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Nov 1, 1996 \hfill LBNL-38803 \\
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{\large \bf Nonlocal Character of Quantum Theory}
\footnote{This work was supported by the Director, Office of Energy
Research, Office of High Energy and Nuclear Physics, Division of High
Energy Physics of the U.S. Department of Energy under Contract
DE-AC03-76SF00098.}
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Henry P. Stapp\\
{\em Theoretical Physics Group\\
Lawrence Berkeley Laboratory\\
University of California\\
Berkeley, California 94720}
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\begin{abstract}
According to the usual conception of causality, the truth of a statement that
refers only to events confined to an earlier time cannot depend upon an
unconstrained choice to be made by an experimenter at a later time. According
to the usual ideas of relativity this causality condition should be valid in
all Lorentz frames. It is shown here that this concept of relativistic
causality is incompatible with certain simple predictions of quantum theory.
\end{abstract}
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{\bf Disclaimer}
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{\it Lawrence Berkeley Laboratory is an equal opportunity employer.}
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\noindent {\bf 1. Introduction}
Certain generalizations [1] of the works of Einstein, Rosen, and Podolsky [2]
and John Bell [3] show that the predictions of quantum theory are incompatible
with a property called ``local realism''. The word ``local'' refers here to a
putative causality condition that claims that, in any Lorentz frame, the truth
of a statement pertaining only to outcomes of experiments localized in one
region at a given time cannot depend upon which experiment is freely chosen and
performed in a spatially separated region at a later time. The word ``realism''
signifies that those demonstrations depend on the validity of an Einstein-type
conception of physical reality, or perhaps on an assumption of determinism or
of hidden variables. But if any of these three reality concepts is used then
locality is not placed in jeopardy, for orthodox quantum philosophy claims the
failure of all three reality concepts in the realm of quantum phenomena. I
intend to show here, in a way perhaps simpler than before [4,5], that the
locality property itself is incompatible with certain predictions of quantum
theory, without assuming determinism, hidden variables, or Einstein reality.
\noindent {\bf 2. Hardy-type Experiment}
Use of a Hardy-type experiment [6,7,8] allows it to be shown that the putative
locality condition itself is incompatible with the predictions of quantum
theory. The essential features of the Hardy-type experimental set up are first
that it involves two regions that are spatially separated from each other. Here
they are called {\bf R} and {\bf L}. In {\bf R} there are two alternative
possible experiments, called $R1$ and $R2$, and in {\bf L} there are two
alternative possible experiments, called $L1$ and $L2$. In some Lorentz frame
the experimenter's choice in {\bf R} occurs {\it later} than the appearance and
recording of the observable outcomes of the experiments in {\bf L}. According
to the usual causality concept it is reasonable to entertain the notion that
the truth of a statement that refers only to observable properties in {\bf L}
at an earlier time cannot depend upon which measurement will be freely
chosen and performed in {\bf R} at a later time. According to the relativistic
causality concept there is an analogous condition with {\bf R} and {\bf L}
interchanged. But in this second case ``earlier'' and ``later''are specified
by using a different Lorentz frame.
What is under consideration here, then, is a set of just four possible
experimental set ups that are specified to be identical except for the
unconstrained bivalent choices of each of the two experimenters, and the
various differences that can arise from differences in these two choices. If
the defining conditions of two of these possibilities differ only by the choice
to be made in one region at a later time then our putative locality condition
claims that outcomes in the spatially separated region at the earlier time must
be {\it the same} in these two cases: what appears earlier cannot be one
outcome if the later free choice is Choice 1, but a different outcome if the
later free choice is Choice 2.
The logical structure of the Hardy experiment is shown in Figure 1, along with
the four pertinent predictions of quantum theory.
\begin{figure}[h]
\vspace{-2in}
\epsfxsize=6in
\epsfbox{fig1.ps}
\vspace{-2in}
\caption{The logical structure of the Hardy
experiment is represented, together with the four pertinent predictions of
quantum theory. The three solid paths between pairs of labelled points
represent predictions that have conditional probability equal to unity. The
dotted line represents a connection that has conditional probabilty equal to
50\%. (The 50\% comes from the Eberhard/Rosselet version: Hardy's optical case
gives 67\%)}
\end{figure}
For example, the solid line from $b$ to $g$ represents the prediction that:\\
if $L1$ is performed in {\bf L} and the outcome there is $b$, then if $R2$ is
performed in {\bf R} the outcome there will be $g$.
