## 1. Can Quantum Mechanical Description of Physical Reality Be Considered Complete?


### 1.1 Setting and prehistory



By 1935 conceptual understanding of the quantum theory was dominated
by Niels Bohr's ideas concerning complementarity. Those ideas centered
on observation and measurement in the quantum domain. According to
Bohr's views at that time, observing a quantum object involves an
uncontrollable physical interaction with a measuring device that
affects both systems. The picture here is of a tiny object banging
into a big apparatus. The effect this produces on the measuring
instrument is what issues in the measurement "result"
which, because it is uncontrollable, can only be predicted
statistically. The effect experienced by the quantum object limits
what other quantities can be co-measured with precision. According to
complementarity when we observe the position of an object, we affect
its momentum uncontrollably. Thus we cannot determine both position
and momentum precisely. A similar situation arises for the
simultaneous determination of energy and time. Thus complementarity
involves a doctrine of uncontrollable physical interaction that,
according to Bohr, underwrites the Heisenberg uncertainty relations
and is also the source of the statistical character of the quantum
theory. (See the entries on the
 Copenhagen Interpretation
 and the
 Uncertainty Principle.)



Initially Einstein was enthusiastic about the quantum theory. By 1935,
however, while recognizing the theory's significant achievements, his
enthusiasm had given way to disappointment. His reservations were
twofold. Firstly, he felt the theory had abdicated the historical task
of natural science to provide knowledge of significant aspects of
nature that are independent of observers or their observations.
Instead the fundamental understanding of the quantum wave function
(alternatively, the "state function", "state
vector", or "psi-function") was that it only treated
the outcomes of measurements (via probabilities given by the Born
Rule). The theory was simply silent about what, if anything, was
likely to be true in the absence of observation. That there could be
laws, even probabilistic laws, for finding things if one looks, but no
laws of any sort for how things are independently of whether one
looks, marked quantum theory as irrealist. Secondly, the quantum
theory was essentially statistical. The probabilities built into the
state function were fundamental and, unlike the situation in classical
statistical mechanics, they were not understood as arising from
ignorance of fine details. In this sense the theory was
indeterministic. Thus Einstein began to probe how strongly the quantum
theory was tied to irrealism and indeterminism.



He wondered whether it was possible, at least in principle, to ascribe
certain properties to a quantum system in the absence of measurement.
Can we suppose, for instance, that the decay of an atom occurs at a
definite moment in time even though such a definite decay time is not
implied by the quantum state function? That is, Einstein began to ask
whether the formalism provides a description of quantum systems that
is complete. Can all physically relevant truths about systems be
derived from quantum states? One can raise a similar question about a
logical formalism: are all logical truths (or semantically valid
formulas) derivable from the axioms. Completeness, in this sense, was
a central focus for the Gottingen school of mathematical logic
associated with David Hilbert. (See entry on
  Hilbert's Program.)
 Werner Heisenberg, who had attended Hilbert's lectures, picked up
those concerns with questions about the completeness of his own,
matrix approach to quantum mechanics. In response, Bohr (and others
sympathetic to complementarity) made bold claims not just for the
descriptive adequacy of the quantum theory but also for its
"finality", claims that enshrined the features of
irrealism and indeterminism that worried Einstein. (See Beller 1999,
Chapters 4 and 9, on the rhetoric of finality and Ryckman 2017,
Chapter 4, for the connection to Hilbert.) Thus complementarity became
Einstein's target for investigation. In particular, Einstein had
reservations about the uncontrollable physical effects invoked by Bohr
in the context of measurement interactions, and about their role in
fixing the interpretation of the wave function. EPR's focus on
completeness was intended to support those reservations in a
particularly dramatic way. 



Max Jammer (1974, pp. 166-181) locates the development of the
EPR paper in Einstein's reflections on a thought experiment he
proposed during discussions at the 1930 Solvay conference. (For more
on EPR and Solvay 1930 see Howard, 1990 and Ryckman, 2017, pp.
118-135.) The experiment imagines a box that contains a clock
set to time precisely the release (in the box) of a photon with
determinate energy. If this were feasible, it would appear to
challenge the unrestricted validity of the Heisenberg uncertainty
relation that sets a lower bound on the simultaneous uncertainty of
energy and time. (See the entry on the
 Uncertainty Principle
 and also Bohr 1949, who describes the discussions at the 1930
conference.) The uncertainty relations, understood not just as a
prohibition on what is co-measurable, but on what is simultaneously
real, were a central component in the irrealist interpretation of the
wave function. Jammer (1974, p. 173) describes how Einstein's thinking
about this experiment, and Bohr's objections to it, evolved into a
different photon-in-a-box experiment, one that allows an observer to
determine either the momentum or the position of the photon
indirectly, while remaining outside, sitting on the box. Jammer
associates this with the distant determination of either momentum or
position that, we shall see, is at the heart of the EPR paper. Carsten
Held (1998) cites a related
 correspondence with Paul Ehrenfest
 from 1932 in which Einstein described an arrangement for the indirect
measurement of a particle of mass *m* using correlations with a
photon established through Compton scattering. Einstein's reflections
here foreshadow the argument of EPR, along with noting some of its
difficulties.



> 
> Thus without an experiment on *m* it is possible to predict
> freely, at will, *either* the momentum *or* the position
> of *m* with, in principle, arbitrary precision. This is the
> reason why I feel compelled to ascribe objective reality to
> *both*. I grant, however, that it is not logically necessary.
> (Held 1998, p. 90)
> 



Whatever their precursors, the ideas that found their way into EPR
were discussed in a series of meetings between Einstein and his two
assistants, Podolsky and Rosen. Podolsky was commissioned to compose
the paper and he submitted it to *Physical Review* in March of
1935, where it was sent for publication the day after it arrived.
Apparently Einstein never checked Podolsky's draft before submission.
He was not pleased with the result. Upon seeing the published version,
Einstein complained that it obscured his central concerns.



> 
> For reasons of language this [paper] was written by Podolsky after
> several discussions. Still, it did not come out as well as I had
> originally wanted; rather, the essential thing was, so to speak,
> smothered by formalism [Gelehrsamkeit]. (Letter from Einstein to Erwin
> Schrodinger, June 19, 1935. In Fine 1996, p. 35.)
> 



Unfortunately, without attending to Einstein's reservations, EPR is
often cited to evoke the authority of Einstein. Here we will
distinguish the argument Podolsky laid out in the text from lines of
argument that Einstein himself published in articles from 1935 on. We
will also consider the argument presented in Bohr's reply to EPR,
which is possibly the best known version, although it differs from the
others in important ways.


