Overcoming the EPR paradox
Ghenadie~N.~Mardari
The EPR paradox reduced the debate between classical realism and quantum mechanics to the problem of non-locality. If non-locality is real, the gap between the two traditions cannot be bridged. If it is not, they can be merged via the principle of contextuality. The reality of non-locality will be finally established when the fair sampling assumption for correlation experiments is verified conclusively. We show that such verification can be provided simply by testing two-channel polarizing beam-splitters for polarization-dependent loss and distortion.
quant-ph0412160.tex

%\documentclass[aps,prl,preprint,groupedaddress,showpacs]{revtex4}
\documentclass[aps,prl,twocolumn,groupedaddress,showpacs]{revtex4}

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\begin{document}

\title{Overcoming the EPR paradox}

\author{Ghenadie~N.~Mardari}
email[]{g.mardari@rutgers.edu}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}
\affiliation{Rutgers~University, 89 George St., New~Brunswick, NJ
08901}

\date{\today}

\begin{abstract}
The EPR paradox reduced the debate between classical realism and
quantum mechanics to the problem of non-locality. If non-locality
is real, the gap between the two traditions cannot be bridged. If
it is not, they can be merged via the principle of contextuality.
The reality of non-locality will be finally established when the
fair sampling assumption for correlation experiments is verified
conclusively. We show that such verification can be provided
simply by testing two-channel polarizing beam-splitters for
polarization-dependent loss and distortion.
\end{abstract}

\pacs{03.65.Ud, 03.65.Ta, 42.50.Xa}

%\keywords{}


\maketitle

The EPR paradox is not a feature of reality. It is a trait of the
Copenhagen interpretation. According to the latter, quantum
mechanical properties are defined by human measurements, and
cannot be assumed to exist prior to observation. If Nature is not
allowed to have properties beyond the measured ones, then no other
assumption could make the theory of quantum mechanics more
inclusive. Yet, Einstein, Podolsky and Rosen \cite{epr} have shown
that this very premise of completeness produces the necessary
conclusion of incompleteness in the case of entanglement. Certain
assumptions, which force us to assume that two particles become
indistinguishable during close interactions, also compel us to
expect that one particle would subsequently have well defined
properties, when the other is measured. If such properties were
independent from the process of measurement, the
self-contradiction of the theory would be obvious. The only way to
escape it would be to show that quantum mechanical properties are
non-local. Somehow, it has to be true that one observation affects
all entities in the Universe that are qualitatively
indistinguishable from the one that is measured. Consequently,
non-locality --- unlike the EPR paradox --- must be a feature of
reality, if it is to provide a loophole for the Copenhagen
interpretation. Though, it is not a necessary feature. The paradox
can just as well be avoided by amending the interpretation which
produced it.

Quantum mechanics is a theory of observables, governed by the
principle of contextuality. Within its scope, physical properties
of objects are inseparable from the effects of devices used to
detect them. In other words, quantum mechanical observables do not
quite refer to the same things as the classical properties with
the same name. For example, angular momentum is observed in terms
of an axis of rotation in classical systems, but only as a
propensity to pass through a filter in quantum systems. Note that
the undefined nature of quantum observables is not necessarily
problematic in classical terms. Hence, the outcome of the toss of
a coin must be undefined prior to the actual result, especially if
the initial conditions are not available without additional
measurements. Yet, this has no consequence on the objective
properties of coins in motion. Therefore, it is not necessarily
obvious that quantum mechanical observables exhaust the properties
of Nature at the fundamental level. This element of the Copenhagen
interpretation demands independent justification. In the context
of the EPR paradox, its validity rests entirely on the
confirmation of non-locality.

