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Quantum Like Decision Theory

Quantum Like Decision Theory. O.G. Zabaleta , C.M. Arizmendi. Angel’s Meeting MdP - May, 28 th 2010. 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.

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Quantum Like Decision Theory

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  1. Quantum Like Decision Theory O.G. Zabaleta, C.M. Arizmendi Angel’s Meeting MdP - May, 28th 2010

  2. 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

  3. Challenges to Classical Probability

  4. 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

  5. Two mutually disjoint events If and

  6. Prisoners’ Dilemma

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

  8. Conjunction Fallacy Two mutually disjoint events

  9. 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.

  10. 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.

  11. Conjunction Fallacy Two mutually disjoint events

  12. Bistable perception - cup or faces

  13. Bistable perception – mother or daughter

  14. The Necker cube Louis Albert Necker (1786-1861)

  15. The mental states state 1 state 2

  16. Rates of perceptive shifts 2 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 t (sec) T=t J.W.Brascamp et.al, Journal of Vision (2005) 5, 287-298

  17. Weak Quantum Theory

  18. 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)

  19. 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.

  20. 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.

  21. The Necker-Zeno Model for Bistable Perception

  22. The quantum Zeno effectB. Misra and E.C.G. Sudarshan (1977)

  23. Quantum Zeno effect w(t) Δt t0 T

  24. The quantum Zeno effectB. Misra and E.C.G. Sudarshan (1977) Dynamics: Dynamics and observation are complementary Observation: Results of observations States:

  25. 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:

  26. Quantum Zeno effect w(t) Δt t0 T

  27. The Necker-Zeno modelH. Atmanspacher, T. Filk, H. Römer, Biol. Cyber. 90, 33 (2004) Mental state 1: Mental state 2: dynamics  „decay“ (continuous change) of a mental state observation  „update“ of one of the mental states Internal dynamics and internal observation are complementary.

  28. 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:

  29. A first test of the Necker-Zeno model Assumption: for long off-times t0 off-time

  30. Necker-Zeno model predictions for the distribution functions probability density Cum. probability J.W.Brascamp et.al, Journal of Vision (2005) 5, 287-298

  31. Refined model g(t), t(t) • 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

  32. Tests for Non-Classicality

  33. 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

  34. Temporal Bell’s inequalitiesA.J. Leggett, A. Garg, PRL 54, 857 (1985) 1 t -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.

  35. H. Atmanspacher, T. Filk JMP 54, 314 (2010) N-(t1,t3) ≤ N-(t1,t2) + N-(t2,t3)

  36. 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 violatedif gτ = π/6 which yields sin2 (g.2τ) = and sin2 (g.τ) =

  37. 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.)

  38. Summary and Challenges • The Necker-Zeno model makes predictions for time scales which can be tested. • The temporal Bell’s inequalities can be tested. • Complementaritybetween the dynamics and observations of mental states is presumably easier to find than complementary observables for mental states.

  39. Summary and Challenges Quantum Decision Theory is the Decision Theory?

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