Download Entropic Dynamics: A hybrid-contextual theory of Quantum Mechanics

arXiv:1704.01571v1 [quant-ph] 4 Apr 2017
Entropic Dynamics: A hybrid-contextual theory of
Quantum Mechanics
Kevin Vanslette
[email protected]
Department of Physics, University at Albany (SUNY),
Albany, NY 12222, USA.
Abstract
The Bell-KS theorem and the more recent ψ-epistemic no-go theorems of QM are
discussed in the context of Entropic Dynamics. In doing so we find that the BellKS theorem allows for, a perhaps overlooked, hybrid-contextual model of QM in which
one set of commuting observables (position in this case) is non-contextual and all other
observables are contextual. Entropic Dynamics is in a unique position as compared
to other foundational theories of QM because it derives QM using the inference rules
of standard probability theory. In this formalism, position is the preferred basis from
which inferences about other contextual operators are made. This leads us to the
interpretation that ED is a hybrid-contextual model of QM which is consistent with
the Bell-KS theorem.
1
Introduction
Quantum Mechanics is an odd mix of the predictable and unpredictable. On one hand, it
is hugely successful in its ability to predict the set of eigenvalues, expectation values, and
operators for a particle-system of interest. On the other hand, each measurement holds
some amount of unpredictability, quantified by a probability distribution, except for a few
trivial cases. This unpredictable nature leaves space for many interpretations of QM to
coexist inharmoniously within the community - a community, no doubt, easily bothered by
disharmony of any-type.
A goal then is to reduce and organize this disharmony by ruling out interpretations and
foundational theories of QM which disagree with the predictable findings of QM. This is
done by first making a few reasonable assumptions an interpretation and/or theory of QM
may obey, and then by showing these assumptions lead to contradictions in the formalism,
construct a no-go theorem. This is the basis of the Bell inequalities [1], the Bell-KS theorem
[2, 3, 4], as well as the findings of Pusey-Barret-Rudolph (PBR) [5, 6] about the epistemic
interpretation of the wavefunction. Later, the final results are sometimes tabulated in 2 by
2 tables like:
1
contextual
noncontextual
ψ-ontic
A
C
ψ-epistemic
B
D
and are followed by statements like, “theories of type “C” or “D” are ruled out by the Bell-KS
theorem and “B” is ruled out by PBR”, where a brash reader may be inclined to conclude
that QM is a theory of type “A” ( e.g. Bohmian mechanics). The 2 by 2 is by nature an
over simplification; it fails to span the entire universe of discourse of plausible theories and
interpretations of QM.
In particular, this paper will show Entropic Dynamics (ED) to be a theory of QM which
lies on the line between theories “B” and “D” while not being ruled out by any of the discussed
no-go theorems as it violates (or partially violates in the case of the Bell-KS theorem) the
proposed contradictory assumptions. It comes to no surprise that no-go theorems concede
that theories which violate their assumptions are not necessarily ruled out (often stated
outright within the no-go). New insights and discussion of how each no-go theorem is
handled/violated within Entropic Dynamics will be presented. This also show areas in which
other theories and interpretations of QM may slip between the cracks, reopening the perhaps
presumed closed universe of discourse, for a few edge theories/interpretations of QM.
In the following section we will review the pertinent aspects of Entropic Dynamics and a
few no-go theorems followed by discussions on how they are handled or do not apply to the
ED framework. Finally there are closing remarks.
2
Entropic Dynamics
Entropic Dynamics is an application of inference and probabilistic techniques from which
laws of physics may be extracted. In this case we will show the constraints and assumptions
needed to derive Quantum Mechanics from the first principles of inference [7, 8, 9, 10, 11].
We look at a condensed version of ED which pertain to some of the quantum no-go theorems.
An extended review can be found at [7].
