q a / Q h = q a / q h = h a ( h ) , (cid:229) a | Q a | =
1; (36)n each local chart U ( h ) and extended to the atlas ∪ U ( h ) . The resultant h a ∈ HP K are precisely the inhomogeneous coordinatesof HP K . In U ( ) , h a ≡ ( q ) − q a = ( Q ) − Q a , ∀ ≤ a ≤ K . It follows that the expression for the state expressed in termsof local HP K coordinates and the fiber (which is the unit quaternion ˆ q ≡ q / (cid:12)(cid:12) q (cid:12)(cid:12) ∈ SU ( ) = S ) is thus | Y i = K (cid:229) a , b = Tr ( P − ˆ q h a ) | a i + Tr ( P + ( i s ) ˆ q h a ) | a i q Tr ( h b h † b ) . (37)By substituting into the Schrodinger equation, we can similarly obtain the evolution of the non-Abelian phase factorˆ q ( t ) = T e i ¯ h R t H dt ˆ q ( )[ T e − i R t Adt ] † ; (38)wherein A = Adt = h a ( dh † a ) / ( dt ) − ( dh a ) / ( dt ) h † a iTr ( h b h † b ) dt is the associated non-Abelian connection. The connection − A = A HP K = dh a h † a − h a dh † a iTr ( h b h † b ) is also the (quaternionic) Kahler connection of the quaternionic Kahler manifold HP K . The overlap function atdifferent times can be computed to be (analogous to the result of Eq. (9)) h Y ( T ) | Y ( ) i = Tr h † a ( T ) q Tr ( h b ( T ) h † b ( T )) T e − i R T Adt ˆ q †0 ( )[ T e i ¯ h R T H dt ] † P − ˆ q ( ) h a ( ) q Tr ( h g ( ) h † g ( )) ; (39)wherein i ¯ h H ≡ i ¯ h (cid:18) Re (cid:10) Y ⊥ | H | Y (cid:11) h Y | H | Y i + iIm (cid:10) Y ⊥ | H | Y (cid:11) h Y | H | Y i − iIm (cid:10) Y ⊥ | H | Y (cid:11) − Re (cid:10) Y ⊥ | H | Y (cid:11) (cid:19) , and (cid:12)(cid:12) Y ⊥ (cid:11) = K (cid:229) a = Tr ( P − s i Q a ) | a i + Tr ( P + Q a ) | a i . This is the complete and exact result revealing the SU ( ) geometricphase factor of arbitrary finite even-dimensional pure systems. Explicit Example: Generic Four-State Systems and the BPST instanton
A generic pure 4-state system is associated with the quaternionic Hopf fibration, S / SU ( ) = HP = S . Parametrization of S can be achieved in terms of two quaternions S ∼ ( Q , Q ) satisfying (cid:12)(cid:12) Q (cid:12)(cid:12) + (cid:12)(cid:12) Q (cid:12)(cid:12) =
1. These can in turn be explicitlywritten as Q = u cos q , Q = uv sin q , with u = (cid:18) e i ( g + b ) / cos a e i ( g − b ) / sin a − e − i ( g − b ) / sin a e − i ( g + b ) / cos a (cid:19) , v = (cid:18) e i ( g + b ) / cos a e i ( g − b ) / sin a − e − i ( g − b ) / sin a e − i ( g + b ) / cos a (cid:19) (40)being SU ( ) matrices. This yields, | Y i = (cid:229) a , b = Tr ( P − ˆ q h a ) | a i + Tr ( P + ( i s ) ˆ q h a ) | a i q Tr ( h b h † b ) , (41)and the non-Abelian connection corresponds exactly to the BPST instanton[7] gauge potential A HP = dh a h † a − h a dh † a iTr ( h b h † b ) = − i sin q dvv † . The parameter q is related to the distance | x | from the instanton center by sin (cid:16) q (cid:17) = | x | / (cid:16) | x | + L (cid:17) . Thesecond Chern number is computed to be C = − p R S Tr ( F ∧ F ) = p R S sin q d q ∧ Tr ( dvv † ) = p R S Tr ( dvv † ) = . Pure state bipartite qubit-qubit entanglement and the BPST instanton
