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Journal of Physics A Mathematical and Theoretical

A measure-free approach to coherent states

A. Horzela, Franciszek Hugon Szafraniec

Published May 30, 2012• 16 citations

Abstract

In the process of generalizing coherent states, the situation when the measure—which is customarily incorporated in their definition—is indeterminate becomes unavoidable. A more dramatic situation may occur if there is no measure which makes the reproducing kernel Hilbert space, involved in the construction of coherent states, isometrically included in an space. Therefore, it appears that there is a need to redefine coherent states making the definition measure-free. Starting out with the reproducing kernel property, we ensure the basic feature of coherent states—resolution of the identity—to be maintained, cf (15) and remark 2. The only investment in the whole undertaking is a sequence (Φn)dn = 0 satisfying (4); the rest, including the aforesaid resolution of the identity, is a consequence of our choice. The approach is supported by examples which make the circumstances clear under which the sequence (Φn)dn = 0 appears; moment problems or rather orthogonal polynomials are one of them.

DOI: 10.1088/1751-8113/45/24/244018

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In the process of generalizing coherent states, the situation when the measure—which is customarily incorporated in their definition—is indeterminate becomes unavoidable. A more dramatic situation may occur if there is no measure which makes the reproducing kernel Hilbert space, involved in the construction of coherent states, isometrically included in an space. Therefore, it appears that there is a need to redefine coherent states making the definition measure-free. Starting out with the reproducing kernel property, we ensure the basic feature of coherent states—resolution of the identity—to be maintained, cf (15) and remark 2. The only investment in the whole undertaking is a sequence (Φn)dn = 0 satisfying (4); the rest, including the aforesaid resolution of the identity, is a consequence of our choice. The approach is supported by examples which make the circumstances clear under which the sequence (Φn)dn = 0 appears; moment problems or rather orthogonal polynomials are one of them.

We investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer’s G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal P-representation of the density operator.

The support vector machine (SVM) is a popular machine learning classification method which produces a nonlinear decision boundary in a feature space by constructing linear boundaries in a transformed Hilbert space. It is well known that these algorithms when executed on a classical computer do not scale well with the size of the feature space both in terms of data points and dimensionality. One of the most significant limitations of classical algorithms using non-linear kernels is that the kernel function has to be evaluated for all pairs of input feature vectors which themselves may be of substantially high dimension. This can lead to computationally excessive times during training and during the prediction process for a new data point. Here, we propose using both canonical and generalized coherent states to calculate specific nonlinear kernel functions. The key link will be the reproducing kernel Hilbert space (RKHS) property for SVMs that naturally arise from canonical and generalized coherent states. Specifically, we discuss the evaluation of radial kernels through a positive operator valued measure (POVM) on a quantum optical system based on canonical coherent states. A similar procedure may also lead to calculations of kernels not usually used in classical algorithms such as those arising from generalized coherent states.

In continuation of our previous works J. Phys. A: Math. Gen. 35, 9355-9365 (2002), J. Phys. A: Math. Gen. 38, 7851 (2005) and Eur. Phys. J. D 72, 172 (2018), we investigate a class of generalized coherent states for associated Jacobi polynomials and hypergeometric functions, satisfying the resolution of the identity with respect to a weight function expressed in terms of Meijer's G-function. We extend the state Hilbert space of the constructed states and discuss the property of the reproducing kernel and its analytical expansion. Further, we provide the expectation values of observables relevant to this quantum model. We also perform the quantization of the complex plane, compute and analyze the probability density and the temporal stability in these states. Using the completeness relation provided by the coherent states, we achieve the thermodynamic analysis in the diagonal

P

-representation of the density operator.