The Foundations of Probability Theory and Quantum Theory

Transcription

1 The Foundations of Probability Theory and Quantum Theory Kevin H. Knuth Departments of Physics and Informatics University at Albany Kevin H. Knuth Departments of Physics and Informatics University at Albany (SUNY)

2 Partially Ordered Sets A partially ordered set is a set along with a binary ordering relation, eg. S L L S R S L R Parts of a Bridge Photograph by Barbara Maddrell, National Geographic Image Collection

3 Posets versus Lattices Lattices are posets where every join and meet is unique. Poset Lattice (x y) x y (x y)

4 Chains and Antichains Chains are totally ordered Antichains are unordered

5 Examples

6 More Complex Examples Subset Inclusion x y z Partitions a b c x y x z y z a bc b ac c ab x y z abc

7 Lattices are Algebras Structural Viewpoint Operational Viewpoint a b a a b b b a

8 Qualitative to Quantitative Algebra Calculus

9 Classical Inference

10 States apple banana cherry states of the contents of my grocery basket

11 Statements powerset { a, b, c } { a, b } { a, c } { b, c } subset inclusion a b c { a } { b } { c } states of the contents of my grocery basket statements about the contents of my grocery basket statements describe potential states

12 Implication { a, b, c } implies { a, b } { a, c } { b, c } { a } { b } { c } statements about the contents of my grocery basket ordering encodes implication

13 Inference { a, b, c } { a, b } { a, c } { b, c } { a } { b } { c } Quantify to what degree knowing that the system is in one of three states {a, b, c} implies knowing that it is in some other set of states statements about the contents of my grocery basket inference works backwards

14 Quantification

15 Quantification Quantify the partial order by assigning real numbers to the elements { a, b, c } { a, b } { a, c } { b, c } { a } { b } { c } Any quantification must be consistent with the lattice structure. Otherwise, information about the partial order is lost.

16 Local Consistency Any general rule must hold for special cases Look at special cases to constrain general rule x y Enforce local consistency f(x y) f(x) and f(y) f : x xl y R I This implies that: f(x y) S[f(x), f(y)] where S is an unknown function to be determined.

17 Associativity of Join Write the same element two different ways x (y z) (x y) z which implies S[f(x), S[f(y),f(z)]] S[S[f(x), f(y)],f(z)] Note that the unknown function S is nested in two distinct ways, which reflects associativity

18 Associativity Equation S[f(x), S[f(y),f(z)]] S[S[f(x), f(y)],f(z)] The general solution (Aczel 1966; Knuth & Skilling 2012) is: F(S[f(x), f(y)]) F(f(x)) F(f(y)) where F is an arbitrary function. Define v(x) = F(f(x)) so that we have straightforward summation. v(x y) v(x) v(y) Derivation of the Summation Axiom in Measure Theory

19 Valuation VALUATION v : x L If y x then v(y) v(x) R x y x y v(x y) v(x) v(y)

20 General Case x y x y x y z

21 General Case x y x y x y z v(y) v(x y) v(z)

22 General Case x y x y x y z v(y) v(x y) v(z) v(x y) v(x) v(z)

23 General Case x y x y x y z v(y) v(x y) v(z) v(x y) v(x) v(z) v(x y) v(x) v(y) v(x y)

24 Sum Rule v(x y) v(x) v(y) v(x y) v(x) v(y) v(x y) v(x y) symmetric form (self-dual)

25 ), min( ), max( y x y x y x ), ( ) ( ) ( ) ; ( Y X H Y H X H Y X I ) ( ) ( ) ( ) ( i y x p i y p i x p i y x p )), log(lcm( ) log( ) log( )), log(gcd( y x y x y x Sum Rule F E V

26 Lattice Products x = Direct (Cartesian) product of two spaces

27 Direct Product Rule The lattice product is also associative A(BC) (AB) C After the sum rule, the only freedom left is rescaling v((a, b)) v(a) v(b) which is again summation (after taking the logarithm)