A bipartite qubit-qubit system is an example of a composite 4-state system. In general the qubit-qubit system may be writtenas | Y i = c i j | i i| j i , where i , j takes value ± . A quantitative measure of the pure qubit-qubit state entanglement is provided bythe expectation value of the Clauser-Horne-Shimony-Holt operator[8] which is CHSH = ( R + S ) ⊗ T + ( R − S ) ⊗ U ; (42)wherein R = ˆ ~ r · s with ˆ ~ r being a unit spatial vector, and similarly the operators S , T and U . The expectation value of theCHSH operator depends on the state and also the directions of the unit vectors; but the maximum value[9] is correlated to theentanglement by h Y | CHSH | Y i max . = q + | det c | , with 0 ≤ | det c | ≤ ; wherein det c denotes the determinant of the2 × c i j . Comparing with our generic 4-state system, | Y ( t ) i = (cid:229) a = C a | a i = (cid:229) a , b = Tr ( P − ˆ q ( t ) h a ( t )) | a i + Tr ( P + ( i s ) ˆ q ( t ) h a ( t )) | a i q Tr ( h b ( t ) h † b ( t )) , Q = u cos q , Q = uv sin q q a = Re ( z a ) I + Im ( z a ) s i + Re ( z a ) s i + Im ( z a ) s i , the four state coefficients C a of the composite systemcan be computed to be C = cos q a e i ( g + b ) / , C = sin q (cid:16) e i ( g + g + b + b ) / cos a cos a − e i ( g − g − b + b ) / sin a sin a (cid:17) C = cos q a e i ( g − b ) / , C = sin q (cid:16) e i ( g + g + b − b ) / cos a sin a + e i ( g − g − b − b ) / sin a cos a (cid:17) A judicious mapping of the bipartite system to the composite system allows us to correlate the entanglement directly to theinstanton parameter q for arbitrary qubit-qubit 4-state composite systems. An earlier correlation can be found in [10]. Thegeneral unitary basis transformation between the two systems, | Y i = c i j | i i ⊗ | j i = C a | a i , is c i j = h i j | a i C a = U ai j C a , wherein Tr (cid:16) U a U † b (cid:17) = U ai j ( U † b ) ji = U ai j ( U bi j ) ∗ = h i j | a ih b | i j i = h i j | b ih a | i j i = d ab . (44)To wit det c = e i j e kl c ik c jl = ( i s ) i j ( i s ) kl c ik c jl = Tr ( s c T s c ) (45) = Tr ( s ( U a C a ) T s ( U b C b )) = Tr ( s ( e U a ) T s e U b ) | C a || C b | ;wherein we have defined e U a ≡ e i f a U a satisfying Tr ( e U a e U † b ) = e i ( f a − f b ) Tr ( U a U † b ) = e i ( f a − f b ) d ab = d ab . The choice of˜ U a = u a s a √ (no sum over a ) with u = i , u = u = i , u = c = − Tr (( s a ) † s b ) u a u b | C a || C b | = − (( u ) | C | + ( u ) | C | + ( u ) | C | + ( u ) | C | )= ( | C | + | C | − | C | − | C | ) = ( cos q − sin q ) =
12 cos q . (46)This relates the entanglement parameter, det c , to the instanton parameter, q , in a generic qubit-qubit system which is consid-ered as a 4-state total system with the Hilbert space S which is total bundle space of the Hopf fibration S / [ SU ( ) = S ] =[ HP = S ] . Summary
The existence of Hopf fibrations S N + / S = CP N and S K + / S = HP K allows us to treat the Hilbert space of generic finite-dimensional quantum systems as the total bundle space with respectively U ( ) and SU ( ) fibers and complex and quaternionicprojective base manifolds. This alternative method of studying and describing quantum states and their evolution reveals theintimate and exact connection between generic quantum systems and fundamental geometrical objects. Acknowledgement
The research for this work is supported in part by the National Science Council of Taiwan under Grant No. NSC101-2112-M-006-007-MY3 and the National Center for Theoretical Sciences. CS would like to thank C. H. Oh and K. Singh for beneficialdiscussions at the Centre for Quantum Technologies, Nat. U. Singapore, where part of this work was completed.
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