A TREATISE ON THE ANALYTICAL DYNAMICS OF PARTICLES AND RIGID BODIES
1 INTRODUCTION 1 PART I CLASSICAL DYNAMICAL SYSTEMS 5 2 CLASSICAL DYNAMICS AND ERGODIC THEORY 8 3 DYNAMICAL ENTROPY AND INFORMATION 70 4 ALGORITHMIC COMPLEXITY 104 PART II QUANTUM DYNAMICAL SYSTEMS 135 5 QUANTUM MECHANICS OF FINITE DEGREES OF FREEDOM 138 6 QUANTUM INFORMATION THEORY 254 7 QUANTUM MECHANICS OF INFINITE DEGREES OF FREEDOM 316 PART III QUANTUM DYNAMICAL ENTROPIES AND COMPLEXITIES 406 8 QUANTUM DYNAMICAL ENTROPIES 406 9 QUANTUM ALGORITHMIC COMPLEXITIES 480 REFERENCES 514 CONTENTS 1 INTRODUCTION THIS BOOK FOCUSSES UPON QUANTUM DYNAMICS FROM VARIOUS POINTS OF VIEW WHICH ARE CONNECTED BY THE NOTION OF DYNAMICAL ENTROPY AS A MEASURE OF INFORMATION PRODUCTION DURING THE COURSE OF TIME FOR CLASSICAL DYNAMICAL SYSTEMS THE NOTION OF DYNAMICAL ENTROPY WAS IN TRODUCED BY KOLMOGOROV AND DEVELOPED BY SINAI KS ENTROPY AND PROVIDED A LINK AMONG DI ERENT ELDS OF MATHEMATICS AND PHYSICS IN FACT IN THE LIGHT OF THE RST THEOREM OF SHANNON THE KS ENTROPY GIVES THE MAXIMAL COM PRESSION RATE OF THE INFORMATION EMITTED BY ERGODIC INFORMATION SOURCES A THEOREM OF PESIN RELATES IT TO THE POSITIVE LYAPOUNOV EXPONENTS AND THUS TO THE EXPONENTIAL AMPLI CATION OF INITIAL SMALL ERRORS IN A WORD TO CLASSICAL CHAOS FINALLY A THEOREM OF BRUDNO LINKS THE KS ENTROPY TO THE COMPRESS IBILITY OF CLASSICAL TRAJECTORIES BY MEANS OF COMPUTER PROGRAMS NAMELY TO THEIR ALGORITHMIC COMPLEXITY A NOTION INTRODUCED INDEPENDENTLY AND ALMOST SIMULTANEOUSLY BY KOLMOGOROV SOLOMONO AND CHAITIN IN A PREVIOUS BOOK BY THE AUTHOR THE NOTION OF QUANTUM DYNAMICAL EN TROPY ELABORATED BY A CONNES H NARNHOFER AND W THIRRING CNT ENTROPY WAS PRESENTED WITHIN THE CONTEXT OF QUANTUM ERGODICITY AND CHAOS THE CNT ENTROPY IS A PARTICULAR PROPOSAL OF HOW THE KS ENTROPY MIGHT BE EXTENDED FROM CLASSICAL TO QUANTUM DYNAMICAL SYSTEMS AFTER THE APPEARANCE OF THE CNT ENTROPY OTHER PROPOSALS OF QUANTUM DY NAMICAL ENTROPIES APPEARED WHICH IN GENERAL ASSIGN DI ERENT ENTROPY PRODUC TIONS TO THE SAME QUANTUM DYNAMICS THE BASIC REASON IS THAT EACH PROPOSAL IS BUILT ACCORDING TO A DI ERENT VIEW ABOUT WHAT INFORMATION IN QUANTUM SYSTEMS SHOULD MEAN CONCRETELY IT IS A GENERAL FACT THAT IN ORDER TO GAIN INFORMATION ABOUT A SYSTEM AND ITS TIME EVOLUTION ONE HAS TO OBSERVE IT AND A QUANTUM FACT THAT OBSERVATIONS MAY BE INVASIVE AND PERTURBING SHOULD THIS FACT BE CONSIDERED INESCAPABLE AND THUS INCORPORATED IN ANY GOOD QUANTUM DYNAMICAL ENTROPY OR RATHER SHOULD IT BE AVOIDED AS A SOURCE OF SPURIOUS E ECTS THAT HAVE NOTHING TO DO WITH THE ACTUAL QUANTUM DYNAMICS THIS IS AN UNAVOIDABLE QUESTION AND BASED ON THE POSSIBLE ANSWERS ONE IS LED TO DI ERENT NOTIONS OF QUANTUM DYNAMICAL ENTROPIES THESE WILL BE SENSITIVE TO DI ERENT ASPECTS OF THE QUANTUM DYNAMICS AND THUS NOT UNEX PECTEDLY NOT EQUIVALENT THE REAL ISSUE IS WHICH THESE ASPECTS ARE AND WHAT KIND OF INFORMATIONAL MEANING THEY DO POSSES 1 IN VIEW OF THE ROLE OF THE KS ENTROPY IN CLASSICAL CHAOS ONE OF THE PRIN CIPAL