In any inference problem, the first thing that must be specified is the set of microstates
over which we are uncertain about. This defines the universe of discourse (the set of possible
outcomes) for the object(s) at hand. To derive Quantum Mechanics (in flat space) the
universe of discourse spanned by N particles are their positions in a flat Euclidean space X
(metric δab ). Our knowledge of the positions of particles is characterized by a probability
density ρ(x) where x is a coordinate in a 3N dimensional configuration space of particle
coordinates xan , where a = 1, 2, 3 and n = 1, 2, ..., N denote the ath spacial axis of the nth
particle’s position. When convenient we may use the notation xA ≡ xan where A = (n, a).
From the onset, particles have definite yet unknown positions and are treated as “physical”
quantities we are interested in making inferences about. The implication of this statement
to no-go theorems is addressed later in this paper, while its implications in measurement
theory are reviewed in the below section following [12, 13].
Now that the microstates have been specified, we are inclined to ask how the position
of these particles change. In particular, if the particles are located at x, we wish to know
2
how probable it is for x → x′ , that is, we seek a probability of the form P (x′ |x) to quantify
this uncertainty. Not knowing anything about how particles change position gives trivial
dynamics, so we make the following assumptions: 1) particles have a tendency to move
along continuous trajectories and 2) particles may have a tendency to be correlated and
drift. These assumptions are represented by expectation value constraints on P (x′ |x). The
′
first assumption is saturated by making large ∆xan = xna − xan improbable. This is done by
imposing P (x′ |x) have small variances, κn , in particle coordinates,
h∆xan ∆xbn iδab = κn ,
(n = 1, . . . , N)
(1)
(where motion is continuous in the limit κn → 0). The second assumption is imposed by
one additional constraint,
h∆xA i∂A φ =
X
n,a
h∆xan i
∂φ(x)
= κ′ ,
∂xan
(2)
is the “drift” gradient. There are many probabilwhere κ′ is another small constant and ∂φ(x)
∂xa
n
′
ity distributions P (x |x) which satisfy the above expectation value constraints. We therefore
use the method of maximum entropy [8, 9] to find the “least biased” distribution by maximizing the entropy,
S[P, Q] = −
Z
dx′ P (x′ |x) log
P (x′ |x)
Q(x′ |x)
(3)
with respect to the expectation value constraints via the Lagrange multiplier method to find
the following probability distribution,
P (x′ |x) =
h
i
1
1X
exp −
αn δab (△xan − h△xan i)(△xbn − h△xbn i) ,
Z
2 n
(4)
∂φ
′
where h∆xA i = ααn δ ab ∂x
b . The prior distribution Q(x |x) is a very broad normalizable Gausn
sian distribution encoding that before constraints are made, particles may jump anywhere
with near to equal probability and has been absorbed into the normalization constant. There
are explicit and in-depth arguments for the notion of an instant in ED [7, 8] that we need not
explore here; however, in summary, t is introduced as a convenient bookkeeping parameter
to label changes is the probability distribution. In particular t fits neatly into the transition
probability,
′
P (x′ |x, ∆t) =
X mn
1
exp[− (
( ∆xan − h∆xan i) ( ∆xan − h∆xan i)] ,
Z
2η∆t
n,a
(5)
such that the state of knowledge of the positions of particles at a later time t′ is given by
the following convolution equation,
′
′
ρ(x |t ) =
Z
P (x′ |x, ∆t)ρ(x|t) dx,
3
(6)
where ρ(x|t) ≡ ρ(x). Equation (6) may be recast as a differential equation for ρ(x|t) ( an
explacation may be found in [8]),
∂t ρ = −∂A ρv A ,
(7)
where v A is the velocity of the probability flow in configuration space, or current velocity,
′
′
aa
δnn′
v A = mAB ∂B Φ where mAA = m−1
n δ
and Φ = ηφ − η log ρ1/2 .
(8)
Briefly, a non-dissipative diffusion is imposed by letting evolution of ρ and Φ (related to the
drift of the probability and the phase of the wavefunction) be imposed by conjugate pair of
(what are later recognized as) Hamilton’s equations,
∂t ρ =
δ H̃
δΦ
and ∂t Φ = −
δ H̃
.
