Quantum Stochastic Calculus with Infinite Degrees of Freedom and Its Applications.

Date of Submission

February 2011

Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Sinha, Kalyan Bidhan (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

Based on the canonical commutation relations between the creation, con- servation ( number ) and annihilation operators of a free field on a boson Fock space, Hudson and Parthasarathy (21) have developed a quantum stochastic calculus, of which a detailed exposition may be found in Meyer [30] and Parthasarathy [38). In order to handle the problem of dilating uniformly continuous quantum dynamical semigroups in the algebra of all bouded op- erators in a Hilbert space, Hudson and Parthasarathy (22] had already noted the importance of extending their calculus when an infinite number of in- dependent creation and annihilation processes are used as integrators. Also, any attempt to extend the results of Meyer (29] and Parthasarathy and Sinha (37) on the realisation of classical Markov processes in the Accardi-Frigerio- Lewiss framework of quantum stochastic processes involves the investigation of quantum stochastic differential equations ( q.s.d.e 's ) with infinite degrees of freedom.The aim of the present thesis is to present a brief exposition of the Hudson- Parthasarathy calculus in Fock space when a possibly infinite number of basic integrators are involved and apply it to the study of the following two basic problems:(a) Under what conditions on Z = {zj, i,j e 3} the following quantum stochastic differential equationdV(t) = EV(t)Z}A{(+), V(0) = I (0.1) %3D kESwhere A(t), i,j e 3 = SU {0} are the basic integrators in the boson-Fock space I(L(R+, K)) with respect to an orthonormal basis {ei, i e S} forthe Hilbert space K and Z = (Zj, i,j e 3} is a family of densely defined operators in the initial Hilbert space Ho, admits a contractive, isometric, co- isometric or unitary operator valued adapted process V = {V(t), t 2 0} as a solution ?(b) Given a family e = {0}, i,j e 3} of structure maps on an initial +-algebra Ao C B(Ho) under what conditions, a quantum stochastic flow I = {I, t 2 0} in the sense of Evans and Hudson (12] exists and satisfies a q.s.d.e of the form:To( z) = 1, dI(2) =E 1(0}(z))dA{(t) ije3 (0.2)on Hoge(M) for all r € A, ?Here is a brief summary of our results:In Theorem 2.12 of Section 2 the first problem is given a complete solution in the form of necessary and sufficient conditions when the family Z = {Z}, i,j e 3} satisfies the inequalitiesfor all f € Ho,j e 3, ; being a positive constants. This generalises and sharpens the previously known results of Hudson- Parthasarathy [21], Mohari-Sinha [33] and Mohari-Parthasarathy [31]. In particular we prove that (0.1) admits a unique contractive solution if and only if Z e Zz where Z5 = (Z, ((Z) + (Z{) + E(z})*Z} ))ises < 0, for all S C 3 , <). kESThis class plays an important role in dealing with (0.1), in the spirit of semi- group theory developed as in Yosida [39], for unbounded dissipative coeffi- cients ( See Section 5 ). As in Mohari-Parthasarathy (32] the role of Journé's time reversal principle ( See Theorem 2.11 ) in the proof seems to be of special interest.When V = {V(t), t 2 0} is the unitary solution in the discussion above the quantum stochastic process j(x) = V (4)(z®)V(t)", t 20 satisfies (0.2) for a family of bounded structure maps L= {Cj, i,j e3} with B(Ho) as theinitial algebra. The expressions for the Lj are presented in order to motivate applications to classical Markov processes in Section 3.In Section 3, following Mohari-Sinha (33) we present a theory for quantum stochastic ( QS ) flows with countably infinite degrees of freedom for the noise when e = {6, i, j e 3} is a family of bounded srtructure maps obeying the regularity conditionsfor all z € Ao, f€ Ho where for each j e 3, I; is a countable index set and {Dj, i € I;} is a family of bounded operators in Ho. As in Parthasarathy and Sinha (37) we show that the family (J(z), I E Ao, t 2 0} is commutative whenever A, is abelian. This enables us to conclude that in the vacuum state of the Fock space, {J, t2 0} describes a classical Markov process with the bounded infinitesimal generator 66. Finally we apply this theory to show that continuous time Markov chains with countablely many state space can be understood as commutative QS fiows on the abelian algebra of functions on the state space. This extends the previous studies by Meyer [29] and Parthasarathy and Sinha (37).


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