Quantum Zermelo problem for general energy resource bounds

A solution to the quantum Zermelo problem for control Hamiltonians with general energy resource bounds is provided. The solution is found to be adiabatic irrespective of the energy resource, and includes as a particular case the result in [Phys. Rev. Lett. 114, 100502 (2015)] for a Hilbert-Schmidt norm equal to one. Interestingly, the energy resource of the control Hamiltonian and the control time define a pair of conjugate variables that minimize the energy-time uncertainty relation. The resulting control protocol is applied to a single qubit as well as to a two-interacting qubit system represented by a Heisenberg spin dimer. For this low-dimensional systems, it is found that physically realizable control Hamiltonians exist only for certain, quantized, energy resources.


I. INTRODUCTION
On a fundamental level, nature requires a quantum description rather than a classical one [1].Nonetheless, quantum-classical correspondence arguments are still in fashion because of their usefulness to understand and explain the behavior of quantum systems [2], and also to device new strategies to tackle quantum problems, as it is the case of optimal control strategies [3][4][5][6][7].In general, control scenarios are related either with the way of constructing a so-called control Hamiltonian or with the procedure aimed at getting an appropriate initial ansatz that, with time, evolves into the desired final quantum state.
Additionally, in the last years it has also attracted much attention the problem of finding optimal unitary operators, Û (t f , t i ), which lead a given initial state |ψ i , at t i to another different, but previously fixed, final state |ψ f , at t f , in the shortest possible time, ∆T = t f − t i , under some constraining conditions.Since finding the optimal unitary operator is equivalent to finding an optimal Hamiltonian, two different routes can be explored.On the one hand, if the constraint implies a bound in the energy resource, then the optimal Hamiltonian is going to be time-independent, and can easily be constructed by noting that the corresponding unitary transformation Û describes the shortest time-evolution.On the other hand, the constraint might imply a search for a timedependent Hamiltonian minimizing the time-evolution, which means that it has to be determined and characterized by variational approaches [8].
The latter case is particularly relevant to those situations where the evolution of the quantum system is either predetermined or inherently affected by an external field out of our control (for instance, in problems within the scope of the quantum technologies).Yet it would be desirable to take the system from one state to another one that does not correspond to the natural evolution of such a system.That is, if Û0 describes such a natural evolution, it is of much interest to devise a method or protocol that warrants the evolution from |ψ i to |ψ f in the least time, provided that |ψ f = Û0 |ψ i .Appealing to the aforementioned quantum-classical correspondence, this is actually the quantum analog of the well-known classical Zermelo navigation problem [9,10].
Brody and Meier [11] have investigated this problem in the field of quantum processing.More specifically, assuming the quantum system is acted by a bare background Hamiltonian, Ĥ0 , these authors determined a method to obtain a time-optimal control Hamiltonian, Ĥc (t), such that its combined action with Ĥ0 , i.e., Ĥ(t) = Ĥ0 + Ĥc (t), (1) generates a time-optimal unitary evolution from |ψ i to |ψ f .The protocol devised by these authors thus includes three key elements: (i) A time-independent, bare background Hamiltonian, Ĥ0 , which describes the natural evolution of the quantum system.
These features thus define the quantum counterpart of the classical Zermelo navigation problem [9,10].
Moved by the possibility to apply the above ideas to more general quantum scenarios, here we present a further development of the quantum Zermelo methodology, suggesting a protocol that can be easily adapted to different physical scenarios.In this regard, we have focused on a series of guidelines, which stress the physics behind the approach to the detriment of a more abstract conceptualization of the problem.That is, we have tried to answer questions such as whether it is possible to build a control Hamiltonian without entering too much formal aspects, but just well known theory.And, if so, we also wanted to know how it looks like and whether it works optimally.Interestingly, by proceeding this way, we have been able to reach a general form for the condition specified by (2), where the l.h.s.equals to a general constant k, which, in turn, is related to the minimum time necessary to take the system from the initial to the final quantum state that we wish, circumventing the unwanted effects of the bare Hamiltonian.Accordingly, a general protocol is presented to determine Ĥc , which here we have tested with a series of well-known quantum systems, such as the harmonic oscillator, entanglement swapping with Bell states, or spin-flip in a Heisenberg dimer.It is worth stressing that, in all cases, although the least time is going to depend on the system Hamiltonian, the condition itself is totally independent of it, which we associate with the fact that the evolution of the quantum state keeps a one-to-one analogy with the geometrical evolution along a meridian joining both states on the Bloch's sphere, as already pointed out by Brody et al. [12].