δρ
(9)
The “ensemble Hamiltonian” H̃ is chosen so that the first equation above reproduces (7)
in which case the second equation in (9) becomes a Hamilton-Jacobi equation. A more
complete specification of H̃ is,
H̃[ρ, Φ] =
Z
#
"
1
1 AB
ρm ∂A Φ∂B Φ + ρV + ξmAB ∂A ρ∂B ρ ,
dx
2
ρ
(10)
where the “kinetic” (∂Φ)2 terms reproduces (7), and the potential terms, V (x) and a “quantum” potential, are forcibly independent of Φ as they originate from integration constants in
the solution for H̃ satisfying (7). The form of the potential terms is suggested by information
geometry in the presence of this inference framework [7]. The parameter ξ = h̄2 /8 defines
the value of Planck’s constant h̄. The formulation of ED is now complete: combining ρ and
Φ into a single complex function, Ψ = ρ1/2 exp(iΦ/h̄) with h̄ = (8ξ)1/2 allows us to rewrite
(9) as a linear Schrödinger equation,
ih̄
X h̄2
∂Ψ
=−
▽2n Ψ + V Ψ .
∂t
2m
n
n
(11)
At this point we can adopt the standard Hilbert space formalism to represent the epistemic
state Ψ(x) as a vector,
|Ψi =
Z
dx Ψ(x)|xi with Ψ(x) = hx|Ψi .
(12)
The expression of |Ψi in other basis is regarded as potentially convenient ways of expressing
position space wavefunctions.
2.1
Unitary Measurement Devices
A natural question is, “If position is the only definite quantity, how are other operators in
Quantum Mechanics measured?”. This question was originally addressed in [12] and more
4
recently is addressed in [13] as well as how the notions of von Neuman, weak measurements,
and Weak Values fit into ED. Simply put, the solution is that position is the only beable
of the theory and thus all other quantities must be inferred from position - they are the
epistemic quantities called the inferables of the theory.
P
Consider the state |Ψi expanded in the basis of an operator Â, |Ψi = a ca |ai. In ED,
the expansion into the basis |ai is (perhaps) a convenient stand-in for the position space
wavefunctions,
|Ψi =
X
ca |ai =
a
X
a
ca
Z
ψa (x)|xi dx =
Z
Ψ(x)|xi dx.
(13)
A post-selection of |Ψi into a state ha| can be done with probability |ca |2 . Using a unitary
measurement device ÛA |ai , ti = |xi , t′ i which maps states |ai at time t to the position on
a screen xa at a later time t′ (not to be confused with the x’s at t of the original state in
(13)) allows for the inference of |ai by making detections of xa at a later time. An example
of such a unitary measurement device is a periodic crystal lattice or prism which diffracts
“momentum” states to position states. That is we have,
UA |Ψ, ti =
X
|xa , t′ iha, t|Ψ, ti =
a
X
ca |xa , t′ i,
(14)
a
such that p(xa |t′ ) = p(a|t) [12], which we claim to be an active (coordinate) transformation of
the system consistent with the SE. The actual detection of the location of a particle usually
is facilitated with another detection device such as a photo-plate, CCD camera, or bubble
chamber. In such instances, the probability of x given a detection at D is given by,
P ′(x) = q(x|D) =
p(x)q(D|x)
q(x, D)
=
,
q(D)
q(D)
(15)
where q(D|x) is the likelihood function which accounts for the accuracy of the measurement
device [13]. In the present case, after a detection at D,
P ′ (xa |t′ ) = q(xa |D, t′ ) =
p(xa |t′ )q(D|xa , t′ )
p(a|t)q(D|a, t)
=
= q(a|D, t) = P ′ (a|t),
q(D)
q(D)
(16)
effectively “collapsing” the wavefunction, which is to say the final state of the system is
known with certainty for sharply peaked likelihood functions.
