Minimum time for the evolution to a nonorthogonal quantum state and upper bound of the geometric efficiency of quantum evolutions

We present a simple proof of the minimum time for the quantum evolution between two arbitrary states. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the efficiency inequality even when the system passes only through nonorthogonal quantum states.


I. INTRODUCTION
There are several works on the minimum time for the quantum evolution to an orthogonal state. A first list of works on a lower bound on the orthogonality time based on the energy spread includes the investigations by Mandelstam and Tamm [1], Fleming [2], Anandan and Aharonov [3], and Vaidman [4]. In a second type of explorations, there are works like the one by Margolous and Levitin [5] that express a lower bound on the orthogonalization time based on the average energy of the system. In Ref. [6], instead, Levitin and Toffoli proposed a lower bound that involves both the energy spread and the average energy of the system. These lower bounds have been extended from isolated to non-isolated systems [7] and, moreover, from pure to mixed quantum states in the presence of entanglement [8,9] as well. For a recent expository review of the minimum evolution time and quantum speed limit inequalities that includes generalizations to mixed system states and open multiparticle systems, we refer to Ref. [10] and references therein.
It is known that the minimum time T for the Schrödinger evolution to an orthogonal state for a system with a time-independent energy uncertainty ∆E is given by T = h/ (4∆E) with h denoting the Planck constant. This finding was proved, for example, by Vaidman in Ref. [4] without using any geometric argument, while it was proved by Anandan and Aharonov in Ref. [3] with the help of more elaborate geometric considerations within the framework of the projective Hilbert space geometry. In both Refs. [4] and [3], the expression for the minimum time was restricted to evolutions to an orthogonal quantum state. More specifically, Anandan and Aharonov derived a rather interesting inequality relating the time interval ∆t of the quantum evolution to the time-averaged uncertainty in energy ∆E (during the time interval ∆t) in Ref. [3], In particular, they stated that the equality sign in Eq. (1) holds if and only if the system moves along a geodesic in the projective Hilbert space. To quantify geodesic motion in the projective Hilbert space, they also introduced a notion of efficiency denoted as ε (this symbol ε will be replaced with η (geometric) QM in this manuscript), with ε ≤ 1 containing the inequality in Eq. (1) as a special case. However, they stated without proof that ε ≤ 1 is valid more generally even when the system does not pass through orthogonal states.
In this paper, inspired by the results presented in Refs. [3,4], we find the expression for the minimum time for the evolution to an arbitrary nonorthogonal quantum state. Moreover, based on this first result, we provide a quantitative justification of the validity of the inequality ε ≤ 1 even when the system passes only through nonorthogonal states.
The layout of the remaining of the paper is as follows. In Section II, we present a derivation of the expression for the minimum time for the evolution to an arbitrary nonorthogonal quantum state. In Section III, exploiting the result obtained in Section II, we provide a quantitative justification of the validity of the inequality ε ≤ 1 even when the system passes only through nonorthogonal states. In Section IV, we discuss in an explicit manner the concepts of minimum evolution time and quantum geometric efficiency in two physical examples. Finally, our final considerations appear in Section V.

II. MINIMUM TIME WITHOUT GEOMETRIC ARGUMENTS
In this Section, we provide a proof that the minimum time T AB for the quantum evolution between two arbitrary states |A and |B is equal to T AB = ℏ cos −1 [| A|B |] /∆E where ∆E denotes the constant energy uncertainty of the system. This proof follows closely the proof presented by Vaidman in Ref. [4] where, however, the quantum evolution was restricted to an orthogonal quantum state.
