Quantum Zermelo problem for general energy resource bounds
J. M. Bofill, A. S. Sanz, G. Albareda, I. P. R. Moreira, W. Quapp
QQuantum Zermelo problem for general energy resource bounds
Josep Maria Bofill,
1, 2, ∗ ´Angel S. Sanz, † Guillermo Albareda,
4, 2, ‡ Ib´erio de P.R. Moreira,
5, 2, § and Wolfgang Quapp ¶ Departament de Qu´ımica Inorg`anica i Org`anica, Secci´o de Qu´ımica Org`anica Institut de Qu´ımica Te`orica i Computacional (IQTCUB),Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain Department of Optics, Faculty of Physical Sciences, Universidad Complutense de Madrid,Pza. Ciencias 1, Ciudad Universitaria – 28040 Madrid, Spain Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science,Luruper Chaussee 149, 22761 Hamburg, Germany Departament de Ci`encia de Materials i Qu´ımica F´ısica,Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain Mathematisches Institut, Universit¨at Leipzig, PF 100920, D-04009 Leipzig, Germany (Dated: May 18, 2020)A solution to the quantum Zermelo problem for control Hamiltonians with general energy resourcebounds is provided. The solution is found to be adiabatic irrespective of the energy resource, andincludes as a particular case the result in [Phys. Rev. Lett. , 100502 (2015)] for a Hilbert-Schmidtnorm equal to one. Interestingly, the energy resource of the control Hamiltonian and the controltime define a pair of conjugate variables that minimize the energy-time uncertainty relation. Theresulting control protocol is applied to a single qubit as well as to a two-interacting qubit systemrepresented by a Heisenberg spin dimer. For this low-dimensional systems, it is found that physicallyrealizable control Hamiltonians exist only for certain, quantized, energy resources.
I. INTRODUCTION
On a fundamental level, nature requires a quantumdescription rather than a classical one [1]. Nonetheless,quantum-classical correspondence arguments are still infashion because of their usefulness to understand and ex-plain the behavior of quantum systems [2], and also todevice new strategies to tackle quantum problems, as itis the case of optimal control strategies [3–7]. In gen-eral, control scenarios are related either with the way ofconstructing a so-called control Hamiltonian or with theprocedure aimed at getting an appropriate initial ansatzthat, with time, evolves into the desired final quantumstate.Additionally, in the last years it has also attractedmuch attention the problem of finding optimal unitaryoperators, ˆ U ( t f , t i ), which lead a given initial state | ψ i (cid:105) ,at t i to another different, but previously fixed, final state | ψ f (cid:105) , at t f , in the shortest possible time, ∆ T = t f − t i ,under some constraining conditions. Since finding theoptimal unitary operator is equivalent to finding an op-timal Hamiltonian, two different routes can be explored.On the one hand, if the constraint implies a bound in theenergy resource, then the optimal Hamiltonian is goingto be time-independent, and can easily be constructedby noting that the corresponding unitary transformationˆ U describes the shortest time-evolution. On the other ∗ jmbofi[email protected] † a.s.sanz@fis.ucm.es ‡ [email protected] § [email protected] ¶ [email protected] hand, the constraint might imply a search for a time-dependent Hamiltonian minimizing the time-evolution,which means that it has to be determined and character-ized by variational approaches [8].The latter case is particularly relevant to those situa-tions where the evolution of the quantum system is eitherpredetermined or inherently affected by an external fieldout of our control (for instance, in problems within thescope of the quantum technologies). Yet it would be de-sirable to take the system from one state to another onethat does not correspond to the natural evolution of sucha system. That is, if ˆ U describes such a natural evolu-tion, it is of much interest to devise a method or protocolthat warrants the evolution from | ψ i (cid:105) to | ψ f (cid:105) in the leasttime, provided that | ψ f (cid:105) (cid:54) = ˆ U | ψ i (cid:105) . Appealing to theaforementioned quantum-classical correspondence, this isactually the quantum analog of the well-known classicalZermelo navigation problem [9, 10].Brody and Meier [11] have investigated this problemin the field of quantum processing. More specifically,assuming the quantum system is acted by a bare back-ground Hamiltonian, ˆ H , these authors determined amethod to obtain a time-optimal control Hamiltonian,ˆ H c ( t ), such that its combined action with ˆ H , i.e.,ˆ H ( t ) = ˆ H + ˆ H c ( t ) , (1)generates a time-optimal unitary evolution from | ψ i (cid:105) to | ψ f (cid:105) . The protocol devised by these authors thus includesthree key elements:(i) A time-independent, bare background Hamilto-nian, ˆ H , which describes the natural evolution ofthe quantum system. a r X i v : . [ qu a n t - ph ] M a y (ii) A time-dependent control ˆ H c ( t ) satisfying at anytime the energy resource boundtr (cid:16) ˆ H c ( t ) (cid:17) = 1 . (2)(iii) The background Hamiltonian ˆ H is not energeti-cally dominant, i.e.,tr (cid:16) ˆ H (cid:17) < tr (cid:16) ˆ H c ( t ) (cid:17) . (3)These features thus define the quantum counterpart ofthe classical Zermelo navigation problem [9, 10].Moved by the possibility to apply the above ideas tomore general quantum scenarios, here we present a fur-ther development of the quantum Zermelo methodology,suggesting a protocol that can be easily adapted to dif-ferent physical scenarios. In this regard, we have focusedon a series of guidelines, which stress the physics behindthe approach to the detriment of a more abstract con-ceptualization of the problem. That is, we have tried toanswer questions such as whether it is possible to builda control Hamiltonian without entering too much formalaspects, but just well known theory. And, if so, we alsowanted to know how it looks like and whether it worksoptimally. Interestingly, by proceeding this way, we havebeen able to reach a general form for the condition speci-fied by (2), where the l.h.s. equals to a general constant k ,which, in turn, is related to the minimum time necessaryto take the system from the initial to the final quantumstate that we wish, circumventing the unwanted effectsof the bare Hamiltonian. Accordingly, a general proto-col is presented to determine ˆ H c , which here we havetested with a series of well-known quantum systems, suchas the harmonic oscillator, entanglement swapping withBell states, or spin-flip in a Heisenberg dimer. It is worthstressing that, in all cases, although