A Harper’s equation, quantum Fourier transform, tightest bound and their inherent connections with unitary operators

Theoretical formulation of finite-dimensional discrete phase spaces: II. On the uncertainty principle for Schwinger unitary operators

Abstract

We introduce a self-consistent theoretical framework associated with the Schwinger unitary operators whose basic mathematical rules embrace a new uncertainty principle that generalizes and strengthens the Massar-Spindel inequality. Among other remarkable virtues, this quantum-algebraic approach exhibits a sound connection with the Wiener-Kinchin theorem for signal processing, which permits us to determine an effective tighter bound that not only imposes a new subtle set of restrictions upon the selective process of signals and wavelets bases, but also represents an important complement for property testing of unitary operators. Moreover, we establish a hierarchy of tighter bounds, which interpolates between the tightest bound and the Massar-Spindel inequality, as well as its respective link with the discrete Weyl function and tomographic reconstructions of finite quantum states. We also show how the Harper Hamiltonian and discrete Fourier operators can be combined to construct finite ground states which yield the tightest bound of a given finite-dimensional state vector space. Such results touch on some fundamental questions inherent to quantum mechanics and their implications in quantum information theory.

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1 Introduction

Initially introduced by Schwinger [1] for treating finite quantum systems characterized by discrete degrees of freedom immersed in a finite-dimensional complex Hilbert space [2], the unitary operators gained their first immediate application in the formal description of Pauli operators. Ever since, an expressive number of manuscripts [3] proposed similar theoretical frameworks with intrinsic mathematical virtues and concrete applications in a wide family of physical systems — here supported by a finite space of states. With regards to these state spaces, it is worth mentioning that certain algebraic approaches related to quantum representations of finite-dimensional discrete phase spaces were constructed from this context in the past [4], and tailored in order to properly describe the quasiprobability distribution functions [5] in complete analogy with their continuous counterparts [6]. Thus, applications associated with the discrete distribution functions covering different topics of particular interest in physics — e.g., quantum information theory and quantum computation [7, 8], as well as the qualitative description of spin-tunneling effects [9], open quantum systems [10] and magnetic molecules [11], among others — emerge from these approaches as a natural extension of an important robust mathematical tool.

Although the efforts in constructing a sound theoretical framework to deal with finite-dimensional discrete phase spaces have recently achieved great advances (e.g., see Ref. [12]), certain fundamental questions particularly associated with the factorization properties of finite spaces [13], uncertainty principle [14] and property testing [15] for the unitary operators still remain without satisfactory answers in the literature (indeed, some of them represent open problems which do not share the same rhythm of progress). In this paper, we focus on the problem of deriving a general uncertainty principle for Schwinger unitary operators in physics. In what follows, we discuss the relevance of such a principle and, subsequently, briefly review the results obtained by Massar and Spindel [14] on this specific subject.

To begin with, it is necessary to remember that, through an original algebraic approach which encompasses the description of finite quantum systems, Weyl [16] was the first to describe quantum kinematics as an Abelian group of ray rotations in the system space. According to Weyl: “The kinematical structure of a physical system is expressed by an irreducible Abelian group of unitary ray rotations in system space. The real elements of the algebra of this group are the physical quantities of the system; the representation of the abstract group by rotations of system space associates with each such quantity a definite Hermitian form which ‘represents’ it.” With respect to the particular case of finite state vector space, one of Weyl’s most significant achievements was that the observation of pairs of unitary rotation operators obey special commutation relations (bringing, as a result, the roots of unity) which are the unitary counterparts of the fundamental Heisenberg relations. Moreover, it is worth mentioning that such a ray representation of the Abelian group of rotations can be connected with some representations of the generalized Clifford algebra, this fact being thoroughly explored by Ramakrishnan and coworkers [17] through extensive studies of certain physical problems. Still within the aforementioned Weyl approach for quantum kinematics, let us briefly mention that some authors have also addressed the problem of discussing quantum mechanics in finite-dimensional state vector spaces, where the coordinate and momentum operators (characterized by discrete spectra) play an essential role in this context [18].