APPLICATIONS OF THE QUANTUM DYNAMICAL ENTROPIES HAS BEEN TO THE PHE NOMENOLOGY OF QUANTUM CHAOS THE SCOPE HAS NOW BECOME WIDER QUANTUM COMPRESSION THEOREMS AND RECENT ATTEMPTS AT FORMULATING A NON COMMUTATIVE ALGORITHMIC COMPLEXITY THEORY MOTIVATE THE STUDY OF WHETHER AND HOW THE DI ERENT QUANTUM DYNAMICAL ENTROPIES ARE RELATED TO THESE NEW CONCEPTS IN PARTICULAR A BETTER UNDERSTANDING OF THE MANY FACETS OF INFORMATION IN QUAN TUM SYSTEMS MAY COME FROM CLARIFYING THE RELATIONS OF THE VARIOUS QUANTUM DYNAMICAL ENTROPIES AMONG THEMSELVES AND THEIR BEARING ON QUANTUM COM PRESSION SCHEMES AND THE ALGORITHMIC REPRODUCIBILITY OF QUANTUM DYNAMICS THE ISSUE AT STAKE CAN BE CONVENIENTLY CONVEYED BY AN EXAMPLE THE SIM PLEST CLASSICAL ERGODIC INFORMATION SOURCE EMITS BITS INDEPENDENTLY OF EACH OTHER WITH PROBABILITIES 1 2 FOR BOTH 0 AND 1 THE KS ENTROPY IS LOG 2 AND REPRESENTS 1 THE INFORMATION RATE OF A CLASSICAL SOURCE EMITTING INDEPENDENT BITS 2 THE LYAPOUNOV EXPONENT OF THE CLASSICAL DYNAMICAL SYSTEM CONSISTING IN THROWING A FAIR COIN 3 THE ALGORITHMIC COMPLEXITY OF ALMOST EVERY RESULTING SEQUENCE OF TAILS AND HEADS THE QUANTUM COUNTERPART OF SUCH AN INFORMATION SOURCE IS A SO CALLED QUAN TUM SPIN CHAIN THAT IS A ONE DIMENSIONAL LATTICE CARRYING A 2 2 MATRIX ALGEBRA AT EACH OF ITS IN NITELY MANY SITES EACH SITE CARRIES A SO CALLED QUBIT THE DYNAMICS OF SUCH A SYSTEM IS JUST THE SHIFT FROM ONE SITE TO THE OTHER AND THE IN NITE DIMENSIONAL ALGEBRA OF OPERATORS IS EQUIPPED WITH A TRANSLATION INVARIANT STATE THESE NON COMMUTATIVE STRUCTURES HAVE RECENTLY BECOME OF PRIMARY IMPORTANCE IN THE BOOSTING ELD OF QUANTUM INFORMATION WHAT IS RELEVANT IS THAT ONE CAN CONSTRUCT SUBALGEBRAS OF QUANTUM SPIN CHAINS CHAR ACTERIZED BY VARYING DEGREES OF NON COMMUTATIVITY BETWEEN THEIR OPERATORS DEPENDING ON THAT DEGREE THE CNT ENTROPY VARIES BETWEEN ZERO AND LOG 2 WHILE ANOTHER QUANTUM DYNAMICAL ENTROPY THE AFL ENTROPY OF ALICKI FANNES AND LINDBLAD IS ALWAYS LOG 2 THE CNT ENTROPY THUS APPEARS TO BE SENSITIVE TO THE AMOUNT OF NON COMMUTATIVITY BETWEEN OPERATORS WHEREAS THE AFL ENTROPY IS APPARENTLY INDEPENDENT OF THAT STRUCTURAL ALGEBRAIC PROPERTY BECAUSE OF ITS UNIFYING PROPERTIES THE KS ENTROPY CAN BE TAKEN AS A GOOD INDICATOR OF CLASSICAL RANDOMNESS AND COMPLEXITY ONE WOULD THEN LIKE TO ASSIGN A SIMILAR ROLE TO THE QUANTUM DYNAMICAL ENTROPIES DOES THIS MEAN THAT IN ACCORDANCE WITH THE CNT ENTROPY BEHAVIOR QUANTUM DYNAMICAL SYSTEMS HAVE VARYING DEGREES OF COMPLEXITY OR RANDOMNESS DEPENDING ON THE DEGREE OF NON COMMUTATIVITY OR ACCORDING TO THE AFL ENTROPY THE ALGEBRAIC STRUCTURAL PROPERTIES HAVE NO BEARING ON DYNAMICAL RANDOMNESS OR COMPLEXITY WHICH ARE RATHER RELATED TO THE STATISTICS OF SUCH SYSTEMS NAMELY TO THEIR SHIFT INVARIANT STATE MORE CONCRETELY ONE MAY ASK WHICH ONE OF THE TWO QUANTUM DYNAMICAL ENTROPIES IS CLOSER TO THE ACTUAL QUANTUM INFORMATIONAL STRUCTURE OF THESE A TREATISE ON THE ANALYTICAL DYNAMICS OF PARTICLES AND RIGID BODIES 2 QUANTUM SOURCES REGARDING THIS ISSUE OF PARTICULAR