We begin with some preliminary remarks. Consider an operatorQ and a normalized quantum state |ψ . The statê Q |ψ can be decomposed as,Q where α 1 , α 2 ∈ C and |ψ ⊥ |ψ ⊥ . Then, we get The stateQ |ψ in Eq. (3) is physically equivalent to the state |α 1 | e −iϕ |ψ + |α 2 | |ψ ⊥ with ϕ def = ϕ 2 − ϕ 1 given that global phases have no relevance in quantum mechanics. Therefore, we conclude from Eq. (3) that we can write the stateQ |ψ asQ with α ∈ C and β ∈ R + . If we assume thatQ is also Hermitian, we have that is, α ∈ R. Moreover, remaining in the working condition withQ being Hermitian, the dispersion ∆Q 2 of the operatorQ becomes that is, Finally, using Eqs. (5) and (7),Q |ψ in Eq. (4) can be decomposed aŝ with ψ|ψ ⊥ = 0. For an alternative derivation of Eq. (8), we refer to Appendix A. Eq. (8) will play a key role in our derivation of the minimum time expression. At this point, we state the problem we wish to address. We want to find an explicit expression of the minimum time for the evolution from the normalized state |A = |ψ (0) to the state |B = |ψ (T AB ) with A|B = 0 in the working condition that the dispersion of the Hamiltonian operatorĤ is constant, Before continuing our proof, we would like to emphasize at this point that our demonstration works equally well for time-dependent HamiltoniansĤ (t) with non constant energy uncertainty ∆E (t) where ∆E 2 (t) 2 and |ψ (t) is assumed to be normalized to one. However, as we shall see, an expression of the minimum evolution time can be only obtained in an implicit manner when taking into consideration systems specified by this type of Hamiltonians.
Returning to our proof, we point out for the sake of completeness that since stationary states are quantum states with no energy uncertainty (that is, ∆E = 0 [11]), we shall limit our considerations to quantum evolutions of nonstationary states [2]. Following the work by Vaidman [4,12], the minimum time of the evolution to a nonorthogonal state can be found by evaluating the maximum of the absolute value of the rate of change of the modulus squared of the quantum overlap ψ (t) |A with |ψ (t) being an intermediate state between |A and |B satisfying the time-dependent Schrödinger equation, Let us consider the quantity d | ψ (t) |A | 2 /dt. Observe that, Therefore, the rate of change in time of the modulus squared of the quantum overlap ψ (t) |A becomes where ψ = dψ dt = d|ψ dt . Using Eqs. (8) withQ =Ĥ together with Eq. (10), we obtain Using Eq. (13), Eq. (12) becomes that is, For a given value of ∆E and | ψ|A |, the absolute value of the RHS of Eq. (15) achieves its maximum number when | A|ψ ⊥ | is maximum by observing that To find the maximum of | A|ψ ⊥ |, we proceed as follows. In general, the resolution of the identity for an n-dimensional Hilbert space H is given by1 with ψ i |ψ j = δ ij for any 1 ≤ i, j ≤ n. For clarity of exposition and without loss of generality, we assume here a resolution of the identity on the full Hilbert space H that can be recast as1 where, Since A|A = 1, from Eqs. (18) and (19) we obtain Therefore, we get from Eq. (20) that that is, | A|ψ ⊥ | is maximum when ψ ⊥⊥ |A = 0. Its maximum value equals, For completeness, we note that in the general scenario where one employs Eq. (17), | A|ψ ⊥ | is maximum when ψ i |A = 0 for any i = 3,..., n with |ψ 1 and |ψ 2 corresponding to |ψ and |ψ ⊥ , respectively. Before continuing our proof, we would like to remark at this point that it is straightforward to see that the subscript "max" in Eq. (22) is not strictly necessary if we employ a different resolution of the identity operator1 on the full Hilbert space with orthogonal decomposition given by H def = H ψ ⊕ H ψ ⊥ as used in Appendix A, and not by Returning to our proof, we note from Eqs. (15) and (22) that the maximum of the absolute value of the rate of change of the quantum overlap depends only on ∆E and | A|ψ | and is given by Finally, to get the fastest evolution to a nonorthogonal state, we impose For the sake of completeness, we point out that this constraint entails imposing a phase relationship between ψ|A = | ψ|A | e i ϕ ψ and that is, Thus, we get from Eq. (26) thatθ Integrating Eq. (27), we obtain Recalling that | ψ (t) |A | def = cos [θ (t)] with θ (0) = 0 since |ψ (0) def = |A , after some simple algebra, we finally get from Eq. (28) that is, The quantity T AB in Eq. (29) denotes the minimum time interval needed for the evolution (unitary Schrödinger evolution with the assumption of constant dispersion of the Hamiltonian operator) from |A to |B with the two states being nonorthogonal. As a side remark, we point out that when |A and |B are orthogonal, Eq. (29) yields Eq. (31) is the result that was originally obtained,without use of geometrical reasoning, by Vaidman in Ref. [4]. As a final remark, we point out that Eq. (29) can be recast as Eq. (32) reduces to the optimal time expression obtained by Bender and collaborators in Ref. [13] when setting |A As pointed out right below Eq. (9), our demonstration leading to Eq. (28) works equally well for time-dependent HamiltoniansĤ (t). In particular, Eq. (28) remains valid if we replace a constant ∆E with a time-dependent ∆E (t). In this case, integration of Eq. (28) yields where ∆E TAB is the time-averaged uncertainty during the time interval T AB defined as, and, therefore, we expect that a closed form analytical expression of the minimum evolution time T AB cannot be generally obtained in an explicit manner in such more realistic time-dependent scenarios.

