DOCSLIB.ORG

Quantum Probability 1982-2017

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Probability 1982-2017 Quantum Probability 1982-2017 Hans Maassen, Universities of Nijmegen and Amsterdam Nijmegen, June 23, 2017. 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. Two historic events 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. Two historic events 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. Two historic events 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. Two historic events 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ 400 years time difference; Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. How are these two events related? ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : From probability: ∗ `We understand that random variables form an algebra, ∗ but what is this funny non-commutativity? ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework ∗ `We understand that random variables form an algebra, ∗ but what is this funny non-commutativity? ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework From probability: ∗ but what is this funny non-commutativity? ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework From probability: ∗ `We understand that random variables form an algebra, ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework From probability: ∗ `We understand that random variables form an algebra, ∗ but what is this funny non-commutativity? From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics.
Load more

Recommended publications
  • The Probability Set-Up.Pdf
    CHAPTER 2 The probability set-up 2.1. Basic theory of probability We will have a sample space, denoted by S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample space would be all possible pairs made up of the numbers one through six. An event is a subset of S. Another example is to toss a coin 2 times, and let S = fHH;HT;TH;TT g; or to let S be the possible orders in which 5 horses nish in a horse race; or S the possible prices of some stock at closing time today; or S = [0; 1); the age at which someone dies; or S the points in a circle, the possible places a dart can hit. We should also keep in mind that the same setting can be described using dierent sample set. For example, in two solutions in Example 1.30 we used two dierent sample sets. 2.1.1. Sets. We start by describing elementary operations on sets. By a set we mean a collection of distinct objects called elements of the set, and we consider a set as an object in its own right. Set operations Suppose S is a set. We say that A ⊂ S, that is, A is a subset of S if every element in A is contained in S; A [ B is the union of sets A ⊂ S and B ⊂ S and denotes the points of S that are in A or B or both; A \ B is the intersection of sets A ⊂ S and B ⊂ S and is the set of points that are in both A and B; ; denotes the empty set; Ac is the complement of A, that is, the points in S that are not in A.
    [Show full text]
  • Determinism, Indeterminism and the Statistical Postulate
    Tipicality, Explanation, The Statistical Postulate, and GRW Valia Allori Northern Illinois University [email protected] www.valiaallori.com Rutgers, October 24-26, 2019 1 Overview Context: Explanation of the macroscopic laws of thermodynamics in the Boltzmannian approach Among the ingredients: the statistical postulate (connected with the notion of probability) In this presentation: typicality Def: P is a typical property of X-type object/phenomena iff the vast majority of objects/phenomena of type X possesses P Part I: typicality is sufficient to explain macroscopic laws – explanatory schema based on typicality: you explain P if you explain that P is typical Part II: the statistical postulate as derivable from the dynamics Part III: if so, no preference for indeterministic theories in the quantum domain 2 Summary of Boltzmann- 1 Aim: ‘derive’ macroscopic laws of thermodynamics in terms of the microscopic Newtonian dynamics Problems: Technical ones: There are to many particles to do exact calculations Solution: statistical methods - if 푁 is big enough, one can use suitable mathematical technique to obtain information of macro systems even without having the exact solution Conceptual ones: Macro processes are irreversible, while micro processes are not Boltzmann 3 Summary of Boltzmann- 2 Postulate 1: the microscopic dynamics microstate 푋 = 푟1, … . 푟푁, 푣1, … . 푣푁 in phase space Partition of phase space into Macrostates Macrostate 푀(푋): set of macroscopically indistinguishable microstates Macroscopic view Many 푋 for a given 푀 given 푀, which 푋 is unknown Macro Properties (e.g. temperature): they slowly vary on the Macro scale There is a particular Macrostate which is incredibly bigger than the others There are many ways more to have, for instance, uniform temperature than not equilibrium (Macro)state 4 Summary of Boltzmann- 3 Entropy: (def) proportional to the size of the Macrostate in phase space (i.e.