3
Hidden Variables, Realism, and Non-locality
The subject of hidden variables, realism, and non-locality has been touched upon in [8, 12]
and it will be further explored here. In Bell’s landmark paper [1], he found a contradiction
between QM and hidden variable theories which claimed local realism. It was accomplished
by considering a hidden variable λ, which if known, would give the outcome of an experiment
5
(an eigenvalue of an operator) with certainty a0 = A(λ = λ0 ). By integrating over the
probability of a hidden variable,
hA(λ)B(λ)i =
Z
p(λ)A(λ)B(λ) dλ,
(17)
he showed that such expectation values do not always agree with the expectations values of
QM with a quite general p(λ).
In ED there is no such hidden variable. The particle dynamics are non-deterministic
as can be seen by the Brownian like paths particles take due to the form of the transition
probability P (x′ |x) in (5). Even if the initial conditions of particle positions are known exactly (or with near precision), ρ(x) ∼ δ(x − x0 ), equation (6) is inevitably nondeterministic
for time steps ∆t > 0. The process which is deterministic in ED is the evolution of the
probability distribution as it follows the HJ-like equations from (9) given the appropriate
guides the probability disboundary/initial conditions are known. The drift gradient ∂φ(x)
∂xa
n
tribution, our knowledge or expectation value, over particle locations rather than guiding
each particle at every point. The solution is to realize that Φ(x) is an epistemic parameter
that is coupled to ρ(x) for each x in a complicated way through the HJ-like equations (9).
The nonlocal nature of probability as a means for quantifying knowledge (of the future, past,
or present) accounts for the nonlocal behavior of Quantum Mechanics in ED. As collapse is
an epistemic change in the system, each observer assigns distributions which coincide with
their current state of knowledge of the system.
4
Bell-KS type Theorems
The Bell-KS theorem sheds light on the incompatibility of hidden variable theories and
Quantum Mechanics [2, 3]. Years later Mermin demonstrated what is considered to be
simplest expression of what is usually an algebra and geometry intensive Bell-KS theorem
[4].
The class of hidden variable theories the Bell-KS theorem excludes have the following
reasonable conditions: The value of an operator is definite yet unknown, such that we may
assign a preexisting value, called its valuation (one of its eigenvalues), such that at any time,
v(Â) = a.
(18)
It is also assumed that functional relationships between operators f (Â, B̂, Ĉ, ...) = 0 hold
throughout the valuation process,
v(f (Â, B̂, Ĉ, ...)) = f (v(Â), v(B̂), v(Ĉ), ...) = 0.
(19)
It should be noted that the considered operators must commute v(ÂB̂) = v(Â)v(B̂) = v(B̂ Â)
when taking valuations. Mermin demonstrates the contradiction of equations (18) and (19)
with Quantum Mechanics by considering the Mermin Square shown in below:
ZI IX
IZ XI
ZZ XX
6
ZX
XZ
YY
Each table entry is an observable from the 2-qubit Pauli Group consisting of a joint eigenbasis
consisting of 4 eigenvectors. As a notational convenience we will omit tensor products when
there is no room for confusion and let X = σx and tensor products XZIY = σx(1) ⊗ σz(2) ⊗
I (3) ⊗ σy(4) , for example (following the notational structure in [14]). The product of the
operators along a given row or column is the rank 4 identity I(4) = II (in our notation)
with the exception of the last row which is −II. Consider the valuation of the first row,
v(ZI IX ZX) = v(II) = 1.
(20)
v(ZI IX ZX) = v(ZI)v(IX)v(ZX) = 1.
(21)
Supposing (19) is true then
The valuation of the ijth element in the table is v(ij) = ±1, and therefore (19) imposes a
constraint on the individual valuations v(ZI)v(IX)v(ZX) = 1 which is only satisfied if either
0 or 2 of the valuations are −1. This cuts the universe of discourse from 23 = 8 possibilities
to 4. Let Ri be the ith row and Ci the ith column such that above R1 =ZI IX ZX.