III. EFFICIENCY WITH GEOMETRIC ARGUMENTS
Recall that in the geometric formulation of quantum mechanical Schrödinger's evolution, one can consider a measure of efficiency that quantifies the departure of an effective (non-geodesic evolution, in general) from an ideal geodesic evolution. Such a geodesic evolution is characterized by paths of shortest length that connect initial and final quantum states |A and |B , respectively. In particular, under this scheme one can define an efficiency in geometric quantum mechanics that takes into account a quantum mechanical evolution of a state vector |ψ (t) in an N -dimensional complex Hilbert space specified by Schrödinger equation, with 0 ≤ t ≤ T AB , ℏ ≤ 1 for such a quantum system can be defined as [3,14], [∆E (t ′ )] /ℏdt ′ is the distance along the (real) actual dynamical trajectory traced by the state vector |ψ (t) with 0 ≤ t ≤ T AB and finally, ∆E represents the uncertainty in the energy of the system. We emphasize that the numerator in Eq. (36) is the angle between the state vectors |ψ (0) and |ψ (T AB ) and is equal to the Wootters distance ds Wootters [15], Furthermore, the denominator in Eq. (36) represents the integral of the infinitesimal distance ds along the evolution curve in the projective Hilbert space [3], Curiously, Anandan and Aharonov proved that the infinitesimal distance ds in Eq. (38) is connected to the Fubini-Study infinitesimal distance ds Fubini-Study by the following condition, with O dt 3 denoting an infinitesimal quantity equal or higher than dt 3 . Eqs. (38) and (39) imply that s is proportional to the time integral of the uncertainty in energy ∆E of the system and specifies the distance along the quantum evolution of the system in the projective Hilbert space as measured by the Fubini-Study metric. We point out that when the actual dynamical curve coincides with the shortest geodesic path connecting the initial and final states, ∆s is equal to zero and the efficiency η (geometric) QM in Eq. (36) equals one. Obviously, π is the shortest possible distance between two orthogonal states in the projective Hilbert space. In general, however, s ≥ π for such a pair of orthogonal pure states.
Given the important role played by energy uncertainty in the geometry of quantum evolutions, one may wonder whether or not there is some sort of quantum mechanical uncertainty relation in this geometric framework. We recall that the standard quantum mechanical uncertainty relation is given by [16], Eq. (40) mirrors the intrinsic randomness of the outcomes of quantum experiments. Precisely, if one repeats several times the same state preparation scheme and then measures the operators x or p, the observations collected for x and p are specified by standard deviations ∆x and ∆p whose product ∆x∆p is greater than ℏ/2. Gaussian wave packets, in particular, are characterized by a minimum position-momentum uncertainty defined by ∆x∆p = ℏ/2. In the geometry of quantum evolutions, there is an analogue of Eq. (40) where, for instance, Gaussian wave packets are replaced by geodesic paths in the projective Hilbert space. Indeed, take into consideration the time-averaged uncertainty in energy ∆E T ⊥ AB during a time interval T ⊥ AB defined as [3], The quantity T ⊥ AB in Eq. (41) defines the orthogonalization time, that is, the time interval during which the system passes from an initial state |A def = |ψ (0) to a final state |B def = ψ T ⊥ AB where B|A = δ AB . Employing Eqs. (38) and (41) and remembering that the shortest possible distance between two orthogonal quantum states in the projective Hilbert space is π, we obtain Specifically, the equality in Eq. (42) holds only when the quantum evolution is a geodesic evolution. Therefore, geodesic paths represent minimum time-averaged energy uncertainty trajectories just as Gaussian wave packets specify minimum position-momentum uncertainty wave packets. Summarizing, when a quantum evolution exhibits minimum uncertainty ∆E T ⊥ AB T ⊥ AB = h/4, unit efficiency η (geometric) QM = 1 is achieved. This, in turn, occurs only if the physical systems moves along a geodesic path in the projective Hilbert space. Interestingly, the Anandan-Aharonov timeenergy uncertainty relation in Eq. (42) is related to the statistical speed of evolution ds FS /dt of the physical system with ds 2 FS being the Fubini-Study infinitesimal line element squared. Precisely, since ds FS /dt is proportional to ∆E, the system moves rapidly wherever the uncertainty in energy assumes large values [17].