28 Context and Bi-Valuations BI-VALUATION w : x, il I R Bi-Valuation w(x i) Context i is explicit v i (x) Measure of x with respect to Context i Valuation v(x) Context i is implicit Bi-valuations generalize lattice inclusion to degrees of inclusion

29 Context Explicit Sum Rule w(x i) w(y i) w(x y i) w(x y i) Direct Product Rule w((a, b) (i, j)) w(a i) w(b j)

30 Associativity of Context =

31 Chain Rule c b w(a c) w(a b) w(b c) a

32 Lemma w(x x) w(y x) w(x y x) w(x y x) Since x x and x x y, w(x x) = 1 and w(x y x) = 1 x y x y w(y x) w(x y x) x y

33 Extending the Chain Rule w(x y z x) w(x y x)w(x y z x y) x y z x y y z x y z

34 Extending the Chain Rule w(x y z x) w(x y x)w(x y z x y) w(y z x) w(y x)w(z x y) x y z x y y z x y z

35 Extending the Chain Rule w(x y z x) w(x y x)w(x y z x y) w(y z x) w(y x)w(z x y) x y z x y y z x y z

36 Extending the Chain Rule w(x y z x) w(x y x)w(x y z x y) w(y z x) w(y x)w(z x y) x y z x y y z x y z

37 Extending the Chain Rule w(x y z x) w(x y x)w(x y z x y) w(y z x) w(y x)w(z x y) x y z x y y z x y z

38 Constraint Equations Sum Rule w(x i) w(y i) w(x y i) w(x y i) Direct Product Rule w((a, b) (i, j)) w(a i) w(b j) Product Rule w(y z x) w(y x)w(z x y)

39 Bayes Theorem Commutativity of the product leads to Bayes Theorem w(x y i) w(y x i) w(x i) w(y i) w(x y) w(y x) w(x i) w(y i) Bayes Theorem involves a change of context.

40 Bayesian Probability Theory Sum Rule p(x y i) p(x i) p(y i) p(x y i) Direct Product Rule p(a, b i, j) p(a i) p(b j) Product Rule p(y z x) p(y x) p(z x y) Bayes Theorem p(x i) p(x y) p(y x) p(y i)

41 Inference { a, b, c } { a, b } { a, c } { b, c } { a } { b } { c } statements Given a quantification of the join-irreducible elements, one uses the constraint equations to consistently assign any desired bi-valuations (probability)

42 Quantum Inference

43 The Goal of QM Here we focus on experimental setups that lead to measurement sequences The ultimate goal is to compute the probability of statements about these measurement sequences

44 The QMical Question To what degree does knowing that we observe any one of a large set of possible measurement sequences imply that we know that we observe a particular measurement sequence?

45 Measurement Sequences Three measurements are made in succession with outcomes x i, x 1, x f, respectively x i x 1 x f A = [x i, x 1, x f ] There is no explicit notion of time, only an notion of an order in which events occur.

46 A = [x i, x L, x f ] Slit Experiment

47 Slit Experiment A = [x i, x L, x f ] B = [x i, x R, x f ]

48 Relating Measurement Sequences C = [x i, (x L,x R ), x f ] A = [x i, x L, x f ] B = [x i, x R, x f ]

49 Parallel Combination C A B A B = C

50 A = [x i, x 1 ] Combining in Series

51 Combining in Series A = [x i, x 1 ] B = [x 1, x f ]

52 Combining in Series A = [x i, x 1 ] B = [x 1, x f ] C = [x i, x 1, x f ] =

53 Series Combination A B C = A B = C

54 Algebraic Relations A B = B A A (B C) = (A B) C Commutativity of Associativity of A (B C) = (A B) C Associativity of A (B C) = (A B) (A C) (B C) A = (B A) (C A) Distributivity over

55 measurement sequences form a poset Think of slits as filters Knuth 2003

56 measurement sequences form a poset This is in contrast to the antichain of states in classical inference Think of slits as filters Knuth 2003

57 Double-Slit Experiment x f L R x i L LR R powerset {L, R, LR} {L, R} {L, LR} {R, LR} {L} {R} {LR}

58 Quantification

59 The Pair Postulate Each sequence of measurement outcomes obtained in a given experiment is represented by a pair of real numbers, where the probability associated with this sequence is a continuous, nontrivial function of both components of this real number pair. A a a a 1 2

60 Why Pairs? The goal is to quantify the poset of measurement sequences. A scalar quantification simply ranks elements on a one-dimensional scale. This will be done with statements when we compute probabilities. But for now, we wish to maintain some of the richness of the poset structure.

