The Logic of Quantum Measurements The Logic of Quantum Measurements - PowerPoint PPT Presentation

The Logic of Quantum Measurements The Logic of Quantum Measurements Data Synthesis San Diego, CA in terms of Conditional Events in terms of Conditional Events Philip Calabrese Philip Calabrese Workshop on Conditionals, Information and Inference (WCII'04) KI-2004 - 27th German Conference on Artificial Intelligence University of Ulm, September 20~24, 2004

Outline of Viewgraphs Outline of Viewgraphs Data Synthesis San Diego, CA • Why Quantum Measurements require a non-Boolean logic such as Hilbert space (3-10) • Measurement disrupts conditions for other measurements • Compatible measurement conditions & Boolean algebra • Conditional Event Algebra, the Algebra of Boolean Fractions (11-16) • Expressing Quantum Logic with Conditional Event Algebra to (17-24) • Deduction with Uncertain Conditionals (25-27) • Consequence Logics, Quantum Logic, and Deductively Closed Sets of Conditionals (28-30) • Summary, Conclusions and References (31-35)

Principles of Quantum Measurement Principles of Quantum Measurement Data Synthesis San Diego, CA • Quanta – discrete mass & energy levels, not continuous variables; E = hf, Δ m = Δ E/c 2 , where h is Plank’s constant • Quantum measurement of a variable A as having value v also disturbs the system of associated variables and determines new probabilities for their values. • Measuring a particle’s position alters its velocity & vice versa • Heisenberg Indeterminacy Principle; Δ x Δ v ≥ h/m → Measurements are therefore also state-change operators. → Some pairs of measurements are therefore incompatible because their implicit apparatus conditions are inconsistent. → Non-Boolean logic needed → Hilbert Space (Yikes!), or … → Conditional Event Algebra

Non-Boolean Logic for QM Non-Boolean Logic for QM Data Synthesis San Diego, CA • H. Putnam ( 1976) : “The whole function of the linear spaces used in quantum mechanics is to provide a convenient mathematical representation of the lattice of physical propositions….” “… (the) lattice of physical propositions is not ‘Boolean’; in particular, distributive laws fail.” • J. S. Bell (1966): “The misuse of the word ‘measurements’ makes it easy to forget” that the results of quantum measurements “have to be regarded as the joint product of ‘system’ and ‘apparatus’, the complete experimental set-up”. Forgetting this has led people to expect that “the ‘results of measurements’ should obey some simple logic in which the apparatus is not mentioned. The resulting difficulties soon show that any such logic is not ordinary logic. It is my impression that the whole vast subject of ‘Quantum Logic’ has arisen in this way from the misuse of a word.”

Context and Experimental Arrangements Context and Experimental Arrangements Data Synthesis San Diego, CA • 1962: A. Messiah: “… evidence obtained under different experimental conditions cannot be comprehended within a single picture.” • 1976: T. Fine: “When … the operators representing observables do not commute then the order of performance of the ‘joint’ measurement affects the outcome, and thus there is said to be no joint observation.” • 1980: D. Bohm: “ … the non-commutativity of two operators is to be interpreted as a mathematical representation of the incompatibility of the arrangements of apparatuses needed to define the corresponding quantities experimentally.” • 2001: J. Hilgevoord: “Since a measuring instrument cannot be rigidly fixed to the spatial reference frame and, at the same time, be movable relative to it, the experiments which serve to precisely determine the position and momentum of an object are mutually exclusive.”

Changing Contexts & Boolean Fractions Changing Contexts & Boolean Fractions Data Synthesis San Diego, CA • 1976: G.M. Hardegree: Quantum logic lacks a “conditional operation by means of which the modus ponens deduction scheme can be incorporated into quantum logic.” • But the algebra of Boolean fractions has a conditional operation; and it also supports deduction using modus ponens [Cal87, p228; Cal02]. • 1985: L.E. Ballentine: “…beware of probability statements expressed as P(X) instead of P(X|C). The second argument may be safely omitted only if the conditional event or information is clear from the context and constant throughout the problem. This is not the case in the double slit example.”