the least time is go-ing to depend on the system Hamiltonian, the conditionitself is totally independent of it, which we associate withthe fact that the evolution of the quantum state keeps aone-to-one analogy with the geometrical evolution alonga meridian joining both states on the Bloch’s sphere, asalready pointed out by Brody et al. [12].The work is organized as follows. The theory is pre-sented, developed and discussed in the next section. Tobe self-contained, both a brief account on the classicalZermelo problem as well as on the Brody and Meier ap-proach are also included, which serves to contextualizethe work. Afterwards, we start the development of ourapproach, which also includes a discussion on the adia-baticity of the solution of the quantum Zermelo problem.In Sec. III, we develop the applications mentioned above,showing how the least-time condition arises in each case.Finally, a series of concluding remarks are exposed inSec. IV. II. THEORYA. Classical Zermelo problem
The classical Zermelo navigation problem can bestated as follows. Given the actual position of a ship, x (cid:62) = ( x , x ), on the surface of an unlimited sea andundergoing the local action of current and/or wind, char-acterized by a position-dependent vector field, w (cid:62) ( x ) =( w ( x ) , w ( x )), one expects to find the optimal controlvelocity, v (cid:62) = v ˆ u (cid:62) = v ( u , u ), that should constantlyact on the ship so that it reaches its destination in theleast time. Here, ˆ u is a unit vector in the direction of v and v denotes its modulus.As it was noticed by Zermelo [9] and Carath´eodory[10], the solution to this problem can be obtained byconstructing the geometrical form of the indicatrix thatallows to obtain the Hamiltonian function and, from it,all extremal curves of the problem. Accordingly, the ab-solute velocity of the ship, namely v = ˙ x /F , must satisfythe equation ˙ x F − w ( x ) = v u , (4)where ˙ x is the derivative of the coordinates with respectto an arbitrary evolution parameter. The time employedby the ship in its full journey is calculated from the in-tegral of the F -function with respect to the arbitraryparameter along the extremal curve. Hence, F becomesthe basic function of this variational problem.Equation (4) allows us to determine the F -function asa positive root of the equation (cid:20) ˙ x F − w ( x ) (cid:21) (cid:62) (cid:20) ˙ x F − w ( x ) (cid:21) = v , (5)whenever such a root exists. Equation (5) is the indica-trix of the classical Zermelo navigation problem, whichdescribes a circle of radius v with center at w ( x ). Theset of points satisfying the circle condition correspond tothe end points of the vector ˙ x /F . As seen below, in thequantum analog for this problem, Brody and Meier [11]found the solution by determining the geodesics of theRanders metric derived from the form of the F -function. B. Quantum Zermelo approach
As noted by Brody et al. [12], it is also possible tofind a direct quantum counterpart of the Zermelo navi-gation problem. To this end, consider some initial andfinal quantum states, | ψ i (cid:105) and | ψ f (cid:105) , respectively, for agiven physical system, which is being acted by a time-independent background Hamiltonian ˆ H . The quantumZermelo problem consists in finding a control Hamilto-nian, ˆ H c ( t ), such that the total Hamiltonian (1) describesa unitary transformation leading from | ψ i (cid:105) to | ψ f (cid:105) in theleast time. Notice that, by appealing to the classicalanalog, the classical vector field describing the wind orcurrent corresponds, in the quantum counterpart, to theunitary operator generated by ˆ H . Furthermore, in thisquantum problem, it is assumed that the energy associ-ated with the transformation from ψ i to ψ f is not onlylimited, but it has also to be totally consumed at the endof the process. Thus, the speed evolution generated bythe control Hamiltonian, ˆ H c ( t ), is related to the energyvariance of ˆ H c ( t ), according to the Anandan-Aharonovrelation [13]. Over the full process, the speed evolutiontakes the maximum attainable value and it is fixed.Based on such constraints, one aims to built an optimalunitary transformation that satisfies them all. Accord-ingly, consider the time-evolution of the unitary operator,ˆ U ( t, t i ), governed by the Schr¨odinger equation, which inthe Heisenberg representation reads as i d ˆ U ( t, t i ) dt = ˆ H ( t ) ˆ U ( t, t i )= (cid:104) ˆ H + ˆ H c ( t ) (cid:105) ˆ U ( t, t i ) , (6)with (cid:126) = 1 (in natural units). The time-evolutionoperator ˆ U ( t, t i ) is required to satisfy the initial con- dition ˆ U ( t i , t i ) ≡ I ( I denotes the identity operator)as well as the unitarity condition ˆ U † ( t, t i ) ˆ U ( t, t i ) =ˆ U ( t, t i ) ˆ U † ( t, t i ) = I , which ensures the norm preserva-tion along the whole evolution.For simplicity and convenience, considering the totaltime lasted in the evolution of the system, ∆ T = t f − t i ,with t i ≤ t ≤ t f , and then defining the dimensionlessevolution parameter s = ( t − t i ) / ∆ T , the time-evolutionoperator can be recast as ˆ U ( t, t i ) = ˆ U T ( s ), and its time-derivative as d ˆ U ( t, t i ) /dt = (1 / ∆ T ) d ˆ U ∆ T ( s ) /ds [14]. Us-ing the above notation and multiplying Eq. (6) from theright by ˆ U † ∆ T ( t, t i ) we get i ∆ T d ˆ U ∆ T ( s ) ds ˆ U † ∆ T ( s ) − ˆ H = ˆ H c ( s ) , (7)which strongly resembles the classical Eq. (4), with ∆ T playing the role of F .In order to further stress the quantum-classical anal-ogy, Eq. (7) is now multiplied by itself. Then, the traceover the full resulting evolution equation gives rise to theequationtr (cid:16) ˆ X ( s ) ˆ X ( s ) (cid:17) − T tr (cid:16) ˆ H ˆ X ( s ) (cid:17) + (∆ T ) tr (cid:16) ˆ H (cid:17) = (∆ T ) tr (cid:16) ˆ H c ( s ) (cid:17) = k (∆ T ) , (8)with ˆ X ( s ) = i d ˆ U ∆ T ( s ) ds ˆ U † ∆ T ( s ) (9)arising from the constraint on the energy resource bound [see condition (ii) above], and k being an arbitrary constant.Equation (8) can thus be seen as the quantum counterpart of Eq. (5). Solving for ∆ T [11], we finally find∆ T { ˆ X ( s ) } = − tr (cid:16) ˆ X ( s ) ˆ H (cid:17) + (cid:114)(cid:104) tr (cid:16) ˆ X ( s ) ˆ H (cid:17)(cid:105) + (cid:104) k − tr (cid:16) ˆ H (cid:17)(cid:105) tr (cid:16) ˆ X ( s ) ˆ X ( s ) (cid:17) k − tr (cid:16) ˆ H (cid:17) , (10)which constitutes the so-called Finslerian norm of ˆ X ( s )[15–17]. As it can be noticed, the positivity of ∆ T , asgiven by Eq. (10), warrants the above condition (iii), withtr (cid:16) ˆ H c ( t ) (cid:17) = k –Brody and Meier found the “optimalpath” ˆ X ( s ) that minimizes the integral over the timegiven by Eq. (10), namely (cid:82) (cid:104) ∆ T { ˆ X ( s ) } (cid:105) ds , in theparticular case k = 1.The question now is whether one can approach thesame problem from a more physical viewpoint, that is,from a more familiar quantum formulation, which, inturn, might serve also to confer more generality to theprocess. The answer is affirmative, as we show nowby considering notions already existing within the time-dependent perturbation theory [14], which is also closer to treatments typically considered in the theory of openquantum systems [18]. To see that, let us introduce theunitary time-evolution operator, ˆ U ( t, t i ), correspondingto ˆ H , solution to the equation i d ˆ U ( t, t i ) dt = ˆ H ˆ U ( t, t i ) , (11)with initial condition ˆ U ( t i , t i ) ≡ I . The solution is wellknown, ˆ U ( t, t i ) = e − i ˆ H ( t − t i ) . (12)Now, in order to determine ˆ U ( t, t i ), we consider the sep-arable ansatz ˆ U ( t, t i ) = ˆ U ( t, t i ) ˆ U c ( t, t i ) , (13)where the time-evolution operator ˆ U c ( t, t i ) is requiredto be unitary and satisfying the unitarity conditionˆ U † c ( t, t i ) ˆ U c ( t, t i ) = I . This constraint, in turn, impliesthat ˆ U ( t, t i ) also satisfies the unitarity condition, as itcan easily be shown.In order to determine ˆ U c ( t, t i ), we now proceed as fol-lows. First, we substitute Eq. (13) into Eq. (6), and thenmake the ˆ U † ( t, t i ) to act on the left of the resulting ex-pression, which renders the equation i d ˆ U c ( t, t i ) dt = ˆ U † ( t, t i ) ˆ H c ( t ) ˆ U ( t, t i ) ˆ U c ( t, t i ) , (14)with initial condition ˆ U c ( t i , t i ) ≡ I , and where we havemade use of Eq. (11) to simplify it. Now, as it can benoticed, on the r.h.s., ˆ U c is acted by the control Hamil-tonian operator in the interaction picture [19],ˆ H (cid:48) c ( t ) = ˆ U † ( t, t i ) ˆ H c ( t ) ˆ U ( t, t i )= e i ˆ H ( t − t i ) ˆ H c ( t ) e − i ˆ H ( t − t i ) . (15)Since the control Hamiltonian ˆ H c is to be determined, wecan make a guess on the particular functional form forits interaction picture, namely that ˆ H (cid:48) c corresponds toˆ H c at t i , so that it becomes time-independent. Althoughthis might look counterintuitive, if we recall the pictureprovided by Brody et al. [12] of the evolution along theBloch sphere when going from one state to the other,the above condition (15), with ˆ H (cid:48) c ( t ) = ˆ H c ( t i ), can beconsidered to be equivalent to continuing the journey onthe back of the sphere, from the final state to the initialone. That is, the condition for a proper control requiresevolution along a meridian, and no other curve, in orderto ensure the equivalence of the two journeys.Therefore, even if the above conditions seems to be astrong constraint, still it is rather reasonable and conve-nient, since it allows us to recast Eq. (15) with a func-tional form analogous to that for the time-evolution op-erator associated with the bare Hamiltonian, Eq. (11),i.e., i d ˆ U c ( t, t i ) dt = ˆ H c ( t i ) ˆ U c ( t, t i ) , (16)with solution ˆ U c ( t, t i ) = e − i ˆ H c ( t i )( t − t i ) . (17)Accordingly, the full Hamiltonian for the quantum Zer-melo problem acquires the final formˆ H ( t ) = ˆ H + e − i ˆ H ( t − t i ) ˆ H c ( t i ) e i ˆ H ( t − t i ) , (18)which corresponds to Eq.(1) in [11].Next, let us see some properties that follow from theabove relationship between ˆ H c ( t ) and ˆ H c ( t i ). Considerthe relationˆ H c ( t ) = e − i ˆ H ( t − t i ) ˆ H c ( t i ) e i ˆ H ( t − t i ) , (19) it readily follows that, if tr (cid:16) ˆ H c ( t i ) (cid:17) is constant, then thesame holds for tr (cid:16) ˆ H c ( t ) (cid:17) , sincetr (cid:16) ˆ H c ( t ) (cid:17) = tr (cid:16) ˆ H c ( t i ) (cid:17) = k, (20)which is satisfied at any time t . Thus, according to (20), d tr (cid:16) ˆ H c (cid:17) /dt = 0 also at any time. Now, differentiationof Eq. (19) with respect to t leads to d ˆ H c ( t ) dt = − i (cid:104) ˆ H , ˆ H c ( t ) (cid:105) , (21)which is a solution to the variational problem, δ (cid:82) (cid:104) ∆ T { ˆ X ( s ) } (cid:105) ds = 0, with ∆ T { ˆ X ( s ) } the same asgiven in Eq.(10) and firstly derived by Brody and Meier[11]. Equation (21) gives the co-adjoint motion and henceit should be solved together with Eq. (18). Besides,from Eq. (21), we also find that tr (cid:16) d ˆ H c ( t ) /dt (cid:17) = 0 and d tr (cid:16) ˆ H c (cid:17) /dt = 2tr (cid:16) ˆ H c ( t ) d ˆ H c ( t ) /dt (cid:17) = 0 by using cyclicpermutation when tracing. Physically, these vanishingvalues imply that the “velocity” of the transition pro-cess remains constant during the whole process, as it isassumed in the problem by definition.From the above formulation, it is now clear thatEq. (21) together with Eq. (6), with ˆ H c ( t ) as given by(19), and ˆ U ( t, t i ) computed from (13), (12) and (17),provide the fundamental solution to the quantum Zer-melo problem [11, 12]. Furthermore, we have seen thatthe condition tr (cid:16) ˆ H c ( t i ) (cid:17) = k arises as a consequence ofEqs. (19) and (21) [20, 21] and generalizes the result inRef. [11]. C. Transition between two specific quantum states
According to the above results, time optimization inthe quantum Zermelo approach is fully determined by theconstruction of the control Hamiltonian ˆ H c ( t i ) providedthe bound condition tr( ˆ H c ( t i )) = k is satisfied, sinceboth the bare Hamiltonian ˆ H and the initial and finalstates, | ψ i (cid:105) and | ψ f (cid:105) , are given. In order to understandthe dynamical transition from | ψ i (cid:105) and | ψ f (cid:105) , and hence tointroduce a protocol to optimize the time lasted in such atransition, let us consider the state reached by | ψ i (cid:105) aftera time t under free evolution, i.e., under the action of thebare background Hamiltonian. This state is given by | ψ ( t ) (cid:105) = ˆ U ( t, t i ) | ψ i (cid:105) . (22)Taking into account Eq. (13), we can introduce the in-termediate state | ψ (cid:48) ( t ) (cid:105) ≡ ˆ U † ( t, t i ) | ψ ( t ) (cid:105) = ˆ U c ( t, t i ) | ψ i (cid:105) . (23)Differentiating this state and its complex conjugate part-ner with respect to time, and then substituting the cor-responding results into Eq. (16) (and the correspondingcomplex conjugate equation), leads to i d | ψ (cid:48) ( t ) (cid:105) dt = ˆ H c ( t i ) | ψ (cid:48) ( t ) (cid:105) , (24a) − i d (cid:104) ψ (cid:48) ( t ) | dt = (cid:104) ψ (cid:48) ( t ) | ˆ H c ( t i ) . (24b)Now, if | ψ i (cid:105) is normalized, then | ψ (cid:48) ( t ) (cid:105) is also normal-ized, as it can readily be inferred from (23). Moreover,if we assume that the control Hamiltonian generates astate vector that is orthogonal to the original one (incompliance with the fact that it has to counterbalancethe effect of the “blowing wind” accounted for the bareHamiltonian), then from (24) we have (cid:104) ψ (cid:48) ( t ) | d | ψ (cid:48) ( t ) (cid:105) dt = d (cid:104) ψ (cid:48) ( t ) | dt | ψ (cid:48) ( t ) (cid:105) = 0 . (25)In order to satisfy both conditions, normalization andorthogonality, also from (24) we notice that ˆ H c ( t i ) hasto display the following functional form [22],ˆ H c ( t i ) = i (cid:20) d | ψ (cid:48) ( t ) (cid:105) dt (cid:104) ψ (cid:48) ( t ) | − | ψ (cid:48) ( t ) (cid:105) d (cid:104) ψ (cid:48) ( t ) | dt (cid:21) , (26)where the r.h.s. shows an explicit dependence on time,although the Hamiltonian is time-independent. Ratherthan an inconsistency, this is just an effect associatedwith the fact that this Hamiltonian has to counterbalanceat every time the effect produced by ˆ H , although the netaction is time-independent, as will be shown below.Notice that the conditions on | ψ (cid:48) ( t ) (cid:105) and itstime-derivative imply that ˆ H c ( t i ) is traceless, i.e.,tr (cid:16) ˆ H c ( t i ) (cid:17) = 0. Moreover, since the variance of the en-ergy is related to the speed of the quantum evolution[13], it can be shown that the orthogonality condition(25) ensures the maximum speed evolution condition forthe control Hamiltonian, since it makes the variance ofthis Hamiltonian, given by the expression (cid:16) ∆ ˆ H c ( t i ) (cid:17) = (cid:104) ψ (cid:48) ( t ) | ˆ H c ( t i ) | ψ (cid:48) ( t ) (cid:105)− (cid:16) (cid:104) ψ (cid:48) ( t ) | ˆ H c ( t i ) | ψ (cid:48) ( t ) (cid:105) (cid:17) = d (cid:104) ψ (cid:48) ( t ) | dt ( I − | ψ (cid:48) ( t ) (cid:105)(cid:104) ψ (cid:48) ( t ) | ) d | ψ (cid:48) ( t ) (cid:105) dt = (cid:13)(cid:13)(cid:13)(cid:13) dψ (cid:48) ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) , (27)to reach its maximum value. Actually, we have that2 (cid:16) ∆ ˆ H c ( t i ) (cid:17) = tr( ˆ H c ( t i )) = k, (28)which is a consequence of the fact that the control Hamil-tonian is traceless [12, 21]. From Eqs. (27) and (28), we find the following relation (cid:13)(cid:13)(cid:13)(cid:13) dψ (cid:48) ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) = k . (29)At any time, this relation is satisfied by the ansatz | ψ (cid:48) ( t ) (cid:105) = cos (cid:104)(cid:112) k/ t − t i ) (cid:105) | ψ (cid:48) ( t i ) (cid:105) + sin (cid:104)(cid:112) k/ t − t i ) (cid:105)(cid:112) k/ d | ψ (cid:48) ( t i ) (cid:105) dt , (30)with time-derivative given by d | ψ (cid:48) ( t ) (cid:105) dt = − (cid:112) k/ (cid:104)(cid:112) k/ t − t i ) (cid:105) | ψ (cid:48) ( t i ) (cid:105) + cos (cid:104)(cid:112) k/ t − t i ) (cid:105) d | ψ (cid:48) ( t i ) (cid:105) dt . (31)This ansatz, in turn, satisfies the above normalizationand orthogonality conditions. Notice here that the ex-pression d | ψ (cid:48) ( t i ) (cid:105) /dt has to be understood as the time-derivative of | ψ (cid:48) ( t ) (cid:105) evaluated at t = t i .In order to further simplify the approach, the aboveexpressions (30) and (31), in terms of the general time-evolved state vector | ψ (cid:48) ( t ) (cid:105) , can be recast in terms ofthe initial and final state vectors, | ψ i (cid:105) and | ψ f (cid:105) , thusproviding an also simpler functional form for the controlHamiltonian (26). To this end, notice that, by virtue ofEq. (23), at t i we have | ψ (cid:48) ( t i ) (cid:105) = | ψ i (cid:105) . Similarly, at t f we findˆ U † ( t f , t i ) | ψ f (cid:105) = ˆ U c ( t f , t i ) | ψ i (cid:105) = | ψ (cid:48) ( t f ) (cid:105) = | ψ (cid:48) f (cid:105) . (32)In order to remove any common support between | ψ (cid:48) f (cid:105) and | ψ i (cid:105) , we need to find the orthonormal form for theformer, which is obtained by applying a Gram-Schmidtorthogonalizing process. Accordingly, the orthonormalform is found to be | ¯ ψ (cid:48) f (cid:105) = ( I − | ψ i (cid:105)(cid:104) ψ i | ) | ψ (cid:48) f (cid:105) = sin (cid:16)(cid:112) k/ T (cid:17)(cid:112) k/ d | ψ (cid:48) ( t i ) (cid:105) dt , (33)where Eq. (30) has been used, with t = t f . Next, wenormalize | ¯ ψ (cid:48) f (cid:105) : | ¯¯ ψ (cid:48) f (cid:105) = 1 (cid:13)(cid:13)(cid:13) ¯ ψ (cid:48) f (cid:13)(cid:13)(cid:13) | ¯ ψ (cid:48) f (cid:105) = ( I − | ψ i (cid:105)(cid:104) ψ i | ) ˆ U † ( t f , t i ) | ψ f (cid:105) (cid:114) − (cid:16) (cid:104) ψ f | ˆ U ( t f , t i ) | ψ i (cid:105) (cid:17) = 1 (cid:112) k/ d | ψ (cid:48) ( t i ) (cid:105) dt . (34)It can be noticed from Eq. (34) that the calculation of | ¯¯ ψ (cid:48) f (cid:105) only includes | ψ i (cid:105) , | ψ f (cid:105) , ˆ H , and the time interval∆ T .The ansatz (30) and its time-derivative, Eq. (31), cannow be recast in terms of the orthonormal state vectors | ψ i (cid:105) and | ¯¯ ψ (cid:48) f (cid:105) , which read as | ψ (cid:48) ( t ) (cid:105) = cos (cid:104)(cid:112) k/ t − t i ) (cid:105) | ψ i (cid:105) + sin (cid:104)(cid:112) k/ t − t i ) (cid:105) | ¯¯ ψ (cid:48) f (cid:105) , (35a) d | ψ (cid:48) ( t ) (cid:105) dt = − (cid:112) k/ (cid:16)(cid:112) k/ t − t i ) (cid:17) | ψ i (cid:105) + (cid:112) k/ (cid:16)(cid:112) k/ t − t i ) (cid:17) | ¯¯ ψ (cid:48) f (cid:105) , (35b)respectively.In order to finally obtain the functional form of thecontrol Hamiltonian, we substitute Eqs. (35) into (26),leading toˆ H c ( t i ) = i (cid:112) k/ (cid:104) | ¯¯ ψ (cid:48) f (cid:105)(cid:104) ψ i | − | ψ i (cid:105)(cid:104) ¯¯ ψ (cid:48) f | (cid:105) , (36) which is the initial optimal control Hamiltonian. Withthe aid of Eq. (35a) in the case t = t f , | ¯¯ ψ (cid:48) f (cid:105) can be recastin terms of | ψ (cid:48) f (cid:105) = | ψ (cid:48) ( t f ) (cid:105) . If the corresponding expres-sion is then substituted into Eq. (36), we shall obtainˆ H c ( t i ) = i (cid:112) k/ (cid:16)(cid:112) k/ T (cid:17) (cid:2) | ψ (cid:48) f (cid:105)(cid:104) ψ i | − | ψ i (cid:105)(cid:104) ψ (cid:48) f | (cid:3) , (37)which is time-independent, as it was stressed above.This is precisely the expression reported by Brody etal. [12] for ˆ H c in the particular case k = 1 /