61 Pairs Represent Sequences Sequences Pairs A B = B A A (B C) = (A B) C A (B C) = (A B) C a b = b a a (b c) = (a b) c a (b c) = (a b) c A (B C) = (A B) (A C) (B C) A = (B A) (C A) a (b c) = (a b) (a c) (b c) a = (b a) (c a)

62 Associativity of a (b c) = (a b) c Aczél and Hosszú 1956 F(a b) = F(a) + F(b)

63 Sum Rule for Pairs Without any loss of generality a 1 a 2 b 1 b 2 = a 1 + b 1 a 2 + b 2 The only freedom left is a real invertible linear transform x x with SV TU S U T x V x 1 2

64 Distributivity of over Using the sum rule and distributivity (a + b) c = (a c) + (b c) a (b + c) = (a b) + (a c) a b has a bilinear multiplicative form a 1 a 2 b 1 b 2 = γ 1 a 1 b 1 + γ 2 a 1 b 2 + γ 3 a 2 b 1 + γ 4 a 2 b 2 γ 5 a 1 b 1 + γ 6 a 1 b 2 + γ 7 a 2 b 1 + γ 8 a 2 b 2

65 Associativity of Using the associativity along with our freedom to apply a linear transform C1 a 1 a 2 b 1 = a 1b 1 a 2 b 2 b 2 a 1 b 2 + a 2 b 1 C2 a 1 a 2 b 1 b 2 = a 1 b 1 a 1 b 2 + a 2 b 1 C3 a 1 a 2 b 1 b 2 = a 1 b 1 a 2 b 2 N1 a 1 a 2 b 1 b 2 = a 1 b 1 a 1 b 2 N2 a 1 a 2 b 1 b 2 = a 1 b 1 a 2 b 1

66 Probability of a Sequence We will map these sequences to statements about sequences, and in doing so, will assign a probability to each sequence of measurements. P(A) Pr(m n,mn 1,,m2 m 1) Our Pair Postulate dictates that this be a continuous real-valued function of the pair that depends non-trivially on both components of the pair. P(A) p(a) Furthermore, since we do not at the outset indicate which component of the pair is which, there must exist a representation such that the probability is symmetric with respect to pair-interchange p(a 1,a2 ) p(a2,a1 )

67 Direct Product Consider A=[m 1,m 2 ] and B=[m 2,m 3 ] so that C=[m 1,m 2,m 3 ] and By the product rule of probability P(C) Pr(m 3,m2 m 1) P(C) Pr(m3 m 2,m 1) Pr(m2 m 1) Measurement m 2 overrides all information obtained from m 1 so that P(C) Pr(m3 m2) Pr(m2 m 1) P(B) P(A) Therefore p(a b) = p(a) p(b)

68 Results from Probability C1 a 1 a 2 b 1 = a 1b 1 a 2 b 2 b 2 a 1 b 2 + a 2 b 1 p(a) = a a 2 2 α 2 C2 a 1 a 2 b 1 b 2 = a 1 b 1 a 1 b 2 + a 2 b 1 p(a) = a 1 α e βa 2 a 1 C3 a 1 a 2 b 1 b 2 = a 1 b 1 a 2 b 2 p(a) = a 1 α a 2 β N1 a 1 a 2 b 1 b 2 = a 1 b 1 a 1 b 2 p(a) = a 1 α N2 a 1 a 2 b 1 b 2 = a 1 b 1 a 2 b 1 p(a) = a 1 α

