Quantum Like Decision Theory Angel’s Meeting MdP - May, 28 th 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

Two mutually disjoint events Ifand

Prisoners’ Dilemma

Defect Cooperate Prisoners’ Dilemma 10, 0 5, 50, 10 1, 1 Nash Equilibrium Neither player can improve his/her position, Row Player

Two mutually disjoint events Conjunction Fallacy

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.

Two mutually disjoint events Conjunction Fallacy

Bistable perception - cup or faces

Bistable perception – mother or daughter

The Necker cube Louis Albert Necker ( )

The mental states state 1state 2

Rates of perceptive shifts 1 2 t (sec) J.W.Brascamp et.al, Journal of Vision (2005) 5, 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 effect B. Misra and E.C.G. Sudarshan (1977)

Quantum Zeno effect Δt t0t0 T w(t)

The quantum Zeno effect B. Misra and E.C.G. Sudarshan (1977) Dynamics: Observation: States: 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 = cos 2 gt. t 0 ~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: w Δt (t) := w(Δt) N = [cos 2 gΔt] N Decay time:

Quantum Zeno effect Δt t0t0 T w(t)

The Necker-Zeno model H. 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  ms t 0 : time scale without updates (“P300”)  300 ms T : average duration of a mental state  2-3 s. Prediction of the Necker-Zeno model:

A first test of the Necker-Zeno model Assumption: for long off-times t 0  off-time

Necker-Zeno model predictions for the distribution functions J.W.Brascamp et.al, Journal of Vision (2005) 5, 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 inequalities J. Bell, Phys. 1, 195 (1964) Let Q 1, Q 2, Q 3, Q 4 be observables with possible results +1 and –1. Let E(i,j)=  Q i Q j  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 inequalities A.J. Leggett, A. Garg, PRL 54, 857 (1985) 1 Let K(t i,t j )=  σ 3 (t i )σ 3 (t j )  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(t 1,t 2 ) + K(t 2,t 3 ) + K(t 3,t 4 ) – K(t 1,t 4 ) |  2. This inequality can be violated in quantum mechanics, e.g., in the quantum Zeno model. t

N - (t 1,t 3 ) ≤ N - (t 1,t 2 ) + N - (t 2,t 3 ) H. Atmanspacher, T. Filk JMP 54, 314 (2010)

p - (t 1,t 3 ) ≤ p - (t 1,t 2 ) + p - (t 2,t 3 ) w ++ (t 1, t 2 ) = |  +|U(t 2 – t 1 )|+  | 2 = cos 2 (g(t 2 – t 1 )) w +- (t 1, t 2 ) = |  +|U(t 2 – t 1 )|-  | 2 = sin 2 (g(t 2 – t 1 )) p - (2τ) ≤ 2p - (τ) Temporal Bell’s inequality violated if gτ = π/6 which yields sin 2 (g.2τ) = and sin 2 (g.τ) =

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?