The previous point can be made even stronger by showing that
Einstein's realism actually demands undefined quantum observables.
For example, a classical description of the photon depends on the
existence of only one plane of linear polarization at the same
time. The measurement of a single particle cannot reveal its
original polarization, because it is likely to rotate during this
interaction. Still, the probability of successful rotation is
known to obey Malus' law, and it can be used to determine the
initial parameters. The catch is to have a large population of
identical particles, in order to divide them in separate samples
for measurements at various angles. Note that quantum mechanics
describes these measurements as non-commuting observables, which
means that only one of them can be defined for the same particle. This
corresponds to the classical description with a single plane of
polarization. If a particle is assumed to switch to a certain
plane during measurement, it is out of the question for it to be
also defined in other planes simultaneously. Consequently, it is
rather desirable to have undefined quantum observables, from a
classical point of view. On the other hand, this also implies that
quantum mechanics is able to provide well-defined values for
unmeasured states, assuming exhaustive detection of otherwise non-commuting
observables with identical quanta. At the population level,
classical and quantum descriptions may be considered equivalent,
with the specification that one classical unobservable is
determined by a full set of quantum observables.

In some cases, non-commuting sets of observables may refer to
several classical variables. The most famous example is the
position-momentum complementarity. Yet, the same explanation
applies here too. The measurement of any single particle for one
variable can be assumed to disturb the original state, preventing
the observation of the concomitant value for the other variable.
If position is measured, momentum --- in the classical sense ---
is not ``made'' undefined. Only its quantum mechanical
representation, as an outcome of the same measurement, is
uncertain. Still, exhaustive measurement with identical particles
may overcome this difficulty, as EPR have shown.

The peculiarity of position and momentum is that single
measurements are sufficient to produce reliable information about
one of the two variables prior to measurement, at least in
principle. This is not the case for other observables, such as
polarization, where single measurements are inconclusive
indicators of original states. As a corollary, classical and
non-local interpretations lead to identical predictions for
position-momentum entanglement, but not for polarization
entanglement. In a classical scenario, the existence of an
original plane of polarization determines the likely measurement
outcomes in all other planes, but these are only detectable as
distributions. The original symmetry of identical particles does
not hold for the outcomes of all individual measurements. In a
non-classical scenario, the existence of a well-defined initial
symmetry is technically ruled out. So, conservation must apply to
post-measurement parameters, even if it is not clear how. As soon
as a particle is measured, its indistinguishable partners must
collapse to the same value by virtue of non-locality. Consequently,
there should be many ways to produce straightforward tests of
non-locality. In particular, correlation measurements for
polarization-entangled photons, as suggested by Aharonov and Bohm
\cite{bah}, appear to be perfectly fitted to the task. As it is
known, correlation measurements were indeed performed, and found
to agree with the predictions of the Copenhagen interpretation.
The problem was that measurements were not ideal. Only a small
share of particles could be properly detected in experiments with
single photons. So, validation rests on the demonstration of fair
sampling. This point calls for additional technical details about
the experiments.

Consider a source that produces pairs of entangled photons with
parallel planes of linear polarization. The task is to determine
if they have this state before or after human observations. Each
photon can be analyzed with a separate polarimeter, but only in
pre-established fixed planes of detection. Parallel measurement
implies that each photon can only pass through one of two
orthogonal channels, \textit{e.g.} horizontal or vertical. If
pairs of photons tend to trigger coincidence counts at parallel
outputs (both horizontal, or both vertical), they are described as
correlated. If they produce coincidence counts at orthogonal
outputs, they are anticorrelated. If there is no clear tendency
either way, they are uncorrelated. Single (non-coincident)
detection events are not taken into account in actual settings,
because their origin is not clear. Yet, most pairs trigger single
or no counts. So, fair sampling is an assumption about the subset
that generates the coincidence counts. It must be representative
of the whole population.

