#### Date of Submission

6-30-1993

#### Date of Award

3-2-1994

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

#### Supervisor

Parthasarthy, K. R. (TSMU-Delhi; ISI)

#### Abstract (Summary of the Work)

In classical probability theory, based on Kolmogorov consistency theorem, one can associate a Markov process to any one parameter semigroup of stochastic matrices or transition probability operators. It is indeed the foundation for the theory of Markov processes. Here a quantum version of this theorem has been established. This effectively answers some of the questions raised by P. A. Meyer in his book (see page 220 of (Me).It is widely agreed upon that irreversible dynamics in the quantum setting is de- scribed by contractive semigroups of completely positive maps on C" algebras ([Kr). (AL]). In other words these semigroups, known as quantum dynamical semigroups, are non-commutative Markov kernels. We take the view point that a quantum Markov process consists of a filtration' and a time indexed family of representations of a C* algebra reflecting the Markov property with respect to a suitable conditional expectation. See (AFL), [Ku2), [Se), and (Sal] for other approaches.In the modest approach taken here conditioning means truncating operators to subspaces and so naturally enough filtrations are increasing families of subspaces (or projections). This leads us to the notion of weak Markov flows. This notion is quite powerful and encompasses atleast three kinds of dilations in its fold. Classical Markov processes, Sz. Nagy dilation of contraction semigroups on Hilbert spaces([SzF), [Da3), [EL]), and Evans-Hudson flows of Fock space stochastic calculus([P1). [Me]) are included here in a natural way. The main theorem in Section 4 shows that every quantum dynamical semigroup can be realized as expectation semigroup of a weak Markov flow. Moreover there is uniqueness upto unitary equivalence under a natural irreducibility coadition. From the operator algebraic point of view this theorem generalizes Stinespring's famous theorem((St), [P1]) showing that every completely positive map on a ; algebra can be dilated to a representation of the algebra. Similar generalization under varying stringent conditions on the semigroup and the algebra may be found in ([Em), (Vi-S]).The Markov process corresponding to a semigroup of substochastic matrices or nonconservative transition probability operators has an exit time which may be in-terpreted as a stop time at which the trajectory of the process goes out of the state space or hits a boundary. There are many ways of continuing the process after the exit time in such a manner that the Markov property and stationarity of transition probabilities are retained. Feller's study of this problem ([Fel,2,3) based on resol- vents of semigroups and Chung's pathwise approach ([C1,2), [Dy}) are well-known. Now it is natural to ask as to what happens when we have quantum Markov processes relating to quantum dynamical semigroups. The study in this direction was initiated by Davies (Da2] in the semigroup level. We are able to follow the footsteps of Feller and Chung to successfully return from the boundary and continue along a duplicate of the original flow as and when we reach the boundary to have a new quantum Markov flow. Of course, here the exits are governed by quantum stop times. Our investigations indicate the possibility of developing an extensive quantum boundary theory.

#### Control Number

ISILib-TH170

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Bhat, B. V. Rajarama Dr., "Markov Dilation of Nonconservative Quantum Dynamical Semigroups and Quantum Boundary Theory." (1994). *Doctoral Theses*. 231.

https://digitalcommons.isical.ac.in/doctoral-theses/231

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843009