### 1.2 The argument in the text



The EPR text is concerned, in the first instance, with the logical
connections between two assertions. One asserts that quantum mechanics
is incomplete. The other asserts that incompatible quantities (those
whose operators do not commute, like the *x*-coordinate of
position and linear momentum in direction *x*) cannot have
simultaneous "reality" (i.e., simultaneously real values).
The authors assert the disjunction of these as a first premise (later
to be justified): one or another of these must hold. It follows that
if quantum mechanics were complete (so that the first assertion
failed) then the second one would hold; i.e., incompatible quantities
cannot have real values simultaneously. They take as a second premise
(also to be justified) that if quantum mechanics were complete, then
incompatible quantities (in particular coordinates of position and
momentum) could indeed have simultaneous, real values. They conclude
that quantum mechanics is incomplete. The conclusion certainly follows
since otherwise (if the theory were complete) one would have a
contradiction over simultaneous values. Nevertheless the argument is
highly abstract and formulaic and even at this point in its
development one can readily appreciate Einstein's disappointment.



EPR now proceed to establish the two premises, beginning with a
discussion of the idea of a complete theory. Here they offer only a
necessary condition; namely, that for a complete theory "every
element of the physical reality must have a counterpart in the
physical theory." The term "element" may remind one
of Mach, for whom this was a central, technical term connected to
sensations. (See the entry on
 Ernst Mach.)
 The use in EPR of elements of reality is also technical but
different. Although they do not define an "element of physical
reality" explicitly (and, one might note, the language of
elements is not part of Einstein's usage elsewhere), that expression
is used when referring to the values of physical quantities
(positions, momenta, and so on) that are determined by an underlying
"real physical state". The picture is that quantum systems
have real states that assign values to certain quantities. Sometimes
EPR describe this by saying the quantities in question have
"definite values", sometimes "there exists an
element of physical reality corresponding to the quantity".
Suppose we adapt the simpler terminology and call a quantity on a
system *definite* if that quantity has a definite value; i.e.,
if the real state of the system assigns a value (an "element of
reality") to the quantity. The relation that associates real
states with assignments of values to quantities is functional so that
without a change in the real state there is no change among values
assigned to quantities. In order to get at the issue of completeness,
a primary question for EPR is to determine when a quantity has a
definite value. For that purpose they offer a minimal sufficient
condition (p. 777):



> 
> If, without in any way disturbing a system, we can predict with
> certainty (i.e., with probability equal to unity) the value of a
> physical quantity, then there exists an element of reality
> corresponding to that quantity.
> 



This sufficient condition for an "element of reality" is
often referred to as the EPR
 *Criterion of Reality*.
 By way of illustration EPR point to those quantities for which the
quantum state of the system is an eigenstate. It follows from the
Criterion that at least these quantities have a definite value;
namely, the associated eigenvalue, since in an eigenstate the
corresponding eigenvalue has probability one, which we can determine
(predict with certainty) without disturbing the system. In fact,
moving from eigenstate to eigenvalue to fix a definite value is the
only use of the Criterion in EPR.



With these terms in place it is easy to show that if, say, the values
of position and momentum for a quantum system were definite (were
elements of reality) then the description provided by the wave
function of the system would be incomplete, since no wave function
contains counterparts for both elements. Technically, no state
function--even an improper one, like a delta function--is a
simultaneous eigenstate for both position and momentum; indeed, joint
probabilities for position and momentum are not well-defined in any
quantum state. Thus they establish the first premise: either quantum
theory is incomplete or there can be no simultaneously real
("definite") values for incompatible quantities. They now
need to show that if quantum mechanics were complete, then
incompatible quantities could have simultaneous real values, which is
the second premise. This, however, is not easily established. Indeed
what EPR proceed to do is odd. Instead of assuming completeness and on
that basis deriving that incompatible quantities can have real values
simultaneously, they simply set out to derive the latter assertion
without any completeness assumption at all. This
"derivation" turns out to be the heart of the paper and
its most controversial part. It attempts to show that in certain
circumstances a quantum system can have simultaneous values for
incompatible quantities (once again, for position and momentum), where
these are definite values; that is, they are assigned by the real
state of the system, hence are "elements of reality".



They proceed by sketching an iconic thought experiment whose
variations continue to be important and widely discussed. The
experiment concerns two quantum systems that are spatially distant
from one another, perhaps quite far apart, but such that the total
wave function for the pair links both the positions of the systems as
well as their linear momenta. In the EPR example the total linear
momentum is zero along the *x*-axis. Thus if the linear
momentum of one of the systems (we can call it Albert's) along the
*x*-axis were found to be *p*, the *x*-momentum
of the other system (call it Niels') would be found to be
[?]*p*. At the same time their positions along *x*
are also strictly correlated so that determining the position of one
system on the *x*-axis allows us to infer the position of the
other system along *x*. The paper constructs an explicit wave
function for the combined (Albert+Niels) system that embodies these
links even when the systems are widely separated in space. Although
commentators later raised questions about the legitimacy of this wave
function, it does appear to guarantee the required correlations for
spatially separated systems, at least for a moment (Jammer 1974, pp.
225-38; see also Halvorson 2000). In any case, one can model the
same conceptual situation in other cases that are clearly well defined
quantum mechanically (see
 Section 3.1).



At this point of the argument (p. 779) EPR make two critical
assumptions, although they do not call special attention to them. (For
the significance of these assumptions in Einstein's thinking see
Howard 1985 and also section 5 of the entry on
 Einstein.)
 The first assumption (*separability*) is that at the time when
the systems are separated, maybe quite far apart, each has its own
reality. In effect, they assume that each system maintains a separate
identity characterized by a real physical state, even though each
system is also strictly correlated with the other in respect both to
momentum and position. They need this assumption to make sense of
another. The second assumption is that of *locality*. Given
that the systems are far apart, locality supposes that "no real
change can take place" in one system as a direct consequence of
a measurement made on the other system. They gloss this by saying
"at the time of measurement the two systems no longer
interact." Note that locality does not require that nothing at
all about one system can be disturbed directly by a distant
measurement on the other system. Locality only rules out that a
distant measurement may directly disturb or change what is counted as
"real" with respect to a system, a reality that
separability guarantees. On the basis of these two assumptions they
conclude that each system can have definite values ("elements of
reality") for both position and momentum simultaneously. There
is no straightforward argument for this in the text. Instead they use
these two assumptions to show how one could be led to assign position
and momentum eigenstates to one system by making measurements on the
other system, from which the simultaneous attribution of elements of
reality is supposed to follow. Since this is the central and most
controversial part of the paper, it pays to go slowly here in trying
to reconstruct an argument on their behalf.



Here is one attempt. (Dickson 2004 analyzes some of the modal
principles involved and suggests one line of argument, which he
criticizes. Hooker 1972 is a comprehensive discussion that identifies
several generically different ways to make the case.) Locality affirms
that the real state of a system is not affected by distant
measurements. Since the real state determines which quantities are
definite (i.e., have assigned values), the set of definite quantities
is also not affected by distant measurements. So if by measuring a
distant partner we can determine that a certain quantity is definite,
then that quantity must have been definite all along. As we have seen,
the
 Criterion of Reality
 implies that a quantity is definite if the state of the system is an
eigenstate for that quantity. In the case of the strict correlations
of EPR, measuring one system triggers a reduction of the joint state
that results in an eigenstate for the distant partner. Hence any
quantity with that eigenstate is definite. For example, since
measuring the momentum of Albert's system results in a momentum
eigenstate for Niels', the momentum of Niels' system is definite.
Likewise for the position of Niels' system. Given separability, the
combination of locality and the Criterion establish a quite general
lemma; namely, *when quantities on separated systems have strictly
correlated values, those quantities are definite*. Thus the strict
correlations between Niels' system and Albert's in the EPR situation
guarantee that both position and momentum are definite; i. e., that
each system has definite position and momentum simultaneously.