    [Show full text]
  • Topic 1: Basic Probability Definition of Sets
    Topic 1: Basic probability ² Review of sets ² Sample space and probability measure ² Probability axioms ² Basic probability laws ² Conditional probability ² Bayes' rules ² Independence ² Counting ES150 { Harvard SEAS 1 De¯nition of Sets ² A set S is a collection of objects, which are the elements of the set. { The number of elements in a set S can be ¯nite S = fx1; x2; : : : ; xng or in¯nite but countable S = fx1; x2; : : :g or uncountably in¯nite. { S can also contain elements with a certain property S = fx j x satis¯es P g ² S is a subset of T if every element of S also belongs to T S ½ T or T S If S ½ T and T ½ S then S = T . ² The universal set ­ is the set of all objects within a context. We then consider all sets S ½ ­. ES150 { Harvard SEAS 2 Set Operations and Properties ² Set operations { Complement Ac: set of all elements not in A { Union A \ B: set of all elements in A or B or both { Intersection A [ B: set of all elements common in both A and B { Di®erence A ¡ B: set containing all elements in A but not in B. ² Properties of set operations { Commutative: A \ B = B \ A and A [ B = B [ A. (But A ¡ B 6= B ¡ A). { Associative: (A \ B) \ C = A \ (B \ C) = A \ B \ C. (also for [) { Distributive: A \ (B [ C) = (A \ B) [ (A \ C) A [ (B \ C) = (A [ B) \ (A [ C) { DeMorgan's laws: (A \ B)c = Ac [ Bc (A [ B)c = Ac \ Bc ES150 { Harvard SEAS 3 Elements of probability theory A probabilistic model includes ² The sample space ­ of an experiment { set of all possible outcomes { ¯nite or in¯nite { discrete or continuous { possibly multi-dimensional ² An event A is a set of outcomes { a subset of the sample space, A ½ ­.
    [Show full text]
  • Quantum Theory Needs No 'Interpretation'
    Quantum Theory Needs No ‘Interpretation’ But ‘Theoretical Formal-Conceptual Unity’ (Or: Escaping Adán Cabello’s “Map of Madness” With the Help of David Deutsch’s Explanations) Christian de Ronde∗ Philosophy Institute Dr. A. Korn, Buenos Aires University - CONICET Engineering Institute - National University Arturo Jauretche, Argentina Federal University of Santa Catarina, Brazil. Center Leo Apostel fot Interdisciplinary Studies, Brussels Free University, Belgium Abstract In the year 2000, in a paper titled Quantum Theory Needs No ‘Interpretation’, Chris Fuchs and Asher Peres presented a series of instrumentalist arguments against the role played by ‘interpretations’ in QM. Since then —quite regardless of the publication of this paper— the number of interpretations has experienced a continuous growth constituting what Adán Cabello has characterized as a “map of madness”. In this work, we discuss the reasons behind this dangerous fragmentation in understanding and provide new arguments against the need of interpretations in QM which —opposite to those of Fuchs and Peres— are derived from a representational realist understanding of theories —grounded in the writings of Einstein, Heisenberg and Pauli. Furthermore, we will argue that there are reasons to believe that the creation of ‘interpretations’ for the theory of quanta has functioned as a trap designed by anti-realists in order to imprison realists in a labyrinth with no exit. Taking as a standpoint the critical analysis by David Deutsch to the anti-realist understanding of physics, we attempt to address the references and roles played by ‘theory’ and ‘observation’. In this respect, we will argue that the key to escape the anti-realist trap of interpretation is to recognize that —as Einstein told Heisenberg almost one century ago— it is only the theory which can tell you what can be observed.