Although both Weyl and Schwinger’s theoretical approaches have made seminal and complementary contributions upon the scope of unitary operators in finite physical systems, the relevance of a general uncertainty principle for such operators has not been clearly discussed or even mentioned with due emphasis in the past. Reflecting on this, Massar and Spindel [14] have recently established a first uncertainty principle for the discrete Fourier transform [19] whose range of applications in physics covers, among other topics, the Pauli operators, the coordinate and momentum operators with finite discrete spectra, the modular variables, as well as signal processing. Furthermore, their result can also be employed to determine a modified discrete version of the Heisenberg-Kennard-Robertson (HKR) uncertainty principle which resembles the generalized uncertainty principle (GUP) in the quantum-gravity framework [12]. However, if one adopts an essentially pragmatic point of view, certain natural questions arise: “Can Massar-Spindel inequality be recognized as a ‘generalized uncertainty principle’ for all finite quantum states?” If not, “What is the reliable starting point for obtaining a realistic description of this generalized uncertainty principle?”

The main goal of this paper is to present a self-consistent theoretical framework for the Schwinger unitary operators which embodies, within other virtues, an important set of convenient inherent mathematical properties that allows us to construct suitable answers for the aforementioned questions. This theoretical framework, constituted of numerical and analytical results, can be interpreted as a “generalized version” of that one by Massar and Spindel, with immediate applications in quantum information theory and quantum computation, as well as in foundations of quantum mechanics. Next, we emphasize certain essential points of our particular construction process: (i) Numerical computations related to a huge number of randomly generated finite states demonstrate the existence of a nontrivial hierarchical relation among the different bounds, the Massar-Spindel inequality being considered in such a case as a zeroth-order approximation. (ii) The existence of a tightest bound for different dimensions of state vector space leads us to produce a sufficient number of formal results related to the Hermitian trigonometric operators (defined through well-known specific combinations of unitary operators) and their corresponding Robertson-Schrödinger (RS) uncertainty principles [20], which culminates in the formulation of a new inequality which takes into account the quantum correlation effects. This tighter bound represents a new and important paradigm for signal processing with straightforward implications on finite quantum states [21] and discrete approaches in GUP [22]. (iii) Numerical and analytical approaches [23, 24] confirm the special link between the ground state inherent to the Harper Hamiltonian and the tightest bound for any Hilbert space dimensions. Finally, (iv) the connection with tomographic measurements of finite quantum states via discrete Weyl function represents, in this case, a tour de force in our investigative journey on unitary operators that allows to join both the Weyl and Schwinger quantum-algebraic approaches in an elegant way.

This paper is structured as follows. In Section 2, we fix a preliminary mathematical background on the Schwinger unitary operators, which allows us to discuss the implications and limitations of the Massar-Spindel inequality for signal processing. In Section 3, we introduce four Hermitian trigonometric operators through effective combinations of the Schwinger unitary operators. Together, these operators provide a self-consistent quantum-algebraic framework, leading us to determine a new tighter bound for Massar-Spindel inequality. Section 4 is dedicated to discuss certain important aspects of the tightest bounds and their respective kinematical link with the Harper Hamiltonian. In addition, Section 5 presents an elegant mathematical procedure for measuring a particular family of expectation values — here mapped upon finite-dimensional discrete phase spaces and related to the unitary operators under investigation — via discrete Weyl function. Section 6 contains our summary and conclusions. Finally, Appendix A concerns the Harper functions and their respective connection with the tightest bounds verified in the numerical calculations.

2 Preliminaries

In order to make the presentation of this section more clear and self-contained, we begin by reviewing some essential mathematical prerequisites related to the Schwinger unitary operators. Only then we establish the Massar-Spindel inequality and its inherent limitations.