EPR point out that position and momentum cannot be measured
simultaneously. So even if each can be shown to be definite in
distinct contexts of measurement, can both be definite at the same
time? The lemma answers "yes". What drives the argument is
locality, which functions logically to decontextualize the reality of
Niels' system from goings on at Albert's. Accordingly, measurements
made on Albert's system are probative for features corresponding to
the real state of Niels' system but not determinative of them. Thus
even without measuring Albert's system, features corresponding to the
real state of Niels' system remain in place. Among those features are
a definite position and a definite momentum for Niels' system along
some particular coordinate direction.



In the penultimate paragraph of EPR (p. 780) they address the problem
of getting real values for incompatible quantities simultaneously.



> 
> Indeed one would not arrive at our conclusion if one insisted that two
> or more physical quantities can be regarded as simultaneous elements
> of reality only when they can be simultaneously measured or predicted.
> ... This makes the reality [on the second system] depend upon the
> process of measurement carried out on the first system, which does not
> in any way disturb the second system. No reasonable definition of
> reality could be expected to permit this.
> 



The unreasonableness to which EPR allude in making "the reality
[on the second system] depend upon the process of measurement carried
out on the first system, which does not in any way disturb the second
system" is just the unreasonableness that would be involved in
renouncing locality understood as above. For it is locality that
enables one to overcome the incompatibility of position and momentum
measurements of Albert's system by requiring their joint consequences
for Niels' system to be incorporated in a single, stable reality
there. If we recall
 Einstein's acknowledgment to Ehrenfest
 that getting simultaneous position and momentum was "not
logically necessary", we can see how EPR respond by making it
*become* necessary once locality is assumed.



Here, then, are the key features of EPR.


* EPR is about the interpretation of state vectors ("wave
functions") and employs the standard state vector reduction
formalism (von Neumann's "projection postulate").
* The Criterion of Reality
 affirms that the eigenvalue corresponding to the eigenstate of a
system is a value determined by the real physical state of that
system. (This is the Criterion's only use.)
* (Separability)
 Spatially separated systems have real physical states.
* (Locality)
 If systems are spatially separate, the measurement (or absence of
measurement) of one system does not directly affect the reality that
pertains to the others.
* (EPR Lemma) If quantities on separated systems have strictly
correlated values, those quantities are definite (i.e., have definite
values). This follows from separability, locality and the Criterion.
No actual measurements are required.
* (Completeness)
 If the description of systems by state vectors were complete, then
definite values of quantities (values determined by the real state of
a system) could be inferred from a state vector for the system itself
or from a state vector for a composite of which the system is a
part.
* In summary, separated systems as described by EPR have definite
position and momentum values simultaneously. Since this cannot be
inferred from any state vector, the quantum mechanical description of
systems by means state vectors is incomplete.



The EPR experiment with interacting systems accomplishes a form of
indirect measurement. The direct measurement of Albert's system
yields information about Niels' system; it tells us what we
would find if we were to measure there directly. But it does this
at-a-distance, without any physical interaction taking place between
the two systems. Thus the thought experiment at the heart of EPR
undercuts the picture of measurement as necessarily involving a tiny
object banging into a large measuring instrument. If we look back at
Einstein's reservations about complementarity, we can appreciate
that by focusing on an indirect, non-disturbing kind of measurement
the EPR argument targets Bohr's program for explaining central
conceptual features of the quantum theory. For that program relied on
uncontrollable interaction with a measuring device as a necessary
feature of any measurement in the quantum domain. Nevertheless the
cumbersome machinery employed in the EPR paper makes it difficult to
see what is central. It distracts from rather than focuses on the
issues. That was Einstein's complaint about Podolsky's
text in his June 19, 1935 letter to Schrodinger.
Schrodinger responded on July 13 reporting reactions to EPR that
vindicate Einstein's concerns. With reference to EPR he
wrote:



> 
> I am now having fun and taking your note to its source to provoke the
> most diverse, clever people: London, Teller, Born, Pauli, Szilard,
> Weyl. The best response so far is from Pauli who at least admits that
> the use of the word "state" ["Zustand"] for
> the psi-function is quite disreputable. What I have so far seen by way
> of published reactions is less witty. ... It is as if one person
> said, "It is bitter cold in Chicago"; and another
> answered, "That is a fallacy, it is very hot in Florida."
> (Fine 1996, p. 74)
> 


### 1.3 Einstein's versions of the argument



If the argument developed in EPR has its roots in the 1930 Solvay
conference, Einstein's own approach to issues at the heart of EPR has
a history that goes back to the 1927 Solvay conference. (Bacciagaluppi
and Valentini 2009, pp. 198-202, would even trace it back to
1909 and the localization of light quanta.) At that 1927 conference
Einstein made a short presentation during the general discussion
session where he focused on problems of interpretation associated with
the collapse of the wave function. He imagines a situation where
electrons pass through a small hole and are dispersed uniformly in the
direction of a screen of photographic film shaped into a large
hemisphere that surrounds the hole. On the supposition that quantum
theory offers a complete account of individual processes then, in the
case of localization, why does the whole wave front collapse to just
one single flash point? It is as though at the moment of collapse an
instantaneous signal were sent out from the point of collapse to all
other possible collapse positions telling them not to flash. Thus
Einstein maintains (Bacciagaluppi and Valentini 2009, p. 488),



> 
> the interpretation, according to which |ps|2 expresses the
> probability that *this* particle is found at a given point,
> assumes an entirely peculiar mechanism of action at a distance, which
> prevents the wave continuously distributed in space from producing an
> action in two places on the screen.
> 



One could see this as a tension between local action and the
description afforded by the wave function, since the wave function
alone does not specify a unique position on the screen for detecting
the particle. Einstein continues,



> 
> In my opinion, one can remove this objection only in the following
> way, that one does not describe the process solely by the
> Schrodinger wave, but that at the same time one localizes the
> particle during propagation.
> 



