11124

10240

كارشناسي ارشد

رياضي محض(آناليز)

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1394

نه، 73ص.

محمود منجگاني

فريد بهرامي

به فارسي و انگليسي

رسول نصراصفهاني، مهدي نعمتي

1395/01/15

رياضي

ID10240

Conditional expectation and Bayes rule for quantum random variables and positive operator valued measures JAFAR SAFDARIAN j safdarian@math iut ac ir February 19 2016 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Seyed Mahmoud Manjegani manjgani@cc iut ac irAdvisor Dr Farid Bahrami fbahrami@cc iut ac ir2000 MSC Primary 47N30 Secondary 46N50 81P15 81P16Keywords Hilbert space Positive operator valued measure Quantum random variable Conditional expectation C Convexity StateAbstract This M Sc thesis is based on the following paperFarenick D R and Kozdron M Conditional expectation and Bayes rule for quantum randomvariables and positive operator valued measures J Math Phys 53 2012 042 201 Suppose that X denote a compact Hausdorff space and H is a d dimensional Hilbert space Let e1 en be an orthonormal basis for H The vector space of all operators acting on H is denoted by B H and B H denotes the cone of positive operators Let O X be the algebra of Borel sets of X A positive operator valued measure on X O X isa function O X B H with values in B H such X and for every countable 0 that collection Ek k N with Ej Ek for j k we have k N Ek k N Ek The set of 1 X we mean those all positive operator valued measures is denoted by P OV MH X By P OV MHP OV MH X satisfying X 1 A quantum random variable is a function X B H with values in B H If P OV MH X 1then the quantum expectation of relative to the quantum probability measure is the operator denoted by E and defined by E X d In this thesis first we introduce positive operator valued measures and quantum random variables to define thequantum expected value E In so doing we are led to theorems for a change of quantum measure as wellas a change of quantum variables we also introduce a quantum coditional expectation which results in quantumversions of some standard identities for Radon Nikodym derivatives This allows us to formulate and prove aquantum analogue of Bayes rule Let f X be a sub algebra of O X A function X B H is said to be f X measurable if forevery density operator the complex valued function X C given by x T r x is f X measurable Each P OV MH X gives rise to measure on X O X via E T r E E O X If X B H is a quantum random variable and if P OV MH X then issaid to be integrable if for every density operator the complex valued function X C defined by d d T r d x 1 2 x d x 1 2 is integrable If 1 2 P OV MH X F X then 2 is absolutely continuous with respect to 1 which we denote by 2 ac 1 if 2 E 0 for every E F X with 1 E 0 If 1 2 P OV MH X F X and H is of finite dimension then the following statements are equivalent1 2 ac 1 2 There exists a bounded 1 integrable F X measurable function X F X B H unique up to sets of 1 measure zero such that 2 E d 1 E

محمود منجگاني

فريد بهرامي

رسول نصراصفهاني، مهدي نعمتي