1 Introduction
The information geometry approach fisher ; frieden ; jaynes ; rao ; rao2 ; weinhold ; ruppeiner ; amari ; ingarden ; janyszek
studies the differential geometric structure of statistical models. A statistical model consists of a family of probability distribution functions (PDFs) parameterized by continuous variables. In order to endow these models with a geometric structure, it is necessary to define the Fisher–Rao metric
rao, which in turn, is linked with the concepts of entropy and Fisher information. Once we have a statistical manifold, the main goal of the information geometry approach is to characterize the family of PDFs using geometric quantities, like the geodesic equations, the Riemann curvature tensor, the Ricci tensor or the scalar curvature.
The geometrization of thermodynamics and statistical mechanics are some of the most important achievements in this field, expressed mainly by the foundational works of Gibbs gibbs , Hermann hermann , Weinhold weinhold , Mrugała mrugala , Ruppeiner ruppeiner2 , and Caratheódory carateodory . These investigations lead to the Weinhold and Ruppeiner geometries, where a a Riemannan metric tensor in the space of thermodynamic parameters is provided and a notion of distance between macroscopic states is obtained. However, the utility of information geometry is not only limited to those areas. For instance, it has been applied in quantum mechanics leading to a quantum generalization of the Fisher–Rao metric bures , and recently, also in nuclear plasmas geert ; geert2 . Moreover, generalized extensions of the information geometry approach to the non-extensive formulation of statistical mechanics tsallis have been also considered abe ; naudts ; portesi ; portesi2 . Applications of information geometry to chaos can be also performed by considering complexity on curved manifolds cafaro ; cafaro2 ; cafaro3 ; cafaro4 ; cafaro5 , leading to a criterion for characterizing global chaos on statistical manifolds: the more negative is the curvature, the more chaotic is the dynamics; from which some consequences concerning dynamical systems have been explored nacho1 . More generally, the curvature has been proved to be a quantifier which measures interactions in thermodynamical systems, where the positive or negative sign corresponds to repulsive or attractive correlations, respectively ruppeiner .
Motivated by previous works of some of us nacho1 ; nacho2 , we propose a generalization of the Gaussian model which we call the Hermite–Gaussian model, and we show its relation with the one-dimensional quantum harmonic oscillator. The application of information geometry techniques to the study of quantum harmonic oscillators can be useful in many applications. For example, the translational modes in a quantum ion trap are quantum harmonic oscillators that need to be characterized and controlled in order to avoid coherence losses. Our contribution may serve as a tool for the characterization of unknown parameters in those scenarios. Furthermore, the present work can be also considered as a continuation of the recent Cafaro’s program cafaro ; cafaro2 ; cafaro3 ; cafaro4 ; cafaro5 of global characterization of dynamics on curved statistical manifolds generated by Gaussian models.
The paper is organized as follows. In Section II, we review the main features of the information geometry approach. In Section III, we present the Hermite–Gaussian model, we obtain the Fisher–Rao metric and the scalar curvature for this model, and we show its relation with the one-dimensional quantum harmonic oscillator. Moreover, we use this model to characterize some families of states of the quantum harmonic oscillator. We focus in three different families of states: Hamiltonian eigenstates, mixtures of eigenstates and superposition of eigenstates. Finally, in Section IV, we present the conclusions and some future research directions.
2 Information geometry
The information geometry approach studies the differential geometric structure possessed by families of probability distribution functions (PDFs). In this section we introduce the general features of this approach, which will be used in the next sections. The presentation is based on the book of S. Amari and H. Nagaoka amari .
2.1 Statistical models
Information geometry applies techniques of differential geometry to study properties of families of probability distribution functions parameterized by continuous variables. These families are called statistical models. More specifically, a statistical model is defined as follows. We consider the probability distribution functions defined on , i.e., the functions which satisfy
(1) |
When is a discrete set the integral must be replaced by a sum. A statistical model is a family of probability distribution function on , whose elements can be parameterized by appealing to a set of real variables, i.e.,
(2) |
with an injective mapping. The dimension of the statistical model is given by the number of real variables used to parameterized the family .