In fact Einstein himself had tried this very route in May of 1927
where he proposed a way of "localizing the particle" by
associating spatial trajectories and velocities with particle
solutions to the Schrodinger equation. (See Belousek 1996 and
Holland 2005; also Ryckman 2017.) Einstein abandoned the project and
withdrew the draft from publication, however, after finding that
certain intuitive independence conditions were in conflict with the
product wave function used by quantum mechanics to treat the
composition of independent systems. The problem here anticipates the
more general issues raised by EPR over separability and composite
systems. This proposal was Einstein's one and only flirtation with the
introduction of hidden variables into the quantum theory. In the
following years he never embraced any proposal of that sort, although
he hoped for progress in physics to yield a more complete theory, and
one where the observer did not play a fundamental role. "We
believe however that such a theory ["a complete description of
the physical reality"] is possible" (p. 780). Commentators
have often mistaken that remark as indicating Einstein's predilection
for hidden variables. To the contrary, after 1927 Einstein regarded
the hidden variables project -- the project of developing a more
complete theory by starting with the existing quantum theory and
adding things, like trajectories or real states -- an improbable
route to that goal. (See, for example, Einstein 1953a.) To improve on
the quantum theory, he thought, would require starting afresh with
quite different fundamental concepts. At Solvay he acknowledges Louis
de Broglie's pilot wave investigations as a possible direction to
pursue for a more complete account of individual processes. But then
he quickly turns to an alternative way of thinking, one that he
continued to recommend as a better framework for progress, which is
not to regard the quantum theory as describing individuals and their
processes at all and, instead, to regard the theory as describing only
ensembles of individuals. Einstein goes on to suggest difficulties for
any scheme, like de Broglie's and like quantum theory itself, that
requires representations in multi-dimensional configuration space.
These are difficulties that might move one further toward regarding
quantum theory as not aspiring to a description of individual systems
but as more amenable to an ensemble (or collective) point of view, and
hence not a good starting point for building a better, more complete
theory. His subsequent elaborations of EPR-like arguments are perhaps
best regarded as *no-go* arguments, showing that the existing
quantum theory does not lend itself to a sensible realist
interpretation via hidden variables. If real states, taken as hidden
variables, are added into the existing theory, which is then tailored
to explain individual events, the result is either an incomplete
theory or else a theory that does not respect locality. Hence, new
concepts are needed. With respect to EPR, perhaps the most important
feature of Einstein's reflections at Solvay 1927 is his insight that a
clash between completeness and locality already arises in considering
a single variable (there, position) and does not require an
incompatible pair, as in EPR.



Following the publication of EPR Einstein set about almost immediately
to provide clear and focused versions of the argument. He began that
process within few weeks of EPR, in the June 19 letter to
Schrodinger, and continued it in an article published the
following year (Einstein 1936). He returned to this particular form of
an incompleteness argument in two later publications (Einstein 1948
and Schilpp 1949). Although these expositions differ in details they
all employ composite systems as a way of implementing indirect
measurements-at-a-distance. None of Einstein's accounts contains the
 Criterion of Reality
 nor the tortured EPR argument over when values of a quantity can be
regarded as "elements of reality". The Criterion and these
"elements" simply drop out. Nor does Einstein engage in
calculations, like those of Podolsky, to fix the total wave function
for the composite system explicitly. Unlike EPR, none of Einstein's
arguments makes use of simultaneous values for complementary
quantities like position and momentum. He does not challenge the
uncertainty relations. Indeed with respect to assigning eigenstates
for a complementary pair he tells Schrodinger "ist mir
wurst"--literally, it's sausage to me; i.e., he couldn't
care less. (Fine 1996, p. 38). These writings probe an incompatibility
between affirming locality and separability, on the one hand, and
completeness in the description of individual systems by means of
state functions, on the other. His argument is that we can have at
most one of these but never both. He frequently refers to this dilemma
as a "paradox".



In the letter to Schrodinger of June 19, Einstein points to a
simple argument for the dilemma which, like the argument from the 1927
Solvay Conference, involves only the measurement of a single variable.
Consider an interaction between the Albert and Niels systems that
establishes a strict correlation between their positions. (We need not
worry about momentum, or any other quantity.) Consider the evolved
wave function for the total (Albert+Niels) system when the two systems
are far apart. Now assume a principle of locality-separability
(Einstein calls it a *Trennungsprinzip*--separation
principle): Whether a determinate physical situation holds for Niels'
system (e.g., that a quantity has a particular value) does not depend
on what measurements (if any) are made locally on Albert's system. If
we measure the position of Albert's system, the strict correlation of
positions implies that Niels' system has a certain position. By
locality-separability it follows that Niels' system must already have
had that position just before the measurement on Albert's system. At
that time, however, Niels' system alone does not have a state
function. There is only a state function for the combined system and
that total state function does not single out an existing position for
Niels' system (i.e., it is not a product one of whose factors is an
eigenstate for the position of Niels' system). Thus the description of
Niels' system afforded by the quantum state function is incomplete. A
complete description would say (definitely yes) if a quantity of
Niels' system had a certain value. (Notice that this argument does not
even depend on the reduction of the total state function for the
combined system.) In this formulation of the argument it is clear that
locality-separability conflicts with
 the eigenvalue-eigenstate link,
 which holds that a quantity of a system has a value if and only if
the state of the system is an eigenstate (or a proper mixture of
eigenstates) of that quantity with that value as eigenvalue. The
"only if" part of the link would need to be weakened in
order to interpret quantum state functions as complete descriptions.
(See the entry on
 Modal Interpretations
 and see Gilton 2016 for a history of the eigenvalue-eigenstate
link.)



This argument rests on the ordinary and intuitive notion of
completeness as not omitting relevant truths. Thus, in the argument,
the description given by the state function of a system is judged
incomplete when it fails to attribute a position to the system in
circumstances where the system indeed has a position. Although this
simple argument concentrates on what Einstein saw as the essentials,
stripping away most technical details and distractions, he frequently
used another argument involving more than one quantity. (It is
actually buried in the EPR paper, p. 779, and a version also occurs in
the June 19, 1935 letter to Schrodinger. Harrigan and Spekkens,
2010 suggest reasons for preferring a many-variables argument.) This
second argument focuses clearly on the interpretation of quantum state
functions in terms of "real states" of a system, and not
on any issues about simultaneous values (real or not) for
complementary quantities. It goes like this.



Suppose, as in EPR, that the interaction between the two systems links
position and also linear momentum, and that the systems are far apart.
As before, we can measure either the position or the momentum of
Albert's system and, in either case, we can infer (respectively) a
position or a momentum for Niels' system. It follows from the
reduction of the total state function that, depending on whether we
measure the position or the momentum of Albert's system, Niels' system
will be left (respectively) either in a position eigenstate or in a
momentum eigenstate. Suppose too that separability holds, so that
Niels' system has some real physical state of affairs. If locality
holds as well, then the measurement of Albert's system does not
disturb the assumed "reality" for Niels' system. However,
that reality appears to be represented by quite different state
functions, depending on which measurement of Albert's system one
chooses to carry out. If we understand a "complete
description" to rule out that one and the same physical state
can be described by state functions with distinct physical
implications, then we can conclude that the quantum mechanical
description is incomplete. Here again we confront a dilemma between
separability-locality and completeness. Many years later Einstein put
it this way (Schilpp 1949, p. 682);



> 
> [T]he paradox forces us to relinquish one of the following two
> assertions:
>   
> 
> (1) the description by means of the psi-function is complete
>   
> 
> (2) the real states of spatially separate objects are independent of
> each other.
> 