2. It canbe shown now that the variance of ˆ H c ( t i ) for any | ψ (cid:48) ( t ) (cid:105) effectively remains constant in time, that is, ∆ ˆ H c ( t i ) = (cid:113) (cid:104) ψ (cid:48) ( t ) | ˆ H c ( t i ) | ψ (cid:48) ( t ) (cid:105) = (cid:112) k/ H c ( t i ) = 1 √ (cid:16) | ψ i (cid:105) − i | ¯¯ ψ (cid:48) f (cid:105) , | ψ i (cid:105) + i | ¯¯ ψ (cid:48) f (cid:105) (cid:17) (cid:18) − (cid:112) k/ (cid:112) k/ (cid:19) √ (cid:32) (cid:104) ψ i | + i (cid:104) ¯¯ ψ (cid:48) f |(cid:104) ψ i | − i (cid:104) ¯¯ ψ (cid:48) f | (cid:33) . (38)With this expression at hand, Eq. (17) takes the explicit formˆ U c ( t, t i ) 1 √ (cid:16) | ψ i (cid:105) − i | ¯¯ ψ (cid:48) f (cid:105) , | ψ i (cid:105) + i | ¯¯ ψ (cid:48) f (cid:105) (cid:17) (cid:32) e i √ k/ t − t i ) e − i √ k/ t − t i ) (cid:33) √ (cid:32) (cid:104) ψ i | + i (cid:104) ¯¯ ψ (cid:48) f |(cid:104) ψ i | − i (cid:104) ¯¯ ψ (cid:48) f | (cid:33) . (39)The time interval ∆ T is then evaluated by consideringthe transformation indicated in Eq. (32), i.e., the time-operator ˆ U c ( t f , t i ) that takes | ψ i (cid:105) to | ψ (cid:48) f (cid:105) in the shortesttime. As it can be noticed in the above expression, therelationship between | ψ i (cid:105) and | ψ (cid:48) f (cid:105) is given in terms of anargument. If φ is the angle between the initial and finalstate vectors, then the angular evolution is related with∆ T through the expression φ ≡ cos − (cid:16) (cid:104) ψ i | ˆ U † ( t f , t i ) | ψ f (cid:105) (cid:17) = ∆ T ∆ ˆ H c ( t i )= (cid:112) k/ T, (40)in compliance with what is stated in the literature on thegeometry of the state vector evolution [13, 23].From the above discussion, we then extract as a con-clusion that, in order to make the state vector to evolve inthe shortest time from | ψ i (cid:105) to | ψ f (cid:105) when there is the influ-ence of a background Hamiltonian ˆ H , we need to deter-mine the time-optimal unitary transformation, ˆ U ( t f , t i ),which includes the following steps:1. Given ˆ H , | ψ i (cid:105) , | ψ f (cid:105) and k (the energy bound),compute the time interval ∆ T recursively bymeans of Eq. (40), and the unitary transformationˆ U ( t f , t i ) by means of Eq. (12). 2. With | ψ i (cid:105) , | ψ f (cid:105) , and ˆ U ( t f , t i ), compute | ¯¯ ψ (cid:48) f (cid:105) bymeans of Eq. (34).3. Compute ˆ U c ( t f , t i ) using | ψ i (cid:105) , | ¯¯ ψ (cid:48) f (cid:105) , k , and ∆ T ,according to Eq. (39).4. Using ˆ U ( t f , t i ) and ˆ U c ( t f , t i ), compute the time-optimal quantum Zermelo unitary transformation,ˆ U ( t f , t i ), according to Eq. (13).This protocol will ensure that the unitary transformationˆ U ( t f , t i ) transforms | ψ i (cid:105) into | ψ f (cid:105) in the least time. D. Adiabaticity of the quantum ZermeloHamiltonian
Let us now comment on the adiabaticity associatedwith the quantum Zermelo Hamiltonian. Consider theSchr¨odinger equation (6), which in general has not a sta-tionary solution. Now, if we assume that ˆ H ( t ) changesslowly in time (or it is even constant in time), the sys-tem, when started from a stationary state of ˆ H ( t i ), willpass through the stationary states corresponding to ˆ H ( t ).This is what the adiabatic theorem of quantum mechan-ics says [14, 24, 25].As it can be noticed from the above discussion, Eq. (13)is the solution of Eq. (6), where ˆ U ( t, t i ) and ˆ U c ( t, t i ) aregiven by Eqs. (12) and (17), respectively. Therefore, wecan rewrite ˆ U ( t, t i ) asˆ U ( t, t i ) = ˆ U ( t, t i ) ˆ U c ( t, t i )= e − i [ ˆ H + ˆ H c ( t i )]( t − t i ) = e − i ˆ H ( t i )( t − t i ) , (41)with ˆ H ( t i ) = ˆ H + ˆ H c ( t i ). Let { φ j ( t i ) } Nj =1 denote theorthonormal set of eigenfunctions of ˆ H ( t i ) and { h j } Nj =1 the corresponding set of eigenvalues, with N being thedimension of the space. The eigenvalues are time-independent, since tr (cid:16) d ˆ H ( t ) /dt (cid:17) = tr (cid:16) d ˆ H c ( t ) /dt (cid:17) = 0,as proven above. Thus, taking into account the spec-tral decomposition of ˆ H ( t i ) = (cid:80) Nj =1 h j | φ j ( t i ) (cid:105)(cid:104) φ j ( t i ) | ,Eq. (41) can be recast asˆ U ( t, t i ) = N (cid:88) j =1 e − ih j ( t − t i ) | φ j ( t i ) (cid:105)(cid:104) φ j ( t i ) | . (42)The action of this operator on an eigenfunction | φ k ( t i ) (cid:105) with eigenvalue h k givesˆ U ( t, t i ) | φ k ( t i ) (cid:105) = exp( − ih k ( t − t i )) | φ k ( t i ) (cid:105) = | φ k ( t ) (cid:105) . (43)But ˆ U ( t, t i ) | φ k ( t i ) (cid:105) is also the solution of Eq. (6) withthe initial state | φ k ( t i ) (cid:105) . Therefore, the solution (43) isgoing to coincide with the eigenfunction | φ k ( t ) (cid:105) of ˆ H ( t )up to a phase factor.Let us now consider an arbitrary wave function | ψ ( t i ) (cid:105) .Acting on the left of (43) with (cid:104) ψ ( t i ) | ˆ U † ( t, t i ), and takinginto account that ˆ U † ( t, t i ) ˆ U ( t, t i ) = I , we obtain (cid:104) ψ ( t i ) | φ k ( t i ) (cid:105) = (cid:104) ψ ( t ) | φ k ( t ) (cid:105) . (44)Thus, if the system is initially represented by the wavefunction | ψ ( t i ) (cid:105) = (cid:80) k c k ( t i ) | φ k ( t i ) (cid:105) where c k ( t i ) = (cid:104) φ k ( t i ) | ψ ( t i ) (cid:105) , then the probability that the system is inthe stationary state | φ k ( t ) (cid:105) at any time t is constant, i.e., ddt |(cid:104) ψ ( t ) | φ k ( t ) (cid:105)| = 0. This result proves that the dynam-ical transformation governed by Eq.(6) taking ˆ U ( t, t i ) asthat given in Eq. (13) satisfies the adiabatic theorem ofquantum mechanics. That is, for a system initially pre-pared in an eigenstate (e.g., the ground state) of the full(underlying plus control) time-dependent Hamiltonian,the time evolution governed by the Schr¨odinger equa-tion will keep the actual state of the system in the cor-responding instantaneous ground state (or other eigen-state). Therefore, considering that the control Hamilto-nian in Eq. (19) provides the least time to go from onequantum state to another, the solution to the quantumZermelo problem can be understood as the optimal (intime) adiabatic transformation.The fact that the quantum evolution of Eq.(6) satisfiesthe most trivial version of the adiabatic theorem provides us with another way to determine the time-optimal uni-tary quantum Zermelo transformation, ˆ U ( t f , t i ). Thisnew way is1. Given ˆ H , | ψ i (cid:105) , | ψ f (cid:105) , and k compute the time in-terval ∆ T recursively using Eq. (40). Compute theunitary transformation, ˆ U ( t f , t i ), by Eq. (12).2. With | ψ i (cid:105) , | ψ f (cid:105) , and ˆ U ( t f , t i ), compute ¯¯ ψ (cid:48) f throughEq. (34).3. Compute ˆ H c ( t i ) using | ψ i (cid:105) , | ¯¯ ψ (cid:48) f (cid:105) , k , and ∆ T , ac-cording to Eq. (36).4. Using ˆ H and ˆ H c ( t i ), compute the ˆ H ( t i ) = ˆ H +ˆ H c ( t i ). Compute the quantum Zermelo unitarytransformation, ˆ U ( t f , t i ), according to Eq. (41). III. APPLICATIONSA. Harmonic oscillator
