AN EPISTEMIC MODEL OF A QUANTUM STATE WITH ONTIC PROBABILITY AMPLITUDE

1. AN EPISTEMIC MODEL OF A QUANTUM STATE WITH ONTIC PROBABILITY AMPLITUDE ARUN K PATI, PARTHA GHOSE & A K RAJAGOPAL

2. WHAT IS AN ONTOLOGICAL MODEL? • THEORY BE FORMULATED OPERATIONALLY, i.e. THE PRIMITIVES OF DESCRIPTION ARE PREPARATIONS AND MEASUREMENTS • IN AN ONTOLOGICAL MODEL OF AN OPERATIONAL THEORY THE PRIMITIVES ARE PROPERTIES OF THE MICROSCOPIC SYSTEMS

3. A PREPARATION PPREPARES A SYTEM WITH CERTAIN PROPERTIES AND A MEASUREMENT M REVEALS THOSE PROPERTIES • A COMPLETE SPECIFICATION OF THE PROPERTIES OF A SYSTEM IS CALLED AN ‘ONTIC STATE’ AND IS DENOTED BY λ • THE ONTIC STATE SPACE IS DENOTED BY Λ

4. EVEN WHEN AN OBSERVER KNOWS THE PREPARATION PROCEDURE P, SHE MAY NOT KNOW THE EXACT ONTIC STATE THAT IS PRODUCED, AND ASSIGNS OVER Λ A PROBABILITY DISTRIBUTION μ(ψ|λ) >0 AND AN ‘INDICATOR FUNCTION’ ξ (ψ|λ) TO EACH STATE ψ SUCH THAT THE BORN RULE IS REPRODUCED:

5. BORN RULE ∫ d λξ (φ|λ) μ(ψ|λ) = |< φ| ψ>|2 ∫ d λμ(ψ|λ) = 1 BY DEFINITION AN INDICATOR/RESPONSE FUNCTION SATISFIES THE CONDITION ξ (ψ|λ) = 1 FOR ALL λ IN Λψ = 0 ELSEWHERE

6. SCHEMATIC VIEWS OF THE ONTIC STATE SPACE FOR 3 MODELS

7. SCHEMATIC REPRESENTATIONS OF PROBABILITY DISTRIBUTIONS ASSOCIATED WITH ψ IN 3 MODELS

8. TWO DISTINCTIONS AND THREE CLASSES OF ONTOLOGICAL MODELS

9. THEOREM-1 Ψ-epistemic ontological models that satisfy the Born probability rule with distributions μ(ψ|λ)> 0 in open sets Λψfor all ψ are inconsistent with the Schrödinger evolution

10. Ψ-EPISTEMIC STATES OVERLAP

11. PROOF Let |ψ> and |φ> be two distinct nonorthogonal epistemic states corresponding to an ontic state λ in the overlap region Δ= μ(ψ|λ)∩ μ(φ|λ) with Δ an open interval in Λ. Consider the quantum state ψ(t) at time t which satisfies the Schrödinger equation. Putting |φ> = | ψ (t)> in the Born rule ∫ d λξ (φ|λ) μ(ψ|λ) = |< φ| ψ>|2 ∫ d λμ(ψ|λ) = 1 one gets ∫ d λξ (ψ(t)|λ) μ(ψ(t)|λ) = 1

12. Now, by definition the indicator function ξ (ψ(t)|λ) = 1 for all λ in Λψ = 0 elsewhere Hence ∫ d λξ (ψ(t+ dt)|λ) μ(ψ(t)|λ) = ∫ d λ [ξ (ψ(t)| λ) + dξ (ψ| λ) + ½ d2ξ (ψ| λ) +…] μ(ψ(t)|λ) = 1 because dnξ (ψ| λ) = 0. This contradicts the Born rule and completes the proof. The theorem is a consequence of the Ψ-epistemic states having continuous Hamiltonian evolution but not the indicator or response functions.

13. ALTERNATIVE ONTOLOGICAL MODEL Quantum mechanics has been riddled with the measurement problem and nonlocality, features that one would like to avoid in an ontological model. This objective can be met by (i) assigning a complex projective Hilbert space structure CP(H) to the ontic state space Λ in which the projective Hilbert space CP(H)qmis embedded, and (ii) changing the definition of Ψ-epistemic from the one given by HS.

14. FROM PROBABILITIES TO AMPLITUDES: HIDDEN STATES ‘Ψ has an ontic character if and only if a variation of Ψ implies a variation of reality, and an epistemic character if and only if a variation of Ψ does not necessarily imply a variation of reality.’ Ontic models: Ψ ↔ λ Epistemic models must avoid such a relationship

15. HS choice: multiple distinct quantum states compatible with the same ontic state λ • Alternative choice: define a quantum state Ψ as an average over multiple distinct ontic states |λ> with a probability amplitude that can change on obtaining new information about the ontic state: Bayesian approach

16. NEW ONTIC SPACE: HIDDEN STATES

17. MINIMALIST EPISTEMIC MODEL |ψ> = ∫ λψ dλ |λ> A(λ|Pψ) with < ψ| ψ> = ∫ λψ dλ |A(λ|Pψ)|2 = ∫ λψ dλ P(λ|Pψ) = 1 Thus, |<φ| ψ>|2 = ( ∫ λψ∩λφ dλ A*(λ|Pψ) A(λ|Pψ ))2

18. LOCALITY: EINSTEIN’S 1927 ARGUMENT

19. EINSTEIN’S 1927 ARGUMENT In ψ-ontic models let Ψ = (1/√2) [ψa + ψb] p(1a Λ 1b |ψ) = p (1a|ψ) p(1b|1a,ψ) = p(1a|ψ) p(1b|ψ) locality = ¼ THIS CONTRADICTS THE STANDARD QM PREDICTION p(1aΛ 1b |ψ) = 0

20. LOCALITY IN THE MINIMALIST EPISTEMIC MODEL THEOREM-2 In an ontological model, ψ-complete and locality are incompatible, while in the minimalist epistemic model, ψ-epistemic and locality are compatible

21. PROOF Let λ = {λa, λb} in λψand λa∩ λb = empty set Theprobability of simultaneous detection of the particle at a and b is p(1aΛ 1b| λa Λλb) = p(1a| λa Λλb) p(1b|1a,, λa Λλb)

22. In this model, the locality condition requires p(1a| λa Λλb) = p(1a| λa) p(1b|1a, λa Λλb) = p(1b|λa ) = 0 Hence, p(1aΛ 1b| λa Λλb) = 0 which is consistent with the stand QM prediction