Quantum Like Decision Theory
Angel’s Meeting MdP - May, 28th 2010
O.G. Zabaleta, C.M. Arizmendi
Can we apply ideas and part of the mathematical formalism of (quantum) physics to describe
perceptions and decisions?
Not: consciousness as an immediate quantum phenomenon
Challenges to Classical Probability
Savage (1954) Sure Thing Principle
• If option A is preferred over B under the state of the world X
• And option A is also preferred over B under the complementary state ~X
• Then option A should be preferred over B even when it is unknown whether state X or ~X
)|1()()|1()()1( AbPAPAbPAPbP
}1,1{ b AA ,
Two mutually disjoint events
If 1)|1( AbP and 1)|1( AbP
1)()()1( APAPbP
Prisoners’ Dilemma
Defect
Defect
Cooperate
Cooperate
Prisoners’ Dilemma
10, 0
5, 5 0, 10
1, 1
Nash Equilibrium Neither player can improve his/her position,
Row Player
)|()()|()()( AbPAPAbPAPbP
}1,1{ b AA , Two mutually disjoint events
Conjunction Fallacy
)|1()()1( AbPAPbP
The Linda Problem
Story: Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
The Linda Problem
Which is more probable?
1. Linda is a bank teller.2. Linda is a bank teller and is active in the feminist movement
85% of those asked chose option 2.
)|()()|()()( AbPAPAbPAPbP
}1,1{ b AA , Two mutually disjoint events
Conjunction Fallacy
)|1()()1( AbPAPbP
AAs
ieAbPAPAbPAPbP )2sin(Re)|()()|()()(
Bistable perception - cup or faces
Bistable perception – mother or daughter
The Necker cubeLouis Albert Necker (1786-1861)
The mental states
state 1 state 2
Rates of perceptive shifts
1
2
t (sec)0 2 4 6 8 18 20 22 24 26 28 30 32 34 3610 12 14 16
J.W.Brascamp et.al, Journal of Vision (2005) 5, 287-298
T=t
Weak Quantum Theory
Observables and States
• Observable: A mathematical object „representing“ a measurable quantity.
• State: A functional (mapping) which associates to each observable a number (expectation value).
• Succesive observations: Product of observables• Commuting: Compatibility• Non-commuting: Complementarity• Complementarity: violate classical concepts
(reality and causation)
Sketch of the axioms of weak QT
• The exist states {z} and observables {A}. Observables act on states (change states).
• Observables can be multiplied (related to successive observations).
• Observables have a “spectrum”, i.e., measurements yield definite results.
• There exists an “identity” observable: the trivial “measurement” giving always the same result.
Complementarity
• Two observables A and B are complementary if they do not commute AB BA .
• Two (sets of) observables A and B are complementary, if they do not commute and if they generate the observable algebra .
• Two (sets of) observables A and B are complementary, if they do not commute on states AB z BA z.
• Two (sets of) observables A and B are complementary, if the eigenstates (dispersion-free states) have a maximal distance.
The Necker-Zeno Model for Bistable Perception
The quantum Zeno effectB. Misra and E.C.G. Sudarshan (1977)
Quantum Zeno effect
Δtt0
T
w(t)
Ttt0
The quantum Zeno effectB. Misra and E.C.G. Sudarshan (1977)
Dynamics:
gtgt
gtgttUgH t
cossini
sinicose)(
01
10 iH
Observation:
10
013
States:
0
1
1
0
Dynamics and observation are complementary
Results of observations
The quantum Zeno effect
The probability that the system is in state |+ at t=0 and still in state |+ at time t is: w(t) = |+|U(t)|+|2 = cos2gt .
t0~1/g is the time-scale of unperturbed time evolution.
The probability that the system is in state |+ at t=0 and is measured to be in state |+ N times in intervals Δt and still in state |+ at time t=N·Δt is given by:
wΔt(t) := w(Δt)N = [cos2gΔt]N
Decay time:
ttgtNg ee 222
t
t
tgtT
20
2
1
Quantum Zeno effect
Δtt0
T
w(t)
Ttt0
The Necker-Zeno modelH. Atmanspacher, T. Filk, H. Römer, Biol. Cyber. 90, 33
(2004)
Mental state 2:Mental state 1:
dynamics „decay“ (continuous change) of a mental state
observation „update“ of one of the mental states
Internal dynamics and internal observation are complementary.
Time scales in the Necker Zeno model
• Δt : internal „update“ time. Temporal separability of stimuli 25-70 ms
• t0 : time scale without updates (“P300”) 300 ms
• T : average duration of a mental state 2-3 s.
Prediction of the Necker-Zeno model:
Ttt0
A first test of the Necker-Zeno model
Tt0
Assumption: for long off-times t0 off-time
Necker-Zeno model predictions for the distribution functions
J.W.Brascamp et.al, Journal of Vision (2005) 5, 287-298
probability density
Cum. probability
Refined model
Modification of
- g g(t)
the „decay“-parameter is smaller in the beginning:
- t t(t)
the update-intervals are shorter in the beginning
Increased attention?
t
g(t), t(t)
Tests for Non-Classicality
Bell‘s inequalitiesJ. Bell, Phys. 1, 195 (1964)
Let Q1, Q2, Q3, Q4 be observables with possible results +1 and –1.
Let E(i,j)=QiQj
Then the assumption of “local realism” leads to
–2 E(1,2) + E(2,3) + E(3,4) – E(4,1) +2
Temporal Bell’s inequalitiesA.J. Leggett, A. Garg, PRL 54, 857 (1985)
-1
1
Let K(ti,tj)=σ3(ti)σ3(tj) be the 2-point correlation function for a measurement of the state, averaged over a classical ensemble of “histories”. Then the following inequality holds:
|K(t1,t2) + K(t2,t3) + K(t3,t4) – K(t1,t4)| 2 .
This inequality can be violated in quantum mechanics, e.g., in the quantum Zeno model.
t
N-(t1,t3) ≤ N-(t1,t2) + N-(t2,t3)
H. Atmanspacher, T. Filk JMP 54, 314 (2010)
p-(t1,t3) ≤ p-(t1,t2) + p-(t2,t3)
w++(t1, t2) = |+|U(t2 – t1)|+|2 = cos2 (g(t2 – t1))
w+-(t1, t2) = |+|U(t2 – t1)|-|2 = sin2 (g(t2 – t1))
p-(2τ) ≤ 2p-(τ)
Temporal Bell’s inequality violated if gτ = π/6 which yields
sin2 (g.2τ) = and sin2 (g.τ) = 43
41
ms157
Caveat
• The derivation of temporal Bell‘s inequalities requires the assumption of „non invasive“ measurements.
(This corresponds to locality in the standard case: the first measurement has no influence on the second measurement.)
Summary and Challenges
• The Necker-Zeno model makes predictions for time scales which can be tested.
• The temporal Bell’s inequalities can be tested.
• Complementarity between the dynamics and observations of mental states is presumably easier to find than complementary observables for mental states.
Summary and Challenges
Quantum Decision Theory is the Decision Theory?
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