ASYMPTOTIC THEORY OF QUANTUM STATISTICAL INFERENCE

By I. LUZAO

PART I HYPOTHESIS TESTING 1 1 INTRODUCTION TO PART I 3 2 STRONG CONVERSE AND STEIN S LEMMA IN QUANTUM HYPOTHESIS TESTING 12 3 THE PROPER FORMULA FOR RELATIVE ENTROPY AND ITS ASYMPTOTICS IN QUANTUM PROBABILITY 27 4 STRONG CONVERSE THEOREMS IN QUANTUM INFORMATION THEORY 48 5 ASYMPTOTICS OF QUANTUM RELATIVE ENTROPY FROM A REPRESENTATION THEORETICAL VIEWPOINT 50 6 QUANTUM BIRTHDAY PROBLEMS GEOMETRICAL ASPECTS OF QUANTUM RANDOM CODING 59 PART II QUANTUM CRAM R RAO BOUND IN MIXED STATES MODEL 73 7 INTRODUCTION TO PART II 75 8 A NEW APPROACH TO CRAM R RAO BOUNDS FOR QUANTUM STATE ESTIMATION 84 9 ON FISHER INFORMATION OF QUANTUM STATISTICAL MODELS 97 10 ON THE PARAMETER ESTIMATION PROBLEM FOR QUANTUM STATISTICAL MODELS 109 11 A GENERALIZATION OF THE SIMULTANEOUS DIAGONALIZATION OF HERMITIAN MATRICES AND ITS RELATION TO QUANTUM ESTIMATION THEORY 117 CONTENTS 12 A LINEAR PROGRAMMING APPROACH TO ATTAINABLE CRAM R RAO TYPE BOUNDS 134 13 STATISTICAL MODEL WITH MEASUREMENT DEGREE OF FREEDOM AND QUANTUM PHYSICS 146 14 ASYMPTOTIC QUANTUM THEORY FOR THE THERMAL STATES FAMILY 154 15 STATE ESTIMATION FOR LARGE ENSEMBLES 162 PART III QUANTUM CRAM R RAO BOUND IN PURE STATES MODEL 199 16 INTRODUCTION TO PART III 201 17 QUANTUM FISHER METRIC AND ESTIMATION FOR PURE STATE MODELS 204 18 GEOMETRY OF QUANTUM ESTIMATION THEORY 213 19 AN ESTIMATION THEORETICAL CHARACTERIZATION OF COHERENT STATES 271 20 A GEOMETRICAL APPROACH TO QUANTUM ESTIMATION THEORY 289 PART I HYPOTHESIS TESTING 1 CHAPTER 1 INTRODUCTION TO PART I 1 QUANTUM HYPOTHESIS TESTING 1 1 SETTING OF QUANTUM HYPOTHESIS TESTING IN THE QUANTUM HYPOTHESIS TESTING IT IS REQUIRED TO DECIDE WHETHER THE GIVEN HYPOTHESIS WHICH IS OFTEN CALLED THE NULL HYPOTHESIS FOR THE DENSITY IS TRUE OR NOT ESPECIALLY THE FOLLOWING SETTING IS THE SIMPLEST AND IS CALLED SIMPLE HYPOTHESIS TESTING THE NULL HYPOTHESIS FOR THE TRUE DENSITY IS GIVEN AS A DENSITY AND THE ALTERNATIVE HYPOTHESIS WHICH IS ASSUMED TO BE VALID IF THE NULL HYPOTHESIS IS NOT VALID IS ALSO GIVEN AS ANOTHER DENSITY EVEN IN THIS SIMPLE SETTING NO QUANTUM VERSION OF THE FUNDAMENTAL ASYMPTOTIC THEORY HAD BEEN ESTABLISHED UNTIL 1990 IN THE FOLLOWING SECTIONS WE BRIE Y REVIEW THE HISTORY OF THE DEVELOPMENT OF ITS QUANTUM VERSION WITH AN OUTLINE OF ITS FORMULATION FOR A MORE RIGOROUS FORMULATION PLEASE REFER TO SECTION 1 OF CHAP 2 IN THIS PROBLEM WE FOCUS ON TWO ERROR PROBABILITIES THE RST ERROR PROB ABILITY IS THE PROBABILITY OF INCORRECTLY DECIDING TO BE THE TRUE STATE WHEN THE TRUE STATE IS AND THE SECOND ERROR PROBABILITY IS THE OPPOSITE ERROR PROBABILITY IN GENERAL THESE TWO ERROR PROBABILITIES DO NOT HAVE THE SAME IMPORTANCE USUALLY WE OPTIMIZE THE SECOND ERROR PROBABILITY UNDER THE CONSTRAINT THAT THE RST ERROR PROBABILITY IS LESS THAN A GIVEN THRESHOLD IN THE CLASSICAL VERSION OF THIS PROBLEM I E THE TEST OF TWO PROBABILITY DISTRIBUTIONS P AND Q THE LIKELIHOOD RATIO P Q PLAYS AN IMPORTANT ROLE IN PARTICULAR AS MENTIONED IN NEYMAN PEARSON LEMMA THE OPTIMAL TEST IS GIVEN BY A LIKELIHOOD RATIO TEST WHICH IS DE NED BY THE LIKELIHOOD RATIO IN THE QUANTUM VERSION HOLEVO INTRODUCED A QUANTUM COUNTERPART OF A LIKELIHOOD RATIO TEST AND OBTAINED THE QUANTUM NEYMAN PEARSON LEMMA I 1 FOLLOWING THIS RESULT HELSTROM PRESENTED IT IN A MORE FAMILIAR FORM I 2 HOWEVER WE CANNOT DERIVE EXPLICIT FORMS OF THOSE ERROR PROBABILITIES BY ONLY USING THIS METHOD 3