2.1 Schwinger unitary operators

{pf*}

Definition (Schwinger). Let be a pair of unitary operators defined in a -dimensional state vector space, and denote their respective orthonormal eigenvectors related by the inner product with . The general properties

together with the fundamental relations

constitute an important set of mathematical rules that are related to the generalized Clifford algebra [17]. Here, the discrete labels obey the arithmetic modulo and represents a symmetrical finite Fourier kernel. A compilation of results and properties associated with and can be found in Ref. [1].

In the following, let describe a set of physical systems labeled by a finite space of states, whereas and denote the variances related to the respective unitary operators and  — in this case, and represent the mean values of and defined in a -dimensional state vectors space. Since refers to a normalized density operator, the Cauchy-Schwarz inequality allows us to prove that and are restricted to the closed interval ; consequently, both the variances are trivially bounded by . Indeed, the upper and lower bounds are promptly reached when one considers the localized bases and , i.e., for a given and ; otherwise, if and . Furthermore, note that and are invariant under phase transformations, namely, and for any .

2.2 Massar-Spindel inequality

This inequality is based on the Wiener-Kinchin theorem for signal processing and provides a constraint between the values of (correlation function) and (discrete Fourier transform of the intensity time series). According to Massar and Spindel: ‘This kind of constraint should prove useful in signal processing, as it constrains what kinds of signals are possible, or what kind of wavelet bases one can construct.’ We state this result in the theorem below (proved in the supplementary material from Ref. [14]), for then proceeding with a numerical study of its content and first implications.

{pf*} Theorem (Massar and Spindel). Let and denote two unitary operators such that and with . The variances and — here defined for a given quantum state and limited to the closed interval — satisfy the inequality (1) where . The saturation is reached for localized bases.

This theorem leads us, in principle, to consider the different possibilities of connections between the Schwinger unitary operators (defined in the previous subsection) and . A first immediate link yields the relation and for , which implies in the apparent violation of Eq. (1) since . The second connection establishes the alternative relation and with , this result being responsible for validating the Massar-Spindel inequality. It is important to mention that the apparent problem detected in the first situation can be properly circumvented making and with fixed. That is, the modified bound

holds for any and integer .

For the sake of simplicity and convenience, let us now introduce the shift operator with for a given arbitrary unitary operator and density operator , which leads to define1 its non-unitary counterpart as follows [25]:

(2)

Following, it is straighforward to show that both the variances and are related through the expressions

(3)

which substituted into inequality (1) for and gives

(4)

Figure 1: Plots of versus for different values of  —  (a), (b), (c), and (d) — with approximately randomly generated states, illustrating the Massar-Spindel inequality for the Schwinger unitary operators. Note that characterizes a particular visualization window which leads us to guess the existence of a tightest bound (see dashed curves) for each value of dimension , since the distance between the saturation curve (solid line) and the cloud of points — generated by numerical calculations — increases for . The dot-dashed curves showed in pictures (b,c,d) represent the intermediate inequality , this result being considered as a first approximation to our initial intents.

To illustrate Eq. (4) and corroborate the analytic results obtained by Massar and Spindel, let us introduce the parameters and such that . This particular inequality defines a region in the two-dimensional space limited by the rectangular hyperbola that preserves the original equation . Figure 1 shows the plots of versus for approximately states of randomly generated , within the visualization window , with (a) , (b) , (c) , and (d) . Note that, excepting picture (a), all the subsequent cases exhibit a gap between the distribution of states and the rectangular hyperbola, the size of such a gap being dependent on the value of (such an evidence motivates the search for a tightest bound that corroborates the numerical calculations). In fact, the dashed lines in (b,c,d) describe hyperbolic curves of the form , where the value of was chosen as the smallest value of amongst the ones computed with the randomly sampled states. Later in this paper, a more rigorous procedure for obtaining such values will be outlined. Finally, let us briefly mention that the dot-dashed lines exhibited in these pictures correspond to the intermediate result , which will be properly demonstrated in the next section.