The work is organized as follows.The theory is presented, developed and discussed in the next section.To be self-contained, both a brief account on the classical Zermelo problem as well as on the Brody and Meier approach are also included, which serves to contextualize the work.Afterwards, we start the development of our approach, which also includes a discussion on the adiabaticity 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 in Sec.IV.

A. Classical Zermelo problem
The classical Zermelo navigation problem can be stated as follows.Given the actual position of a ship, x = (x 1 , x 2 ), on the surface of an unlimited sea and undergoing the local action of current and/or wind, characterized by a position-dependent vector field, w (x) = (w 1 (x), w 2 (x)), one expects to find the optimal control velocity, v = v û = v(u 1 , u 2 ), that should constantly act on the ship so that it reaches its destination in the least time.Here, û is a unit vector in the direction of v and v denotes its modulus.
As it was noticed by Zermelo [9] and Carathéodory [10], the solution to this problem can be obtained by constructing the geometrical form of the indicatrix that allows to obtain the Hamiltonian function and, from it, all extremal curves of the problem.Accordingly, the absolute velocity of the ship, namely v = ẋ/F , must satisfy the equation where ẋ is the derivative of the coordinates with respect to an arbitrary evolution parameter.The time employed by the ship in its full journey is calculated from the integral of the F -function with respect to the arbitrary parameter along the extremal curve.Hence, F becomes the basic function of this variational problem.Equation (4) allows us to determine the F -function as a positive root of the equation whenever such a root exists.Equation ( 5) is the indicatrix of the classical Zermelo navigation problem, which describes a circle of radius v with center at w(x).The set of points satisfying the circle condition correspond to the end points of the vector ẋ/F .As seen below, in the quantum analog for this problem, Brody and Meier [11] found the solution by determining the geodesics of the Randers metric derived from the form of the F -function.

B. Quantum Zermelo approach
As noted by Brody et al. [12], it is also possible to find a direct quantum counterpart of the Zermelo navigation problem.To this end, consider some initial and final quantum states, |ψ i and |ψ f , respectively, for a given physical system, which is being acted by a timeindependent background Hamiltonian Ĥ0 .The quantum Zermelo problem consists in finding a control Hamiltonian, Ĥc (t), such that the total Hamiltonian (1) describes a unitary transformation leading from |ψ i to |ψ f in the least time.Notice that, by appealing to the classical analog, the classical vector field describing the wind or current corresponds, in the quantum counterpart, to the unitary operator generated by Ĥ0 .Furthermore, in this quantum problem, it is assumed that the energy associated with the transformation from ψ i to ψ f is not only limited, but it has also to be totally consumed at the end of the process.Thus, the speed evolution generated by the control Hamiltonian, Ĥc (t), is related to the energy variance of Ĥc (t), according to the Anandan-Aharonov relation [13].Over the full process, the speed evolution takes the maximum attainable value and it is fixed.
Based on such constraints, one aims to built an optimal unitary transformation that satisfies them all.Accordingly, consider the time-evolution of the unitary operator, Û (t, t i ), governed by the Schrödinger equation, which in the Heisenberg representation reads as with = 1 (in natural units).The time-evolution operator Û (t, t i ) is required to satisfy the initial con-dition Û (t i , t i ) ≡ I (I denotes the identity operator) as well as the unitarity condition Û † (t, t i ) Û (t, t i ) = Û (t, t i ) Û † (t, t i ) = I, which ensures the norm preservation along the whole evolution.