We shall start the application of the protocol above de-scribed with the paradigmatic harmonic oscillator actedby an external field. In particular, we are going to con-sider a two-level transition, which for simplicity is goingto be considered to be the ground and the first excitedone, which can be denoted as | (cid:105) = (cid:18) (cid:19) , | (cid:105) = (cid:18) (cid:19) , (45)respectively. Let us consider the transition from theground state to the excited one, so that | ψ i (cid:105) = | (cid:105) and | ψ f (cid:105) = | (cid:105) . Of course, these states are under the actionof the harmonic oscillator Hamiltonian,ˆ H = (cid:126) ω (cid:18) ˆ a † ˆ a + 12 (cid:19) , (46)with frequency ω . So, in principle, if they are isolated,their only time-dependence is in terms of a phase factor; ifthey form a linear superposition, there will be a periodictransition from one to the other, with frequency equalto the oscillator frequency, since Ω = ( E − E ) / (cid:126) = ω .Besides, it is interesting to note that the creation andannihilation operators included in (46), in terms of thestates (45), can be written asˆ a = | (cid:105)(cid:104) | = (cid:18) (cid:19) , ˆ a † = | (cid:105)(cid:104) | = (cid:18) (cid:19) . (47)The minimum control time is ∆ T = π/ √ k and thecontrol Hamiltonian in (37) can be written asˆ H c ( t i ) = i (cid:114) k (cid:104) e − πi (cid:126) ω/ √ k ˆ a † − e πi (cid:126) ω/ √ k ˆ a (cid:105) = i (cid:114) k (cid:18) π (cid:126) ω √ k (cid:19) (cid:0) ˆ a † − ˆ a (cid:1) + (cid:114) k (cid:18) π (cid:126) ω √ k (cid:19) (cid:0) ˆ a † + ˆ a (cid:1) . (48)In order this Hamiltonian to be assimilated by a standardexternal driving force, one needs to check the followingcondition ˆ H c ( t i ) = − (cid:114) (cid:126) ω (cid:0) ˆ a † + ˆ a (cid:1) E , (49)where E is the amplitude of the external electric drivingfield. A simple inspection allows us to realize that theabove equation is fulfilled only if cos(3 π (cid:126) ω/ √ k ) = 0,which leads to the conclusion k = (3 / (cid:126) ω ) n + 1 / = (cid:15) f n + 1 / , (50)with n ∈ Z .Therefore, given a frequency ω , the maximum k isgiven by k = 2 (cid:15) f , (51)which corresponds to the minimum control time∆ T = π | (cid:15) f | . (52)As it will seen below, these results are in compliancewith those for the Heisenberg spin chain, thus paving theway to intuitively consider that there might an underly-ing common pattern for any quantum system in the formof k that provides a physical (implementable) control. B. Entanglement swapping
Let us now consider entanglement swapping with max-imally entangled states of a Bell basis [26–28], where thetwo entangled qubits are assumed to be spatially distant,a paradigm with special interest in quantum informationand quantum computation [29, 30]. More specifically,here we consider two spins interacting via anisotropictime-independent J j -couplings, with ( j = x, y, z ), actedby local, controllable magnetic fields B ( i ) ( t ), with ( i =1 , z -direction. Thus, we choose toconsider the following two-qubit Heisenberg Hamiltonian[21], ˆ H = − (cid:88) j J j ˆ σ (1) j ˆ σ (2) j + (cid:88) i =1 B ( i ) ˆ σ ( i ) z , (53) to be the quantum Zermelo Hamiltonian ˆ H ( t ). Here, weuse the tensor products ˆ σ (1) j = ˆ σ j ⊗ I and ˆ σ (2) j = I ⊗ ˆ σ j ,with I being the unit operator of dimension 2 ×
2, andˆ σ ( i ) j the Pauli matrices [14].A simpler ansatz for ˆ H was already reported in [31],where only a fixed coupling, J , was considered. Here, weare going to associate the first term in (53) with the non-controlled, time-independent background Hamiltonian,ˆ H , and the second term with the time-dependent con-trol Hamiltonian, ˆ H c ( t ), satisfying the energy resourcebound, tr (cid:16) ˆ H c ( t ) (cid:17) = k . In this case, the computationalbasis set is provided by the factorizable state vectors (cid:12)(cid:12) (cid:105) = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) = (cid:0) (cid:1) (cid:62) , (54a) (cid:12)(cid:12) (cid:105) = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) = (cid:0) (cid:1) (cid:62) , (54b) (cid:12)(cid:12) (cid:105) = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) = (cid:0) (cid:1) (cid:62) , (54c) (cid:12)(cid:12) (cid:105) = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) = (cid:0) (cid:1) (cid:62) . (54d)In this basis, ˆ H reads as [21]ˆ H = − J z − J − J z − J + − J + J z − J − − J z , (55)where J ± = J x ± J y . The diagonal form for ˆ H isˆ H = − ( J z + J − ) | Φ + (cid:105)(cid:104) Φ + | − ( J z − J − ) | Φ − (cid:105)(cid:104) Φ − | +( J z − J + ) | Φ + (cid:105)(cid:104) Φ + | + ( J z + J + ) | Φ − (cid:105)(cid:104) Φ − | , (56)which allows us to rearrange the above basis set in termsof the Bell basis of maximally entangled states, namely | Φ + (cid:105) = 1 √ | (cid:105) + | (cid:105) ) , (57a) | Φ − (cid:105) = 1 √ | (cid:105) − | (cid:105) ) , (57b) | Φ + (cid:105) = 1 √ | (cid:105) + | (cid:105) ) , (57c) | Φ − (cid:105) = 1 √ | (cid:105) − | (cid:105) ) . (57d)Now the question is how to reach one of these basis vec-tors from another of them, for instance, the | ψ f (cid:105) = | Φ − (cid:105) state from the | ψ i (cid:105) = | Φ + (cid:105) state, in the shortest timeusing the optimal-time Zermelo unitary transformation,Eq. (13).The first term of the unitary time transformationEq. (13), namely, ˆ U ( t, t i ) is easily obtained from thespectral decomposition of ˆ H given in Eq.(56),ˆ U ( t, t i ) = e i ( J z + J − )∆ t | Φ + (cid:105)(cid:104) Φ + | + e i ( J z − J − )∆ t | Φ − (cid:105)(cid:104) Φ − | + e − i ( J z − J + )∆ t | Φ + (cid:105)(cid:104) Φ + | + e − i ( J z + J + )∆ t | Φ − (cid:105)(cid:104) Φ − | , (58)with ∆ t = t − t i . The calculation of the second termof Eq. (13), ˆ U c ( t ), is a bit more subtle. As men-tioned above, we are interested in the transformation of | Φ + (cid:105) into | Φ − (cid:105) via the unitary transformation | Φ − (cid:105) =ˆ U ( t, t i ) ˆ U c ( t, t i ) | Φ + (cid:105) in the shortest time possible. Asexplained above, in the previous section, ˆ U c ( t, t i ) trans-forms | Φ + (cid:105) into ˆ U † ( t, t i ) | Φ − (cid:105) = | Φ (cid:48)− (cid:105) (see Eq. (23)). Ac-cordingly, in the present case, | Φ (cid:48)− (cid:105) = | Φ − (cid:105) exp( i ( J z − J − )∆ t ), where we have made use of (58). The interme-diate state | Φ (cid:48)− (cid:105) satisfies the relations (cid:104) Φ (cid:48)− | Φ (cid:48)− (cid:105) = 1, and (cid:104) Φ (cid:48)− | Φ + (cid:105) = 0, hence ˆ H c ( t i ) will have the functional formˆ H c ( t i ) = i (cid:112) k/ (cid:0) | Φ (cid:48)− m (cid:105)(cid:104) Φ + | − | Φ + (cid:105)(cid:104) Φ (cid:48)− m | (cid:1) = i (cid:112) k/ (cid:104) e i ( J z − J − )∆ T | Φ − (cid:105)(cid:104) Φ + |− e − i ( J z − J − )∆ T | Φ + (cid:105)(cid:104) Φ − | (cid:105) , (59)where as noted before, ∆ T = t f − t i , is the minimumtime interval to be determined, and | Φ (cid:48)− m (cid:105) = | Φ (cid:48)− (cid:105) for∆ t = ∆ T .The next task consists in transforming the ˆ H c ( t ) formspecified in the second term of Eq. (53) into the ˆ H c ( t i )form of Eq. (59), as it was also done in the case of theHarmonic oscillator. In the basis set (54), the controlHamiltonian ˆ H c ( t ) reads asˆ H c ( t ) = B + B − − B −