It appears that the central point of EPR was to argue that any
interpretation of quantum state functions that attributes real
physical states to systems faces these alternatives. It also appears
that Einstein's different arguments make use of different notions of
completeness. In the first argument completeness is an ordinary notion
that amounts to not leaving out any relevant details. In the second,
completeness is a technical notion which has been dubbed
"bijective completeness" (Fine 1996 ): no more than one
quantum state should correspond to a real state. These notions are
connected. If completeness fails in the bijective sense, and more than
one quantum state corresponds to some real state, we can argue that
the ordinary notion of completeness also fails. For distinct quantum
states will differ in the values they assign to certain quantities.
(For example, the observable corresponding to the projector on a state
takes value 1 in one case but not in the other.) Hence each will omit
something that the other affirms, so completeness in the ordinary
sense will fail. Put differently, ordinary completeness implies
bijective completeness. (The converse is not true. Even if the
correspondence of quantum states to real states were one-to-one, the
description afforded by a quantum state might still leave out some
physically relevant fact about its corresponding real state.) Thus a
dilemma between locality and "completeness" in Einstein's
versions of the argument still implicates ordinary completeness. For
if locality holds, then his two-variable argument shows that bijective
completeness fails, and then completeness in the ordinary sense fails
as well.



As we have seen, in framing his own EPR-like arguments for the
incompleteness of quantum theory, Einstein makes use of
 separability
 and
 locality,
 which are also tacitly assumed in the EPR paper. Using the language
of "independent existence" he presents these ideas clearly
in an article that he sent to Max Born (Einstein 1948). 



> 
> It is ... characteristic of ... physical objects that they
> are thought of as arranged in a space-time continuum. An essential
> aspect of this arrangement ... is that they lay claim, at a
> certain time, to an existence independent of one another, provided
> these objects "are situated in different parts of space".
> ... The following idea characterizes the relative independence of
> objects (A and B) far apart in space: external influence on A has no
> direct influence on B. (Born, 1971, pp. 170-71)
> 



In the course of his correspondence with Schrodinger, however,
Einstein realized that assumptions about separability and locality
were not necessary in order to get the incompleteness conclusion that
he was after; i.e., to show that state functions may not provide a
complete description of the real state of affairs with respect to a
system. Separability supposes that there is a real state of affairs
and locality supposes that one cannot influence it immediately by
acting at a distance. What Einstein realized was that separability was
already part of the ordinary conception of a macroscopic object. This
suggested to him that if one looks at the local interaction of a
macro-system with a micro-system one could avoid having to assume
either separability or locality in order to conclude that the quantum
description of the whole was incomplete with respect to its
macroscopic part. 



This line of thought evolves and dominates over problems with
composite systems and locality in his last published reflections on
incompleteness. Instead he focuses on problems with the stability of
macro-descriptions in the transition to a classical level from the
quantum. 



> 
> the objective describability of individual macro-systems (description
> of the "real-state") can not be renounced without the
> physical picture of the world, so to speak, decomposing into a fog.
> (Einstein 1953b, p. 40. See also Einstein 1953a.)
> 



In the August 8, 1935 letter to Schrodinger Einstein says that he
will illustrate the problem by means of a "crude macroscopic
example".



> 
> The system is a substance in chemically unstable equilibrium, perhaps
> a charge of gunpowder that, by means of intrinsic forces, can
> spontaneously combust, and where the average life span of the whole
> setup is a year. In principle this can quite easily be represented
> quantum-mechanically. In the beginning the psi-function characterizes
> a reasonably well-defined macroscopic state. But, according to your
> equation [i.e., the Schrodinger equation], after the course of a
> year this is no longer the case. Rather, the psi-function then
> describes a sort of blend of not-yet and already-exploded systems.
> Through no art of interpretation can this psi-function be turned into
> an adequate description of a real state of affairs; in reality there
> is no intermediary between exploded and not-exploded. (Fine 1996, p.
> 78)
> 



The point is that after a year either the gunpowder will have
exploded, or not. (This is the "real state" which in the
EPR situation requires one to assume separability.) The state
function, however, will have evolved into a complex superposition over
these two alternatives. Provided we maintain the eigenvalue-eigenstate
link, the quantum description by means of that state function will
yield neither conclusion, and hence the quantum description is
incomplete. For a contemporary response to this line of argument, one
might look to the program of decoherence. (See
 Decoherence.)
 That program points to interactions with the environment which may
quickly reduce the likelihood of any interference between the
"exploded" and the "not-exploded" branches of
the evolved psi-function. Then, breaking the eigenvalue-eigenstate
link, decoherence adopts a perspective according to which the (almost)
non-interfering branches of the psi-function allow that the gunpowder
is indeed either exploded or not. Even so, decoherence fails to
identify which alternative is actually realized, leaving the quantum
description still incomplete. Such decoherence-based interpretations
of the psi-function are certainly "artful", and their
adequacy is still under debate (see Schlosshauer 2007, especially
Chapter 8).



The reader may recognize the similarity between Einstein's
 exploding gunpowder example
 and Schrodinger's cat (Schrodinger 1935a, p. 812). In the
case of the cat an unstable atom is hooked up to a lethal device that,
after an hour, is as likely to poison (and kill) the cat as not,
depending on whether the atom decays. After an hour the cat is either
alive or dead, but the quantum state of the whole atom-poison-cat
system at this time is a superposition involving the two possibilities
and, just as in the case of the gunpowder, is not a complete
description of the situation (life or death) of the cat. The
similarity between the gunpowder and the cat is hardly accidental
since Schrodinger first produced the cat example in his reply of
September 19, 1935 to Einstein's August 8 gunpowder letter. There
Schrodinger says that he has himself constructed "an
example very similar to your exploding powder keg", and proceeds
to outline the cat (Fine 1996, pp. 82-83). Although the
"cat paradox" is usually cited in connection with the
problem of quantum measurement (see the relevant section of the entry
on
 Philosophical Issues in Quantum Theory)
 and treated as a paradox separate from EPR, its origin is here as an
argument for incompleteness that avoids the twin assumptions of
separability and locality. Schrodinger's development of
"entanglement", the term he introduced for the
correlations that result when quantum systems interact, also began in
this correspondence over EPR -- along with a treatment of what he
called quantum "steering" (Schrodinger 1935a, 1935b;
see
 Quantum Entanglement and Information).


## 2. A popular form of the argument: Bohr's response



The literature surrounding EPR contains yet another version of the
argument, a popular version that--unlike any of
Einstein's--features the
 Criterion of Reality.
 Assume again an interaction between our two systems linking their
positions and their linear momenta and suppose that the systems are
far apart. If we measure the position of Albert's system, we can infer
that Niels' system has a corresponding position. We can also predict
it with certainty, given the result of the position measurement of
Albert's system. Hence, in this version, the Criterion of Reality is
taken to imply that the position of Niels' system constitutes an
element of reality. Similarly, if we measure the momentum of Albert's
system, we can conclude that the momentum of Niels' system is an
element of reality. The argument now concludes that since we can
choose freely to measure either position or momentum, it
"follows" that both must be elements of reality
simultaneously.