For simplicity and convenience, considering the total time lasted in the evolution of the system, ∆T = t f − t i , with t i ≤ t ≤ t f , and then defining the dimensionless evolution parameter s = (t − t i )/∆T , the time-evolution operator can be recast as Û (t, t i ) = ÛT (s), and its timederivative as d Û (t, t i )/dt = (1/∆T )d Û∆T (s)/ds [14].Using the above notation and multiplying Eq. ( 6) from the right by Û † ∆T (t, t i ) we get which strongly resembles the classical Eq. ( 4), with ∆T playing the role of F .In order to further stress the quantum-classical analogy, Eq. ( 7) is now multiplied by itself.Then, the trace over the full resulting evolution equation gives rise to the equation tr X(s) X(s) − 2∆T tr Ĥ0 X(s) + (∆T ) 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 which constitutes the so-called Finslerian norm of X(s) [15][16][17].As it can be noticed, the positivity of ∆T , as given by Eq. ( 10), warrants the above condition (iii), with tr Ĥ2 c (t) = k -Brody and Meier found the "optimal path" X(s) that minimizes the integral over the time given by Eq. ( 10), namely The question now is whether one can approach the same problem from a more physical viewpoint, that is, from a more familiar quantum formulation, which, in turn, might serve also to confer more generality to the process.The answer is affirmative, as we show now by considering notions already existing within the timedependent perturbation theory [14], which is also closer to treatments typically considered in the theory of open quantum systems [18].To see that, let us introduce the unitary time-evolution operator, Û0 (t, t i ), corresponding to Ĥ0 , solution to the equation with initial condition Û0 (t i , t i ) ≡ I.The solution is well known, Now, in order to determine Û (t, t i ), we consider the separable ansatz where the time-evolution operator Ûc (t, t i ) is required to be unitary and satisfying the unitarity condition Û † c (t, t i ) Ûc (t, t i ) = I.This constraint, in turn, implies that Û (t, t i ) also satisfies the unitarity condition, as it can easily be shown.
In order to determine Ûc (t, t i ), we now proceed as follows.First, we substitute Eq. ( 13) into Eq.( 6), and then make the Û † 0 (t, t i ) to act on the left of the resulting expression, which renders the equation with initial condition Ûc (t i , t i ) ≡ I, and where we have made use of Eq. ( 11) to simplify it.Now, as it can be noticed, on the r.h.s., Ûc is acted by the control Hamiltonian operator in the interaction picture [19], Since the control Hamiltonian Ĥc is to be determined, we can make a guess on the particular functional form for its interaction picture, namely that Ĥ c corresponds to Ĥc at t i , so that it becomes time-independent.Although this might look counterintuitive, if we recall the picture provided by Brody et al. [12] of the evolution along the Bloch sphere when going from one state to the other, the above condition (15), with Ĥ c (t) = Ĥc (t i ), can be considered to be equivalent to continuing the journey on the back of the sphere, from the final state to the initial one.That is, the condition for a proper control requires evolution along a meridian, and no other curve, in order to ensure the equivalence of the two journeys.
Therefore, even if the above conditions seems to be a strong constraint, still it is rather reasonable and convenient, since it allows us to recast Eq. ( 15) with a functional form analogous to that for the time-evolution operator associated with the bare Hamiltonian, Eq. ( 11), i.e., with solution Accordingly, the full Hamiltonian for the quantum Zermelo problem acquires the final form which corresponds to Eq.( 1) in [11].
Next, let us see some properties that follow from the above relationship between Ĥc (t) and Ĥc (t i ).Consider the relation it readily follows that, if tr Ĥ2 c (t i ) is constant, then the same holds for tr Ĥ2 c (t) , since which is satisfied at any time t.Thus, according to (20), dtr Ĥ2 c /dt = 0 also at any time.Now, differentiation of Eq. ( 19) with respect to t leads to which is a solution to the variational problem, given in Eq.( 10) and firstly derived by Brody and Meier [11].Equation ( 21) gives the co-adjoint motion and hence it should be solved together with Eq. ( 18).Besides, from Eq. ( 21), we also find that tr d Ĥc (t)/dt = 0 and dtr Ĥ2 c /dt = 2tr Ĥc (t)d Ĥc (t)/dt = 0 by using cyclic permutation when tracing.Physically, these vanishing values imply that the "velocity" of the transition process remains constant during the whole process, as it is assumed in the problem by definition.

