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

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The Logic of Quantum Measurements The Logic of Quantum Measurements 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

n Artificial Intelligence

University of Ulm, September 20~24, 2004

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Outline of Viewgraphs Outline of Viewgraphs

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)

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Principles of Quantum Measurement Principles of Quantum Measurement

Quanta – discrete mass & energy levels, not continuous variables;

E = hf, Δm = ΔE/c2, 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; Δx Δv ≥ 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

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Non-Boolean Logic for QM Non-Boolean Logic for QM

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.”

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Context and Experimental Arrangements Context and Experimental Arrangements

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.”

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Changing Contexts & Boolean Fractions Changing Contexts & Boolean Fractions

1976: G.M. Hardegree: Quantum logic lacks a “conditional
peration by means of which the modus ponens deduction

scheme can be incorporated into quantum logic.”

But the algebra of Boolean fractions has a conditional
peration; 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.”

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Two-Slit Experiment & Changing Context Two-Slit Experiment & Changing Context

“The probability of detection at X in the first case (only slit no. 1

pen) should be written P(X|C1), where the conditional information

C1 includes (at least) the state function ψ1 for the particle beam and the state S1 (only slit no. 1 open). In the second case (only slit no. 2 open) the probability should be written as P(X|C2), where C2 includes the state function ψ2 and the screen state S2 (only slit no. 2 open). In the third case (both slits open) the probability is of the form P(X|C3), where C3 includes the state function ψ12 (approximately equal to ψ1 + ψ2 but this fact plays no role in our argument) and the screen state S3 (both slits open). We observe from experiment that P(X|C3) ≠ P(X|C1) + P(X|C2).”

(L.E. Ballentine. 1985)

X position

S1 S2

Particle Source Screen

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Compatibility & Boolean Subdomains Compatibility & Boolean Subdomains

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-
rtholattice generated by any commuting family of

projections is a Boolean algebra.”

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Strange Quantum Phenomena Strange Quantum Phenomena

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]

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Standard Quantum Mechanics Standard Quantum Mechanics

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 Ap onto it from H such that ∀h ∈ H, Ap(h) = the member (vector) in A nearest to h.

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Truth Values and Operations Truth Values and Operations for Conditional Events for Conditional Events

(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”.

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Operations Defined on Boolean Fractions Operations Defined on Boolean Fractions

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).

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Non-Monotonicity - Intuitive in Non-Monotonicity - Intuitive in Compound Conditionals Compound Conditionals

In Boolean algebra, P(A and B) ≤ P(A) for any two events A, B.

But if A = (a|b) and C = (c|d), this may not hold.

Non-monotonicities in logic and probability sound unintuitive when

described in the abstract (E. Adams [Ada66] and Schay [Sch68].)

Consider the following compound conditional bet in the game of rolling

a single normal die once: “Given the roll is less than 5 it will be less than 4, and given the roll is greater than 3 it will be greater than 4”.

The component conditionals have conditional probabilities 3/4 and 2/3

respectively, but the conjunction has conditional probability 5/6.

This can be calculated using the operations on conditionals, but just

check the six possible outcomes of the die roll: They each apply to

ne or the other conditional, but not to both except for “4”, which

yields the only “false” for both. Since the lone false instance contributes “false” to both component conditionals, combining them into one conditional lowers the overall probability of false when the context is expanded.

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Non-Commutativity of Conditioning Non-Commutativity of Conditioning

Let (a|b) represent the act of measuring proposition “a” under

experimental condition “b”, and similarly for conditional (c|d).

If b ∧ d = bd ≠ 0, then a and c can be simultaneously measured:

((a|b) ∧ (c|d) | bd) = (a|bd) ∧ (c|bd) = (ac | bd)

But if meeting both conditions is impossible, then conditioning by b

allows measurement of a: ((a|b) ∧ (c|d) | b) = (a|b) ∧ (c|0) = (a|b),

r conditioning by d allows measurement of c:

((a|b) ∧ (c|d) | d) = (a|0) ∧ (c|d) = (c|d)

But after conditioning by b, additional conditioning by d to measure c

yields ((ac | b) | d) = (ac|bd) = (ac|0) = U, undefined (and not verifiable)

2-valued logic cannot represent this situation.

