**Nonlocality:** In quantum mechanics the term "nonlocality" refers to
an apparent failure of a certain relativity-theory-based → locality
assumption. This assumption is that no information about which
experiment is freely chosen and performed in one space-time region can
be present in a second space-time region unless a point traveling at the
speed of light (or less) can reach the second region from the first.
This assumption is valid in relativistic *classical* physics. Yet
quantum theory permits the existence of certain experiments in which
this locality assumption seems to fails. Einstein called the
faster-than-light effect evidently entailed by conventional (Copenhagen)
quantum theory "spooky action at a distance".

The simplest of the experiments pertinent to this issue involve two
measurements performed in two space-time regions that lie so far apart
that nothing traveling at the speed of light or less can pass from
either of these two regions to the other. The experimental arrangements
are such that an experimenter in each region\-\--or perhaps some device
that he has set up\-\--is able to choose between two alternative
possible measurements. The locality assumption then demands, for each
region, that the truth of statements exclusively about the outcomes of
the possible measurements performed in that region be independent of
which experiment is "freely chosen" in the other (faraway) region.

The first actual experiment exhibiting these features was carried out by
Aspect, Grangier, and Roger \[Aspect 1981\]. Dozens of other such
experiments have been carried out since,

and the validity of the quantum predictions appears to be borne out.

The significance of this nonlocality property of quantum theory is
clouded by several considerations. The first is that although the
conventional quantum precepts do appear to entail the need for some sort
of *sub rose,* behind-the-scenes, faster-than-light transfer of
information \[→ Einstein Locality\], this effect cannot be used to send
a superluminal *signal*: no one can use this effect to transfer,
superluminally, information that he or she possesses to a faraway
colleague. This limitation on *signal* velocity, together with other
relativistic features of the actually verifiable predictions of the
theory, allows relativistic quantum field theory to be called
"relativistic" in spite of the apparently entailed faster-than-light
transfer of information.

It might seem contradictory to assert first that locality fails, and
hence that information about which experiment is freely chosen and
performed in a first region is present in a second region, yet then to
assert that the experimenter in the first region cannot use this feature
to send information to a colleague in the second region. The resolution
of the puzzle is that the dependence of faraway measurable properties on
the choice made by the nearby experimenter arises only via nature's
choice of the *outcome* of the nearby experiment. The faraway colleague,
lacking all knowledge about which outcome occurs in the sender's region,
must treat that outcome as unknown. This leads to a quantum theoretical
averaging over these outcomes that exactly eliminates all dependence
upon the sender's free choice of anything that the receiving colleague
can observe.

A second clouding consideration is this: in order to analyze the
consequences of the non-dependence of some property upon a free choice
one must consider, theoretically, or logically, within one argument, the
consequences of various alternative choices. But, in the cases of
interest, only one of the alternative possibilities can actually occur
in any one existing empirical/experimental situation. Thus the argument
needed to demonstrate the existence of faster-than-light transfer of
information requires some sort of counterfactual

reasoning that involves considering in one argument the predictions
about outcomes of several experiments that cannot all be actually
performed.

A logical opening to counterfactual argumentation is provided by the
precepts of quantum theory themselves. Bohr often emphasized the freedom
of experimenters to choose which experiment is actually performed. This
*freedom to choose* is important in quantum theory for the following
reason: the quantum state (or wave function) of a physical system
provides the basis for predictions about outcomes of *whichever*
experiment is freely chosen and performed: predictions for various
alternative possible choices are given by the theory, even though only
one of the alternatives can be realized physically. On the other hand,
the structure of the quantum mathematics entails that the *outcomes* of
certain pairs of measurements, between which the experimenter is
considered free to choose, cannot be simultaneously represented within
this mathematics. This theoretical limitation upon the theoretically
representable outcomes is reconciled with the claim of the pragmatic or
epistemological completeness of quantum theory by noting that whenever
the outcomes of the two measurements cannot be theoretically represented
simultaneously then the two experiments also cannot be physically
performed simultaneously. Hence the theoretical and physical limitations
match, and completeness can be claimed.

The validity of this way of arguing for the completeness of the theory
was brought into question by a 1935 paper by Einstein, Podolsky , and
Rosen. . Because these authors were endeavoring to prove an internal
inconsistency of the quantum precepts, they were careful not to *assume*
that, contrary to the precepts of quantum theory, the outcomes of
mutually incompatible measurements were simultaneously well defined. On
the contrary, they used the quantum prohibition on well defined values
of mutually incompatible properties to deduce that they could influence
by their nearby choice which of two faraway mutually incompatible
properties was real. Thus what they actually

proved was that Copenhagen precepts entailed the existence of
faster-than-light transfer of information, though not faster-than-light
signaling.

