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?