When statistical models are applied to physical systems, the interpretation of and is the following. represents the microscopic variables of the system, which are typically difficult to determine, for instance the positions of the particles of a gas. represents the macroscopic variables of the system, which can be easily measured. The set is called the microspace and the variables are the microvariables. The set is called the macrospace and the variables are the macrovariables.
Given a physical system, we can define many statistical models. First, we have to choose the microvariables to be considered, and then we have to choose the macrovariables which parametrized the PDFs defined on the microspace. All statistical models are equally valid, but no all of them are equally useful. In general, the choice of the statistical model would be based in pragmatic considerations.
2.2 Geometric structure of statistical models
In order to apply differential geometry to statistical models, it is necessary to endow them with a metric structure. This is accomplished by means of the Fisher–Rao metric
(3) |
The metric tensor gives to the macrospace a geometrical structure. Therefore, the family results to be a statistical manifold, i.e., a differential manifold whose elements are probability distribution functions.
From the Fisher–Rao metric, we can obtain the line element between two nearby PDFs with parameters and
Using the metric tensor we can obtain the geodesic equations for the macrovariables along with relevant geometrical quantities, like the Riemann curvature tensor, the Ricci tensor or the scalar curvature.
Geodesic equations: | (4) | ||||
Christoffel symbols: | (5) | ||||
Riemman curvature tensor: | (6) | ||||
Ricci tensor: | (7) | ||||
Scalar curvature: | (8) |
The comma in the sub-indexes denotes the partial derivative operation (of first and second orders), is the inverse of , and is a parameter that characterizes the geodesic curves.
Moreover, the Fisher–Rao metric gives information about the estimators of the macrovariables. Given an unbiased estimator
of the parameters , i.e., , the Cramér–Rao bound gives a lower bound for the covariance matrix of T,(9) |
where the matrix inequality means that the matrix
is positive semi-definite. In particular, this relation gives bounds for the variance of the unbiased estimators
,(10) |
This bound is important when looking for optimal estimators. In what follows, we present an important statistical model used in the geometry information approach, the Gaussian model.
2.3 Gaussian model
One of the most relevant statistical models used in the geometry information approach is the Gaussian model. The reason for that is the wide versatility of this model for describing multiple phenomena. The Gaussian model is obtained by choosing the family
as the set of multivariate Gaussian distributions. For instance, if
are the microvariables and there is no correlations between them, then are the set of macrovariables, where and correspond to the mean value and the variance of the microvariable .If we consider only one microvariable , the Gaussian model is given by the following probability distribution functions
(11) |
which are parameterized by the mean value
and the standard deviation
. From equations (3) to (8), one can obtain the Fisher–Rao metric and the scalar curvature of this model,(12) |
(13) |
The Gaussian model is a curved manifold with constant curvature. In some contexts, the negative value of the curvature is interpreted as modeling attractive interactions, like in an ideal gas ruppeiner .
In the next section, we are going to introduce a generalization of the Gaussian model, based on the eigenstates of the harmonic oscillator Hamiltonian.
3 Hermite–Gaussian model
We propose a generalization of the Gaussian model, called the Hermite-Gaussian model, which is motivated by the quantum harmonic oscillator. Given the microspace and the macrospace , we define for each the Hermite–Gaussian model as the family of probability distribution functions given by
(14) |
In particular, if , the Gaussian model is recovered. The Fisher–Rao metric of the Hermite–Gaussian model takes the form
(15) |
and its explicit formula is the following (see B)
(16) |
Taking into account that the scalar curvature is given by
(17) |
we can express the Fisher–Rao metric in terms of
(18) |
From the Fisher–Rao metric, we can compute the Cramér–Rao bound for unbiased estimators of the parameters and . The lower covariance matrix of any pair of unbiased estimators of the parameters , is given by
(19) |
For the covariance of the estimators we obtain
(20) | ||||
(21) |
In what follows, we show the connection between the Hermite–Gaussian model and the quantum harmonic oscillator. We use these model to characterize the PDFs given by quantum states of the harmonic oscillator. We focus on Hamiltonian eigenstates, mixtures of eigenstates and superposition of eigenstates.