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Distributivity and Conditionals Distributivity and Conditionals

Theorem on Distributivity.

(a|b) ∧ [(c|d) ∨ (e|f)] = (a|b)(c|d) ∨ (a|b)(e|f) if and only if (ab)(e'f) ≤ d and (ab)(c'd) ≤ f. That is, conjunction distributes over disjunction just in case the truth of the outside conditional and the falsity of one of the inside conditionals implies the other inside conditional is applicable.

Corollary on Distributivity.

(a|b) ∨ [(c|d) ∧ (e|f)] = [(a|b) ∨ (c|d)] ∧ [(a|b) ∨ (e|f)] if and only if (a'b)(ef) ≤ d and (a'b)(cd) ≤ f. That is, disjunction distributes over conjunction if and only if whenever the outside conditional is false and one inside conditional is true then the other inside conditional is applicable.

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Superposition formulas Superposition formulas

Disjunction. For any conditionals (a|b) & (c|d),

(a|b) ∨ (c|d) = (a|b)(b| b ∨ d) ∨ (c|d)(d| b ∨ d), and P((a|b) ∨ (c|d)) = P(a|b)P(b| b ∨ d) + P(c|d)P(d| b ∨ d) – P(ac|bd)P(bd | b ∨ d).

Conjunction.

(a|b) ∧ (c|d) = (a|b)(bd'| b ∨ d) ∨ (c|d)(b'd| b ∨ d) ∨ (acbd | b ∨ d), and P((a|b) ∧ (c|d)) = P(a|bd')P(bd'| b ∨ d) + P(c|b'd)P(b'd| b ∨ d) + P(acbd | b ∨ d).

If (a|bd) = (c|bd) then (a|b) ∨ (c|d) = (a|b) ∧ (c|d).
If (abcd | b ∨ d) = (0 | b ∨ d) then

P((a|b) ∨ (c|d)) = P(a|b)P(b | b ∨ d) + P(c|d)P(d | b ∨ d).

If also a=c, then (a | b ∨ d) = (a|b)(b | b ∨ d) ∨ (a|d)(d | b ∨ d), and

P(a | b ∨ d) = P(a|b)P(b | b ∨ d) + P(a|d)P(d | b ∨ d).

For a partition of u = ∨iui, this is just P(a|u) = Σi P(a|ui)P(ui|u)

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Orthogonality for Conditional Events Orthogonality for Conditional Events

Definition. Two conditionals (a|b) and (c|d) are orthogonal

(disjoint) if (a|b) ∧ (c|d) = (0 | b ∨ d). Notation: (a|b) ⊥ (c|d).

Theorem. Two conditionals are orthogonal, (a|b) ⊥ (c|d), if and
nly if ab ≤ c'd and cd ≤ a'b.
Proof: This follows from (a|b) ∧ (c|d) = (ab(c ∨ d') ∨ cd(a ∨ b') |

b ∨ d) = (0 | b ∨ d) iff (ab(c ∨ d') = 0 and cd(a ∨ b') = 0.]

Whenever one conditional is true the other conditional must be

false, and not (merely) inapplicable.

If ω is any instance where (a|b) is applicable but an orthogonal

conditional is not applicable at ω, then (a|b) must be false at ω.

Three or more conditionals form an orthogonal set of conditionals

(OSC) by being pairwise orthogonal.

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Simultaneous Verifiability in Logic Simultaneous Verifiability in Logic

Definition. Simultaneous Verifiability (V. S. Varadarajan)

Two members a and c of a logic are simultaneously verifiable, a ↔ c, if there are members a1, c1, and e of the logic such that a1 ∧ c1 = 0 and a1 ∧ e = 0 and e ∧ c1 = 0, and a = a1 ∨ e and c = e ∧ c1.