Of course no such conclusion follows from our freedom of choice. It is
not sufficient to be able to choose at will which quantity to measure;
for the conclusion to follow from the Criterion alone one would need
to be able to measure both quantities at once. This is precisely the
point that Einstein recognized in his
 1932 letter to Ehrenfest
 and that EPR addresses by assuming locality and separability. What is
striking about this version is that these principles, central to the
original EPR argument and to the dilemma at the heart of Einstein's
versions, are obscured here. Instead this version features the
Criterion and those "elements of reality". Perhaps the
difficulties presented by Podolsky's text contribute to this reading.
In any case, in the physics literature this version is commonly taken
to represent EPR and usually attributed to Einstein. This reading
certainly has a prominent source in terms of which one can understand
its popularity among physicists; it is Niels Bohr himself.



By the time of the EPR paper many of the early interpretive battles
over the quantum theory had been settled, at least to the satisfaction
of working physicists. Bohr had emerged as the
"philosopher" of the new theory and the community of
quantum theorists, busy with the development and extension of the
theory, were content to follow Bohr's leadership when it came to
explaining and defending its conceptual underpinnings (Beller 1999,
Chapter 13). Thus in 1935 the burden fell to Bohr to explain what was
wrong with the EPR "paradox". The major article that he
wrote in discharging this burden (Bohr 1935a) became the canon for how
to respond to EPR. Unfortunately, Bohr's summary of EPR in that
article, which is the version just above, also became the canon for
what EPR contained by way of argument.



Bohr's response to EPR begins, as do many of his treatments of the
conceptual issues raised by the quantum theory, with a discussion of
limitations on the simultaneous determination of position and
momentum. As usual, these are drawn from an analysis of the
possibilities of measurement if one uses an apparatus consisting of a
diaphragm connected to a rigid frame. Bohr emphasizes that the
question is to what extent can we trace the interaction between the
particle being measured and the measuring instrument. (See Beller
1999, Chapter 7 for a detailed analysis and discussion of the
"two voices" contained in Bohr's account. See too
Bacciagaluppi 2015.) Following the summary of EPR, Bohr (1935a, p.
700) then focuses on the Criterion of Reality which, he says,
"contains an ambiguity as regards the meaning of the expression
'without in any way disturbing a system'." Bohr
agrees that in the indirect measurement of Niels' system achieved when
one makes a measurement of Albert's system "there is no question
of a mechanical disturbance" of Niels' system. Still, Bohr
claims that a measurement on Albert's system does involve "an
influence on the very conditions which define the possible types of
predictions regarding the future behavior of [Niels'] system."
The meaning of this claim is not at all clear. Indeed, in revisiting
EPR fifteen years later, Bohr would comment, 



> 
> Rereading these passages, I am deeply aware of the inefficiency of
> expression which must have made it very difficult to appreciate the
> trend of the argumentation (Bohr 1949, p. 234).
> 



Unfortunately, Bohr takes no notice there of Einstein's later versions
of the argument and merely repeats his earlier response to EPR. In
that response, however inefficiently, Bohr appears to be directing
attention to the fact that when we measure, for example, the position
of Albert's system conditions are in place for predicting the position
of Niels' system but not its momentum. The opposite would be true in
measuring the momentum of Albert's system. Thus his "possible
types of predictions" concerning Niels' system appear to
correspond to which variable we measure on Albert's system. Bohr
proposes then to block the EPR Criterion by counting, say, the
position measurement of Albert's system as an "influence"
on the distant system of Niels. If we assume it is an influence that
disturbs Niels' system, then the Criterion could not be used, as in
Bohr's version of the argument, in producing an element of reality for
Niels' system that challenges completeness.



There are two important things to notice about this response. The
first is this. In conceding that Einstein's indirect method for
determining, say, the position of Niels' system does not mechanically
disturb that system, Bohr departs from his original program of
complementarity, which was to base the uncertainty relations and the
statistical character of quantum theory on uncontrollable physical
interactions, interactions that were supposed to arise inevitably
between a measuring instrument and the system being measured. Instead
Bohr now distinguishes between a genuine physical interaction (his
"mechanical disturbance") and some other sort of
"influence" on the conditions for specifying (or
"defining") sorts of predictions for the future behavior
of a system. In emphasizing that there is no question of a robust
interaction in the EPR situation, Bohr retreats from his earlier,
physically grounded conception of complementarity.



The second important thing to notice is how Bohr's response needs to
be implemented in order to block the argument of EPR and Einstein's
later arguments that pose a dilemma between principles of locality and
completeness. In these arguments the locality principle makes explicit
reference to the reality of the unmeasured system: the reality
pertaining to Niels' system does not depend on what measurements (if
any) are made locally on Albert's system. Hence Bohr's suggestion that
those measurements influence conditions for specifying types of
predictions would not affect the argument unless one includes those
conditions as part of the reality of Niels' system. This is exactly
what Bohr goes on to say, "these conditions constitute an
inherent element of the description of any phenomena to which the term
'physical reality' can be properly attached" (Bohr
1935a, p. 700). So Bohr's picture is that these
"influences", operating directly across any spatial
distances, result in different physically real states of Niels' system
depending on the type of measurement made on Albert's. (Recall EPR
warning against just this move.) 



The quantum formalism for interacting systems describes how a
measurement on Albert's system reduces the composite state and
distributes quantum states and associated probabilities to the
component systems. Here Bohr redescribes that formal reduction using
EPR's language of influences and reality. He turns ordinary local
measurements into "influences" that automatically change
physical reality elsewhere, and at any distance whatsoever. This
grounds the quantum formalism in a rather magical ontological
framework, a move quite out of character for the usually pragmatic
Bohr. In his correspondence over EPR, Schrodinger compared ideas
like that to ritual magic. 



> 
> This assumption arises from the standpoint of the savage, who believes
> that he can harm his enemy by piercing the enemy's image with a
> needle. (Letter to Edward Teller, June 14, 1935, quoted in
> Bacciagaluppi 2015)
> 



It is as though EPR's talk of "reality" and its elements
provoked Bohr to adopt the position of Moliere's doctor who, pressed
to explain why opium is a sedative, invents an inherent dormative
virtue, "which causes the senses to become drowsy."
Usually Bohr sharply deflates any attempt like this to get behind the
formalism, insisting that "the appropriate physical
interpretation of the symbolic quantum-mechanical formalism amounts
only to predictions, of determinate or statistical character"
(Bohr 1949, p. 238).



Could this portrait of nonlocal influences automatically shaping a
distant reality be a by-product of Bohr's "inefficiency of
expression"? Despite Bohr's seeming tolerance for a breakdown of
locality in his response here to EPR, in other places Bohr rejects
nonlocality in the strongest terms. For example in discussing an
electron double slit experiment, which is Bohr's favorite model for
illustrating the novel conceptual features of quantum theory, and
writing only weeks before the publication of EPR, Bohr argues as
follows.