Mermin showed his square indeed leads to a contradiction when considering the product of
the row and column valuations,
Y
v(Ri )v(Ci ) = v(II)5v(−II) = −1,
(22)
i
whereas applying (19) to each row and column, v(Ri ) =
Y
i
v(Ri )v(Ci ) →
YY
i
Q
j
v(ij), one finds
v(ij)2 = 1,
(23)
j
a contradiction due to the fact that not all of the elements in Mermin’s square commute.
Quantum Mechanical formalism and experiment agrees with (22) and not with (23) and thus
(19) must be thrown out. Bell makes the good point that it may be overconstraining for the
valuation to produce identical values when different sets of commuting observables are being
considered, just to refute it by noting that if a space-like separated observer would have
choice of a commuting set of observables that in principle he/she could change during the
time of flight of the measurement. A hidden variable theory would then have to explain this
nonlocal change in the valuation meaning that the Bell-KS theory only refutes local hidden
variables theories.
4.1
Interpreting the Contradiction: Contextuality
The standard interpretation of the contradiction by Bell, Kochen, Specker, Mermin and
others is that quantum mechanical observables are contextual, meaning that the operator’s
“aspect”, “character”, or “value” depend on the remaining set of commuting observables
under which it is considered. Any observable which does not depend on the remaining set
of commuting observables is called noncontextual, which, for example, are the individual
observables v(ij) from the Mermin square from (23).
7
In more recent years the interpretation of the Bell-KS theorem, which in principle would
rule out all local hidden variable theories obeying (18) and (19), has been under attack, in
essence, for having a more restrictive interpretation than the theorem merits. In particular
the work by [16, 17, 18] opens a loop hole because in practice infinite measurement precision is impossible and thus the Bell-KS theorem is “nullified” in their language, although
Appleby (and others) find the “nullified” critique to be too harsh of a criticism [15]. In an
attempt to clear notions, de Ronde [19] has perhaps muddied the water further (for those
unaware of their preconceptions) by pointing out that epistemic and ontic contextualty are
consistently being scrambled into a omelet when perhaps the yoke and egg whites should
be cooked separately. He defines ontic contextuality as the formal algebraic inconsistency of
the operator and valuation formalism of Quantum Mechanics within the Bell-KS theorem having nothing to do with measurement1 . The epistemic counterpart is more aligned with
the principles of Bohr in that Quantum Mechanics involves an interaction between system
and measurement apparatus whose outcomes are inevitably communicated in classical terms
- the context is given by the measurement device. The difference is subtle but, as noted,
ontic contextuality is defined to be independent of the differing interpretations of quantum
mechanics where epistemic contextuality need not be.
4.2
Hybrid-contextual theories
Here we point out another loophole in the interpretation of the Bell-KS theorem. Strictly
speaking the theorem discards realist theories in which all of the considered operators are
treated ontically variables through their valuation. This leaves open the possibility for hybrid
contextual theories in which only a subset of commuting observables are definite yet unknown,
or noncontextual, while other variables are contextual. To date the only theory of Quantum
Mechanics known to the author that seems to fit this description is Entropic Dynamics (ED)
[7].
From the outset of ED, we wish to make inferences about the definite yet unknown
position of a particle. Because particle positions constitute the (noncontextual) beables of
the theory, all other variables are classified as “inferables” of the theory [13]. Inferables are
potentially useful purely epistemic parameters, which in ED ultimately encode our knowledge
of particle positions. Position takes the form of a “preferred basis” in quantum mechanical
language.
The only operators required to undergo valuation in ED are the 3N-particle position
coordinates with the corresponding 3N operators X̂ (n) , we have,
(n)
(n)
v(x̂i ) = xi .
(24)
Position operators obey (19) such that
(n)
(m)
(n)
(m)
v(f (x̂i , x̂j , ...)) = f (v(x̂i ), v(x̂j ), ...) = 0,
1
(25)
This statement might be out of context in Entropic Dynamics where operators are considered to be
nothing more than“inferables” in the theory which help quantify our knowledge of positions and vice-versa
[13].
