## 1. Introduction

One of the most important problems in quantum control [1, 2], as well in quantum information processing and quantum computing [3, 4], is the transition from an initial state to a target state in minimum time. In fact, the existence of the *Decoherence* phenomenon in quantum systems, characterized by extremely short coherence times, presents serious difficulties on implementing quantum information devices [3, 5].

Indeed, one of the difficulties of the operational nature in quantum systems lies in the fact that they are very sensitive to the presence of the external environment, which often destroys its main quantum characteristics, which are essential, for example, for the implementation of systems processing quantum information, as well as for the viability of quantum computing. This is the problem of quantum decoherence. Thus, it is widely desirable for expedients and methods of optimal control of time minimization, applied to quantum systems, be the most efficient possible accordingly, whether of analytical nature, or of algorithmic and computational nature. However, there are physical limitations inherent by quantum dynamics, which relate to the minimum time, physically possible, so that a transition from quantum states occurs.

Therefore, it is natural to ask, what is the shortest physically possible time for a quantum state to evolve to another? The answer is provided by *time-energy uncertainty relations*. Opposing the famous quantum uncertainty relations regarding non-commuting operators, for example, position and momentum, time-energy uncertainty relations have a different mathematical and physical nature; they are deeply rooted in quantum dynamics.

The Soviet physicists Leonid I Mandelstam and Igor E. Tamm carried out the first successful theoretical approach addressing this issue, in the 1940s [6]. Under the hypothesis of time-independent Hamiltonian, Mandelstam and Tamm deduced a quantum dynamical inequality for time-energy, which sets up the temporal lower bound for a quantum state transition.

It is our goal, in the first section of this Chapter, to follow the theoretical steps of [6], translating the deductions to the modern quantum mechanical formalism, and to perform detailed analyses of the dynamical issues. Thereafter, we wish to apply the Mandelstam-Tamm time-energy inequality to a quantum system of interest, for example, the Fahri-Gutmann model of digital quantum computation [7], in order to obtain an analytical expression for the minimal time required for a state transition in such a quantum system. These analyses allow us to ultimately introduce a quantitative measure for the performance of time-optimal quantum controls [8].

In a subsequent section, we shall drop the time-independent Hamiltonian hypothesis of the original time-energy uncertainty relation and generalize it in the case of a time-dependent Hamiltonian

## 2. Minimum time for quantum state transitions

Before tackling the control problem in quantum systems with respect to minimum time, that is, to make a given state transfer from a prescribed initial state to a desired target state as quickly as possible, one must take into account a fundamental issue of physical nature, which can be stated as follows. Given an initial state

This problem of a theoretical nature has been widely studied and it is closely linked to dynamical characterizations derived from time-energy uncertainty relations. Such relationships differ fundamentally from the Heisenberg’s uncertainty principle (e.g., the simultaneous measurement uncertainty of the position and the linear momentum of a quantum particle), which comes from the *incompatibility* between the quantum observables, a physical fact that arises, within the mathematical and theoretical framework of quantum mechanics, from the *non-commutability* of the quantum observables involved in the measurement process. The time-energy uncertainty, on the other hand, finds its roots in quantum dynamics, as we are about to see.

From a historical point of view, since the so-called “old quantum theory,” pioneered by Max Planck, Albert Einstein, Niels Bohr, among others, comprising the first two decades of the last century and firmly established, was the Planck-Einstein equation, which relates the energy and frequency of a photon through the Planck’s fundamental constant *h*:

As we know, such relationship is fundamental not only for the pioneering and groundbreaking Planck’s hypothesis of quantization of radiation emission by an ideal black body, written in 1900 but also for the explanation given by Einstein for the photoelectric effect in 1905. Moreover, the atomic model introduced by Bohr in 1913, which explained the stability of the hydrogen atom by means of quantized energy levels corresponding to the stable possible orbits for the electron; postulated in addition that, when jumping to an energy level (or orbit) more or less energetic, the electron absorbed or emitted a quantum of energy, respectively, following the relationship (**2.1**), corresponding to a photon with frequency

So, given a trivial variation in frequency at a given time interval, such as

providing the same account of an “uncertainty” **2.1**), we are led to the following:

This “uncertainty relation” deduction is eminently heuristic, although expression (**2.3**) still has experimental support within the physical conditions one has evoked to get it. Nevertheless, if one tries to generalize it as something of the form:

because the Hamiltonian corresponds to the energy of a mechanical system (classical or quantum), one will run into serious difficulties. If *H* represents a quantum observable in the current quantum mechanical sense, in the same way Paul Dirac had formally stated in the very first edition of his famous treatise [10], dating back to 1930, we can no longer identify the energy with the frequency of the monochromatic radiation times the Planck’s constant. Moreover, expression (**2.4**) becomes inherently invalid and devoid of meaning for any quantum system when properly mathematically treated.

The Soviet physicists Leonid I. Mandelstam and Igor E. Tamm in the 1940s carried out the first successful theoretical approach addressing this quantum dynamical issue, the statement of a meaningful time-energy uncertainty relation [6].