    [Show full text]
  • Non-Archimedean Probability (NAP) Theory
    Non-Archimedean Probability Vieri Benci∗ Leon Horsten† Sylvia Wenmackers‡ October 29, 2018 Abstract We propose an alternative approach to probability theory closely re- lated to the framework of numerosity theory: non-Archimedean prob- ability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and zero- and unit-probability events pose no particular epistemological problems. We use a non-Archimedean field as the range of the probability func- tion. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by a different type of infinite additivity. Mathematics subject classification. 60A05, 03H05 Keywords. Probability, Axioms of Kolmogorov, Nonstandard models, De Finetti lottery, Non-Archimedean fields. Contents 1 Introduction 2 1.1 Somenotation .......................... 3 2 Kolmogorov’s probability theory 4 2.1 Kolmogorov’saxioms.. .. .. .. .. .. .. 4 2.2 Problems with Kolmogorov’s axioms . 6 3 Non-Archimedean Probability 7 arXiv:1106.1524v1 [math.PR] 8 Jun 2011 3.1 The axioms of Non-Archimedean Probability . 7 3.2 Analysisofthefourthaxiom . 10 3.3 First consequences of the axioms . 11 3.4 Infinitesums ........................... 13 ∗Dipartimento di Matematica Applicata, Universit`adegli Studi di Pisa, Via F. Buonar- roti 1/c, Pisa, ITALY and Department of Mathematics, College of Science, King Saud University, Riyadh, 11451, SAUDI ARABIA. e-mail: [email protected] †Department of Philosophy, University of Bristol, 43 Woodland Rd, BS81UU Bristol, UNITED KINGDOM. e-mail: [email protected] ‡Faculty of Philosophy, University of Groningen, Oude Boteringestraat 52, 9712 GL Groningen, THE NETHERLANDS. e-mail: [email protected] 1 4 NAP-spaces and Λ-limits 14 4.1 Fineideals............................
    [Show full text]
  • Interpretations of Probability First Published Mon Oct 21, 2002; Substantive Revision Mon Dec 19, 2011
    Open access to the Encyclopedia has been made possible, in part, with a financial contribution from the Australian National University Library. We gratefully acknowledge this support. Interpretations of Probability First published Mon Oct 21, 2002; substantive revision Mon Dec 19, 2011 ‘Interpreting probability’ is a commonly used but misleading characterization of a worthy enterprise. The so-called ‘interpretations of probability’ would be better called ‘analyses of various concepts of probability’, and ‘interpreting probability’ is the task of providing such analyses. Or perhaps better still, if our goal is to transform inexact concepts of probability familiar to ordinary folk into exact ones suitable for philosophical and scientific theorizing, then the task may be one of ‘explication’ in the sense of Carnap (1950). Normally, we speak of interpreting a formal system , that is, attaching familiar meanings to the primitive terms in its axioms and theorems, usually with an eye to turning them into true statements about some subject of interest. However, there is no single formal system that is ‘probability’, but rather a host of such systems. To be sure, Kolmogorov's axiomatization, which we will present shortly, has achieved the status of orthodoxy, and it is typically what philosophers have in mind when they think of ‘probability theory’. Nevertheless, several of the leading ‘interpretations of probability’ fail to satisfy all of Kolmogorov's axioms, yet they have not lost their title for that. Moreover, various other quantities that have nothing to do with probability do satisfy Kolmogorov's axioms, and thus are interpretations of it in a strict sense: normalized mass, length, area, volume, and other quantities that fall under the scope of measure theory, the abstract mathematical theory that generalizes such quantities.
    [Show full text]
  • CURRICULUM VITAE June, 2016 Hu, Bei-Lok Bernard Professor Of
    CURRICULUM VITAE June, 2016 Hu, Bei-Lok Bernard Professor of Physics, University of Maryland, College Park 胡悲樂 Founding Fellow, Joint Quantum Institute, Univ. Maryland and NIST Founding Member, Maryland Center for Fundamental Physics, UMD. I. PERSONAL DATA Date and Place of Birth: October 4, 1947, Chungking, China. Citizenship: U.S.A. Permanent Address: 3153 Physical Sciences Complex Department of Physics, University of Maryland, College Park, Maryland 20742-4111 Telephone: (301) 405-6029 E-mail: [email protected] Fax: MCFP: (301) 314-5649 Physics Dept: (301) 314-9525 UMd Physics webpage: http://umdphysics.umd.edu/people/faculty/153-hu.html Research Groups: - Gravitation Theory (GRT) Group: http://umdphysics.umd.edu/research/theoretical/87gravitationaltheory.html - Quantum Coherence and Information (QCI) Theory Group: http://www.physics.umd.edu/qcoh/index.html II. EDUCATION Date School Location Major Degree 1958-64 Pui Ching Middle School Hong Kong Science High School 1964-67 University of California Berkeley Physics A.B. 1967-69 Princeton University Princeton Physics M.A. 1969-72 Princeton University Princeton Physics Ph.D. III. ACADEMIC EXPERIENCE Date Institution Position June 1972- Princeton University Research Associate Jan. 1973 Princeton, N.J. 08540 Physics Department Jan. 1973- Institute for Advanced Study Member Aug. 1973 Princeton, N.J. 08540 School of Natural Science Sept.1973- Stanford University Research Associate Aug. 1974 Stanford, Calif. 94305 Physics Department Sept.1974- University of Maryland Postdoctoral Fellow Jan. 1975 College Park, Md. 20742 Physics & Astronomy Jan. 1975- University of California Research Mathematician Sept.1976 Berkeley, Calif. 94720 Mathematics Department Oct. 1976- Institute for Space Studies Research Associate May 1977 NASA, New York, N.Y.