Definition. Simultaneous Verifiability for Conditionals. Let (a|b)

and (c|d) be two conditionals. Then (a|b) ↔ (c|d) means there exist 3

rthogonal conditionals (α|β), (χ|δ) and (e|f) such that

(a|b) = (α|β) ∨ (e|f) and (c|d) = (χ|δ) ∨ (e|f). a1 c1 a c e

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Simultaneous Verifiability of Conditionals Simultaneous Verifiability of Conditionals

Theorem (Calabrese). In the conditional event logic two

conditionals (a|b) and (c|d) are simultaneously verifiable, (a|b) ↔ (c|d), if and only if ab ≤ d and cd ≤ b. That is, two conditionals are simultaneously verifiable if and only if the truth of one conditional implies the applicability of the other conditional.

Corollary. (a|b) ↔ (c|d) iff (a|b) ∧ (c|d) = (abcd | b ∨ d)

b d bd c a ab cd b d bd c a b'cd abd'

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Orthogonal Expansion Theorem Orthogonal Expansion Theorem

If (a|b) and (c|d) are simultaneously verifiable, then {(ac'|b), (ac|bd), (a'c|d)} are three

rthogonal

conditionals for which (a|b) = (ac'|b) ∨ (ac|bd), (c|d) = (ac|bd) ∨ (a'c|d), and so (a|b) ∨ (c|d) = (ac'|b) ∨ (ac|bd) ∨ (a'c|d), with P((a|b) ∨ (c|d)) = P(ac'|b)P(b|b ∨ d) + P(ac|bd)P(bd|b ∨ d) + P(a'c|d)P(d|b ∨ d).

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Quantum Falsifiability & Compatibility Quantum Falsifiability & Compatibility

Definition. (a|b) & (c|d) are simultaneously falsifiable if and only if

their negations, (a'|b) & (c'|d), are simultaneously verifiable.

Corollary. In the conditional event logic two conditionals (a|b) &

(c|d) are simultaneously falsifiable if and only if a'b ≤ d and c'd ≤ b.

Corollary. Two conditionals (a|b) & (c|d) are both simultaneously

verifiable and simultaneously falsifiable if and only if b = d. Then (a|b) & (c|d) are said to be compatible.

Theorem. (P. Calabrese): Two conditionals (a|b) and (c|d) are in a

common Boolean sub-algebra of the algebra of conditionals if and

nly if b = d.
Corollary. Two conditionals (a|b) & (c|d) are compatible if and
nly if they are in a common Boolean sub-algebra.

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Complete Set of Orthogonal Conditionals Complete Set of Orthogonal Conditionals

Definition. Complete Set of Orthogonal Conditionals (COSC).

A set {(ai|bi)} of pairwise orthogonal conditionals is complete if (c|d) ⊥ (ai|bi) for all i ⇒ (c|d) = (0|d). That is, if only zero conditionals are orthogonal to all of the members

f an OSC, then it is complete.
Lemma. If J = {(ai|bi)} is a set of pairwise orthogonal conditionals

then ∨i aibi ≤ ∧ibi and ∧ibi - ∨iaibi = ∧i ai'bi.

Completeness Characterization Theorem. An orthogonal set of

conditionals J is complete if and only if ∧i ai'bi = 0.

Corollary. An orthogonal set of conditionals J = {(ai|bi)} is

complete if and only if ∨i (ai|bi) = (∧i bi | ∨i bi).

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Complete Set of Orthogonal Conditionals(2) Complete Set of Orthogonal Conditionals(2)

Expansion Theorem. Let J = {(ai | bi)} be an OSC and set b =∨ibi,

then P(∨i (ai | bi)) = ∑i P(ai | bi)P(bi | b).

Representation Theorem. If J = {(ai | bi)} is a COSC then any

conditional (e|f) = ∨i (e|f)(ai|bi) if and only if ∨i bi ≤ f.

Completion Theorem. Every OSC can be completed.

Proof: If ∧i ai'bi ≠ 0 then (e|f) = (∧i ai'bi | ∧i bi) is a non-zero conditional that completes J.

Restriction Theorem. If J is a COSC, and f is any proposition,

then (J | f) = {(ai | bif)} is also a COSC.

Second Representation Theorem. If (e|f) is any conditional and J

= {(ai|bi)} is any COSC, then (e|f) = ∨i (ai | bif)(e|f) = ∨i (e(ai|bi) | f)

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Complete Set of Orthogonal Conditionals(3) Complete Set of Orthogonal Conditionals(3)

In Hilbert space, the “amplitude” or logical representative of starting

in state d and going to state c by way of some events bi is 〈cd〉 = ∑i 〈cbi〉〈 bid 〉 and P〈cd〉 = ∑i 〈cbi〉〈 bid 〉2.