We observe that if |A and |B are orthogonal and, in addition, ∆E is constant, we have from Eq. (36) that (43) Therefore, from Eq. (43), we get In particular, we obtain that is, We remark that the geodesic constraint in Eq. (46) is not a time-energy uncertainty condition. Instead, it simply states that distance = speed × time. Indeed, using the Anandan-Aharonov relation that states that the (angular) speed v of a unitary evolution is proportional to the energy uncertainty ∆E [3,18], v = (2∆E) /ℏ, the condition ∆E · T ⊥ AB = h/4 can be recast as s 0 = vT ⊥ AB with s 0 def = π. Furthermore, if |A and |B are nonorthogonal and, in addition, ∆E is constant, we have from Eq. (36) that Therefore, Eq. (47) yields that is, Eq. (49) generalizes the inequality ∆E · T ⊥ AB ≥ h/4 in Eq. (44) and is valid more generally even when the quantum system does not pass through orthogonal states. Moreover, the inequality in Eq. (49) is in agreement with Eq. (29) derived in the previous Section without any geometrical consideration. The derivation of Eq. (49) provides a simple quantitative justification of the verbal statement made by Anandan and Aharonov in Ref. [3] concerning the validity of the inequality ε ≤ 1 extended to a system that does not pass through orthogonal states. In particular, we have from Eq. (49) that that is, We emphasize that the inequality η Interestingly, we also point out that the energy dispersion ∆E (as defined in Eq. (9)) of a constant Hamiltonian op-eratorĤ describing a two-levels quantum system with spectral decomposition given byĤ with respect to the normalized initial state |A is given by, Quantum States Time-Energy Inequality Constraint Optimal Quantum Evolution Condition orthogonal once |A is decomposed as α 1 |E 1 + α 2 |E 2 with α 1 , α 2 ∈ C. From Eq. (52), we note that the maximum value of ∆E is obtained for |α 1 | = |α 2 | where |α 1 | def = | E 1 |A | and |α 2 | def = | E 2 |A |, respectively. Moreover, this maximum value equals ∆E max def = (E 2 − E 1 ) /2. Therefore, the minimum evolution time T AB from an initial state |A to a final state |B becomes Finally, when the quantum evolution is between orthogonal initial and final states |A and |B , Eq. (53) yields Clearly, from Eqs. (51) and (53) we observe that the travel time T AB depends on the Hamiltonian H through the energy uncertainty ∆E. Specifically, T AB can be made arbitrarily small if ∆E can be made arbitrarily large. However, in typical physical scenarios specified by a finite-dimensional Hilbert space with temporally bounded energy eigenvalues {E n (t)} [19], the dispersion of the Hamiltonian operator is upper bounded. Specifically, it happens that ∆E ≤ E max if for any n ∈ N and for any t, one imposes |E n (t)| ≤ E max for some E max ∈ R + . Thus, the minimum time travel is lower bounded with T min In Table I we summarize our results for optimal quantum evolution conditions between both orthogonal, and nonorthogonal states. In the next section, we present two explicit examples.

IV. APPLICATIONS
In this section, we explicitly discuss the notions of minimum evolution time and quantum geometric efficiency in two examples.

A. Time-independent scenario
In the first scenario, we consider a physical system characterized by a time-independent Hamiltonian with ǫ > 0 denoting the strength of the Hamiltonian and σ x being the usual Pauli matrix. The unitary evolution operator U (t) def = e − i ℏ Ht corresponding to the Hamiltonian in Eq. (54) is given by, with I being the 2 × 2 identity matrix. Let us consider the quantum mechanical evolution from |A is physically equivalent to |1 . To begin, we observe that the path γ t traced by the state vector |ψ (t) = cos ǫ ℏ t |A + sin ǫ ℏ t |B is a geodesic path. Indeed, a simple calculation shows that |ψ (t) can be recast as a quantum geodesic line [20], where N ξ def = [1 − 2ξ (1 − ξ)] −1/2 is a normalization constant, while ξ (t) denotes a strictly monotonic function of t given by with 0 ≤ ξ (t) ≤ 1. The geodesic nature of the path γ t can also be explained in terms of the energy spread and the efficiency concepts. Indeed, we have ∆E 2 = ǫ 2 , T (effective) AB = π 2ǫ ℏ, and A|B = 0. Therefore, the minimum evolution constraint condition ∆E · T Finally, the quantum motion occurs with unit geometric efficiency η (geometric) QM = 1.