00 0 0 − B + , (60)where B ± = B (1) ± B (2) (the time-dependence in B (1) and B (2) has been dropped for simplicity). As it can benoticed tr (cid:16) ˆ H c ( t ) (cid:17) = 0, but tr (cid:0) H c ( t ) (cid:1) is not constant intime, because the ˆ H c ( t ) form in Eq. (60) does not involvetime-unitarity. Hence, next we have to transform theˆ H c ( t ) in Eq. (60) into the form given by Eq. (19) with anappropriated choice of ˆ H c ( t i ), according to Eq. (36). Theprojection of H c ( t i ) given by Eq. (36) onto the subspacespanned by | ψ i (cid:105) and | ¯¯ ψ (cid:48) f (cid:105) results in two vanishing diag-onal elements and two off-diagonal elements with zeroreal part, where their imaginary part is equal to (cid:112) k/ H c ( t ) from Eq.(59) onto thesubspace spanned by | Φ + (cid:105) and | Φ (cid:48)− m (cid:105) . In this new rep-resentation, we have (cid:104) Φ + | ˆ H c ( t ) | Φ + (cid:105) = (cid:104) Φ (cid:48)− m | ˆ H c ( t ) | Φ (cid:48)− m (cid:105) = 0 , (61)whereas (cid:104) Φ (cid:48)− m | ˆ H c ( t ) | Φ + (cid:105) = B + e i ( J z − J − )∆ T = B + [cos [( J z − J − )∆ T ]+ i sin [( J z − J − )∆ T ]] , (62) where, effectively, we noticeRe (cid:104) Φ (cid:48)− m | ˆ H c ( t ) | Φ + (cid:105) = B + cos [( J z − J − )∆ T ] = 0 , (63a)Im (cid:104) Φ (cid:48)− m | ˆ H c ( t ) | Φ + (cid:105) = B + sin [( J z − J − )∆ T ] = (cid:112) k/ . (63b)On the other hand, from Eq. (40),cos − (cid:0) (cid:104) Φ + | Φ (cid:48)− m (cid:105) (cid:1) = π/ T (cid:112) k/ , (64)which renders ∆ T = π (cid:112) k/ . (65)Substituting the value ∆ T into the real part, we have (cid:112) k/ J z − J − , while if the substitution is made intothe imaginary part, then B + = (cid:112) k/
2, since B + (cid:54) = 0.Furthermore, the control variable B + decouples from theothers, namely B + = B cos[2( µt + ν )], where B , µ and ν are time-independent constants. Taking µ = ν =0, B = J z − J − = (cid:112) k/ T = ( π/ J z − J − ) − =( π/ B ) − , we reach the final form for ˆ H c ( t i ), whichreads as ˆ H c ( t i ) = B ( | Φ − (cid:105)(cid:104) Φ + | + | Φ + (cid:105)(cid:104) Φ − | )= B ( | Ψ + (cid:105)(cid:104) Ψ + | − | Ψ − (cid:105)(cid:104) Ψ − | )= B σ z ⊗ I + I ⊗ ˆ σ z )= B (cid:16) ˆ σ (1) z + ˆ σ (2) z (cid:17) , (66)where | Ψ + (cid:105) = ( | Φ + (cid:105) + | Φ − (cid:105) ) / √ | Ψ − (cid:105) = ( | Φ + (cid:105) −| Φ − (cid:105) ) / √
2. With this, the corresponding unitary trans-formation is given byˆ U c ( t, t i ) = e − iB ∆ t | Ψ + (cid:105)(cid:104) Ψ + | + e iB ∆ t | Ψ − (cid:105)(cid:104) Ψ − | . (67)Finally, using Eqs. (58) and (67), we obtain the time-optimal quantum Zermelo unitary transformation thatleads the Bell basis vector | Φ + (cid:105) into | Φ − (cid:105) , namely | Φ − (cid:105) = ˆ U z ( t, t i ) | Φ + (cid:105) = ˆ U ( t, t i ) ˆ U c ( t, t i ) | Φ + (cid:105) = 12 (cid:104) e i ( J z + J − )∆ t | Φ + (cid:105) (cid:2) e − iB ∆ t + e iB ∆ t (cid:3) + e i ( J z − J − )∆ t | Φ − (cid:105) (cid:2) e − iB ∆ t − e iB ∆ t (cid:3)(cid:105) , (68)with 0 ≤ ∆ t ≤ ∆ T . As it can be noticed, once thejourney is complete, i.e., ∆ t = ∆ T , the Bell state | Φ − (cid:105) is reached.It is worth noting that in the basis set (57), the quan-tum Zermelo Hamiltonian acquires the form0ˆ H z ( t i ) = ˆ H + ˆ H c ( t i )= − ( J z + J − ) | Φ + (cid:105)(cid:104) Φ + | − ( J z − J − ) | Φ − (cid:105)(cid:104) Φ − | + ( J z − J + ) | Φ + (cid:105)(cid:104) Φ + | + ( J z + J + ) | Φ − (cid:105)(cid:104) Φ − | + B [ | Φ − (cid:105)(cid:104) Φ + | + | Φ + (cid:105)(cid:104) Φ − | ]= (cid:0) | Φ + (cid:105) , | Φ − (cid:105) , | Φ + (cid:105) , | Φ − (cid:105) (cid:1) − ( J z + J − ) B B − ( J z − J − ) 0 00 0 ( J z − J + ) 00 0 0 ( J z + J + ) (cid:104) Φ + |(cid:104) Φ − |(cid:104) Φ + |(cid:104) Φ − | . (69)This Hamiltonian has been obtained using ˆ H and ˆ H c ( t i )as given by Eqs. (56) and (66), respectively. Notice that B = ( J z − J − ), as it has been proven and explainedabove. The eigenvectors (69) can also be computed andread as v (cid:62) = 1 N ( α − β, , , , (70a) v (cid:62) = 1 N ( α + β, , , , (70b) v (cid:62) = (0 , , , , (70c) v (cid:62) = (0 , , , , (70d)where α = − J − /B and β = √ α + 1, while N = (cid:112) ( α − β ) + 1 and N = (cid:112) ( α + β ) + 1 are norm fac-tors. The corresponding eigenvalues are h (1) z = − J z − B β, (71a) h (2) z = − J z + B β, (71b) h (3) z = J z − J + , (71c) h (4) z = J z + J + , (71d)Thus in the quantum Zermelo Hamiltonian, the set ofeigenvalues and eigenvectors are time-independent as ex-pected. C. Spin-flip in a Heisenberg dimer
In Sec. III B we have assumed the functional form ofa Zeeman coupling for the control Hamiltonian, eventhough the algorithm presented in Sec. II does not as-sume any particular form for this Hamiltonian. One maythen wonder what would be the resulting control Hamil-tonian if its form is not imposed a priori .Let us thus consider that the initial and final states, | ψ i (cid:105) and | ψ f (cid:105) , respectively, are orthonormal. It is theneasy to notice that Eq. (40) reads as∆ T = π √ k , (72)i.e., the time needed to reach a target state is inverselyproportional to the square root of k . Actually, since k is related to energy, this relation is just a reminiscenceof the time-energy uncertainty relation: the larger theamount of energy put into play to optimally guide the vector state to its final destination, the shortest the timeemployed in the journey, and vice versa. Now, given ∆ T ,it is then easy to find a general expression for the controlHamiltonian H c ( t i ), as seen in Sec. II, H c ( t i ) = i (cid:114) k (cid:16) e πi(cid:15) f / √ k | ψ f (cid:105)(cid:104) ψ i |−| ψ i (cid:105)(cid:104) ψ f | e − πi(cid:15) f / √ k (cid:17) , (73)where (cid:15) f is the energy of the final state ψ f .To gain some insight into the structure of the abovecontrol Hamiltonian, we pick up the particular case con-sidered in the previous section, viz., the case where ini-tial and final states are maximally entangled Bell states.Thus, with the