IEEE International Symposium on Information Theory (ISIT)

Coherent-state representations for the functionals of Wiener process and Poisson process

Hisanao Ogura

Published Dec 17, 2002• 0 citations

Abstract

Coherent-state representations by analogy with that of quantum optics are introduced in the functional analysis of the Wiener and Poisson processes, and their relations to the Wiener-Ito expansions are discussed.>

DOI: 10.1109/isit.1994.395056

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Coherent‐state representations for functionals of the wiener process and the poisson process

Hisanao Ogura

Electronics and Communications in Japan (Part III Fundamental Electronic Science)Jan 1, 19940 citations

Wiener and Poisson process regularization for coherent-state path integrals

Robert Alicki, John R. Klauder

Journal of Mathematical PhysicsSep 1, 19934 citations

The poisson process in quantum stochastic calculus

Shayanthan Pathmanathan

Jan 1, 20021 citation

Fock Space and the Poisson Process

Sureka Pathmanathan, G. F. Vincent-Smith

Proceedings of the London Mathematical SocietyNov 1, 20040 citations

Coherent states of the electromagnetic field

L. Mandeļ, E. Wolf

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Coherent-State Representations for the Photon Density Operator

Kevin E. Cahill

Physical ReviewJun 21, 196586 citations

IEEE International Symposium on Information Theory (ISIT)

Coherent-state representations for the functionals of Wiener process and Poisson process

Hisanao Ogura

Published Dec 17, 2002• 0 citations

Publisher

Abstract

DOI: 10.1109/isit.1994.395056

Related papers

Coherent‐state representations for functionals of the wiener process and the poisson process

Hisanao Ogura

Abstract By an analogy between the theories of the Wiener functional and photon quantum field, such as the Wiener‐Itô expansion and the Fock‐space representation, a “coherent‐state” representation is introduced as a functional integral of nonorthogonal, overcomplete basis, corresponding to the coherent‐state representation of a photon field. A discrete‐state version is given to the “coherent‐state” representation in a form convenient for practical application. Analogously, a “coherent‐state” representation is established for functional of the Poisson process, where a remarkable analogy exists between the two. The Wiener‐Itô representation is derived from the coherent‐state representation by interpreting the coherent state as a generating functional of the orthogonal functionals.Abstract By an analogy between the theories of the Wiener functional and photon quantum field, such as the Wiener‐Itô expansion and the Fock‐space representation, a “coherent‐state” representation is introduced as a functional integral of nonorthogonal, overcomplete basis, corresponding to the coherent‐state representation of a photon field. A discrete‐state version is given to the “coherent‐state” representation in a form convenient for practical application. Analogously, a “coherent‐state” representation is established for functional of the Poisson process, where a remarkable analogy exists between the two. The Wiener‐Itô representation is derived from the coherent‐state representation by interpreting the coherent state as a generating functional of the orthogonal functionals.

Electronics and Communications in Japan (Part III Fundamental Electronic Science)Jan 1, 19940 citations

Wiener and Poisson process regularization for coherent-state path integrals

Robert Alicki, John R. Klauder

By introducing a suitable continuous-time regularization into a formal phase-space path integral it follows that the propagator is given by the limit of well-defined functional integrals involving standard stochastic processes and their associated probability measures. Such regularizations require pinning of both coordinate and momenta variables, and automatically lead to coherent-state representations. It is found that each standard independent increment process, involving a superposition of a Wiener and a Poisson process, is associated with a specific, generally non-Gaussian, fiducial vector with which the coherent states are defined.By introducing a suitable continuous-time regularization into a formal phase-space path integral it follows that the propagator is given by the limit of well-defined functional integrals involving standard stochastic processes and their associated probability measures. Such regularizations require pinning of both coordinate and momenta variables, and automatically lead to coherent-state representations. It is found that each standard independent increment process, involving a superposition of a Wiener and a Poisson process, is associated with a specific, generally non-Gaussian, fiducial vector with which the coherent states are defined.

Journal of Mathematical PhysicsSep 1, 19934 citations

The poisson process in quantum stochastic calculus

Shayanthan Pathmanathan

Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$ , the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$ . A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ , using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$ , we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$ , where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$ -field with respect to $\mathbb{P}$ , of a $C^*$ -algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$ . The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$ . This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$ , we show that if $M_t = \int_0^t F_s dX_s$ , then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$ , $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$ , is also proved.Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$, the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$. A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$, using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$, we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$, where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$-field with respect to $\mathbb{P}$, of a $C^*$-algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$. The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, we show that if $M_t = \int_0^t F_s dX_s$, then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$, $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$, is also proved.

