#### Title

### Bures Distance for Completely Positive Maps and Cp-H-Extendable Maps Between Hilbert C*- Modules.

#### Date of Submission

11-28-2013

#### Date of Award

11-28-2014

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

#### Supervisor

Bhat, B. V. Rajarama (TSMU-Bangalore; ISI)

#### Abstract (Summary of the Work)

Completely positive (CP-) maps are special kinds of positivity preserving maps on C âˆ— -algebras. W.F. Stinespring [Sti55] obtained a structure theorem for CP-maps showing that they are closely connected with âˆ—-homomorphisms. W. Arveson and other operator algebraists quickly realized the importance of these maps. Presently the role of the theory of CP-maps in our understanding of C âˆ— -algebras and von Neumann algebras is well recognised. It has been argued by physicists that CPmaps are physically more meaningful than just positive maps due to their stability under ampliations. From quantum probabilistic point of view CP-maps are quantum analogues of stochastic or sub-stochastic transition probability maps. Therefore one begins with such maps in order to construct quantum Markov processes. Recently there has been lot of interest in quantum computation and quantum information theory and here trace preserving, unital CP-maps play the role of quantum channels. This justifies detailed study of CP-maps and related concepts.Often it is the structure theorems that makes a theory worth studying. GNStheorem and Stinespringâ€™s theorem are the basic structure theorems for CP-maps. Our main tool to study CP-maps is the theory of Hilbert C âˆ— -modules. They are objects similar to Hilbert spaces. Close connections between CP-maps and Hilbert C âˆ— -modules are well-known ([Kas80, Mur97, Pas73]).Given CP-maps Ï•1 and Ï•2 between unital C âˆ— -algebras A and B, by a common representation module for them we mean a Hilbert A-B-module E where they can be represented, that is, there exists xi âˆˆ E such that Ï•i(Â·) = hxi ,(Â·)xii. We define Î² as the infimum of the norm differences x1 âˆ’ x2 taken over all common representation modules E and representing vectors xi âˆˆ E, and call it Bures distance. We show the existence of a sort of universal module where we can take infimum to compute the Bures distance, and thereby prove that Î² is a metric when the CP-maps under consideration map to a von Neumann algebra or to an injective C âˆ— -algebra. However, Î² is not a metric when the range algebra is a general C âˆ— -algebra. The definition of Bures distance is abstract and does not give us indications as to how to compute it for concrete examples. We show that Bures distance can be computed using intertwiners between two (minimal) GNS-constructions of CP-maps. We also prove a rigidity theorem, showing that GNS-representation modules ([Pas73]) of CP-maps which are close to the identity map contain a copy of the original algebra.If Ï• : A â†’ B is a linear map, then by a Ï•-map we mean a linear map T : E â†’ F from a Hilbert A-module E into a Hilbert B-module F such that T(x1), T(x2)i = Ï•(x1, x2) for all xi âˆˆ E, that is, T preserves the inner product up to the linear map Ï•. We prove that if E is full and if Ï• is bounded linear, then Ï• will be automatically CP. Moreover,

#### Control Number

ISILib-TH414

#### 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

K, Sumesh Dr., "Bures Distance for Completely Positive Maps and Cp-H-Extendable Maps Between Hilbert C*- Modules." (2014). *Doctoral Theses*. 308.

https://digitalcommons.isical.ac.in/doctoral-theses/308

## 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:28843368