3 A Hierarchy of Tighter Bounds

In the first part of this paper, we established a basic theoretical framework related to the Schwinger unitary operators where the Massar-Spindel inequality and its inherent limitations occupied an important place in our discussion on uncertainty principles for physical systems labeled by a finite space of states. At this moment, let us clarify some fundamental points raised by those results: (i) the aforementioned state spaces consist of -dimensional Hilbert spaces; (ii) the Massar-Spindel inequality represents a “zeroth-order approximation” in the hierarchy of uncertainty principles; and finally, (iii) the results obtained from the numerical calculations reveal certain unexplored intrinsic properties of some finite quantum states [26, 27] with potential applications in quantum information theory and quantum computation. In this second part, we begin the construction of a solid algebraic framework based on the RS uncertainty principle, which leads us, in a first moment, to determine a self-consistent set of results for the unitary operators and that permits to generalize the Massar-Spindel inequality. Indeed, these results represent an important tool in our search for tighter bounds (see numerical evidence exhibited in Fig. 1), whose intermediate uncertainty principles will constitute a hierarchical relation between the Massar-Spindel inequality and the tightest bound.

3.1 Quantum-algebraic framework

Let denote a pair of Hermitian operators defined in a -dimensional state vectors space which obey the RS uncertainty principle [20]

(5)

where represents the covariance function, and corresponds to the commutator (anticommutator) between and .2 Next, since and are generally non-Hermitian operators, let us consider the cartesian decomposition of an arbitrary non-Hermitian operator into its ‘real’ and ‘imaginary’ parts as follows [28]:

Note that and represent two Hermitian operators that comply with the RS uncertainty principle and allow to introduce, in particular, the cosine and sine operators through the Schwinger unitary operators.

{pf*}

Definition. Let denote four Hermitian operators written in terms of simple combinations associated with and , i.e., , , , and . The commutation relations involving these operators exhibit a direct connection with certain anticommutation relations:

where the parameter was previously defined in Eq. (1). These results lead us, in principle, to conclude that partial information on two particular elements of the set is not complete, since the complementary elements are also necessary to fully characterize the commutator . In this sense, let us now consider the four RS uncertainty principles below:

(6)
(7)
(8)
(9)

In addition, the extra result resembles a well-known mathematical property associated with the trigonometric functions cosine and sine. For this reason, these Hermitian operators will be henceforth termed ‘cosine’ and ‘sine’ operators, whose respective variances can be shown to obey the mathematical identity

To make complete this definition, it is interesting to observe that certain combinations of and also yield the additional result

which proves itself useful in our subsequent calculations.

Next, by means of mathematical remarks, we establish an important set of other useful results for the sine and cosine operators, whose relevance is intrinsically connected with the hierarchy relations involving the uncertainty principles related to and which generalize the Massar-Spindel inequality. It is worth mentioning that some proofs demand a logical sequence of algebraic manipulations to be detailed in the text.

{pf*}

Remark 1. Although the sine and cosine operators are genuinely defined for unitary operators, we shall employ, in due time, the same definition for a general operator as well. In this general case, it can be easily demonstrated that the following properties hold: , , and . Note that for a normal operator (which satisfies ), the covariance function between and matches the variance of , namely, .

{pf*}

Remark 2. Let us initially consider the sum of all aforementioned RS uncertainty principles for the sine and cosine operators, as well as the connection between the commutation and anticommuation relations of such Hermitian operators. Adequate algebraic manipulations allow, in principle, to obtain an inequality for the product with remarkable mathematical features, i.e.,

(10)

where

Note that and show a nontrivial dependence on the unitary operators and , which will be properly discussed in the subsequent remarks. Moreover, if compared with Massar-Spindel inequality (1), such a result yields subtle additional corrections that will depend explicitly on the dimension of the state space.