> 
> If we only imagine the possibility that without disturbing the
> phenomena we determine through which hole the electron passes, we
> would truly find ourselves in irrational territory, for this would put
> us in a situation in which an electron, which might be said to pass
> through this hole, would be affected by the circumstance of whether
> this [other] hole was open or closed; but ... it is completely
> incomprehensible that in its later course [the electron] should let
> itself be influenced by this hole down there being open or shut. (Bohr
> 1935b)
> 



It is uncanny how closely Bohr's language mirrors that of EPR. But
here Bohr defends locality and regards the very contemplation of
nonlocality as "irrational" and "completely
incomprehensible". Since "the circumstance of whether this
[other] hole was open or closed" does affect the possible types
of predictions regarding the electron's future behavior, if we expand
the concept of the electron's "reality", as he appears to
suggest for EPR, by including such information, we do
"disturb" the electron around one hole by opening or
closing the other hole. That is, if we give to "disturb"
and to "reality" the very same sense that Bohr appears to
give them when responding to EPR, then we are led to an
"incomprehensible" nonlocality, and into the territory of
the irrational (like Schrodinger's savage).



There is another way of trying to understand Bohr's position.
According to one common reading (see
 Copenhagen Interpretation),
 after EPR Bohr embraced a relational (or contextual) account of
property attribution. On this account to speak of the position, say,
of a system presupposes that one already has put in place an
appropriate interaction involving an apparatus for measuring position
(or at least an appropriate frame of reference for the measurement;
Dickson 2004). Thus "the position" of the system refers to
a relation between the system and the measuring device (or measurement
frame). (See
 Relational Quantum Mechanics,
 where a similar idea is developed independently of measurements.) In
the EPR context this would seem to imply that before one is set up to
measure the position of Albert's system, talk of the position of
Niels' system is out of place; whereas after one measures the position
of Albert's system, talk of the position of Niels' system is
appropriate and, indeed, we can then say truly that Niels' system
"has" a position. Similar considerations govern momentum
measurements. It follows, then, that local manipulations carried out
on Albert's system, in a place we may assume to be far removed from
Niels' system, can directly affect what is meaningful to say about, as
well as factually true of, Niels' system. Similarly, in the double
slit arrangement, it would follow that what can be said meaningfully
and said truly about the position of the electron around the top hole
would depend on the context of whether the bottom hole is open or
shut. One might suggest that such relational actions-at-a-distance are
harmless ones, perhaps merely "semantic"; like becoming
the "best" at a task when your only competitor--who
might be miles away--fails. Note, however, that in the case of
ordinary relational predicates it is not inappropriate (or
"meaningless") to talk about the situation in the absence
of complete information about the relata. So you might be the best at
a task even if your competitor has not yet tried it, and you are
definitely not an aunt (or uncle) until one of your siblings gives
birth. But should we say that an electron is nowhere at all until we
are set up to measure its position, or would it be inappropriate
(meaningless?) even to ask? 



If quantum predicates are relational, they are different from many
ordinary relations in that the conditions for the relata are taken as
criterial for the application of the term. In this regard one might
contrast the relativity of simultaneity with the proposed relativity
of position. In relativistic physics specifying a world-line fixes a
frame of reference for attributions of simultaneity to events
regardless of whether any temporal measurements are being made or
contemplated. But in the quantum case, on this proposal, specifying a
frame of reference for position (say, the laboratory frame) does not
entitle one to attribute position to a system, unless that frame is
associated with actually preparing or completing a measurement of
position for that system. To be sure, analyzing predicates in terms of
occurrent measurement or observation is familiar from neopositivist
approaches to the language of science; for example, in Percy
Bridgman's operational analysis of physical terms, where the actual
applications of test-response pairs constitute criteria for any
meaningful use of a term (see
 Theory and Observation in Science ).
 Rudolph Carnap's later introduction of reduction sentences (see the
entry on the
 Vienna Circle)
 has a similar character. Still, this positivist reading entails just
the sort of nonlocality that Bohr seemed to abhor.



In the light of all this it is difficult to know whether a coherent
response can be attributed to Bohr reliably that would derail EPR. (In
different ways, Dickson 2004 and Halvorson and Clifton 2004 make an
attempt on Bohr's behalf. These are examined in Whitaker 2004 and Fine
2007. See also the essays in Faye and Folse 2017.) Bohr may well have
been aware of the difficulty in framing the appropriate concepts
clearly when, a few years after EPR, he wrote,



> 
> The unaccustomed features of the situation with which we are
> confronted in quantum theory necessitate the greatest caution as
> regard all questions of terminology. Speaking, as it is often done of
> disturbing a phenomenon by observation, or even of creating physical
> attributes to objects by measuring processes is liable to be
> confusing, since all such sentences imply a departure from conventions
> of basic language which even though it can be practical for the sake
> of brevity, can never be unambiguous. (Bohr 1939, p. 320. Quoted in
> Section 3.2 of the entry on the
>  Uncertainty Principle.)
>  


## 3. Development of EPR


### 3.1 Spin and The Bohm version



For about fifteen years following its publication, the EPR paradox was
discussed at the level of a thought experiment whenever the conceptual
difficulties of quantum theory became an issue. In 1951 David Bohm, a
protege of Robert Oppenheimer and then an untenured
Assistant Professor at Princeton University, published a textbook on
the quantum theory in which he took a close look at EPR in order to
develop a response in the spirit of Bohr. Bohm showed how one could
mirror the conceptual situation in the EPR thought experiment by
looking at the dissociation of a diatomic molecule whose total spin
angular momentum is (and remains) zero; for instance, the dissociation
of an excited hydrogen molecule into a pair of hydrogen atoms by means
of a process that does not change an initially zero total angular
momentum (Bohm 1951, Sections 22.15-22.18). In the Bohm
experiment the atomic fragments separate after interaction, flying off
in different directions freely to separate experimental wings.
Subsequently, in each wing, measurements are made of spin components
(which here take the place of position and momentum), whose measured
values would be anti-correlated after dissociation. In the so-called
singlet state of the atomic pair, the state after dissociation, if one
atom's spin is found to be positive with respect to the orientation of
an axis perpendicular to its flight path, the other atom would be
found to have a negative spin with respect to a perpendicular axis
with the same orientation. Like the operators for position and
momentum, spin operators for different non-orthogonal orientations do
not commute. Moreover, in the experiment outlined by Bohm, the atomic
fragments can move to wings far apart from one another and so become
appropriate objects for assumptions that restrict the effects of
purely local actions. Thus Bohm's experiment mirrors the entangled
correlations in EPR for spatially separated systems, allowing for
similar arguments and conclusions involving locality, separability,
and completeness. Indeed, a late note of Einstein's, that may have
been prompted by Bohm's treatment, contains a very sketchy spin
version of the EPR argument - once again pitting completeness
against locality ("A coupling of distant things is
excluded." Sauer 2007, p. 882). Following Bohm (1951) a paper by
Bohm and Aharonov (1957) went on to outline the machinery for a
plausible experiment in which entangled spin correlations could be
tested. It has become customary to refer to experimental arrangements
involving determinations of spin components for spatially separated
systems, and to a variety of similar set-ups (especially ones for
measuring photon polarization), as "EPRB"
experiments--"B" for Bohm. Because of technical
difficulties in creating and monitoring the atomic fragments, however,
there seem to have been no immediate attempts to perform a Bohm
version of EPR.


### 3.2 Bell and beyond



That was to remain the situation for almost another fifteen years,
until John Bell utilized the EPRB set-up to construct a stunning
argument, at least as challenging as EPR, but to a different
conclusion (Bell 1964). Bell considers correlations between
measurement outcomes for systems in separate wings where the
measurement axes of the systems differ by angles set locally. In his
original paper, essentially using the lemma from EPR governing strict
correlations, Bell shows that correlations measured in different runs
of an EPRB experiment satisfy a system of constraints, known as the
Bell inequalities. Later demonstrations by Bell and others, using
related assumptions, extend this class of inequalities. In certain of
these EPRB experiments, however, quantum theory predicts correlations
that violate particular Bell inequalities by an experimentally
significant amount. Thus Bell shows (see the entry on
 Bell's Theorem)
 that the quantum statistics are inconsistent with the given
assumptions. Prominent among these is an assumption of locality,
similar to the locality assumptions tacitly assumed in EPR and
(explicitly) in the one-variable and many-variable arguments of
Einstein. One important difference is that for Einstein locality
restricts factors that might influence the (assumed) real physical
states of spatially separated systems (separability). For Bell,
locality is focused instead on factors that might influence outcomes
of measurements in experiments where both systems are measured. (See
Fine 1996, Chapter 4.) These differences are not usually attended to
and Bell's theorem is often characterized simply as showing that
quantum theory is nonlocal. Even so, since assumptions other then
locality are needed in any derivation of the Bell inequalities
(roughly, assumptions guaranteeing a classical representation of the
quantum probabilities; see Fine 1982a, and Malley 2004), one should be
cautious about singling out locality (in Bell's sense, or Einstein's)
as necessarily in conflict with the quantum theory, or refuted by
experiment.



Bell's results have been explored and deepened by various theoretical
investigations and they have stimulated a number of increasingly
sophisticated and delicate EPRB-type experiments designed to test
whether the Bell inequalities hold where quantum theory predicts they
should fail. With a few anomalous exceptions, the experiments appear
to confirm the quantum violations of the inequalities. (Brunner *et
al* 2014 is a comprehensive technical review.) The confirmation is
quantitatively impressive, although not fully conclusive. There are a
number of significant requirements on the experiments whose failures
(generally downplayed as "loopholes") allow for models of
the experimental data that embody locality (in Bell's sense),
so-called local realist models. One family of "loopholes"
(sampling) arises from possible losses (inefficiency) between emission
and detection and from the delicate coincidence timing required to
compute correlations. All the early experiments to test the Bell
inequalities were subject to this loophole, so all could be modeled
locally and realistically. (The prism and synchronization models in
Fine 1982b are early models of this sort. Larsson 2014 is a general
review.) Another "loophole" (locality) concerns whether
Niels' system, in one wing, could learn about what measurements are
set to be performed in Albert's wing in time to adjust its behavior.
Experiments insuring locality need to separate the wings and this can
allow losses or timing glitches that open them to models exploiting
sampling error. Perversely, experiments to address sampling may
require the wings to be fairly close together, close enough generally,
it turns out, to allow information sharing and hence local realist
models. There are now a few experiments that claim to close both
loopholes together. They too have problems. (See Bednorz 2017 for a
critical discussion.)



There is also a third major complication or "loophole". It
arises from the need to ensure that causal factors affecting
measurement outcomes are not correlated with the choices of
measurement settings. Known as "measurement independence"
or sometimes "free choice", it turns out that even
statistically small violations of this independence requirement allow
for local realism (Putz and Gisin 2016). Since connections between
outcomes and settings might occur anywhere in the causal past of the
experiment, there is really no way to insure measurement independence
completely. Suitably random choices of settings might avoid this
loophole within the time frame of the experiment, or even extend that
time some years into the past. An impressive, recent experiment pushes
the time frame back about six hundred years by using the color of
Milky Way starlight (blue or red photons) to choose the measurement
settings. (Handsteiner *et al* 2017). Of course traveling
between the Milky Way and the detectors in Vienna a lot of starlight
is lost (over seventy per cent), which leaves the experiment wide open
to the sampling loophole. Moreover, there is an obvious common cause
for settings and outcomes (and all); namely, the big bang. With that
in mind one might be inclined to dismiss free choice as not serious
even for a "loophole". It may seem like an *ad hoc*
hypothesis that postulates a cosmic conspiracy on the part of Nature
just to the save the Bell inequalities. Note, however, that ordinary
inefficiency can also be modeled locally as a violation of free
choice, because an individual measurement that produces no usable
result can just as well be regarded as not currently available. Since
inefficiency is not generally counted as a violation of local
causality or a restriction on free will, nor as a conspiracy (well,
not a cosmic one), measurement dependence should not be dismissed so
quickly. Instead, one might see measurement dependent correlations as
normal limitations in a system subject to dynamical constraints or
boundary conditions, and thus use them as clues, along with other
guideposts, in searching for a covering local theory. (See Weinstein
2009.)



Experimental tests of the Bell inequalities continue to be refined.
Their analysis is delicate, employing sophisticated statistical models
and simulations. (See Elkouss and Wehner 2016 and Graft 2016.) The
significance of the tests remains a lively area for critical
discussion. Meanwhile the techniques developed in the experiments, and
related ideas for utilizing the entanglement associated with EPRB-type
interactions, have become important in their own right. These
techniques and ideas, stemming from EPRB and the Bell theorem, have
applications now being advanced in the field of quantum information
theory -- which includes quantum cryptography, teleportation and
computing (see
 Quantum Entanglement and Information).



To go back to the EPR dilemma between
 locality
 and completeness, it would appear from the Bell theorem that
Einstein's preference for locality at the expense of completeness may
have fixed on the wrong horn. Even though the Bell theorem does not
rule out locality conditions conclusively, it should certainly make
one wary of assuming them. On the other hand, since Einstein's
 exploding gunpowder argument
 (or Schrodinger's cat), along with his later arguments over
macro-systems, support incompleteness without assuming locality, one
should be wary of adopting the other horn of the dilemma, affirming
that the quantum state descriptions are complete and
"therefore" that the theory is nonlocal. It may well turn
out that both horns need to be rejected: that the state functions do
not provide a complete description and that the theory is also
nonlocal (although possibly still separable; see Winsberg and Fine
2003). There is at least one well-known approach to the quantum theory
that makes a choice of this sort, the de Broglie-Bohm approach
 (Bohmian Mechanics).
 Of course it may also be possible to break the EPR argument for the
dilemma plausibly by questioning some of its other assumptions (e.g.,
 separability,
 the reduction postulate, the
 eigenvalue-eigenstate link,
 or measurement independence). That might free up the remaining
option, to regard the theory as both local and complete. Perhaps some
version of the
 Everett Interpretation
 would come to occupy this branch of the interpretive tree, or perhaps
 Relational Quantum Mechanics.