This prediction can be written symbolically as:
$$
(L1\wedge b) \Longrightarrow (R2 \Longrightarrow g),
$$
where $\Longrightarrow$ stands for {\it implies}, the strict conditional, and
$\wedge$ stands for conjunction. Equivalently, this prediction can be written
in the form of line $9$ below.
\vspace{.5in}
Informally, the general line of argument goes as follows. Suppose we consider a
frame in which the measurement in region {\bf L} is chosen and performed
earlier than the measurement in region {\bf R}. Suppose experiment L2 is
chosen in {\bf L}. Then the LOC condition that the outcome at the earlier time
cannot be one outcome if the later choice in {\bf R} is to perform experiment
R2, but a different outcome if the later choice is to perform experiment R1,
combined with the two QM predictions $ g \rightarrow c$ and $c \rightarrow f$,
allows one to conclude, under this condition that the earlier measurement is L2,
that $g \rightarrow f$: if under the condition that R2 is performed the outcome
$g$ appears in {\bf R} then we know, on theoretical grounds, that if the later
choice had been to perform measurement R1, instead of measurement R2, then the
outcome in {\bf R} would necessarily have been $f$. This is because the outcome
$c$ at the earlier time entailed by QM and the premise $g$ cannot be
altered by changing the choice of which measurement is performed in {\bf R}
at the later time.
Following the symbolic conventions of modal logic we express this conclusion
in the following way:
$$
(L2\wedge R2\wedge g)\Longrightarrow
[ R1 \Box \rightarrow (L2\wedge R1\wedge f)],
$$
which is equation 5 of the proof. If says that:\\ If L2 and R2 are performed
and the result g appears in {\bf R} then if R1 had been performed there,
instead of R2, then the outcome f would have appeared.
Statements of this general kind are commonplace in physics: theory often
allows one to deduce from the outcome of certain measurements on a system
what the outcome of some alternative possible measurement would necessarily be.
That is why modal logic really had to be developed: to cope with the fact that
statements of this general kind are endemic in science.
The conclusion obtained above can be formulated as a statement that if
measurement L2 is performed then $g \rightarrow f$, where the consequence
is a statement referring only observables in region {\bf R}. [See equation 6
below for the complete statement.]
Now the truth of this statement cannot depend on how we select the frame of
reference, which is, according to the ideas of relativity theory, merely a
notational convention. So let us shift to the other frame, where the
measurements in region {\bf L} are chosen and performed later. The statement
$g \rightarrow f$ should still be true. But then the truth of this statement,
which refers only to observables in {\bf R}, cannot, according to the putative
locality principle LOC, depend on whether measurement L2 or L1 is chosen to
be performed at the later time. Yet if this implication $g \rightarrow f$
holds also under the condition that L1 is performed in {\bf L} then we can
combine that implication with $b \rightarrow g$ to get $b \rightarrow f$,
which contradicts the prediction of QM that $b$ entails $e$ some nonzero
fraction of the time.\\
\noindent {\bf 3. The Proof}
I state the steps of the proof in logical symbols, and also in words for the
first few steps in order to make clear the meanings of the symbols. Later, I
justify each step on the basis of the predictions of quantum mechanics (QM) or
the locality condition described above (LOC), or simple logic (LOGIC).
In the proof I shall use the symbol $R1\Box\rightarrow$. It is taken from modal
logic, and is to be read: ``If $R1$ is performed, instead of $R2$, then...''.
It is to be used in situations where there is an unconstrained (free) choice
between the two alternative possible measurements $R1$ and $R2$. The form
$$(X\wedge R2)\Longrightarrow [R1 \Box \rightarrow Y]$$ means that the truth of
statement $X$ under the condition that $R2$ is chosen entails that $Y$ would be
true if $R1$ were to be measured instead of $R2$.