The correlation coefficients for ideal settings may be determined
by a simplified formula such as this:
\begin{equation}
r=\frac{P_{+ +} +P_{- -} -P_{+ -} - P_{- +}}{P_{+ +} +P_{- -} +
P_{+ -} + P_{- +}},
\end{equation}
where $P_{+ +}$ and $P_{- -}$ denote the probability of identical
outcomes for each pair, and $P_{+ -}$ and $P_{- +}$ denote the
probability of opposite values. For parallel measurements, the
Copenhagen interpretation predicts a perfect 1 for the value of
$r$, as shown above. The realist interpretation is more
complicated, because each photon is assumed to have a plane of
polarization that is inclined at an angle $\theta$ to the plane of
measurement. For every value of $\theta$, correlation is given by:
\begin{widetext}
\begin{equation}
r=\frac{cos^{2}\theta _{1} cos^{2}\theta _{2} + sin^{2}\theta _{1}
sin^{2}\theta _{2} - cos^{2}\theta _{1} sin^{2}\theta _{2} -
sin^{2}\theta _{1} cos^{2}\theta _{2}}{cos^{2}\theta _{1}
cos^{2}\theta _{2} + sin^{2}\theta _{1} sin^{2}\theta _{2} +
cos^{2}\theta _{1} sin^{2}\theta _{2} + sin^{2}\theta _{1}
cos^{2}\theta _{2}}.
\end{equation}
\end{widetext}
Since $\theta _{1}$ is equal to $\theta_{2}$ for parallel
measurements of entangled photons, this simplifies to:
\begin{equation}
r=cos^{2} 2 \theta.
\end{equation}
When a source emits pairs with random polarization, the total
expected correlation is the average of $r$ for all values of
$\theta$. Note that the values of $r$ in equation (3) are always
positive. So, the classical model still predicts a detectable
total correlation ($r_{tot} \rightarrow 0.5$). This happens
because entangled photons with parallel polarization do not
anticorrelate in parallel measurements, regardless of the value of
$\theta$. Nevertheless, this realist interpretation cannot be
reconciled with current experimental data without an assumption of
unfair sampling. To be specific, pairs of photons with $\theta
\rightarrow n\pi/2$, where $n$ is a real integer, must have a
significantly higher likelihood for generating coincidence counts
in actual settings with orthogonal channels. Conversely, a
two-channel polarizing beam splitter must have a testable
propensity for polarization-dependent loss, with the highest
values at $\theta \rightarrow m\pi/4$, where $m$ is an odd real
integer. As a corollary, the debate could be settled merely by
testing the efficiency of two-channel polarizers that are used for
correlation experiments. To the best of our knowledge, such experiments
have yet to be reported.

Fair sampling was apparently not considered a long-run problem
during the early tests of non-locality. Classical realism was
practically discarded when the correlation coefficients were found
to be closer to non-local predictions. Instead, Bohmian mechanics
advanced a different sort of realism to the forefront. The new
approach assumed that local hidden variables (LHV), if real,
should produce well-defined quantum mechanical observables prior
to measurement \cite{bom}. This innovation was ontologically
opposite to the Copenhagen interpretation. Yet, as shown above, it
was not any closer to Einstein's realism, which requires
well-defined original values only for classical unobservables.

Bohm's realism was not verifiable with parallel correlation
measurements, because it was designed to match the non-local
predictions. Nevertheless, Bell's theorem \cite{bel} demonstrated
a verifiable discrepancy for certain non-parallel angles of
measurement. New tests were performed, only to confirm non-local
predictions again. However, even the most famous test of Bell's
theorem \cite{asp} acknowledged that validation is contingent upon
the fair sampling assumption. Aspect \textit{et al.} performed
additional tests to show that their source and detectors were free
of significant bias. What they did not test was the likelihood of
polarization-dependent variation in the transmission efficiency of
their two-channel polarizers. According to Adenier and Khrennikov
\cite{kad,adn}, a set of reasonable assumptions about
polarization-dependent loss and distortion in these devices can
tip the balance in favor of LHV theories. Though, as shown above,
unfair sampling must also invalidate the findings for parallel
measurements. Paradoxically, if non-locality fails, Bohmian LHV
theories must fail too (and with them --- the need for Bell's
theorem). The only approach that could survive such a turn of
events is Einstein's classical realism.

Given the above, we must evaluate the possibility of
polarization-dependent loss in polarizing beam-splitters. If they
were ideal homogenous media, like the fields of Stern-Gerlach
magnets, the description of their outputs would be exhausted by
the principle of superposition. In such a case there would be no
bias. As it happens, the molecular properties of crystals do not
allow for such an unqualified idealization. In real settings, the
output is affected by many factors, such as absorption, frequency
distortion, birefringence. Furthermore, there appears to be a
trade-off between polarization purity and transmission efficiency.
Hence, if the polarization is not pure, more photons pass, but
they may go through the wrong channel. If the quality of
polarization is higher, this entails higher levels of loss in each
channel. Notably, this type of loss is subject to Malus' law and
must be polarization-dependent. Granted, these are mostly
speculative arguments based on anecdotal evidence. The point,
however, is that polarizer efficiency is the last line of defense
for classical realism. Testing the quality of two-channel
polarizers is of utmost scientific importance for this debate.