INTEREST ARE THE YET UN EXPLORED RELATIONS OF THE QUANTUM DYNAMICAL ENTROPIES TO THE QUANTUM ALGO RITHMIC COMPLEXITIES INDEED AS THERE INEQUIVALENT GENERALIZATIONS OF THE KS ENTROPY SO THERE ARE DI ERENT EXTENSIONS OF THE CLASSICAL ALGORITHMIC COMPLEXITY THESE EXTEN SIONS HAVE BEEN MOTIVATED BY THE POSSIBILITY OF A MODEL OF COMPUTATION BASED ON THE LAWS OF QUANTUM MECHANICS AND ON THE THEORETICAL FORMULATION OF THE NOTION OF QUANTUM TURING MACHINES QTMS LIKE CLASSICAL TURING MA CHINES TMS QTMS CONSIST OF A READ WRITE HEAD MOVING ON TAPES WITH SAY BINARY PROGRAMS WRITTEN ON THEM ONLY THE TAPES OF QTMS CAN OCCUR IN LINEAR SUPERPOSITIONS OF THE CLASSICAL CON GURATIONS OF 0 S AND 1 S IN A WORD INPUTS AND OUTPUTS OF QTMS ARE QUBITS SINCE THE VARIOUS QUANTUM DYNAMICAL ENTROPIES WERE PROPOSED INDEPEN DENTLY OF QUANTUM INFORMATION AS TOOLS TO BETTER STUDY THE LONG TIME DY NAMICAL FEATURES OF IN NITE QUANTUM SYSTEMS ONE MAY DOUBT THAT RELATIONS SHOULD EXIST BETWEEN THEM AND QUANTUM INFORMATION ONE NOTICES HOWEVER THAT THE CNT ENTROPY WAS DEVELOPED USING THE NOTION OF ENTROPY OF A SUB ALGEBRA WHICH YEARS LATER INDEPENDENTLY APPEARED IN QUANTUM INFORMATION THEORY AS A MEASURE OF ENTANGLEMENT KNOWN AS ENTANGLEMENT OF FORMATION ALSO THE AFL ENTROPY IS BASED ON TECHNIQUES THAT IN QUANTUM INFORMATION THEORY ARE FUNDAMENTAL TOOLS TO DESCRIBE QUANTUM CHANNELS AND MORE IN GENERAL ALL QUANTUM OPERATIONS THAT MAY A ECT QUANTUM SYSTEMS THE BOOK IS ORGANIZED IN THREE PARTS IN THE RST PART THE RST CHAPTER PRESENTS BASIC NOTIONS OF ERGODIC THEORY THE SECOND GIVES AN OVERVIEW OF ENTROPY IN INFORMATION THEORY THE THIRD ADDRESSES THE NOTION OF KS ENTROPY AND THE CLASSICAL COMPRESSION THEOREMS WHILE ALGORITHMIC COMPLEXITY IS THE SUBJECT OF THE FOURTH CHAPTER THE SECOND PART CONSISTS OF THREE CHAPTERS THE RST O ERS AN OVERVIEW OF ALGEBRAIC QUANTUM MECHANICS WITH PARTICULAR EMPHASIS ON THE NOTIONS OF POSITIVITY AND COMPLETE POSITIVITY OF QUANTUM MAPS AND QUANTUM TIME EVOLUTIONS BOTH REVERSIBLE AND IRREVERSIBLE THE SECOND CHAPTER INTRODUCES THE FUNDAMENTALS OF QUANTUM INFORMATION THE RELATIONS BETWEEN POSITIVE AND COMPLETELY POSITIVE MAPS AND QUANTUM ENTANGLEMENT THE ENTROPY OF A SUBALGEBRA THE ENTANGLEMENT OF FORMATION AND THE ACCESSIBLE INFORMATION OF A QUANTUM CHANNEL THE THIRD CONCERNS IN NITE QUANTUM DYNAMICAL SYS TEMS AND QUANTUM ERGODICITY QUANTUM CHAINS AS QUANTUM SOURCES AND THE QUANTUM COUNTERPARTS TO SHANNON S THEOREMS IN THE RST CHAPTER OF THE THIRD PART A DETAILED INTRODUCTION IS GIVEN TO THE CNT AND AFL ENTROPIES AND TO THEIR USE IN THE STUDY OF DYNAMICAL INFOR MATION PRODUCTION IN QUANTUM SYSTEMS FINALLY THE SECOND AND LAST CHAPTER OF THE BOOK FOCUSSES ON SOME RECENT EXTENSIONS OF ALGORITHMIC COMPLEXITY TO QUANTUM SYSTEMS STARTING WITH A DISCUSSION OF QUANTUM TURING MACHINES AND QUANTUM COMPUTERS AND CONCLUDING WITH AN EXPLORATION OF THE POSSIBLE ROLE PLAYED IN THIS CONTEXT BY THE QUANTUM DYNAMICAL ENTROPIES A TREATISE ON THE ANALYTICAL DYNAMICS OF PARTICLES AND RIGID BODIES 3