Third Representation Theorem. If (c|d) is a conditional and {bi} is

a partition of 1, then (c|d) = ∨i (c | bid)(bid | d) and P(c|d) = ∑i P(c | bid)P(bid | d)

Corollary. If c is conditionally independent of d given bi, for all i,

that is, if P(c | bid) = P(c|bi), then P(c|d) = ∑i P(c | bi)P(bi | d)

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Deduction with Boolean Fractions Deduction with Boolean Fractions

(a|b) ≤? (c|d) What is implied? a ∧ b ≤ c ∧ d Truth (tr) a ∨ b' ≤ c ∨ d' Non-falsity (nf) b ≤ d (b implies d) Applicability (ap) b' ≤ d' (not b implies not d) Inapplicability (ip)

Combinations of these four types of implication are

also examples of implications, deductive relations

Probability implications like P(a ∧ b) ≤ P(c ∧ d)

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Elementary Implications: tr - Implication of Truth (≤tr)

(a|b) ≤tr (c|d) iff ab ≤ cd nf - Implication of Non-Falsity (≤nf) (a|b) ≤nf (c|d) iff (a ∨ b') ≤ (c ∨ d') ap - Implication of Applicability (≤ap) (a|b) ≤ap (c|d) iff b ≤ d ip - Implication of Inapplicability (≤ip) (a|b) ≤ip (c|d) iff b' ≤ d'

Deductive Relations on Conditionals Deductive Relations on Conditionals

Trivial Implications:

1 - Implication of Identity (≤1)

(a|b) ≤1 (c|d) iff (a|b) = (c|d) ec - Implication of Equal Conditions (≤ec) (a|b) ≤ec (c|d) iff b = d 0 - Universal Implication (≤0) (a|b) ≤0 (c|d) for all (a|b) & (c|d)

Four Elementaries Combined

bo - Boolean Deduction (B | fixed b) (≤bo) (a|b) ≤bo (c|d) iff b = d and ab ≤ cd

Two Elementaries Combined:

∨ - Disjunctive Implication (≤∨)

pm - Probabilistically Monotonic Implication (≤pm) ∧ - Conjunctive Implication (≤∧)

Three Elementaries Combined:

m∨ - (Probabilistically) Monotonic and

Applicability Implication (≤m∨) m∧ - (Probabilistically) Monotonic and Inapplicability Implication (≤m∧)

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Deductively Closed Sets of Conditionals Deductively Closed Sets of Conditionals

A subset H of conditionals is deductively closed with respect to a

deductive relation ≤x if H has the following conjunctive and deductive closure properties: 1) If (a|b) ∈ H and (c|d) ∈ H then (a|b) ∧ (c|d) ∈ H, 2) If (a|b) ∈ H and (a|b) ≤x (c|d) then (c|d) ∈ H.

Let Hx(J) denote the smallest deductively closed set (DCS) of

conditionals with respect to ≤x that includes an initial set J of

conditionals. Hx(J) is the set of implications of J with respect to ≤x.
With respect to the deductive relation ≤pm,
(a|b) ≤pm (e|f) does not imply (a|b) ∧ (c|d) ≤pm (e|f);
(a|b) ≤pm (e|f) does not imply (a|b) ≤pm (c|d) ∨ (e|f).
Hpm{(a|b), (c|d)} = Hpm{(a|b), (c|d), (a|b)(c|d)}

= Hpm(a|b) ∪ Hpm(c|d) ∪ Hpm((a|b)(c|d)). [Cal02, pp173-5].

H∧{(a|b), (c|d)} = H∧((a|b)(c|d)).

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Consequence Logics* & Conditional Events

Definition. C(A) is the set of all deductive consequences or

conclusions of a subset A of (possibly non-Boolean) propositions L.

If C has the following two properties then it is said to be a C-logic.