8
for any function f because all position operators mutually commune. No contradiction in the
sense of Mermin can be reached because all of the operators requiring valuation commute.
A question of interest is, how, if everything is to be measured or inferred using a (noncontextual) position basis (section: Unitary Measurement Devices and [12, 13]), is the contextual nature of a set of contextual operators Âij ∈ A non-contradictory? This question
is especially tricky because it mixes the epistemic and ontic palates of contextuality in the
sense of [19], who, as well as [20], quote Mermin , “the whole point of an experimental test
of KS [theorem] misses the point.”. That being said, the contexuality of the operators Âij
is simply expressed through the lack of commutativity between sets of commuting observables, e.g., the lack of commutation of the rows and columns of the Mermin Square and like
Bell-KS proofs. ED perhaps sheds some light onto Mermin’s statement about the lack of an
experimental test.
Suppose Alice prepares a two particle system and sends it to Bob who has a compound
unitary measurement device (14) for each set of commuting observables (each row and column) of the Mermin square (for simplicity). Because Bob can only measure one row or
column for a given pair of particles sent from Alice, him choosing a measurement device
from the ith row Ri or column means he has chosen and applied the unitary measurement
device ÛRi to the incoming state and mapped it to position coordinates from which inferences
can be made. That is, applying ÛRi = ÛAi1 ÛAi2 ÛAi3 picks the ith row,
A11
Ri ∗ A21
A31
A12
A22
A32
A13
A23 −→ ÛRi Ai1
A33
Ai2
Ai3 −→ xi1
xi2
xi3
where after Ri (or similarly Ci ) has been picked, only then can valuations be applied to
the positions and the values of Aij inferred post detection. Because the operators Aij are
contextual, the valuation function never needs to be applied to them. As only one column or
row may be picked at a time by Bob, who seeks to measure a set of commuting observables,
the quantum mechanical expectation values match that as read by Bob (and are therefore
in the form of (22)). Alice, being in the dark, does not know which row Bob will pick and is
free to assign a probability Bob picks the ith row or column, and after learning the chosen
row or column may update her probability of the experimental outcomes.
The (contextual) operators Aij need not be noncontextual in ED because they are epistemic parameters which help characterize our knowledge of the positions of the
particles. In
R
this case one should not claim Aij , one of its eigenvalue aij , or a state |aij i = dx ψaij (x)|xi,
to have a definite existence outside of characterizing our knowledge of the definite yet unknown positions of particles x. That being said, one could claim each Aij may be expanded
in the position basis under which valuations may be made - potentially leading to a noncontextual treatment of Aij ,
?
v(Aij ) → haij |Aij |aij i = aij ,
(26)
giving one of its eigenvalues independent of the basis of Aij . If one attempts to force valuation
9
of a position before measurement (where |xi = |x1 i ⊗ |x2 i... ⊗ |xN i for N particles), one finds
v(Aij ) = v(
XZ
aij
′
dxdx
ψaij (x)ψa∗ij (x′ )|xiaij hx′ |)
=
X
aij
aij |ψaij (x0 )|2 =
→ hx0 |
X
XZ
aij
dxdx′ ψaij (x)ψa∗ij (x′ )|xiaij hx′ | |x0 i
aij ρaij (x0 ) 6= aij .
(27)
aij
Essentially, this calculation states that definite (noncontextual) positions before measurement do not imply definite (noncontextual) values of Aij , and therefore, we are justified in
treating the operators Aij contextually. Hybrid-contextual operators (an operator which is
a product of contextual and noncontextual operators) can be treated analogously.
It should be noted that in Entropic Dynamics, the idea of valuation is very unnatural.
The flexibility of an inference based theory is that it allows us to state, quantify, and represent
how much we do not know about the state of a system through a probability distribution,
upon which we use the rules of inference and probability updating to determine what we do
know. The most natural inferential tool is probability and it proper use clears up several
difficulties within the quantum measurement problem [13] including an interpretation of
Weak Values in the ED framework. The Bell-KS theorems are pivotal no-go theorems of
Quantum Mechanics. What we have learned here is that it is possible for a hybrid contextual
theory to still obey the Bell-KS theorem.
