Abstract
This chapter is about the minimum time evolution between two quantum states considering the dynamics obeying either time-invariant Hamiltonians or time-varying ones. Merit figures are defined to help quantum control designers to define optimization parameters. The expressions are derived from the time-energy uncertainty relations and a practical case is studied as an example.
Keywords
- merit figures
- minimum time
- quantum control
- quantum evolution
- uncertainty relations
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
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
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
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
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”
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
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
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
in which the denominator of the right side of the inequality denotes the average value of
It is appropriate to introduce, at this point, a special notation, Δ
Now, let us consider a projection operator of form
Indeed,
Furthermore, the expectation value
Therefore, making use of the expressions (2.7) and (2.11); then, applying operator
This inequality contains only one-time variable quantity,
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
From (2.14), two expressions relevant to our purposes shall be deduced. Since the average value of the projection operator
where
Finally, from (2.15), it can be deduced that the lowest physically possible time, or more generally, the inferior time limit, Mathematically speaking, one can say it is the
in which
From (2.16), it is immediately noticed that the time
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
where
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
Combining the expressions (2.16) and (2.17), η
or, alternatively as
where
From the dimensional analysis point of view, it is straightforward to verify that η
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
or, in dual form, which minimize the amount of
that is, control actions such that, once elapsed control time
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. Or orthonormal states, once the normalization condition is taken for granted.
where
Let us consider a quantum dynamical evolution, starting from a generic state
with
Adapting the expression (3.1) for such a case, results in
which its associated probability amplitude,
If a state transition from
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
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
which, in fact, consists of an orthonormal basis for the subspace spanned by
Considering
We shall assume that
By defining the auxiliary constant λ as follows:
the Hamiltonian
so that this matrix can be diagonalized as follows:
with the
corresponding to the diagonalization unitary operator built up with the eigenvectors of
After performing some cumbersome, but straightforward calculations, to diagonalize
where we have imposed
Noting that
indeed, it suffices to impose that the value of the function
In the same way, once achieving
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
In quantum control, this is by far the most commonly found and studied case regarding the attempts to find control laws
The time-evolution operator
Therefore, given an initial state
where
In an applied point of view, given the vector representations of In [8], we present an analytical case study of a time-dependent Hamiltonian, namely the Landau-Zener system.
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
wherein, at the first step, we replaced the time derivative of
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
Finally, the minimum time for a generic quantum state transition, between an initial state
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
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Notes
- Mathematically speaking, one can say it is the inf or greatest lower bound (GLB) of the subset S of the physically possible times for quantum states transitions.
- Or orthonormal states, once the normalization condition is taken for granted.
- In [8], we present an analytical case study of a time-dependent Hamiltonian, namely the Landau-Zener system.