choice | ψ i (cid:105) = | Φ + (cid:105) and | ψ f (cid:105) = | Φ − (cid:105) , andhence (cid:15) f = − J z + J − , we have | Φ + (cid:105)(cid:104) Φ − | = 14 (cid:16) ˆ σ (1) z + ˆ σ (2) z (cid:17) − i σ x ⊗ ˆ σ y + ˆ σ y ⊗ ˆ σ x ) , (74a) | Φ − (cid:105)(cid:104) Φ + | = 14 (cid:16) ˆ σ (1) z + ˆ σ (2) z (cid:17) + i σ x ⊗ ˆ σ y + ˆ σ y ⊗ ˆ σ x ) . (74b)Substituting these expressions into the control Hamilto-nian (73), we finally obtain H c ( t i ) = 12 (cid:114) k (cid:20) sin (cid:18) (cid:15) f π √ k (cid:19) (ˆ σ (1) z + ˆ σ (2) z ) − cos (cid:18) (cid:15) f π √ k (cid:19) (ˆ σ x ⊗ ˆ σ y + ˆ σ y ⊗ ˆ σ x ) (cid:21) . (75)From Eq. (75), it is clear that the control Hamiltonianadopts the form of a Zeeman coupling for some partic-ular k -values, and hence it can be implemented in thelaboratory. More specifically, this is the case when thecondition k = (cid:15) f n + 1 / = ( J z − J − ) n + 1 / , (76)is satisfied, with n ∈ Z . Accordingly, given J z , the max-imum k -value is determined from the relation k = 2 (cid:15) f = 2( J z − J − ) , (77)1 FIG. 1. Schematic picture of the paddle-wheel centrosymmet-ric molecular complex Cu (O CCH ) · O, for the crystalstructure of copper(II) acetate monohidrate. which corresponds to the minimum control time,∆ T = π | (cid:15) f | = π | J z − J − | , (78)as it follows from (72). D. The Cu(II) acetate molecular complex
As a realistic application of the time-optimal quantumZermelo navigation, we consider the copper(II) acetatemonohydrate. This complex corresponds to an atiferro-magnetic ( S = S = 1 /
2) coupled spin dimer. As such,this system can be cast in the form of an interactingtwo-qubit described by a dimer Heisenberg spin chain asin the previous section. Our goal is to find the optimaltime for the transition between two maximally entangled(Bell) states to occur for a physically implementable con-trol Hamiltonian in the form of a Zeeman-coupling.The crystal structure of copper(II) acetate mono-hidrate, Cu (O CCH ) · O, has been determined byX-ray powder diffraction [32] and refined by neutrondiffraction at room temperature [33]. The crystal isformed by well-defined and separated molecular entities,as displayed in Fig. 1. This complex has a paddle-wheelcentrosymmetric structure with two equivalent Cu(II)centers at 2 . J , J and J in (53) take thefollowing values: J = 297 .
793 cm − , J = 297 .
753 cm − ,and J = 298 .
453 cm − . Then, with the choice of | ψ i (cid:105) = | Φ + (cid:105) and | ψ f (cid:105) = | Φ − (cid:105) , and hence (cid:15) f = − J z + J − ,the maximum value of k compatible with a Zeeman-likecoupling of the form in (53) is 1 . × , and the min-imum control time corresponds to ∆ T = 27 .
94 fs.
IV. CONCLUSIONS
Given the actual position of a classical particle underthe action of a given time-independent force-field, thereexists an optimal control velocity that, acting constantlyon the particle, allows it to reach another position of in-terest in the least possible time. This problem, known asZermelo navigation problem [9, 10], can be recast in therealm of quantum mechanics by simply substituting theclassical particle by a quantum state. In this context, atime-independent Hamiltonian plays the role of the un-derlying classical force-field, and a time-dependent con-trol Hamiltonian with constant energy resource boundis analogous to the control velocity in the classical nav-igation problem. A first solution to the above quantumZermelo problem was put forth by Brody and Meier fora particular energy resource bound [11]. Here we haveextended this result for general energy resource bounds.From a fundamental point of view, the solution to thequantum Zermelo problem defines a pair of conjugatevariables, viz., the energy resource bound and the controltime, that minimize the energy-time uncertainty. Whilethe time-energy uncertainty relation still arouses contro-versy, in the last decades there has been several attemptstowards its explanation. This effort has led to the in-terpretation of the time-energy uncertainty relation as aso-called quantum speed limit, i.e., the ultimate boundimposed by quantum mechanics on the minimal evolu-tion time between two distinguishable states of a system(see [35] and references therein). Therefore, the solutionto the quantum Zermelo problem attains the quantumspeed limit for any energy resource bound.In the above respect, however, we have proven thatthe solution of the quantum navigation problem does notalways lead to physically implementable control Hamil-tonians. For a single qubit and two interacting qubits,we have shown that energy resources leading to physicallyimplementable control Hamiltonians are not any one, butfollow a well defined mathematical pattern. Specifically,for a orthogonal initial and target states, the resource en-ergy bound of physically implementable control Hamilto-nians does obey a quantization rule that depends, exclu-sively, on the energy of the target state.As a realistic application of the time-optimal quantumZermelo navigation, we have shown results for an acetatemolecular complex. The magnetic behavior of copper(II)acetate monohydrate corresponds to an atiferromagnetic( S S /
2) coupled spin dimer. As such, this sys-tem can be cast in the form of an interacting two-qubitdescribed by a dimer Heisenberg spin chain. Employ-ing available experimental data, we have evaluated theoptimal time for the transition between two maximallyentangled (Bell) states to occur. For a physically imple-mentable control Hamiltonian in the form of a Zeeman-coupling, this time is in the order of a few femtosecond.Finally, we have shown that the evolution ruled bythe Zermelo control Hamiltonian is adiabatic. That is,for a system initially prepared in an eigenstate (e.g., the2ground state) of the full (underlying plus control) time-dependent Hamiltonian, the time evolution governed bythe Schr¨odinger equation will keep the actual state of thesystem in the corresponding instantaneous ground state(or other eigenstate). This result is particularly relevant,as the control Hamiltonian solution to the quantum Zer-melo problem is, by construction, the one that minimizesthe time to go from one state to the other. Therefore, weconclude that the Zermelo control Hamiltonian defines anoptimal adiabatic evolution. This result thus paves theway for the design of novel adiabatic algorithms, wherean initial Hamiltonian whose ground state is easy to pre-pare, leads to a final Hamiltonian whose ground stateencodes the solution to a complex eigenstate problem.
ACKNOWLEDGMENTS
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