In 1964 John Bell published a follow-up to the 1935 paper of Einstein
et. al.. Because it was, specifically, the Copenhagen prohibition
against well defined values for the outcomes of mutually incompatible
measurements that allowed Einstein et.al. to deduce the need for
faster-than-light transfer of information, Bell (1964) inquired whether
dropping that Copenhagen precept could extinguish the need for
faster-than-light information transfer. Bell forthrightly contravened
the Copenhagen ban on determinate outcomes of mutually incompatible
measurements by introducing "deterministic hidden variables". These
hidden variables specify, simultaneously, the outcomes of *all* of the
alternative possible experiments under consideration. Bell then showed
\[→Bell's Theorem\] that, within this deterministic hidden variable
structure, one cannot reconcile the validity of the predictions of
quantum theory (in these experiments) with the locality assumption that
the outcomes in each region be independent of which experiment is
performed in the other (faraway) region

The hidden-variable machinery introduced by Bell is actually
superfluous: all that is really needed is the assumption that in any
given empirical *instance,* prior to the independent choices made by the
experimenters in the two far-apart region, any one of the allowed pairs
of choices *could* occur, and that for each such pair of choice (of
which pair of measurement is performed) some long sequence of N pairs of
numbers represent outcomes that *could* occur in the pair of regions if
N repetitions of the selected pair of measurements were performed. The
existence of such sequences of pairs of numbers specifying possible
outcomes follows from Bell's hidden-variable machinery. But they refer
only to *performable actions* and *observable outcomes*. Thus they can
be stated without bringing in any notions of "microscopic", "invisible",
or other "hidden" variables. The assumption that such a set of pairs of
numbers specifying outcomes exists is called "counterfactual
definiteness". This assumption cannot be consistently reconciled with
the assumed validity of the predictions of quantum theory for each of
the measurement possibilities available to the experimenters, if one
demands also that outcomes in each region be independent of which
experiment is chosen and performed in the faraway region. (Stapp, 1979)

Bell (1971) and others (Clauser 1969 ) went on to consider, instead of
*deterministic* local hidden-variable theories, rather *probabilistic*
local hidden variable theories. But, as shown by Stapp(1978), and
independently by Fine (1982), this change does not substantially change
the situation, because the two detailed formulations are, from a logical
point of view, essentially equivalent.

The locality assumption fails, therefore, under either of these two
opposing conditions on outcomes: either the Copenhagen prohibition of
well defined values of outcomes of mutually incompatible measurements,
or the counterfactual definiteness assumption that for each of the two
times two, or four, possible combinations of measurements available to
the experimenters, some set numbers represents outcomes that could occur
if that pair of measurement were to be selected by the experimenters

In both of these to cases some special conditions pertaining to outcomes
are imposed.

The question thus naturally arises whether locality fails also under the
weaker assumptions that, for some selected experimental situation, the
predictions of quantum theory are valid and the two choices (one made in
each of two very far apart regions, and determining which measurement
will be performed in that region) can be treated as two independent free
variables.

The answer is affirmative! Under experimental conditions described by
Hardy (1993) there are again two far apart experimental space-time
regions, labeled R and L, and in each region an experimenter chooses
between a first or second possible measurement and he observes and
records there whether the first or second possible outcome of the
*single measurement* that he performs occurs. In some specific frame of
reference the space-time region L will be earlier than the space-time
region R. Quantum theory makes four pertinent predictions. The first two
prediction combine with the locality condition that "the outcome
observed and recorded in the earlier space-time region does not depend
upon which measurement is chosen and performed later" to prove, under
the condition that the *first* of the two alternative possible
measurements is chosen in the earlier region, the truth of the following
statement (Stapp 2003):

SR: If performing the first measurement in the later region gives the
first of the two possible outcomes, then performing, instead, the second
measurement would (necessarily) give the first of the two possible
outcomes of that second experiment.

Under the condition that the first measurement is performed in the
earlier region, the first two predictions of quantum theory in the Hardy
case are:

1)  If the first measurement is performed in the later region and the
    first possible outcome appears there, then the first possible
    outcome must have appeared in the earlier region.

2)  If the second measurement is performed in the later region and the
    first possible outcome appeared in the earlier region, then the
    first possible outcome must appear in the later region.

Notice that the first of these two predictions is analogous in form to
the predictions used by Einstein, Podolsky, and Rosen, in their
argument, except that here the possible outcomes are just two in number,
rather that a continuum. But the second prediction, which is again a
prediction with certainty (probability unity), in the idealized limit
that is being considered here, pertains to the case in which the pairing
of measurements in the two regions is different from what it was for the
first prediction. This crossing of the pairings creates a potent new
logical situation.