For completeness sake, let us now establish a first numerical evaluation on the functions and . Figure 2 exhibits the plots of versus for (a) and (b) with the same number of normalized random states used in the previous figure. Since the solid line represents in both the situations, this preliminary numerical search demonstrates a greater contribution coming from than from for most states of low dimension . This fact appeals for a detailed formal investigation on the origins of such contributions, in which each term of and would be examined separately.

Figure 2: Plots of versus with states of randomly generated for two different values of dimension , namely, (a) and (b) . The solid line corresponds to the case in both the pictures, where the limit situation describes the localized bases. Our numerical computations demonstrate that, in principle, the contributions due to are more significant than those coming from for a large number of random states.
{pf*}

Remark 3. Important mathematical properties of are easily attained examining each of its terms separately: for instance, to calculate the modulus squared of , we initially expand the cosine operators and in terms of the Schwinger unitary operators and ; the next step consists in decomposing the resulting terms into their real and imaginary Hermitian parts, whose final expression assumes the form

Once the remaining terms are obtained in the same way, some algebraic simplification gives rise to the following expression for :

However, this result does not represent a convenient form of since the proper summation of with (in this situation, one considers all possible matches of ) permits us to derive a simplified expression for such a function that includes the variances and . Indeed, since and , it turns immediate to prove that

(11)

which is invariant under the transformations and for .

{pf*}

Remark 4. Note that Eq. (11) represents a lower bound for any element of3

this fact being discussed by Massar and Spindel [14] through different mathematical arguments. In fact, the authors demonstrated that for a given choice of phases of the Schwinger unitary operators, the restrictions select a set of finite quantum states for which the inequality

can be formally verified and also numerically tested. Despite the correlations between and do not appear in such an expression, the comparison with is unavoidable in this case, since correlations represent an important quantum effect that deserve our attention. The saturation is reached in both the situations for localized bases.

{pf*}

Remark 5. Let us now decompose into three terms , and , whose different contributions will be formally calculated with the help of the mathematical procedure sketched in the previous remarks. We initially consider the term

responsible for contributions associated with squared mean values of all anticommutation relations involved in through the covariance functions. So, let admit the form

in complete analogy with . Writing the remaining mean values in an analogous way, subsequent computations allow to prove that is connected with through the relation for fixed.

Following, in what concerns the second term

it can be expressed by means of the convenient form

which represents an important formal result in our calculations.

Finally, let us briefly mention that coincides with since it is equivalent to the product ; consequently, the function

(12)

can be immediately obtained. Note that the nontrivial dependence of Eq. (10) on the Schwinger unitary operators is completely justified in these remarks.

This set of mathematical remarks establishes a first solid algebraic framework for the unitary operators and , whose intrinsic virtues lead us to formulate a theorem which generalizes the Massar-Spindel inequality (1) in order to include the quantum correlation effects between the aforementioned operators. In fact, this theorem consists of an initial compilation of efforts in our future search for the tightest bound, where the correlation function has occupied an important place in the investigative process.

{pf*} Theorem. Let and be two unitary operators defined in a -dimensional state vector space which satisfy the commutation relations and for . Moreover, let and represent the respective variances such that . The inequality (13) yields a new bound for , where the quantum correlation effects related to the unitary operators are taken into account. Note that and were precisely defined and formally studied in the previous mathematical remarks, the saturation being attained for localized bases.

Next, let us determine some additional results focused on the Hermitian operators in order to yield a set of specific intermediate inequalities whose hierarchical relations correspond to a solid bridge towards the tightest bound.

3.2 Hierarchical relations

We start this subsection stating a first important result for the sine and cosine operators defined, in turn, in terms of the non-unitary counterparts and . This particular result shows how the mean values of their commutation and anticommutation relations are linked with determined correlation functions, namely,

Following, let us complete this set of results with relations that connect all the covariance functions previously defined with the original framework: