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{\Huge\bf EPR Bingo Revisited}\\[6ex]
{\large\sf H. Roos\\ Institut f\"ur Theoretische Phyik\\[.5ex] Universt\"at
G\"ottingen}\\[.5ex] Bunsenstr.9, D-37073 G\"ottingen\\
{\small e-mail roos@theorie.phyik.uni-goettingen.de}
\end{center}
\vspace{12ex}
{\bf Abstract}\\
There is no information transfer at superluminal speed in
O. Steinmann's EPR bingo (Helv.\ Phys.\ Acta {\bf 69}, 702 (1996)).
{\tt Submitted to Helv.\ Phys. Acta}
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There is a paper by O. Steinmann [1], the ``EPR bingo'', presenting an
EPR-like thought experiment which seems to imply the possibility of
information transfer with superluminal speed. Indeed, the paper is
nicely written and a pleasure to read, strongly suggesting a
paradoxical result. I want to point out that, nevertheless, the result
sketched by O. Steinmann, can be understood in quite natural, even
classical, terms without any violation of causality.
The set-up is as follows: Assume that, in the distant future, all
inhabitable planets and moons of the solar systems are
colonized. There is a professional betting company, accepting bets on
the outcome of an EPR experiment with a spin 0 particle decaying into
two spin 1/2 particles. The spin component in the direction of the
polar star of one of the particles, called $A$, arriving on earth will
be measured, the other, particle $B$, will be neglected. All branch
offices throughout the solar system accept bets on the outcome of the
measurement until shortly before the result can become known at the
place of the branch office. It is understood that the information can
be transferred at best with the velocity of light. Now assume that
there is a clever fellow living on Titan, the largest moon of Saturn,
more than one light hour away from earth, who has gained knowledge of
the source providing the decaying spin 0 particle and thus generating
the particle in question, and, furthermore, that he is able to
intercept the second particle and measure its relevant spin component
at about the same time as the measurement on earth is performed by the
betting company. Knowing his result he can conclude what is measured
on earth; he has thus gained practically instantaneous knowledge of
the outcome of the measurement on which bets are placed, enabling him
to place a substantial bet at the local office which does not yet have
any results. This possibility of gaining money by winning a bet with
probability one suggests the inevitable conclusion that there is
indeed a superluminal transfer of practical information, with
violation of causality. Furthermore, there seems to be a problem
concerning the time $t_m$ at which particle $B$ has ``learned'' that
its state is a product state as far as the spin is concerned, $t_m$
being earlier than the time of measurement of $B$, but possibly also
earlier than the measurement of $A$ in a suitable frame of reference.
My answer is a twofold one. As to the second question, it seems clear
to me that it is not a matter of particle $B$ ``knowing'' its state
(as is implied by O. Steinmann's suggestive formulation) but of the
observer and his knowledge. One should remember that --- in the purely
statistical interpretation of quantum mechanics --- any state refers
to an {\em ensemble}, i.\,e. yielding, in general, only probability
statements as far as a single system is concerned; the state is not a
property of the system but a means of description of its properties by
the observer.\footnote{In this respect I disagree with O. Steinmann
who makes a statement to the contrary in his paper; but this is not a
relevant point as far as possible superluminal speeds are concerned.}
If the guy on Titan knows that a certain spin component has been, or
is going to be, measured on earth, he may describe the (combined)
system by a mixed state, given by the density matrix (acting in ${\cal
H}_A\otimes{\cal H}_B$)
\begin{equation}
\label{e.misch}
\rho = {1\over2}(P_{u^+\otimes u^-}+ P_{u^-\otimes u^+})\;,
\end{equation}
where $u^\pm$ denotes the spin state with spin component up (resp.\
down) relative to the preferred direction, and $P_\varphi$ the projection
onto $\varphi$. This is the appropriate state to be used after a spin
measurement on particle $A$ without noting the result. But he may as
well choose to
describe the system by a pure spin 0 state
\begin{equation}
\label{e.rein}
P_\psi,\;\qquad\; \psi=2^{-{1\over2}}(u^+\otimes u^- - u^-\otimes u^+)
\in{\cal H}_A\otimes{\cal H}_B\;,
\end{equation}
thus neglecting what has possibly happened far away on earth.
It is an easy exercise to check that both states give the same
expectation values for any observable $\idty\otimes O$
referring to particle $B$:
\begin{equation}
\label{e.exp}
\langle O\rangle_{P_\psi}=\langle O\rangle_\rho\;.
\end{equation}
The second part of my answer is that there is no superluminal
information transfer involved; moreover, the bingo sketched in the paper
is actually a classical bingo, not a genuine EPR bingo. There are two
pieces of information which are necessary for the man on Titan in
order two win with certainty. The first is the knowledge of which spin
component will be measured; this is known beforehand since it is part of
the rules of the game, and may well have been transferred arbitrarily
slowly. The second one is the knowledge of the relevant spin component of
particle $B$ which is gained by measurement; it can be viewed as being
transferred with the speed of particle $B$ {\em from
the source of the decaying particle}, not from earth: information about
the outcome of any measurement on particle $B$ can be
obtained as soon as the particle has reached the measuring apparatus.
(Cf.\ also the above remarks comprising equations
(\ref{e.rein})--(\ref{e.exp}).)
Actually the situation can be mapped onto a purely classical one, {\em
because it is known which spin component will be measured}. This
amounts to a choice of two well defined possibilities, allowing the
man on Titan to infer with certainty the result of the measurement on
earth by his own measurement. The same would be the case if there were
two (classical) balls sent from the source which were known to be
either red or green. Everybody will agree that in this case no
velocities larger than that of light are involved; nevertheless, the
Titan man will win having gained information by intercepting the
second ball.
A truly quantum mechanical situation would mean, inadequately
expressed in classical language, that the two balls could, {\em at the
same time}, be either red and green or blue and yellow or any of many
other pairs of colours, depending on what pair of colours is chosen as
the relevant one.
Let us change the rules of the bingo to make it truly EPR: a certain
celestial half sphere is defined as the positive one; it is left open
which spin component $\vec{a}\cdot\vec{S}$ of particle $A$ will be
measured, $\vec{a}$ being a unit vector pointing into a ``positive''
direction which will be chosen at random immediately before the
measurement is carried out. Bets can be placed on whether the result
will be positive or negative. Of course, under these circumstances no
measurement on particle $B$ can give any information, or allow any
conclusion as to the outcome of the first measurement.
Using the inappropriate pseudoclassical language of the colored balls
one might ask whether there is an interaction at arbitrarily large
distances between the two balls making them change stochastically from
red-green to blue-yellow {\em at the same time}. A more appropriate
formulation is given in Steinmann's paper. He says that after a
measurement of, say, the z-component of the spin of particle $A$
resulting in, e.\,g., $S_z^A=+1/2$, the following is an {\em
objective\/} statement: ``A measurement of $S_z^B$ yields the result
$S_z^B=-1/2$ with certainty''; and to him `objective' means that it is
in a sense an objective property of particle $B$, or at least of the
state of the combined system $A$-$B$ which is now in a product state
$P_{u^+\otimes u^-}$. This causes Steinmann to ask at which time
$t_m$ the transition from the entangled spin-0 state to the product
state has occured (i.\,e. at which time particle $B$ has ``learned''
that the change has taken place). I have already given my answer in a
former paragraph; to sum it up: the problem is completely avoided by
the use of a strict statistical interpretation of the concept of
`state'; no objective property is ascribed to a single pair of
particles; the objective statement refers to the result of the
measurement.
To illustrate my point of view once more, let us consider the
following setting. Assume that there is a Big Boss producing a series
of decaying particles in a spin-zero-state giving rise to pairs of
particles $A$ and $B$, and, at the same time, issuing decrees ---
distributed with the speed of light --- as to which is the relevant
spin component to be measured on both particles of a certain pair.
The results of the measurements will be strictly correlated,
independent of {\em which\/} direction is chosen as relevant.
Nevertheless, there is clearly no interaction at a distance
and no superluminal information transfer. It is certainly misleading
to say that the particles are correlating their {\em properties\/} at
all times prior to the measurements; it is the {\em results of the
measurements\/} which are correlated provided the experimentors abide
by the decrees of the Big Boss.
Steinmann's bingo set-up is quite similar: the Big Boss is replaced by
the boss of the betting company who once and for all decided that a
fixed spin direction be the relevant one, thus enabling the Titan guy
to win --- without any need for superluminal information transfer.
\vspace{2ex}
{\Large\bf References}
[1] O. Steinmann, {\em The EPR Bingo},
Helv.\ Physica Acta {\bf 69}, 702 (1996)
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