Inclusion: A ⊆ L ⇒ A ⊆ C(A) Cumulativity: A, B ⊆ L, A ⊆ B ⊆ C(A) ⇒ C(A) = C(B) Idempotence: A ⊆ L, C(C(A)) = C(A) Cautious Monotonicity: A, B ⊆ L, A ⊆ B ⊆ C(A) ⇒ C(A) ⊆ C(B) 2-Loop: A ⊆ C(B), B ⊆ C(A) ⇒ C(A) = C(B)

The set A is consistent if C(A) ≠ L and maximal if adding any other

proposition makes it inconsistent.

For all deductive relations ≤x, Hx satisfies these properties. C = Hx

* [Eng02]; [Leh02]

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Quantum Logic as Consequence Logics Quantum Logic as Consequence Logics

Quantum logics are Consequence Logics. [Eng02, p64]

Starting with a quantum state s and making a measurement of a quantum

bservable A and getting a value λ sends the system into a state t in which A=λ,

and also in general the values of some other variables B, … are now determined with certainty by the state t. (state t)(B=ρ)(A=λ) ∈ C(state s, meas(A))

So B=ρ is a consequence of measuring A in state s and getting A=λ, and also

sending the system into some other state such that B=ρ is certain.

Assume a classical extension set M of models u, v, … of L, and a “satisfaction”

binary relation |= that expresses by u |= a whether “a is true in model u”. Mod(A) = {u ∈ M: u |= a for all a ∈ A}.

u |= (a ∧ b) if and only if u |= a and u |= b,
u |= a' if and only if u |≠ a, that is, “a is not true in model u”
Prop(u) = {a ∈ L : u |= a } = all propositions of L that are true in model u
P(U) = {a ∈ L : u |= a for all u in U} = all propositions true in every model in U

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Quantum Operations as Consequences Quantum Operations as Consequences

Theorem (Lehmann). Let a and b be propositions in L and A ⊆ L.

Assuming the models M of L behave classically,

1) C(A, a ∧ c) = C(A, a, c) - The consequences of (a ∧ c) and A are the same as the consequences of the set {a, c} with A. 2) C(A, a, a') = L - the consequences with A of a and a' are the whole of L. 3) C(A, a') = L ⇒ a ∈ C(A) - if A and a' are inconsistent then a must be a consequence of A.

Quantum logic satisfies 1) and 2) but not 3).
If L is the algebra of conditionals (B|B), A ⊆ L and C = Hx, then

4) Hx(A, (a|b) ∧ (c|d)) = Hx(A, (a|b), (c|d)) only for ≤x, where x ∈ {∧, m∧, bo}. 5) Hx((a|b), (a|b)') = Hx(0|b) = {(c|d): (0|b) ≤x (c|d)} = (B|B) only for x = tr, but for all x, Hx(0|b) includes the whole Boolean sub-algebra (B|b). 6) If Hx(A, (a|b)') includes (B|b), then (a|b) ∈ Hx(A) is true for x ∈ {ap, tr, nf, ∨, pm, m∨} but not for x ∈ {ip, ∧, m∧, bo}.

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Summary & Conclusions Summary & Conclusions

With present methods, measuring the value of a quantum

variable disturbs the system and changes the state of the probabilities for other variables.

Measuring position changes velocity, and vice versa
The quantum logic of changing measurement conditions is not wholly Boolean
Linear algebra was the closest existing algebraic system capable of holding the

relationships of quantum logic

Quantum Logic can be more naturally expressed in terms of Boolean fractions, the

algebra of conditional events, which is also a type of Consequence Logic.

The non-commutativity of quantum operators arises from the non-commutativity
f iterative conditioning when the two conditions are inconsistent.
Quantum anomalies can arise from the non-monotonicity of the non-distributive
perations on, and surprising deductions, from conditionals.
Quantum Orthogonality & Superpositions can be expressed with conditionals.
Simultaneous Verifiability & Compatibility have been completely modeled.
Boolean subspaces of (B|B) correspond to sets of compatible quantum variables.
The quantum operations are expressible with conditional event algebra.

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References References

E. W. Adams, A primer of Probability Logic, CSLI

Publications (1998).

L. E. Ballentine, “Probability theory in quantum mechanics”, Am. J.
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