    [Show full text]
  • Stat 5101 Lecture Notes
    Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000 by Charles J. Geyer January 16, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random Variables .......................... 1 1.1.1 Variables ........................... 1 1.1.2 Functions ........................... 1 1.1.3 Random Variables: Informal Intuition ........... 3 1.1.4 Random Variables: Formal De¯nition ........... 3 1.1.5 Functions of Random Variables ............... 7 1.2 Change of Variables ......................... 7 1.2.1 General De¯nition ...................... 7 1.2.2 Discrete Random Variables . ............... 9 1.2.3 Continuous Random Variables ............... 12 1.3 Random Vectors ........................... 14 1.3.1 Discrete Random Vectors . ............... 15 1.3.2 Continuous Random Vectors . ............... 15 1.4 The Support of a Random Variable . ............... 17 1.5 Joint and Marginal Distributions . ............... 18 1.6 Multivariable Change of Variables . ............... 22 1.6.1 The General and Discrete Cases .............. 22 1.6.2 Continuous Random Vectors . ............... 22 2 Expectation 31 2.1 Introduction .............................. 31 2.2 The Law of Large Numbers ..................... 32 2.3 Basic Properties ........................... 32 2.3.1 Axioms for Expectation (Part I) .............. 32 2.3.2 Derived Basic Properties ................... 34 2.3.3 Important Non-Properties . ............... 36 2.4 Moments ............................... 37 2.4.1 First Moments and Means . ............... 38 2.4.2 Second Moments and Variances ............... 40 2.4.3 Standard Deviations and Standardization ......... 42 2.4.4 Mixed Moments and Covariances .............. 43 2.4.5 Exchangeable Random Variables .............. 50 2.4.6 Correlation .......................... 50 iii iv Stat 5101 (Geyer) Course Notes 2.5 Probability Theory as Linear Algebra ..............
    [Show full text]
  • Chapter 2 Conditional Probability and Independence
    CHAPTER 2 CONDITIONAL PROBABILITY AND INDEPENDENCE INTRODUCTION This chapter introduces the important concepts of conditional probability and statistical independence. Conditional probabilities arise when it is known that a certain event has occurred. This knowledge changes the probabilities of events within the sample space of the experiment. Conditioning on an event occurs frequently, and understanding how to work with conditional probabilities and apply them to a particular problem or application is extremely important. In some cases, knowing that a particular event has occurred will not effect the probability of another event, and this leads to the concept of the statistical independence of events that will be developed along with the concept of conditional independence. 2-1 CONDITIONAL PROBABILITY The three probability axioms introduced in the previous chapter provide the foundation upon which to develop a theory of probability. The next step is to understand how the knowledge that a particular event has occurred will change the probabilities that are assigned to the outcomes of an experiment. The concept of a conditional probability is one of the most important concepts in probability and, although a very simple concept, conditional probability is often confusing to students. In situations where a conditional probability is to be used, it is often overlooked or incorrectly applied, thereby leading to an incorrect answer or conclusion. Perhaps the best starting point for the development of conditional probabilities is a simple example that illustrates the context in which they arise. Suppose that we have an electronic device that has a probability pn of still working after n months of continuous operation, and suppose that the probability is equal to 0.5 that the device will still be working after one year (n = 12).