Two-Slit Experiment & Changing Context Two-Slit Experiment & Changing Context Data Synthesis San Diego, CA “The probability of detection at X in the first case (only slit no. 1 open) should be written P(X|C 1 ), where the conditional information C 1 includes (at least) the state function ψ 1 for the particle beam and the state S 1 (only slit no. 1 open). In the second case (only slit no. 2 open) the probability should be written as P(X|C 2 ), where C 2 includes the state function ψ 2 and the screen state S 2 (only slit no. 2 open). In the third case (both slits open) the probability is of the form P(X|C 3 ), where C 3 includes the state function ψ 12 (approximately equal to ψ 1 + ψ 2 but this fact plays no role in our argument) and the screen state S 3 (both slits open). We observe from experiment that P(X|C 3 ) ≠ P(X|C 1 ) + P(X|C 2 ).” (L.E. Ballentine. 1985) Screen S 1 Particle Source X position S 2

Compatibility & Boolean Subdomains Compatibility & Boolean Subdomains Data Synthesis San Diego, CA • C. Piron (1976) defined two propositions b and c to be compatible if the sub-lattice generated by them and their negations is distributive (Boolean). • P. Suppes (1990): “If we avoid noncommuting variables in quantum mechanics, then probability is classical.” • B. Coecke, D. Moore and A. Wilce (2002): “The sub- ortholattice generated by any commuting family of projections is a Boolean algebra.”

Strange Quantum Phenomena Strange Quantum Phenomena Data Synthesis San Diego, CA • Interference Effects (e.g. the 2-slit experiment) • Very fast, perhaps even instantaneous interference • Particles with associated “wave amplitudes” • “Collapse” of wave when particle is measured • “Non-local” effects; particle “entanglement” • Very fast communication and computation potential ? • “… When will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations?” [ C.A. Fuchs, “Quantum mechanics as quantum information”, 2002]

Standard Quantum Mechanics Standard Quantum Mechanics Data Synthesis San Diego, CA Hilbert Space – a complete, normed, infinite dimensional vector space H over the complex numbers • Unit vectors h, k, …represent pure (atomic) physical states • Closed linear subsets A, B represent propositions • A ∧ B = A ∩ B, the Intersection of subspaces A, B • Not A = A ⊥ , the Orthogonal Complement of subspace A • A ∨ B = Smallest closed subspace including A ∪ B • (A ⇒ B) = (A ⊂ B). Implication is subspace inclusion • Alternate Representation. Each linear closed subset A ⊂ H has an associated orthogonal projection A p onto it from H such that ∀ h ∈ H, A p (h) = the member (vector) in A nearest to h.

Truth Values and Operations Truth Values and Operations for Conditional Events for Conditional Events Data Synthesis San Diego, CA • ( B | B ) = {(a|b): a, b in a Boolean algebra B } is called the set of conditionals , "a given b", of B . • Definition: Equivalence (=) of conditionals (a|b) and (c|d): (a|b) = (c|d) means that b = d and a ∧ b = c ∧ d. ⇒ (a | false) = (c | false) no matter what the truth of a and of c. ⇒ (a|b) has just 3 possible truth states or values: • (true | true); (a|b) is true; a is true and b is true • (false | true); (a|b) is false; a is false and b is true • (true | false); (a|b) is inapplicable-undefined (U); b is false • Notation: “or” ( ∨ ), “and” ( ∧ or juxtaposition), “not” ('), “given” (|), “a given b” is sometimes “a if b” or “if b then a”.

Operations Defined on Boolean Fractions Operations Defined on Boolean Fractions Data Synthesis San Diego, CA • not (a|b) = (not a | b) = (a'| b ). Note: P(not a | b) = 1 - P(a|b). • (a|b) or (c|d) = (ab or cd) | (b or d) = (ab ∨ cd) | (b ∨ d) “Given either conditional is applicable, at least one is true” • (a|b) and (c|d) = (a|b) ∧ (c|d) = [ab(c or d') or (a or b')cd] | (b or d) = [abd' ∨ abcd ∨ b'cd] | (b ∨ d) “Given either conditional is applicable, at least one is true while the other is not false.” • (a | b) given (c | d) = (a | b) | (c | d) = (a | b and (c | d) ) = (a | b ∧ (c or not d) ) “Given b and (c|d) are not false, a is true“ • Note: The original system of unconditioned events are the fractions (a | Ω ) where a is any event and Ω is the universal event. In logical notation these are (a | 1).