AN EPISTEMIC MODEL OF A
QUANTUM STATE WITH ONTIC
PROBABILITY AMPLITUDE
ARUN K PATI, PARTHA
GHOSE & A K
RAJAGOPAL
WHAT IS AN ONTOLOGICAL MODEL?
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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
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A PREPARATION P PREPARES 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
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:
BORN RULE
∫ d λ ξ (φ|λ) μ(ψ|λ) = |< φ| ψ>|2
∫ d λ μ(ψ|λ) = 1
BY DEFINITION AN INDICATOR/RESPONSE FUNCTION
SATISFIES THE CONDITION
ξ (ψ|λ) = 1 FOR ALL λ IN Λψ
= 0 ELSEWHERE
SCHEMATIC VIEWS OF THE ONTIC
STATE SPACE FOR 3 MODELS
SCHEMATIC REPRESENTATIONS OF
PROBABILITY DISTRIBUTIONS
ASSOCIATED WITH ψ IN 3 MODELS
TWO DISTINCTIONS AND THREE
CLASSES OF ONTOLOGICAL MODELS
THEOREM-1
Ψ-epistemic ontological models that
satisfy the Born probability rule with
distributions μ(ψ|λ)> 0 in open sets Λψ for
all ψ are inconsistent with the
Schrödinger evolution
Ψ-EPISTEMIC STATES OVERLAP
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 λ ξ (φ|λ) μ(ψ|λ) = |< φ| ψ>|2
∫ d λ μ(ψ|λ) = 1
one gets
∫ d λ ξ (ψ(t) |λ) μ(ψ(t)|λ) = 1
Now, by definition the indicator function
ξ (ψ(t)|λ) = 1 for all λ in Λψ
= 0 elsewhere
Hence
∫ d λ ξ (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.
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)qm is
embedded, and
(ii) changing the definition of Ψ-epistemic from
the one given by HS.
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
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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
NEW ONTIC SPACE: HIDDEN STATES
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
LOCALITY: EINSTEIN’S 1927
ARGUMENT
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(1a Λ 1b |ψ) = 0
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
PROOF
Let λ = {λa, λb} in λψ and λa∩ λb = empty set
The probability of simultaneous detection of the
particle at a and b is
p(1aΛ 1b | λa Λ λb )
= p(1a | λa Λ λb ) p(1b |1a,, λa Λ λb )
In this model, the locality condition requires
p(1a | λa Λ λb ) = p(1a | λa )
p(1b |1a, λa Λ λb ) = p(1b |λa ) = 0
Hence,
p(1aΛ 1b | λa Λ λb ) = 0
which is consistent with the stand QM prediction