In modal logic the truth of such a statement is generally
supposed to be justified by appeal to the notion of ``closeness of worlds'',
which is not part of physics. Here such statements are justified by
appeal to our putative physics-based causality condition LOC, which brings in
the condition of non-dependence directly in terms of the idea of non-dependence
of earlier results upon which of two alternative possible measurement is freely
chosen later.\\
{\sf Proof}
1. LOC: ``If $L2$ is chosen and performed (at some earlier time) and the
outcome of $L2$ is $c$ and $R2$ is chosen and performed (at a later time),
then under the alternative possible condition that $R1$ is chosen and
performed (at the later time), instead of $R2$, the outcome of $L2$ (at the
earlier time) would still be $c$'': \\ $(L2\wedge c \wedge R2)
\Longrightarrow [R1\Box \rightarrow c].$
2. LOGIC: ``If $L2$ and $R2$ are chosen and performed and the outcome of $L2$
is $c$ then under the alternative possible condition that $R1$ is chosen and
performed ( at the later time), instead of $R2$, then $L2$ and $R1$ would be
performed and the outcome of $L2$ would be $c$'':\\ $(L2\wedge R2\wedge c)
\Longrightarrow [R1\Box \rightarrow (L2\wedge R1\wedge c)].$
3. QM: ``If $L2$ is chosen and performed (at the earlier time) and $R2$ is
chosen and performed (at the later time) and the outcome of $R2$ is $g$,
then $L2$ and $R2$ are performed and the outcome of $L2$ is $c$'':\\
$(L2\wedge R2\wedge g) \Longrightarrow (L2\wedge R2\wedge c).$
4. QM: ``If $L2$ and $R1$ are performed and the outcome of $L2$ is $c$ then
$L2$ and $R1$ are performed and the outcome of $R1$ is $f$'':\\ $ (L2\wedge
R1\wedge c) \Longrightarrow (L2\wedge R1\wedge f).$
5. LOGIC: $(L2\wedge R2\wedge g)\Longrightarrow [ R1 \Box
\rightarrow (L2\wedge R1\wedge f)].$
6. LOGIC: $(L2)\Longrightarrow [(R2\wedge g) \Longrightarrow (R1\Box
\rightarrow f)].$
7. LOC: $(L1) \Longrightarrow [(R2\wedge g) \Longrightarrow (R1\Box
\rightarrow f)].$
8. LOGIC: $(L1\wedge R2) \Longrightarrow [ g \Longrightarrow
(R1 \Box\rightarrow f)].$
9. QM: $(L1\wedge R2) \Longrightarrow [ b\Longrightarrow g].$
10. LOGIC: $(L1\wedge R2) \Longrightarrow [ b\Longrightarrow
(R1 \Box\rightarrow f)].$
11. LOC: $(L1\wedge R2) \Longrightarrow
[R1 \Box\rightarrow (b\Longrightarrow f)].$
12. LOGIC: $R2\Longrightarrow [L1\Longrightarrow [R1\Box \rightarrow
(b\Longrightarrow f)]].$
13. QM: $L1\Longrightarrow [R1 \Longrightarrow \neg (b\Longrightarrow f)].$
14. LOGIC: $L1\Longrightarrow [R1\Box \rightarrow \neg (b\Longrightarrow f)].$
15. LOGIC: $R2\Longrightarrow [L1\Longrightarrow [R1\Box \rightarrow
\neg (b\Longrightarrow f)]].$
The symbol $\neg$ is negates the proposition that follows it.
The conjunction of $12$ and $15$ contradicts the assumption that the
experimenters in regions {\bf R} and {\bf L} are free to choose which
experiments they will perform. Thus the incompatibility of the assumptions
of the proof is established.
\noindent {\bf 4. Justification of each step}
1. The statement $(R2\wedge X)\Longrightarrow [R1\Box\rightarrow Y]$ asserts:
``If $R2$ is performed and $X$ is true then [if $R1$ is performed, instead
of $R2$, then $Y$ is true.]'' The validity of line 1 thus follows from LOC,
which claims that the truth of statements referring only to measurements
performed and outcomes appearing in {\bf L} at the earlier time cannot be
affected by changing in {\bf R}, at the later time, the freely chosen $R2$
to the freely chosen $R1$.
2. This line is just a rewriting of line 2.
3. This is the prediction of QM corresponding to the path from $g$ to $c$ in
Fig. 1.
4. This is the prediction of QM corresponding to the path from $c$ to $f$ in
Fig. 1.
5. This follows from lines 2, 3, and 4 by two syllogisms.
6. This line follows from line 5 by elementary logic.
7. This follows from line 6 and the LOC claim that (also in the second Lorentz
frame, in which the experiments in {\bf L} occur later) a true statement
referring only to experiments and observables that can appear only at an
earlier time in {\bf R} cannot be made false by changing the free choice
made at a later time in {\bf L} from $L2$ to $L1$.