3.1 Hamiltonian Eigenstates
The relation between the Hermite–Gaussian model and the quantum harmonic oscillator is straightforward. We start considering the Hamiltonian of the harmonic oscillator
(22) |
where is the mass, is the frequency, is the equilibrium position of the oscillator, and and are the position and momentum operators. Its eigenstates satisfy the time-independent Schrödinger equation, , with . Moreover, the eigenstates satisfy orthogonality and completeness relations
where is the identity operator.
The wave function of the eigenstate , in the coordinate representation, is given by
(23) |
with , , and . Then, the PDF of the position operator for the eigenstate is .
Therefore, if we consider the eigenstate of an harmonic oscillator with parameters and , the PDF of the position operator is equal to the probability distribution function of the Hermite–Gaussian model, given in equation (14). Moreover, the Fisher–Rao metric and the scalar curvature associated with the probability distribution function are given in equations (16) and (17), respectively.
It is important to remark that the Fisher–Rao metric is diagonal, and the scalar curvature is always negative and decreases with the quantum number , tending to zero in the limit of high quantum numbers. Moreover, from the Cramér–Rao bound we obtain that the minimal variance of the estimators of the parameter grows with and decreases with the eigenstate number, and the minimal variance of estimators of the parameter also grows with but decreases with the square of the eigenstate number. Equivalently, the minimal variance of the estimators of is proportional to the scalar curvature.
3.2 General states
We are going to consider the PDF of the position operator obtained from general states of the harmonic oscillator. Let us consider the basis of the Hamiltonian eigenstates , and a state of the form
(24) |
The probability distribution function of the position operator is given by
(25) |
where is the wave function of the eigenstate , given in equation (23).
For practical reasons, we define the function ,
(26) |
Then, we have , with .
In order to calculate the Fisher–Rao metric associated with , we need the partial derivatives and , which are given by
(27) | ||||
(28) |
with .
Replacing the PDF (25) and the partial derivatives (27) and (28) in the integral of equation (3), and making the change of variable , we obtain the Fisher–Rao metric
where in the first equation we used that , and in the last equation we used that and .
Therefore, we can write the Fisher–Rao metric as follows:
(29) |
where , and are independent of and , and they are given by
From the Fisher–Rao metric and using equations (5) to (8), we can obtain the scalar curvature
(30) |
The Cramér–Rao bound gives the lower covariance matrix of any pair of unbiased estimators of the parameters ,
(31) |
Finally, we can express the variance of in terms of the scalar curvature,
(32) |
Corollary 1: The Fisher–Rao metric for a general state of the harmonic oscillator is independent of the parameter and it only depends on the parameter by a general factor .
Corollary 2: The scalar curvature for a general state of the harmonic oscillator is independent of the parameters and , and it only involves integrals of the dimensionless function and its derivative .
Corollary 3: The lower variance of unbiased estimators of the parameter is proportional to .
3.3 Mixtures of Hamiltonian eigenstates
We consider quantum states which are mixtures of the Hamiltonian eigenstates. Mixtures of eigenstates are particular cases of the states given in equation (24), with i.e., . Therefore, the probability distribution function of the position operator, the Fisher–Rao metric and the scalar curvature can be obtained from the general expressions (25), (29) and (30), considering .
In this case, the PDF of the position operator takes the form
The diagonal elements of the Fisher–Rao metric are zero, and the elements are given in equation (29),
(33) |
with . Since Hermite polynomials are even or odd functions of the variable , are even functions. Then, is also an even function and its derivative is an odd function. Finally, the integrand of equation (33) is an odd function of . Therefore, ,
Finally, the scalar curvature is obtained from equation (30),
As an example, we consider the mixture state . The Fisher–Rao metric is given by
(34) |
where is the Gauss error function, with . The scalar curvature is approximately .
3.4 Superposition of Hamiltonian eigenstates
We consider quantum states which are superpositions of Hamiltonian eigenstates. Superpositions of eigenstates of the form are particular cases of states given in equation (24), with , i.e., . Therefore, the PDF of the position operator, the Fisher–Rao metric and the scalar curvature can be obtained from the general expressions (25), (29) and (30), considering
3.4.1 Even or odd superpositions
In this section we focus on a family of superpositions that yield analytic expressions. If we consider a superposition of eigenstates with only even or odd eigenstates, i.e.,
we obtain that the diagonal elements of the Fisher–Rao metric are zero. The proof is similar to the case of mixtures of eigenstates. The diagonal elements are given in equation (29),
(35) |
with
If the indexes can only take even or odd values, then the product is always an even function of the variable . Then, is also an even function and its derivative is an odd function. Finally, the integrand of equation (35) is an odd function of , and the result of the integral is zero.