Jan 1, 20021 citation

Fock Space and the Poisson Process

Sureka Pathmanathan, G. F. Vincent-Smith

Using the Wiener–Poisson isomorphism, we show that if (Ft)0 ⩽ t ⩽ 1 is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process (Xt)0 ⩽ t ⩽ 1, and if M ^ t is the operator corresponding to multiplication by M t = ∫ 0 t F s d X s , then for any regular self-adjoint quantum semimartingale J = ( J t ) 0 ⩽ t ⩽ 1 , the essentially self-adjoint quantum semimartingale ( M ^ t + J t ) 0 ⩽ t ⩽ 1 satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process to a measure space (M, M, μ) as an isometry I: L2 (M, M, μ) → L2(Ω, F, P) and give a new construction of the generalised Wiener–Poisson isomorphism WI: F+ (L2(M)) → L2 (Ω, F, P) using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures. 2000 Mathematics Subject Classification 60G20, 60G35, 46L53, 81S25.Using the Wiener–Poisson isomorphism, we show that if (Ft)0 ⩽ t ⩽ 1 is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process (Xt)0 ⩽ t ⩽ 1, and if M ^ t is the operator corresponding to multiplication by M t = ∫ 0 t F s d X s , then for any regular self-adjoint quantum semimartingale J = ( J t ) 0 ⩽ t ⩽ 1 , the essentially self-adjoint quantum semimartingale ( M ^ t + J t ) 0 ⩽ t ⩽ 1 satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process to a measure space (M, M, μ) as an isometry I: L2 (M, M, μ) → L2(Ω, F, P) and give a new construction of the generalised Wiener–Poisson isomorphism WI: F+ (L2(M)) → L2 (Ω, F, P) using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures. 2000 Mathematics Subject Classification 60G20, 60G35, 46L53, 81S25.

Proceedings of the London Mathematical SocietyNov 1, 20040 citations

Coherent states of the electromagnetic field

L. Mandeļ, E. Wolf

Introduction In our discussion of the classical electromagnetic field we found it convenient to describe the field by a complex amplitude, both in the frequency and in the time domains. The complex representation is convenient partly because it contains information about the magnitude and about the phase of the electromagnetic disturbance, and partly because of its analytic properties. These features turn out to be particularly useful for the description of the optical coherence properties of the field. We shall see that an analogous quantum state of the field exists, leading to an interesting representation that is also particularly useful for the treatment of optical coherence. This coherent-state representation leads to a close correspondence between the quantum and classical correlation functions. The coherent states of the field come as close as possible to being classical states of definite complex amplitude. We shall find that coherent states turn out to be particularly appropriate for the description of the electromagnetic fields generated by coherent sources, like lasers and parametric oscillators; indeed it turns out that the field produced by any deterministic current source is in a coherent state. In the following sections we shall examine some of the properties of coherent states, and then go on to discuss representations based on these states. Coherent states were first discovered in connection with the quantum harmonic oscillator by Schrödinger (1926), who referred to them as states of minimum uncertainty product.Introduction In our discussion of the classical electromagnetic field we found it convenient to describe the field by a complex amplitude, both in the frequency and in the time domains. The complex representation is convenient partly because it contains information about the magnitude and about the phase of the electromagnetic disturbance, and partly because of its analytic properties. These features turn out to be particularly useful for the description of the optical coherence properties of the field. We shall see that an analogous quantum state of the field exists, leading to an interesting representation that is also particularly useful for the treatment of optical coherence. This coherent-state representation leads to a close correspondence between the quantum and classical correlation functions. The coherent states of the field come as close as possible to being classical states of definite complex amplitude. We shall find that coherent states turn out to be particularly appropriate for the description of the electromagnetic fields generated by coherent sources, like lasers and parametric oscillators; indeed it turns out that the field produced by any deterministic current source is in a coherent state. In the following sections we shall examine some of the properties of coherent states, and then go on to discuss representations based on these states. Coherent states were first discovered in connection with the quantum harmonic oscillator by Schrödinger (1926), who referred to them as states of minimum uncertainty product.

Cambridge University Press eBooksSep 29, 19956 citations

Coherent-State Representations for the Photon Density Operator

Kevin E. Cahill

The "diagonal" $P$ representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the $P$ representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the $P$ representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the $P$ representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.The "diagonal" $P$ representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the $P$ representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the $P$ representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the $P$ representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.

Physical ReviewJun 21, 196586 citations

Given a compensated Poisson process

(X_t)_{t \geq 0}

based on

(\Omega, \mathcal{F}, \mathbb{P})

, the Wiener-Poisson isomorphism

\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})

is constructed. We restrict the isomorphism to

\mathfrak{F}_+(L^2 [0,1])

and prove some novel properties of the Poisson exponentials

\mathcal{E}(f) := \mathcal{W}(e(f))

. A new proof of the result

\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}

is also given. The analogous results for

\mathfrak{F}_+(L^2 (\mathbb{R}_+))

are briefly mentioned. The concept of a compensated Poisson process over

\mathbb{R}_+

is generalised to any measure space

(M, \mathcal{M}, \mu)

as an isometry

I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})

satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism,

\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})

, using Charlier polynomials. Two alternative constructions of

\mathcal{W}_I

are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space

(M, \mathcal{M}, \mu)

, we construct a canonical generalised Poisson process

I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})

, where

\Delta

is the maximal ideal space, with

\mathcal{B}

the completion of its Borel

\sigma

-field with respect to

\mathbb{P}

, of a

C^*

-algebra

\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))

. The Gelfand transform

\mathcal{A} \to \mathfrak{B}(L^2(\Delta))

is unitarily implemented by the Wiener-Poisson isomorphism

\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)

. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for

(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}

is given. If

(F_{\ \! \! t})_{0 \leq t \leq 1}

is a real, bounded, predictable process with respect to a compensated Poisson process

(X_t)_{0 \leq t \leq 1}

, we show that if

M_t = \int_0^t F_s dX_s

, then on

\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}

\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),

and that

(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}

is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if

(J_t)_{0 \leq t \leq 1}

is a regular self-adjoint quantum semimartingale, then

(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}

is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core

\mathsf{E}_{\mathrm{lb}}

, is also proved.

The "diagonal"

P

representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the

P

representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the

P

representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the

P

representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.