C. Transition between two specific quantum states
According to the above results, time optimization in the quantum Zermelo approach is fully determined by the construction of the control Hamiltonian Ĥc (t i ) provided the bound condition tr( Ĥ2 c (t i )) = k is satisfied, since both the bare Hamiltonian Ĥ0 and the initial and final states, |ψ i and |ψ f , are given.In order to understand the dynamical transition from |ψ i and |ψ f , and hence to introduce a protocol to optimize the time lasted in such a transition, let us consider the state reached by |ψ i after a time t under free evolution, i.e., under the action of the bare background Hamiltonian.This state is given by Taking into account Eq. ( 13), we can introduce the intermediate state Differentiating this state and its complex conjugate partner with respect to time, and then substituting the corresponding results into Eq.( 16) (and the corresponding complex conjugate equation), leads to Now, if |ψ i is normalized, then |ψ (t) is also normalized, as it can readily be inferred from (23).Moreover, if we assume that the control Hamiltonian generates a state vector that is orthogonal to the original one (in compliance with the fact that it has to counterbalance the effect of the "blowing wind" accounted for the bare Hamiltonian), then from (24) we have In order to satisfy both conditions, normalization and orthogonality, also from (24) we notice that Ĥc (t i ) has to display the following functional form [22], where the r.h.s.shows an explicit dependence on time, although the Hamiltonian is time-independent.Rather than an inconsistency, this is just an effect associated with the fact that this Hamiltonian has to counterbalance at every time the effect produced by Ĥ0 , although the net action is time-independent, as will be shown below.
Notice that the conditions on |ψ (t) and its time-derivative imply that Ĥc (t i ) is traceless, i.e., tr Ĥc (t i ) = 0.Moreover, since the variance of the energy 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 for the control Hamiltonian, since it makes the variance of this Hamiltonian, given by the expression to reach its maximum value.Actually, we have that which is a consequence of the fact that the control Hamiltonian is traceless [12,21].
From Eqs. ( 27) and ( 28), we find the following relation At any time, this relation is satisfied by the ansatz with time-derivative given by This ansatz, in turn, satisfies the above normalization and orthogonality conditions.Notice here that the expression d|ψ (t i ) /dt has to be understood as the timederivative of |ψ (t) evaluated at t = t i .
In order to further simplify the approach, the above expressions (30) and (31), in terms of the general timeevolved state vector |ψ (t) , can be recast in terms of the initial and final state vectors, |ψ i and |ψ f , thus providing an also simpler functional form for the control Hamiltonian (26).To this end, notice that, by virtue of Eq. ( 23), at t i we have |ψ In order to remove any common support between |ψ f and |ψ i , we need to find the orthonormal form for the former, which is obtained by applying a Gram-Schmidt orthogonalizing process.Accordingly, the orthonormal form is found to be where Eq. ( 30) has been used, with t = t f .Next, we normalize | ψ f : It can be noticed from Eq. ( 34) that the calculation of | ψ f only includes |ψ i , |ψ f , Ĥ0 , and the time interval ∆T .
The ansatz (30) and its time-derivative, Eq. ( 31), can now be recast in terms of the orthonormal state vectors |ψ i and | ψ f , which read as respectively.
In order to finally obtain the functional form of the control Hamiltonian, we substitute Eqs. ( 35) into (26), leading to which is the initial optimal control Hamiltonian.With the aid of Eq. (35a) in the case t = t f , | ψ f can be recast in terms of |ψ f = |ψ (t f ) .If the corresponding expression is then substituted into Eq.(36), we shall obtain which is time-independent, as it was stressed above.This is precisely the expression reported by Brody et al. [12] for Ĥc in the particular case k = 1/2.It can be shown now that the variance of Ĥc (t i ) for any |ψ (t) effectively remains constant in time, that is, ∆ Ĥc (t i ) = The expression of the optimal control Hamiltonian (36) can be written in diagonal form, as With this expression at hand, Eq. ( 17) takes the explicit form The time interval ∆T is then evaluated by considering the transformation indicated in Eq. ( 32), i.e., the timeoperator Ûc (t f , t i ) that takes |ψ i to |ψ f in the shortest time.As it can be noticed in the above expression, the relationship between |ψ i and |ψ f is given in terms of an argument.If φ is the angle between the initial and final state vectors, then the angular evolution is related with ∆T through the expression in compliance with what is stated in the literature on the geometry of the state vector evolution [13,23].