In conclusion, we have seen that the principle of contextuality
has the potential to bridge the gap between classical and quantum
mechanics. The only obstacle is ontological: the possible reality
of non-locality. Yet, the latter cannot be validated or rejected
until the fair sampling assumption for correlation experiments is
verified. Remarkably little is needed for this --- testing the
two-channel polarizers for polarization-dependent loss and
distortion. We must be sure that experiments detect a property of
all photons, and do not just select the particles that happen to
be already polarized in the planes of measurement.


%\bibliography{epr08}
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  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Podolsky}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Rosen}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{47(10)}},
  \bibinfo{pages}{777} (\bibinfo{year}{1935}).

\bibitem[{\citenamefont{Bohm and Aharonov}(1957)}]{bah}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Bohm}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Aharonov}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{108(4)}},
  \bibinfo{pages}{1070} (\bibinfo{year}{1957}).

\bibitem[{\citenamefont{Bohm and Hiley}(1993)}]{bom}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Bohm}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Hiley}},
  \emph{\bibinfo{title}{The undivided universe}}
  (\bibinfo{publisher}{Routledge}, \bibinfo{year}{1993}).

\bibitem[{\citenamefont{Bell}(1987)}]{bel}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Bell}},
  \emph{\bibinfo{title}{Speakable and unspeakable in quantum mechanics}}
  (\bibinfo{publisher}{Cambridge}, \bibinfo{year}{1987}).

\bibitem[{\citenamefont{Aspect et~al.}(1982)\citenamefont{Aspect, Grangier, and
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\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Aspect}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Grangier}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Roger}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{49(2)}},
  \bibinfo{pages}{91} (\bibinfo{year}{1982}).

\bibitem[{\citenamefont{Adenier and Khrennikov}({\natexlab{a}})}]{kad}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Adenier}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khrennikov}},
  \eprint{quant-ph/0306045}.

\bibitem[{\citenamefont{Adenier and Khrennikov}({\natexlab{b}})}]{adn}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Adenier}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khrennikov}},
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\end{thebibliography}

\end{document}

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\bibitem[{\citenamefont{Einstein et~al.}(1935)\citenamefont{Einstein, Podolsky,
  and Rosen}}]{epr}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Einstein}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Podolsky}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Rosen}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{47(10)}},
  \bibinfo{pages}{777} (\bibinfo{year}{1935}).

\bibitem[{\citenamefont{Bohm and Aharonov}(1957)}]{bah}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Bohm}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Aharonov}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{108(4)}},
  \bibinfo{pages}{1070} (\bibinfo{year}{1957}).

\bibitem[{\citenamefont{Bohm and Hiley}(1993)}]{bom}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Bohm}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Hiley}},
  \emph{\bibinfo{title}{The undivided universe}}
  (\bibinfo{publisher}{Routledge}, \bibinfo{year}{1993}).

\bibitem[{\citenamefont{Bell}(1987)}]{bel}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Bell}},
  \emph{\bibinfo{title}{Speakable and unspeakable in quantum mechanics}}
  (\bibinfo{publisher}{Cambridge}, \bibinfo{year}{1987}).

\bibitem[{\citenamefont{Aspect et~al.}(1982)\citenamefont{Aspect, Grangier, and
  Roger}}]{asp}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Aspect}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Grangier}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Roger}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{49(2)}},
  \bibinfo{pages}{91} (\bibinfo{year}{1982}).

\bibitem[{\citenamefont{Adenier and Khrennikov}({\natexlab{a}})}]{kad}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Adenier}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khrennikov}},
  \eprint{quant-ph/0306045}.

\bibitem[{\citenamefont{Adenier and Khrennikov}({\natexlab{b}})}]{adn}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Adenier}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Khrennikov}},
  \eprint{quant-ph/0309010}.

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