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arXiv:0803.2269 · Journal of Physics A Mathematical and Theoretical

Coherent states and Bayesian duality

S. Twareque Ali, Jean‐Pierre Gazeau, Bárbara Heller

Published Aug 5, 2008• 23 citations
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Abstract

We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.

DOI
10.1088/1751-8113/41/36/365302

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We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.
Abstract. Starting with the canonical coherent states, we demonstrate that all the so-called nonlinear coherent states, used in the physical literature, as well as large classes of other generalized coherent states, can be obtained by changes of bases in the underlying Hilbert space. This observation leads to an interesting duality between pairs of generalized coherent states, bringing into play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in each dual pair of families of nonlinear coherent states, at least one family is related to a (generally) non-unitary projective representation of the Weyl-Heisenberg group, which can then be thought of as characterizing the dual pair. 1.
Starting with the canonical coherent states, we demonstrate that all the so-called nonlinear coherent states, used in the physical literature, as well as large classes of other generalized coherent states, can be obtained by changes of bases in the underlying Hilbert space. This observation leads to an interesting duality between pairs of generalized coherent states, bringing into play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in each dual pair of families of nonlinear coherent states, at least one family is related to a (generally) non-unitary projective representation of the Weyl–Heisenberg group, which can then be thought of as characterizing the dual pair.
In a Hilbert space ℋ, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in ℋ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ℋ, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.
Some people hate the very name of statistics, but I find them full of beauty and interest. Sir Francis Galton In quantum mechanics one often encounters sets of non-negative numbers that sum to unity, having a more or less direct interpretation as probabilities. This includes the squared moduli of the coefficients when a pure state is expanded in an orthonormal basis, the eigenvalues of density matrices, and more. Continuous distributions also play a role, even when the Hilbert space is finite dimensional. From a purely mathematical point of view a probability distribution is simply a measure on a sample space, constrained so that the total measure is one. Whatever the point of view one takes on this, the space of states will turn into a convex set when we allow probabilistic mixtures of its pure states. In classical mechanics the sample space is phase space, which is typically a continuous space. This leads to technical complications but the space of states in classical mechanics does share a major simplifying feature with the discrete case, namely that every state can be expressed as a mixture of pure states in a unique way. This was not so in the case of colour space, nor will it be true for the convex set of all states in quantum mechanics. Majorization and partial order Our first aim is to find ways of describing probability distributions; we want to be able to tell when a probability distribution is ‘more chaotic’ or ‘more uniform’ than another. One way of doing this is provided by the theory of majorization .
This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to an electron-phonon model in the condensed matter physics, obeying a f -deformed Heisenberg algebra.The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided.A Glauber-Sudarshan P-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented.Moreover, a NCS quantization of classical phase space observables is performed and illustrated in a concrete example of q-deformed coherent states.Finally, an exposition of quantum optical properties is given.
We propose a reformulation of quantum mechanics as a theory of unresolved uncertainty. This theory of potentiality is formulated in the language of complex-valued measure theory, regarded as a pre-probabilistic counterpart of ordinary probability theory. In this formulation, additivity, conditioning, independence, mixtures, transition kernels, and temporal divisibility retain natural linear forms at the potentiality level, while non-classical probability-level features such as interference arise from the nonlinear Born map. Measurement is described as Bayesian-type conditioning of potentialities on actualized information, and non-selective measurement as the replacement of coherent potentiality by statistical mixtures of conditional potentiality branches. Mixed states, decoherence, composite systems, entanglement, and Bell-type correlations are also given a unified potentiality-level interpretation. The density matrix is interpreted as a coherence kernel whose off-diagonal blocks encode retained phase relations. For pure bipartite states, potentiality independence is shown to be equivalent to factorization of the Born distribution in every pair of local contexts. The resulting formulation is empirically equivalent to standard quantum mechanics, but it makes explicit a pre-probabilistic description of physical reality that is usually implicit in the Hilbert-space formalism.