From the above discussion, we then extract as a conclusion that, in order to make the state vector to evolve in the shortest time from |ψ i to |ψ f when there is the influence of a background Hamiltonian Ĥ0 , we need to determine the time-optimal unitary transformation, Û (t f , t i ), which includes the following steps: 1. Given Ĥ0 , |ψ i , |ψ f and k (the energy bound), compute the time interval ∆T recursively by means of Eq. ( 40), and the unitary transformation Û0 (t f , t i ) by means of Eq. ( 12).
This protocol will ensure that the unitary transformation Û (t f , t i ) transforms |ψ i into |ψ f in the least time.

D. Adiabaticity of the quantum Zermelo Hamiltonian
Let us now comment on the adiabaticity associated with the quantum Zermelo Hamiltonian.Consider the Schrödinger equation (6), which in general has not a stationary solution.Now, if we assume that Ĥ(t) changes slowly in time (or it is even constant in time), the system, when started from a stationary state of Ĥ(t i ), will pass through the stationary states corresponding to Ĥ(t).This is what the adiabatic theorem of quantum mechanics says [14,24,25].
As it can be noticed from the above discussion, Eq. ( 13) is the solution of Eq. ( 6), where Û0 (t, t i ) and Ûc (t, t i ) are given by Eqs. ( 12) and ( 17), respectively.Therefore, we can rewrite Û (t, t i ) as with Ĥ(t i ) = Ĥ0 + Ĥc (t i ).Let {φ j (t i )} N j=1 denote the orthonormal set of eigenfunctions of Ĥ(t i ) and {h j } N j=1 the corresponding set of eigenvalues, with N being the dimension of the space.The eigenvalues are timeindependent, since tr d Ĥ(t)/dt = tr d Ĥc (t)/dt = 0, as proven above.Thus, taking into account the spectral decomposition of Ĥ(t i ) = N j=1 h j |φ j (t i ) φ j (t i )|, Eq. ( 41) can be recast as The action of this operator on an eigenfunction |φ k (t i ) with eigenvalue h k gives ) is also the solution of Eq. ( 6) with the initial state |φ k (t i ) .Therefore, the solution (43) is going to coincide with the eigenfunction |φ k (t) of Ĥ(t) up to a phase factor.
Let us now consider an arbitrary wave function |ψ(t i ) .Acting on the left of (43) with ψ(t i )| Û † (t, t i ), and taking into account that Û † (t, t i ) Û (t, t i ) = I, we obtain Thus, if the system is initially represented by the wave function |ψ , then the probability that the system is in the stationary state |φ k (t) at any time t is constant, i.e., 0. This result proves that the dynamical transformation governed by Eq.( 6) taking Û (t, t i ) as that given in Eq. ( 13) satisfies the adiabatic theorem of quantum mechanics.That is, for a system initially prepared in an eigenstate (e.g., the ground state) of the full (underlying plus control) time-dependent Hamiltonian, the time evolution governed by the Schrödinger equation will keep the actual state of the system in the corresponding instantaneous ground state (or other eigenstate).Therefore, considering that the control Hamiltonian in Eq. ( 19) provides the least time to go from one quantum state to another, the solution to the quantum Zermelo problem can be understood as the optimal (in time) adiabatic transformation.
The fact that the quantum evolution of Eq.( 6) satisfies the most trivial version of the adiabatic theorem provides us with another way to determine the time-optimal unitary quantum Zermelo transformation, Û (t f , t i ).This new way is 1.Given Ĥ0 , |ψ i , |ψ f , and k compute the time interval ∆T recursively using Eq. ( 40).Compute the unitary transformation, Û0 (t f , t i ), by Eq. ( 12).