Again, we obtain that the scalar curvature, given in equation (30), is
3.4.2 Real or imaginary superpositions
Analytic expressions can also be obtained for superpositions of eigenstates that involve only real coefficients, i.e., . In order to compute the Fisher–Rao metric, we need the fuction , given in (26), and it derivative ,
(36) |
where in the last equation we have used the recurrence relations of the Hermite polynomials (38). Replacing expressions (36) in the Fisher–Rao metric (29), and taking into account relations (37) and (38), we obtain
If we consider a superposition of eigenstates with only imaginary coefficients, we obtain a similar result, but replacing the coefficients by its imaginary part, i.e., by Im.
4 Conclusions
In this work we have proposed a generalization of the Gaussian model -namely, the Hermite-Gaussian model- and we have studied many of its properties from the point of view of the information geometry approach. We have shown its relation with the probabilities associated to the one-dimensional quantum harmonic oscillator model and analytic expressions for some particular classes of states were provided. Specifically, we found that for finite mixtures of eigenstates and finite superpositions of (even or odd) eigenstates the Fisher metric is always diagonal. Real and imaginary superpositions of eigenstates do not imply a diagonal Fisher metric and the matrix elements are given in terms of a series sum. An analytic expression for the scalar curvature was only obtained when the Fisher metric is diagonal, being negative and inversely proportional to the element.
Due to the relevance of this model in many applications, our contribution may serve to extend the scope of information geometry techniques into a wider class of physical problems. For example, since in irreversible processes the final (reduced) state of a system (after interacting with the environment) is typically a mixture of their eigenstates, which for the case of Hermite-Gaussian models has a diagonal Fisher metric, the results obtained could be used for determining if the process involved is irreversible or not by simple inspection of the diagonal elements of the Fisher metric. In this context and considering that the states of the system can be expressed by means of Hermite-Gaussian models, if the Fisher metric of the final reduced state results non-diagonal then by Sections 3.3 and 3.4 it is not mixture of harmonic oscillator eigenstates, and thus the process cannot be irreversible.
Acknowledgments
This research was founded by the CONICET, CAPES / INCT-SC (at Universidade Federal da Bahia, Brazil), the National University of La Plata and the University of Buenos Aires.
References
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Appendix A Hermite polynomials
The Hermite polynomials are given by the expression
and their orthogonality relation is
(37) |
An important feature of these polynomials is that if is even, is an even function; and if is odd, is an odd function.
Some relevant recurrence relations are the following:
(38) |
Appendix B Hermite–Gaussian model
For parameters and , the probability distribution of the –Hermite–Gaussian model is
(39) |
In order to obtain the elements of the metric tensor, we need to calculate the partial derivatives of the probability distribution. It easy to show that
(40) | ||||
(41) |
with
(42) |
where we have used the recurrence relations (38). It should be noted that is even an function of , thus is an odd function of .
Also, we will need to express in terms of Hermite polynomials,
where we have used expression (42) and the recurrence relations (38).
b.1 Off–diagonal elements
Since the metric tensor is symmetric, it is enough to calculate the element , given by
(43) |
Replacing expressions (40) and (41) in (43) and doing some easy manipulations, we obtain
(44) |
where in the last equation we changed from variable to the variable . Since and are even and odd functions of , respectively, then the integrand of (44) is an odd function. Therefore, .
b.2 Element
The element is given by
(45) |
Replacing expression (40) in (45), we obtain
(46) |
In the last step, we have changed from variable to the variable . Then, if we replace expressions (39) and (42) in (46) and we rearrange the expression, we obtain
In the last step we have used the orthogonality relation (37). Finally, we obtain .
b.3 Element
The element is given by
(47) |
Replacing expression (41) in (47), we obtain
(48) |
In the last equation we have changed from variable to the variable . Then, if we replace expressions (39) and (B) in (48) and we rearrange the expression, we obtain
In the last step, we have used the orthogonality relation (37). Finally, we obtain
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