Combining these two predictions with the assumption that changing the
choice of which experiment was performed in the later region cannot
affect what already happened earlier in the faraway region entails the
truth of SR.

The second two predictions hold under the condition that the second
measurement is performed in the earlier region. They are:

3)  If the first possible outcome appears in the earlier region and the

> first measurement is performed in the later region, then the first
> possible
>
> outcome will appear in the later region

4)  If the first possible outcome appears in the earlier region the

> second measurement is performed in the later region, and then the
> second
>
> possible outcome will sometimes occur in the later region,

Quantum theory predicts that no matter which of the measurements under
consideration is performed, each possible outcome will occur half the
time. Thus the common premise of 3) and 4) is sometimes satisfied.
Combining these two predictions with the assumption that changing the
choice of which experiment was performed in the later region cannot
affect what already happened earlier in the faraway region entails that
SR. sometimes fails: the assertion SR is false.

The fact that statement SR about outcomes of measurements performable in
the later region is true if the first possible measurement is chosen and
performed in the earlier region but is false if the second possible
measurement is chosen and performed in that earlier region means that
information about which experiment is performed in the earlier region
must be present in the later region. This conclusion contradicts the
locality condition that information about which choice is freely made by
an experimenter in one region cannot be present in a second region
unless the second can be reached from the first by traveling no faster
than light.

David Mermin (1987) gives a rather compelling argument that the
predictions of quantum theory are very mysterious if one tries to deny
the existence of superluminal information transfer. Shimony (1987) and
Jarrett (1987), like most other contributors to the nonlocality issue,
tie their analyses to Bell's theorem, and hence to hidden-variable
reality" assumptions that conflict with the precepts of quantum theory.
Hence it is not clear that it is the locality assumption, rather than
the reality assumption, that fails.

Jarrett and Shimony call by the names "locality" and "parameter
independence", respectively, a certain property that is satisfied by the
predictions of quantum theory, and that is entailed by the requirement
of no superluminal signaling. Using Jarrett's weak definition (i.e.,
weak locality requirement) one would call quantum theory "local".
However, Shimony emphasizes that because entangled states of well
separated bodies exist "there is a peculiar kind of *quantum
nonlocality* in nature. To get to the crux of the matter I have defined
locality to be the requirement of no superluminal transfer of
information about which measurements are chosen and performed by
experimenters, and taken nonlocality to be the failure of that
condition. According to this definition, conventional (Copenhagen)
quantum theory and relativistic quantum field theory are nonlocal,
though in a way that does not allow superluminal signaling.

**Primary/Secondary References.**

Aspect, A., Grangier, P., & Roger, G.,(1981): Experimental tests of
realistic local theories via Bell's Theorem, *Phys. Rev. Lett. 47,
460-463.*

Bell, J. S. (1964) On the Einstein Podolsky Rosen Paradox, *Physics* 1,
195-200.

Bell, J. S. (1971): Introduction to the hidden variable question.
*Foundations of Quantum Mechanics*, Proceedings of the International
School of Physics 'Enrico Fermi',

Course II, New York:Academic pp.279-81.

Clauser, J. F., Horne, M.A., Shimony, A., and Holt, R.A., Proposed
experiment to test local hidden-variable theories, *Phys. Rev. Lett*.,
23, 880-884.

Fine, A. (1982): Hidden variables, joint probabilities, and Bell
inequalities, *Physical Review Letters* 48, 291-295.

Hardy, J. (1993): Nonlocalityfor two particles without inequalitries for
almost all entangled states, *Physical Review Letters 71, 1665-1668.*

Jarrett, .J. P. (1987): Bell's theorem: a guide to the implications, in
in *Philosophical consequences of quantum theory*, eds. J.T. Cushing and
E. McMullin, Notre Dame: Notre Dame U. P..pp. 60-79

Mermin, N. D. (1987): Quantum mysteries for anyone, in *Philosophical
consequences of quantum theory*, eds. J.T. Cushing and E. McMullin,
Notre Dame: Notre Dame U. P.

pp. 49-59.

Shimony, A. ((1987): Our world view and microphysics, in *Philosophical
consequences of quantum theory*, eds. J.T. Cushing and E. McMullin,
Notre Dame: Notre Dame U. P.

pp.. 25-37.

Stapp, H. P. (1978). Non-local character of quantum mechanics.
Epistemological letters, June 1978 (Association F. Gonseth, Case Postal
1081, Bienne, Switzerland)

Stapp, H. P. (1979): Whiteheadian approach to quantum theory and
generalized Bell's theorem. *Foundations of Physics*, 9, 1-25.

Stapp, H. P. (2003): A Bell-type theorem without hidden variables, *Am.
J. Phys*. 72, 30-33.