III. APPLICATIONS A. Harmonic oscillator
We shall start the application of the protocol above described with the paradigmatic harmonic oscillator acted by an external field.In particular, we are going to consider a two-level transition, which for simplicity is going to be considered to be the ground and the first excited one, which can be denoted as respectively.Let us consider the transition from the ground state to the excited one, so that |ψ i = |0 and |ψ f = |1 .Of course, these states are under the action of the harmonic oscillator Hamiltonian, with frequency ω.So, in principle, if they are isolated, their only time-dependence is in terms of a phase factor; if they form a linear superposition, there will be a periodic transition from one to the other, with frequency equal to the oscillator frequency, since Ω = (E 1 − E 0 )/ = ω.Besides, it is interesting to note that the creation and annihilation operators included in (46), in terms of the states (45), can be written as The minimum control time is ∆T = π/ √ 2k and the control Hamiltonian in (37) can be written as In order this Hamiltonian to be assimilated by a standard external driving force, one needs to check the following condition where E 0 is the amplitude of the external electric driving field.A simple inspection allows us to realize that the above equation is fulfilled only if cos(3π ω/2 √ 2k) = 0, which leads to the conclusion with n ∈ Z.Therefore, given a frequency ω, the maximum k is given by which corresponds to the minimum control time As it will seen below, these results are in compliance with those for the Heisenberg spin chain, thus paving the way to intuitively consider that there might an underlying common pattern for any quantum system in the form of k that provides a physical (implementable) control.

B. Entanglement swapping
Let us now consider entanglement swapping with maximally entangled states of a Bell basis [26][27][28], where the two entangled qubits are assumed to be spatially distant, a paradigm with special interest in quantum information and quantum computation [29,30].More specifically, here we consider two spins interacting via anisotropic time-independent J j -couplings, with (j = x, y, z), acted by local, controllable magnetic fields B (i) (t), with (i = 1, 2), pointing along the z-direction.Thus, we choose to consider the following two-qubit Heisenberg Hamiltonian [21], to be the quantum Zermelo Hamiltonian Ĥ(t).Here, we use the tensor products σ(1) j = σj ⊗ I and σ(2) j = I ⊗ σj , with I being the unit operator of dimension 2 × 2, and σ(i) j the Pauli matrices [14].
A simpler ansatz for Ĥ was already reported in [31], where only a fixed coupling, J, was considered.Here, we are going to associate the first term in (53) with the noncontrolled, time-independent background Hamiltonian, Ĥ0 , and the second term with the time-dependent control Hamiltonian, Ĥc (t), satisfying the energy resource bound, tr Ĥ2 c (t) = k.In this case, the computational basis set is provided by the factorizable state vectors In this basis, Ĥ0 reads as where J ± = J x ± J y .The diagonal form for Ĥ0 is which allows us to rearrange the above basis set in terms of the Bell basis of maximally entangled states, namely Now the question is how to reach one of these basis vectors from another of them, for instance, the |ψ f = |Φ − state from the |ψ i = |Φ + state, in the shortest time using the optimal-time Zermelo unitary transformation, Eq. ( 13).
The first term of the unitary time transformation Eq. ( 13), namely, Û0 (t, t i ) is easily obtained from the spectral decomposition of Ĥ0 given in Eq.( 56), Û0 (t, with ∆t = t − t i .The calculation of the second term of Eq. ( 13), Ûc (t), is a bit more subtle.As mentioned above, we are interested in the transformation of |Φ + into |Φ − via the unitary transformation |Φ − = Û0 (t, t i ) Ûc (t, t i )|Φ + in the shortest time possible.As explained above, in the previous section, Ûc (t, t i ) transforms 23)).Accordingly, in the present case, |Φ − = |Φ − exp(i(J z − J − )∆t), where we have made use of (58).The intermediate state |Φ − satisfies the relations Φ − |Φ − = 1, and Φ − |Φ + = 0, hence Ĥc (t i ) will have the functional form where as noted before, ∆T = t f − t i , is the minimum time interval to be determined, and |Φ −m = |Φ − for ∆t = ∆T .The next task consists in transforming the Ĥc (t) form specified in the second term of Eq. ( 53) into the Ĥc (t i ) form of Eq. ( 59), as it was also done in the case of the Harmonic oscillator.In the basis set (54), the control Hamiltonian Ĥc (t) reads as where B ± = B (1) ± B (2) (the time-dependence in B (1)  and B (2) has been dropped for simplicity).As it can be noticed tr Ĥc (t) = 0, but tr H 2 c (t) is not constant in time, because the Ĥc (t) form in Eq. ( 60) does not involve time-unitarity.Hence, next we have to transform the Ĥc (t) in Eq. ( 60) into the form given by Eq. ( 19) with an appropriated choice of Ĥc (t i ), according to Eq. (36).The projection of H c (t i ) given by Eq. (36) onto the subspace spanned by |ψ i and | ψ f results in two vanishing diagonal elements and two off-diagonal elements with zero real part, where their imaginary part is equal to k/2.Analogously, we project the Ĥc (t) from Eq.(59) onto the subspace spanned by |Φ + and |Φ −m .In this new representation, we have where, effectively, we notice On the other hand, from Eq. (40), which renders Substituting the value ∆T into the real part, we have k/2 = J z − J − , while if the substitution is made into the imaginary part, then B + = k/2, since B + = 0. Furthermore, the control variable B + decouples from the others, namely B + = B 0+ cos[2(µt + ν)], where B 0+ , µ and ν are time-independent constants.Taking µ = ν = 0, B 0+ = J z −J − = k/2 and ∆T = (π/2)(J z −J − ) −1 = (π/2)(B 0+ ) −1 , we reach the final form for Ĥc (t i ), which reads as where With this, the corresponding unitary transformation is given by Finally, using Eqs.( 58) and (67), we obtain the timeoptimal quantum Zermelo unitary transformation that leads the Bell basis vector |Φ + into |Φ − , namely with 0 ≤ ∆t ≤ ∆T .As it can be noticed, once the journey is complete, i.e., ∆t = ∆T , the Bell state |Φ − is reached.
It is worth noting that in the basis set (57), the quantum Zermelo Hamiltonian acquires the form This Hamiltonian has been obtained using Ĥ0 and Ĥc (t i ) as given by Eqs. ( 56) and (66), respectively.Notice that B 0+ = (J z − J − ), as it has been proven and explained above.The eigenvectors (69) can also be computed and read as where α = −J − /B 0+ and β = √ α 2 + 1, while N 1 = (α − β) 2 + 1 and N 2 = (α + β) 2 + 1 are norm factors.The corresponding eigenvalues are Thus in the quantum Zermelo Hamiltonian, the set of eigenvalues and eigenvectors are time-independent as expected.
C. Spin-flip in a Heisenberg dimer In Sec.III B we have assumed the functional form of a Zeeman coupling for the control Hamiltonian, even though the algorithm presented in Sec.II does not assume any particular form for this Hamiltonian.One may then wonder what would be the resulting control Hamiltonian if its form is not imposed a priori.
Let us thus consider that the initial and final states, |ψ i and |ψ f , respectively, are orthonormal.It is then easy to notice that Eq. (40) reads as i.e., the time needed to reach a target state is inversely proportional to the square root of k.Actually, since k is related to energy, this relation is just a reminiscence of the time-energy uncertainty relation: the larger the amount of energy put into play to optimally guide the vector state to its final destination, the shortest the time employed in the journey, and vice versa.Now, given ∆T , it is then easy to find a general expression for the control Hamiltonian H c (t i ), as seen in Sec.II, where f is the energy of the final state ψ f .To gain some insight into the structure of the above control Hamiltonian, we pick up the particular case considered in the previous section, viz., the case where initial and final states are maximally entangled Bell states.Thus, with the choice |ψ i = |Φ + and |ψ f = |Φ − , and hence f = −J z + J − , we have Substituting these expressions into the control Hamiltonian (73), we finally obtain From Eq. (75), it is clear that the control Hamiltonian adopts the form of a Zeeman coupling for some particular k-values, and hence it can be implemented in the laboratory.More specifically, this is the case when the condition is satisfied, with n ∈ Z. Accordingly, given J z , the maximum k-value is determined from the relation which corresponds to the minimum control time, as it follows from (72).

D. The Cu(II) acetate molecular complex
As a realistic application of the time-optimal quantum Zermelo navigation, we consider the copper(II) acetate monohydrate.This complex corresponds to an atiferromagnetic (S 1 = S 2 = 1/2) coupled spin dimer.As such, this system can be cast in the form of an interacting two-qubit described by a dimer Heisenberg spin chain as in the previous section.Our goal is to find the optimal time for the transition between two maximally entangled (Bell) states to occur for a physically implementable control Hamiltonian in the form of a Zeeman-coupling.
The crystal structure of copper(II) acetate monohidrate, Cu 2 (O 2 CCH 3 ) 4 •2H 2 O, has been determined by X-ray powder diffraction [32] and refined by neutron diffraction at room temperature [33].The crystal is formed by well-defined and separated molecular entities, as displayed in Fig. 1.This complex has a paddle-wheel centrosymmetric structure with two equivalent Cu(II) centers at 2.6143 Ȧ.
Using the development found in [34] (see p. 503), we start from the Heisenberg spin-chain Hamiltonian for the dimer in (53).It can then be shown that for the Cu(II) acetate molecular complex, J 1 , J 2 and J 3 in (53) take the following values: J 1 = 297.793cm −1 , J 2 = 297.753cm −1 , and J 3 = 298.453cm −1 .Then, with the choice of |ψ i = |Φ + and |ψ f = |Φ − , and hence f = −J z + J − , the maximum value of k compatible with a Zeeman-like coupling of the form in (53) is 1.7815 × 10 5 , and the minimum control time corresponds to ∆T = 27.94fs.

IV. CONCLUSIONS
Given the actual position of a classical particle under the action of a given time-independent force-field, there exists an optimal control velocity that, acting constantly on the particle, allows it to reach another position of interest in the least possible time.This problem, known as Zermelo navigation problem [9,10], can be recast in the realm of quantum mechanics by simply substituting the classical particle by a quantum state.In this context, a time-independent Hamiltonian plays the role of the underlying classical force-field, and a time-dependent control Hamiltonian with constant energy resource bound is analogous to the control velocity in the classical navigation problem.A first solution to the above quantum Zermelo problem was put forth by Brody and Meier for a particular energy resource bound [11].Here we have extended this result for general energy resource bounds.
From a fundamental point of view, the solution to the quantum Zermelo problem defines a pair of conjugate variables, viz., the energy resource bound and the control time, that minimize the energy-time uncertainty.While the time-energy uncertainty relation still arouses controversy, in the last decades there has been several attempts towards its explanation.This effort has led to the interpretation of the time-energy uncertainty relation as a so-called quantum speed limit, i.e., the ultimate bound imposed by quantum mechanics on the minimal evolution time between two distinguishable states of a system (see [35] and references therein).Therefore, the solution to the quantum Zermelo problem attains the quantum speed limit for any energy resource bound.
In the above respect, however, we have proven that the solution of the quantum navigation problem does not always lead to physically implementable control Hamiltonians.For a single qubit and two interacting qubits, we have shown that energy resources leading to physically implementable control Hamiltonians are not any one, but follow a well defined mathematical pattern.Specifically, for a orthogonal initial and target states, the resource energy bound of physically implementable control Hamiltonians does obey a quantization rule that depends, exclusively, on the energy of the target state.
As a realistic application of the time-optimal quantum Zermelo navigation, we have shown results for an acetate molecular complex.The magnetic behavior of copper(II) acetate monohydrate corresponds to an atiferromagnetic (S1 = S2 = 1/2) coupled spin dimer.As such, this system can be cast in the form of an interacting two-qubit described by a dimer Heisenberg spin chain.Employing available experimental data, we have evaluated the optimal time for the transition between two maximally entangled (Bell) states to occur.For a physically implementable control Hamiltonian in the form of a Zeemancoupling, this time is in the order of a few femtosecond.
Finally, we have shown that the evolution ruled by the Zermelo control Hamiltonian is adiabatic.That is, for a system initially prepared in an eigenstate (e.g., the ground state) of the full (underlying plus control) timedependent Hamiltonian, the time evolution governed by the Schrödinger equation will keep the actual state of the system in the corresponding instantaneous ground state (or other eigenstate).This result is particularly relevant, as the control Hamiltonian solution to the quantum Zermelo problem is, by construction, the one that minimizes the time to go from one state to the other.Therefore, we conclude that the Zermelo control Hamiltonian defines an optimal adiabatic evolution.This result thus paves the way for the design of novel adiabatic algorithms, where an initial Hamiltonian whose ground state is easy to prepare, leads to a final Hamiltonian whose ground state encodes the solution to a complex eigenstate problem.