So, let us try to present their theoretical starting point drawing on modern quantum mechanical formalism and its current notation, and finally arrive at the desired time-energy uncertainty relation through rigorous deductions combined with detailed analyses. The goal is also to modify it, in order to obtain variants of it and alternative expressions suitable for some purposes, which will become clear in the sections ahead.

Let

where **2.6**) is nothing but the dynamical evolution equation in the Heisenberg picture for the expectation value of observable **2.5**) and apply the result of (**2.6**), the following inequality is obtained:

This expression provides, therefore, the connection between the standard deviation

Relationship (**2.7**) may be expressed otherwise. Since the modulus of an integral is less than or equal to the integral of the integrand modulus [11], upon performing an integration of (**2.7**) from *t* to *t* + Δ*t*, and taking into account that

in which the denominator of the right side of the inequality denotes the average value of *t*.

It is appropriate to introduce, at this point, a special notation, Δ*T*, for the shortest time, during which the average value of a certain physical quantity is changed by an amount equal to the standard deviation thereof. Thus, Δ*T* can be called standard deviation (uncertainty) of time; making use of this notation, (**2.8**) can be rewritten as follows:

Now, let us consider a projection operator of form

Indeed,

Furthermore, the expectation value **2.10**), it follows that

Therefore, making use of the expressions (**2.7**) and (**2.11**); then, applying operator

This inequality contains only one-time variable quantity, **2.12**) that, for

From (**2.13**), by using basic trigonometric properties and simple algebraic manipulations, it leads to the following expression:

for

Here, an important fact should be noted. Although the projection operator *ket* and the *bra* corresponding to a state ψ_{0}, we could have also defined it, in a more general manner, such as _{t} evolved in time, for any instant of time *t*, so that the idempotency property of

From (**2.14**), two expressions relevant to our purposes shall be deduced. Since the average value of the projection operator **2.14**) shall be rewritten as

where

Finally, from (**2.15**), it can be deduced that the lowest physically possible time, or more generally, the inferior time limit,1
physically needed to perform a transition between quantum states, under the assumption of a time-independent Hamiltonian, is given by the following inequality, which takes into account the uncertainty in determining the energy (dispersion or standard deviation of

in which *t* is the necessary time for a quantum transition of states that its associated probability is

From (**2.16**), it is immediately noticed that the time *t* is always a real number, despite the internal product (“bracket”) of states being, in general, a complex number. Likewise, as any quantum observable, *arccos* function for this specific domain and, therefore, the values of *t*, will always be positive.

Some remarks about technological issues and its terminology are in order here. In quantum control literature, inequality (**2.16**) is also known as the “Bhattacharyya limit,” after a paper by the Indian physicist Kamal Bhattacharyya, in which the author revisits the Mandelstam-Tamm’s time-energy uncertainty relations, in order to apply them in decay or non-decay problems in quantum systems [12].

Inequality (**2.16**) gives a strong motivation to introduce a quantitative measure for the evaluation of the quantum control systems performance, with special interest on the time-optimal quantum control, for example, [1] and [13]. Lets then define the following *figure of merit*,

where *t*_{min} is the shortest physically possible time to obtain a desired transition of quantum states, and *t*_{CQS} is the time by which such transition can be effectively accomplished in the *controlled quantum system*, hence, the notation chosen is [8].

In general, analytical solutions to problems of optimal quantum control are rare, and in most cases, control algorithms and numerical simulations are employed to obtain the desired results [1]. Indeed, in an article by Caneva and other authors [13], the Krotov algorithm is employed, for example, [14] and [15], and applied on the Landau-Zener system, as well as on a theoretical scheme of quantum information transfer in a spin chain (“One dimensional Heisenberg spin chain of length N”). The authors obtained state transition times close to the “Bhattacharyya limit,” which is equivalent to inequality (**2.16**) for both systems studied [13].

Regarding the figure of merit η_{t}, it is easy to notice that η_{t} is a real number and a dimensionless quantity, defined on the interval [0,1]:

Combining the expressions (**2.16**) and (**2.17**), η_{t} can be more explicitly expressed as

or, alternatively as

where

From the dimensional analysis point of view, it is straightforward to verify that η_{t}, as expressed by (**2.18**) and (**2.19**), is consistent with the adimensionality requirement, since the constant

Frequently, in practical applications, it is not always possible to obtain the “exact transition” of quantum states by the use of control algorithms and numerical simulations per se, from an initial state to the desired goal state, such as *quantum fidelity*:

or, in dual form, which minimize the amount of *quantum infidelity*:

that is, control actions such that, once elapsed control time *T*, the probability for the evolved state in time *T* and show that the algorithmic method in question is capable of producing infidelities arbitrarily close to zero [13].

## 3. Application to a particular quantum state transition

Now, consider the application of the time-energy uncertainty relations (**2.15**) and (**2.16**) to a specific and very important quantum state transition, namely the transition between two orthogonal states.2 By rewriting inequality (**2.15**), in order to make explicit the probability associated to the transition from an initial state to a time-evolving state, on the left side of the expression, we have

where

Let us consider a quantum dynamical evolution, starting from a generic state

with *t*_{0} *=* 0 can be imposed without loss of generality.