8. This is just a restatement of line 7.
9. This is the prediction of QM corresponding to the path from $b$ to $g$ in
Fig. 1.
10. This follows from lines 8 and 9 by syllogism.
11. Note that in line 10 the statement $b$ is made under the condition that
$R2$ is performed whereas in line 11 statement $b$ is made under the
condition that $R1$ instead of $R2$ is performed. But then line 11 follows
from line 10 and LOC, for LOC implies that the truth of $b$, which is fixed
in {\bf L} at the earlier time, cannot be altered by changing the choice in
{\bf R} at the later time from $R2$ or $R1$. It follows from this that if
$10$ is true then so is $11$.
12. This is just a re-writing of 11.
13. This line is entailed by the dotted line from $b$ to $e$ in Fig. 1. Under
the condition that $L1$ and $R1$ are performed it is not true that that if
$b$ appears in {\bf L} then $f$ must appear in {\bf R}: $50\%$ of
the time the outcome $e$ appears instead of $f$.
14. If something is true under a condition $R1$ then it is true if $R1$ is
performed instead if the alternative.
15. This is implied by 14.
As mentioned previously, I have used a symbol, $\Box\rightarrow$, that is
similar to one used in modal logic, and which has a verbal translation
identical to the one used in modal logic. But the meaning of this symbol in
modal logic is usually tied to a notion of ``closeness of possible worlds'' that
is not part of physical theory, and for which various definitions can be given.
Consequently, there are many modal logics, and appeal to ``modal logic'' is, by
itself, not sufficient to determine the correctness of arguments [5].
The present proof, although structurally more complex than the proof given in
[4] is logically simpler in that all of the steps that follow from LOGIC are
true in the {\it general} theory of counterfactuals [9], without appeal to the
special rules that define closeness of worlds.\\
The present proof, although dealing with ``instead of'' conditionals, is self
contained and uses, in addition to the general logical principles, only
ordinary ideas from quantum physics. These are, first of all, the idea that
the choices to be made by experimenters can be considered to be free (i.e.,
unconstrained) variables, and then the idea that an outcome that appears
``now'', although not completely fixed by past events, does become fixed
when the experiment is performed, and hence cannot depend upon what an
experimenter will choose to do at a later time. These assumptions, together
with the assumed validity of the predictions of quantum theory, lead to line 6.
The conclusion asserted in line 6 refers only to observables appearing in
region R. It is derived under the supposition that $L2$ is performed earlier.
Then the ideas of special relativity are used to assert that the truth of
statements cannot depend upon which frame of reference is used, and hence, in
particular, that the established validity of the conclusion in line 6 must
continue to be true in a frame in which the choice L2 is made later rather
than earlier.
Then a normal idea of causality is used to claim that if a statement S
pertaining only to possible outcomes of possible experiments confined to a
spacetime region R is known to be true under the supposition that the
experimenter in region L freely chooses {\it at a later time} to perform the
experiment L2, then S must be true also if that experimenter were, at the
later time, to make the other possible choice: the validity of a statement
that relates possible events occurring in the system under study at an earlier
time cannot depend upon an unconstrained choice to be made by an experimenter
at a later time.
There is, however, no apparent problem with the assumption that these causality
conditions hold in some single preferred frame, such as the cosmic black-body
radiation rest frame.
\begin{center}{\bf Acknowledgement}\end{center}
I thank Philippe Eberhard for useful comments and suggestions.
\begin{center}{\bf References}\end{center}
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without inequalities {\it American Journal
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realistic theories, and Lorentz-invariant realistic theories, {\it Physical
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particles without inequalities for almost all entangled states, {\it
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Rosselet, Bell's theorem based on a generalized EPR criterion of reality,
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195-200 (1964).
4. H.P. Stapp, ``Noise-induced reduction of wave packets and faster-than-light
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5. H.P. Stapp, Reply to `Stapp's algebraic argument for nonlocality', {\it
Physical Review A49}, 4257-4260 (1994).
6. Lucien Hardy, ``Quantum mechanics, local realistic theories, and
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9. David Lewis, {\it Counterfactuals,} Blackwell Press, Oxford; {\it
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{\it Synthese 102}, 139-164, 1995.
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