69 Results from Probability C1 a 1 a 2 b 1 = a 1b 1 a 2 b 2 b 2 a 1 b 2 + a 2 b 1 p(a) = a a 2 2 α 2 C2 a 1 a 2 b 1 b 2 = a 1 b 1 a 1 b 2 + a 2 b 1 p(a) = a 1 α e βa 2 a 1 C3 a 1 a 2 b 1 b 2 = a 1 b 1 a 2 b 2 p(a) = a 1 α a 2 β N1 a 1 a 2 b 1 b 2 = a 1 b 1 a 1 b 2 p(a) = a 1 α N2 a 1 a 2 b 1 b 2 = a 1 b 1 a 2 b 1 p(a) = a 1 α symmetry rules out three cases

70 Summation Clearly, when performing measurements in parallel, probabilities do not sum in general. However, if these measurements are sufficiently disjoint, one would expect that at least sometimes, probabilities should sum. This rules out C3 and fixes α=2 in case C1 so that a 1 a 2 b 1 b 2 = a 1b 1 a 2 b 2 a 1 b 2 + a 2 b 1 p(a) = a a 2 2

71 Manipulating Amplitude Pairs Given that quantum measurement sequences are quantified by pairs of real numbers, obey the requisite symmetries of combination in series and parallel and are consistent with probabilities of statements about sequences. we have derived Feynman Rules Born Rule a 1 a 2 b 1 b 2 = a 1 + b 1 a 2 + b 2 p(a) = a a 2 2 a 1 a 2 b 1 b 2 = a 1b 1 a 2 b 2 a 1 b 2 + a 2 b 1

72 Relationships spaces quantification

73 Feynman Mirror Example Feynman Mirror x i x f states quantified by a pair of real numbers: amplitudes a b c [x i, (a,b,c), x f ] [x i, (a,b), x f ] [x i,(a,c),x f ] [x i,(b,c),x f ] [x i, a, x f ] [x i,b,x f ] [x i,c,x f ] assign amplitudes to join-irreducible elements Measurement sequences

74 Feynman Mirror Example Feynman Mirror x i x f states quantified by a pair of real numbers: amplitudes a b c amplitudes combine by rules of complex arithmetic sum and product rules used to compute amplitudes of interest [x i, (a,b,c), x f ] [x i, (a,b), x f ] [x i,(a,c),x f ] [x i,(b,c),x f ] [x i, a, x f ] [x i,b,x f ] [x i,c,x f ] Measurement sequences

75 Feynman Mirror Example Feynman Mirror x i x f states quantified by a pair of real numbers: amplitudes a b c statement lattice is the powerset of states Born Rule gives probabilities of atomic statements [x i, (a,b,c), x f ] [x i, (a,b), x f ] [x i,(a,c),x f ] [x i,(b,c),x f ] [x i, a, x f ] [x i,b,x f ] [x i,c,x f ] Measurement sequences

76 Double-Slit Experiment x f L R x i L LR R powerset {L, R, LR} {L, R} {L, LR} {R, LR} {L} {R} {LR}

77 Double-Slit Experiment x f L R One slit Open Which One?? x i {L, R, LR} z L + z R LR powerset z L 2 + z R 2 {L, R} {L, LR} {R, LR} L R z L z R {L} {R} {LR} z L 2 z R 2 z L +z R 2 Both slits Open

78 Conclusions The complex Feynman formalism is derived from the Pair Postulate, and basic symmetries of combining measurement sequences in series and parallel Quantum Theory is not distinct from Probability Theory, rather, it relies on it! Quantum Inference differs from Classical Inference in that the measurement sequences form posets, whereas classically we deal with mutually exclusive, exhaustive states that form an antichain Quantum Theory and Probability Theory both arise from quantifying fundamental relationships represented by lattices and posets

79 More Information Knuth K.H., Skilling J Foundations of Inference. Axioms 1(1), 38-73; Goyal P., Knuth K.H., Skilling J Why Quantum Theory is Complex, Phys. Rev. A 81, Goyal P., Knuth K.H Quantum theory and probability theory: their relationship and origin in symmetry, Symmetry 3(2): Knuth K.H., Bahreyni N The Physics of Events: A Potential Foundation for Emergent Space- Time. Knuth K.H Inferences about Interactions: Fermions and the Dirac Equation. MaxEnt arxiv: [quant-ph]