5
ψ-epistemic?
Although the main result is above, this paper is a convenient platform to discuss ED and
PBR’s ψ-epistemic state no-go theorem [5]. Their first assumption (paraphrased) 1) is that
a ψ-epistemic state has physical values upon which inferences may be made. ED agrees
with this assumption whole heartedly, and the variables which are “physical” in ED are the
definite yet unknown positions of particles. The second assumption (verbatim) 2) is that
“systems which are prepared independently have independent physical states”.
The second assumption requires further definition: 2a) first - what does it mean for a system to be prepared independently and second 2b) - what does it mean for a system to have
independent physical states? The definition of independence in 2a) seems to be saturated by
the definition of independence in probability theory, namely that if two systems are prepared
independently then their joint probability distribution is factorable into independent probability distributions p(x1 , x2 ) = p(x1 )p(x2 ) and therefore there are no correlations between x1
and x2 . The quantum mechanical analog is that these two states are non-entangled product
states. The definition of independent physical states in 2b) is rather unclear from the outset
but later is defined quantitatively by,
Z
1
D(µ0 , µ1 ) =
|µ0 (λ) − µ1 (λ)|dλ,
(28)
2
or equivalently in our case,
1Z
D[p(x1 ), p(x2 )] =
|p(x1 ) − p(x2 )|δ(x1 − x2 )dx1 dx2 ,
(29)
2
10
such that if D = 1 then p(x1 ) and p(x2 ) are completely disjoint and thus occupy “independent
physical states”. It is easy to see now how the definition of independent in 2a) differs from
the definition of independent in 2b), in-fact the 2a) and 2b) definitions of independent clash
in assumption 2) for any independent joint probability p(x1 , x2 ) = p(x1 )p(x2 ) which are not
entirely disjoint (i.e. “physically” independent 2b)).
Consider the classical example given in [5] for the disjoint energy planes in phase space
for two particles ρE1 (x1 , p1 ) and ρE2 (x2 , p2 ). Here they are disjoint and independent probabilistically so they satisfy assumption 2) and E1 and E2 are deemed “real physical states”
of the system. However, epistemologists would likely agree that a “physical state” should
be defined independently of the knowledge of said physical state. In the present case, if one
R
marginalizes Rover the momentum of both particles, one is left with p(x1 ) = dp1 ρE1 (x1 , p1 )
and p(x2 ) = dp2 ρE2 (x2 , p2 ) in which it is perfectly reasonable to have D[p(x1 ), p(x2 )] < 1
in the case p(x1 ), p(x2 ) overlap and therefore E1 and E2 would no-longer be deemed “real
physical states” of the system by their measure. As a counter example can be given for
the classical case, we need not concern ourselves with the quantum case. In ED, the noncontextual, “physical”, values of the locations of the particles is independent of the state of
knowledge at hand and therefore their no-go theorem simply does not apply to our case.
6
Conclusion
A few hidden variable no-go theorems of QM have been reviewed in the context of ED.
The absence of hidden variables and the nonlocal nature of QM in ED is discussed and
attributed to the nonlocal nature of probability. ED was found to be in concordance with
the Bell-KS theorem due to its hybrid-contextual nature. Position is the only ontic variable
in ED it is treated noncontextually while all other variables are epistemic in nature treated
contextually. The definition of physicality differs from that as presented in [5], and thus
their no-go theorem for epistemic wavefunctions does not apply. This papers shows the
sense in which a foundational QM theory may be hybrid-contextual and still obey the BellKS theorem.
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Acknowledgments:
I would like to thank everyone in the information physics group at UAlbany, especially Ariel
Caticha, Nick Carrara, and Tony Gai who have encouraged me throughout the writing of
this paper. I would also like to thank Mordecai Waegell for our discussions about Entropic
Dynamics and quantum physics in general.
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