    [Show full text]
  • Born's Rule and Measurement
    Born’s rule and measurement Arnold Neumaier Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria email: [email protected] http://www.mat.univie.ac.at/~neum December 20, 2019 arXiv:1912.09906 Abstract. Born’s rule in its conventional textbook form applies to the small class of pro- jective measurements only. It is well-known that a generalization of Born’s rule to realistic experiments must be phrased in terms of positive operator valued measures (POVMs). This generalization accounts for things like losses, imperfect measurements, limited detec- tion accuracy, dark detector counts, and the simultaneous measurement of position and momentum. Starting from first principles, this paper gives a self-contained, deductive introduction to quantum measurement and Born’s rule, in its generalized form that applies to the results of measurements described by POVMs. It is based on a suggestive definition of what constitutes a detector, assuming an intuitive informal notion of response. The formal exposition is embedded into the context of a variaety of quotes from the litera- ture illuminating historical aspects of the subject. The material presented suggests a new approach to introductory courses on quantum mechanics. For the discussion of questions related to this paper, please use the discussion forum https://www.physicsoverflow.org. MSC Classification (2010): primary: 81P15, secondary: 81-01 1 Contents 1 The measurement process 3 1.1 Statesandtheirproperties . .... 5 1.2 Detectors,scales,andPOVMs . .. 8 1.3 Whatismeasured? ................................ 9 1.4 InformationallycompletePOVMs . ... 11 2 Examples 12 2.1 Polarizationstatemeasurements. ...... 13 2.2 Joint measurements of noncommuting quantities .
    [Show full text]
  • Quantum Computation: a Short Course
    Quantum Computation: A Short Course Frank Rioux Emeritus Professor of Chemistry College of St. Benedict | St. John’s University The reason I was keen to include at least some mathematical descriptions was simply that in my own study of quantum computation the only time I really felt that I understood what was happening in a quantum program was when I examined some typical quantum circuits and followed through the equations. Julian Brown, The Quest for the Quantum Computer, page 6. My reason for beginning with Julian Brown’s statement is that I accept it wholeheartedly. I learn the same way. So in what follows I will present mathematical analyses of some relatively simple and representative quantum circuits that are designed to carry out important contemporary processes such as parallel computation, teleportation, data-base searches, prime factorization, quantum encryption and quantum simulation. I will conclude with a foray into the related area of Bell’s theorem and the battle between local realism and quantum mechanics. Quantum computers use superpositions, entanglement and interference to carry out calculations that are impossible with a classical computer. The following link contains insightful descriptions of the non-classical character of superpositions and entangled superpositions from a variety of sources. http://www.users.csbsju.edu/~frioux/q-intro/EntangledSuperposition.pdf To illuminate the difference between classical and quantum computation we begin with a review of the fundamental principles of quantum theory using the computational methods of matrix mechanics. http://www.users.csbsju.edu/~frioux/matmech/RudimentaryMatrixMechanics.pdf The following is an archive of photon and spin vector states and their matrix operators.
    [Show full text]
  • QUANTUM NONLOCALITY and INSEPARABILITY Asher Peres
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server QUANTUM NONLOCALITY AND INSEPARABILITY Asher Peres Department of Physics Technion—Israel Institute of Technology 32 000 Haifa, Israel A quantum system consisting of two subsystems is separable if its den- 0 00 0 00 sity matrix can be written as ρ = wK ρK ⊗ ρK,whereρK and ρK are density matrices for the two subsytems,P and the positive weights wK satisfy wK = 1. A necessary condition for separability is derived and is shownP to be more sensitive than Bell’s inequality for detecting quantum inseparability. Moreover, collective tests of Bell’s inequality (namely, tests that involve several composite systems simultaneously) may sometimes lead to a violation of Bell’s inequality, even if the latter is satisfied when each composite system is tested separately. 1. INTRODUCTION From the early days of quantum mechanics, the question has often been raised whether an underlying “subquantum” theory, that would be deterministic or even stochastic, was viable. Such a theory would presumably involve additional “hidden” variables, and the statistical 1 predictions of quantum theory would be reproduced by performing suit- able averages over these hidden variables. A fundamental theorem was proved by Bell [1], who showed that if the constraint of locality was imposed on the hidden variables (namely, if the hidden variables of two distant quantum systems were themselves be separable into two distinct subsets), then there was an upper bound to the correlations of results of measurements that could be performed on the two distant systems.
    [Show full text]