Adapting the expression (**3.1**) for such a case, results in

which its associated probability amplitude,

If a state transition from **3.2**) becomes zero, being able to define formally the first instant of time *t*, for which such transition takes place as follows:

Therefore, it is straightforward to conclude from (**3.2**) that

Furthermore, the quantitative measure of temporal transfer efficiency can be defined for this specific case, figure of merit

or, more explicitly, taking into account (**3.5**),

where

Finally, it is worthwhile mentioning the transitions or transfers between orthogonal (orthonormal) quantum states are of paramount importance to any schemes or devices, whether theoretical, experimental, or of technological nature, aimed at implementing quantum information processing, or quantum computing. The interested reader is referred to standard and authoritative sources like [3,4].

## 4. An analytical case study: The Fahri-Gutmann system

In this section, an application example is presented for illustrating the ideas and theoretical concepts developed so far. The Fahri-Gutmann system is a digital quantum computing model, which in its turn can be interpreted as a variation of the quantum search algorithm, similar to the well-known Grover’s algorithm [16]. Here, we follow the steps of [17].

Let

where *E _{a}* and

*E*are positive constants.

_{b}As already stressed, there can be computational difficulties to achieve the exact desired transfer of states, that is, from the initial state to the target one over the time evolution. Nevertheless, one can think of formulating the quantum control problem in a less restrictive manner, namely in terms of a state transition, as fast as possible, such that one can ensure maximum fidelity. Translating it into quantum mechanical language, we want to maximize the quantity:

Firstly, let us impose *s* is a real number, since a phase factor can be associated to **4.1**), it is evident that it acts non-trivially only on the subspace spanned by the eigenstates

which, in fact, consists of an orthonormal basis for the subspace spanned by

Considering *E = E _{a} + E_{b}, x = E_{a} – E_{b}*, and defining the quantity:

We shall assume that

By defining the auxiliary constant λ as follows:

the Hamiltonian **4.6**), has its matricial representation, with *E* being an arbitrary constant:

so that this matrix can be diagonalized as follows:

with the *U* matrix given by

corresponding to the diagonalization unitary operator built up with the eigenvectors of **4.7**).

After performing some cumbersome, but straightforward calculations, to diagonalize **4.7**), making use of (**4.8**) and (**4.9**), one finally obtains from (4.5a) and (4.5b) the time transition probability *P _{t}*, which can be regarded as a fidelity measure between the “search state”

where we have imposed *t*_{0} *=* 0 on the time-evolution operator

Noting that *s ≤ μ ≤* 1, it is easy to see that the maximum value of the probability *P _{t}* is

indeed, it suffices to impose that the value of the function **4.10**) to obtain (**4.11**).

In the same way, once achieving **4.10**), it can be conclude that the first instant of time *t*, for which maximum probability (or maximum fidelity), is given by

Thus, for the Fahri-Gutmann’s quantum computing model, the particular figure of merit that quantifies the time transition efficiency can be stated as

where

## 5. Quantum systems with time-dependent Hamiltonians: Two theoretical approaches to minimum time in quantum state transitions

Let us consider, a time-evolution operator *t*_{1} and *t*_{2}, *t*_{1} ≠ *t*_{2}.

In quantum control, this is by far the most commonly found and studied case regarding the attempts to find control laws *u*(*t*) to achieve a certain control goal of interest, which involves, for example, the minimization or maximization of dynamical variables to perform desired state transitions. Here, we are interested in minimizing transfer times between quantum states. Physically speaking, for example, a spin ½ system (e.g., an electron) subjected to a magnetic field which magnitude varies in time, but not in direction.

The time-evolution operator

Therefore, given an initial state

where **5.1**).

In an applied point of view, given the vector representations of **5.1**), expression (**5.2**) will provide the necessary formula to obtain the minimum time value.3

Now, the focus is on another possible formulation addressing this quantum dynamical issue. The goal is to directly employ a time-dependent energy uncertainty (or standard deviation)

with a time-evolving state

From this particular Schrödinger equation, Pfeifer proposed the following expression for a time-dependent energy uncertainty [9]:

in which we can immediately notice the dependency on the time-evolved state **5.4**), some manipulations are carried out to deduce an alternative expression for the time-dependent energy uncertainty

wherein, at the first step, we replaced the time derivative of **5.3**), so that we could eliminate the dependency on

Now, let us consider a time-energy uncertainty relation of the form

Furthermore, to achieve the perspective of our chain of theoretical reasoning so far, we restate it in the following way:

in which the energy uncertainty is now considered as depending on a time-dependent Hamiltonian *t _{0} = 0* in

Finally, the minimum time for a generic quantum state transition, between an initial state *T*. Its final state is achieved or characterized by

or, in terms of the maximum possible quantum fidelity, also by

Finally, taking into account expressions (**5.7**) and (**5.5**), we have

obtaining an alternative expression for the greatest temporal lower bound for a generic transition of states, that occurs in a quantum system with dynamics governed by a time-dependent Hamiltonian