B. Time-dependent scenario
In the second scenario, we take into consideration a physical system specified by means of a time-dependent Hamiltonian, with ǫ > 0, ω > 0, and ℏω 0 > 0 denoting the strength of the external drive, the angular frequency of the external drive, and the energy difference between the two states of the two-state quantum system (that is, ℏω 0 def = E 2 −E 1 > 0). We define the detuning of the driving field from resonance as ∆ def = (ω − ω 0 ). Clearly, σ x , σ y , and σ z are the usual Pauli matrices. The Hamiltonian in Eq. (59) emerges in the context of the near-resonance phenomenon in a two-state quantum system. The unitary evolution operator U (t) corresponding to H (t) in Eq. (59) is [21], where κ def = ǫ 2 + ∆ 2 /4. Let us consider the quantum mechanical evolution from |A As a side remark, we note that near-resonance, |∆| ≪ ǫ and κ → ǫ.
To begin, we note that the path γ t traced by the state vector |ψ (t) = cos κ ℏ t |A + sin κ ℏ t |B with A|B = −i ∆ 2κ = 0 is not a geodesic path. Indeed, following the reasoning outlined in the first example, a simple calculation shows that |ψ (t) cannot be recast as a quantum geodesic line. The non-geodesic nature of the path γ t can also be understood by studying the expression of the energy spread of the system. Indeed, from a simple calculation, we obtain At this point, to show the non-geodesic behavior of the evolution path γ t in this second scenario, we focus on the short-time limit of ∆E 2 (t) in the near-resonance case. First, assuming |∆| ≪ ǫ, ∆E 2 (t) in Eq. (61) reduces to with 0 ≤ t ≤ π 2ǫ ℏ. Before considering the short-time limit, we make a few considerations that motivate the consideration of this limit. The strength of the external drive ǫ is related to the Rabi angular frequency by the relation ǫ = ℏΩ Rabi [22], with Ω Rabi def = eB ⊥ /(2mc). The quantities e and m are the charge and the mass of an electron. The quantity c denotes the speed of light, while B ⊥ is the intensity of the magnetic field originating from the magnetic field components that are in the plane orthogonal to the quantization axis (that is, the z-axis). The energy gap ℏω 0 is related to the Larmor angular frequency, ℏω 0 = ℏΩ Larmor [22], with Ω Larmor In particular, we provided a quantitative justification of the validity of the inequality η (geometric) QM ≤ 1 even when the system passed through nonorthogonal states (see Eq. (49)). A schematic description of our main discussion points appears in Table I. While our investigation is performed in the spirit of the original Vaidman work, we additionally consider here a number of new modifications. Firstly, we extend the reasoning to unitary Schrödinger evolutions between quantum states that are not necessarily orthogonal. Secondly, we provide two explicit and alternative detailed proofs of the clever decomposition ofQ |ψ in Eq. (8) which plays a key role in the main proof itself. Thirdly, we emphasize its generalization to time-dependent Hamiltonian evolutions. Fourthly, and perhaps most importantly, we show the usefulness of the outcomes of the proof in upper bounding the geometric efficiency of quantum evolutions between two arbitrary states, either orthogonal or nonorthogonal. Lastly, we quantitatively present two illustrative examples discussing both time-independent and time-dependent quantum Hamiltonian evolutions in terms of minimum evolution time and geometric efficiency.
As a final remark, we point out that it would be interesting to further deepen our understanding of this geometric efficiency analysis to physical scenarios where the energy uncertainty ∆E is not constant in time. A partial list of scenarios that could be considered includes the su (2; C) time-dependent Hamiltonian evolutions used to describe distinct types of analog quantum search schemes viewed as driving strategies in Ref. [24] and, in addition, the time-dependent Hamiltonian describing the resonance phenomenon in a two-state quantum system used to construct quantum search algorithms by Wilczek and collaborators in Ref. [25] without limiting the analysis to the short-time limit of the near-resonance regime. We hope to address these more applied investigations in future efforts.