### Physics

Author: Avshalom C. Elitzur, Shahar Dolev, Anton Zeilinger

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RPE2.tex

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\begin{document}

\date{\today}

\title{Time-Reversed
EPR and the Choice of Histories in Quantum Mechanics}

\author{Avshalom
C. Elitzur}

\email{avshalom.elitzur@weizmann.ac.il}

\author{Shahar
Dolev}

\email{shahar_dolev@email.com}

\affiliation{Unit
for Interdisciplinary Studies, Bar-Ilan University, 52900 Ramat-Gan, Israel.}

\author{Anton
Zeilinger}

\email{anton.zeilinger@univie.ac.at}

\affiliation{Institut f\”ur Experimentalphysik, Universit\”at Wien,
Boltzmanngasse 5, 1090 Wien, Austria.}

**\begin{abstract}**

When a
single photon is split by a beam splitter, its two ``halves'' can entangle two
distant atoms into an EPR pair. We discuss a time-reversed analogue of this
experiment where two distant sources cooperate so as to emit a single photon.
The two ``half photons,'' having interacted with two atoms, can entangle these
atoms into an EPR pair once they are detected as a single photon. Entanglement
occurs by creating indistinguishabilility between the two mutually exclusive
histories of the photon. This indistinguishabilility can be created either at
the end of the two histories (by ``erasing” the single photon’s path) or at
their beginning (by ``erasing” the two atoms’ positions).

\end{abstract}

\pacs{03.65.Ta,
03.65.Ud, 03.65.Xp, 03.67.-a}

\keywords{quantum
measurement, interaction-free measurement, EPR, delayed-choice, histories,
erasure, time-symmetry, retroactive causality, realism}

\maketitle

**\section
{Introduction}**

As peculiar
as quantum measurement is known to be, its strangeness is even greater when one
tries to determine not merely the state of a system, but its entire {\it
history}. Past events are supposed to be unchangeable, and as such the most
essential aspect of reality. And yet, when a quantum measurement traces a
certain history, it seems to take an active part in the very formation of that
history.

So far,
however, this assertion has been merely philosophical. The most notable
experiment supporting it, namely, the Einstein-Wheeler ``delayed choice’’
experiment (see Sec.~\ref{sec:del_choice}), is equally open to other, less
radical interpretations. Could there be a more straightforward experiment,
showing that the history observed is retroactively affected by observations
carried out much later? In this article we propose a few experiments of this
type and discuss their implications.

**\section
{The Delayed Choice Experiment} **

\label{sec:del_choice}

\begin{figure}

\centering

\includegraphics[scale=0.7]{mzi.eps}

\caption{Mach-Zehnder
Interferometer.}

\label{fig:mzi}

\end{figure}

We
shall begin with the ``delayed choice’’ experiment. Discussing its limitations
will later highlight the advantage of our proposed demonstration of ``choosing
history.’’

Let a
Mach-Zehnder Interferometer (MZI) be large enough such that it takes light a
long time to traverse it (Fig.~\ref{fig:mzi}). Due to interference, every
single photon traversing this MZI must hit detector $C$. Suppose, however,
that, at the last moment, the experimenter decides to pull out $BS_2$. In this
case the photon hits either $C$ or $D$ with equal probability.

What
concerned Einstein about this experiment was that the two options given to the
experimenter’s choice seem to entail two mutually exclusive histories. In the
former case the photon seems to have been, {\it all along,} a wave that has
traversed both MZI arms and then gave rise to interference. In the latter case
the photon must have been -- again, {\it all along} -- a particle: if it has
hit $D$ it must have traversed only the right arm, and conversely for $C$. To
make the result more impressive, Wheeler \cite{Wheeler78} proposed to perform the
experiment on photons coming from outer space, whereby the history thus
``chosen’’ is millions-years long.

However,
the delayed choice experiment is not scientific in the full sense of the word,
as other explanations are possible within interpretations that do not invoke
backward causation. One could, for example, just stick to the observed facts,
refrain from any statement about the unobserved past and explain the experiment
strictly in terms of wave mechanics or ``collapse.’’

Can
there be an experiment that indicates more strongly that past events are
susceptible to the effect of future observation?

**\section
{Interference between Independent Sources} **

\label{sec:interf_indep_src}

Even
more striking than the delayed-choice experiment is an effect that was still
unknown to Einstein, namely, the interference of light coming from different
sources. It was first discovered by Hanbury-Brown and Twiss \cite{HBT57,HBT58},
and later demonstrated at the single-photon level \cite{Pfleegor67,Paul86}
(Fig.~\ref{fig:mandel}). It is odd that, although this experiment offends
classical notions more than most other experiments known today, it has not yet
received appropriate attention. When the radiation involved is of sufficiently
low intensity, then even a single particle seems to ``have originated’’ from
two distant sources.

\begin{figure}

\centering

\includegraphics[scale=0.6]{mandel.eps}

\caption{A
schematic description of Pfleegor-Mandel experiment for interference between
two distinct sources.}

\label{fig:mandel}

\end{figure}

We
shall first point out two variations of this experiments that highlight its
peculiar nature. First, it can have a delayed-choice variant: If the
experimenter chooses at the last moment to pull out the BS, a click at detector
$C$ will indicate that a single photon has emerged from only one source,
namely, the one facing the detector that clicked. If, on the other hand, she
leaves the BS in its place, the interference will again indicate that the
photon ``has been emitted’’ by both sources.

\begin{figure}

\centering

\includegraphics[scale=0.6]{mandel-ifm.eps}

\caption{A
variation of Pfleegor-Mandel experiment, implementing Interaction-Free
Measurement.}

\label{fig:mandel-ifm}

\end{figure}

Next
consider an Interaction-Free Measurement \cite{EV93} variant of this setting
(Fig.~\ref{fig:mandel-ifm}). Assuming that the phase between the sources is fixed
for the time of the experiment, it can be arranged that all the photons will
reach detector $C$. Now, if an object is placed next to one of the sources, it
will occasionally absorb the photon. Therefore, when a photon eventually hits
the detector, it is obvious that it has been emitted only from the other,
unblocked source. But then, in 50\% of the cases, that photon will emerge from
the BS towards the ``dark” detector $D$, thereby indicating that, although it
could have originated from only one source, it has somehow sensed the object
blocking the other source!

How can
two distant sources emit together a single photon? It is instructive to study
this effect as a time-reversed version of the familiar case where a single
photon is split by a BS and then goes to two distant detectors. In that case,
there is an uncertainty as to which detector {\it will} absorb the photon.
Similarly, in our case, there is an uncertainty as to which source {\it has}
emitted the photon.

This
time-symmetry suggests constructing a new experiment. Consider first the
familiar, V-shaped case (one source, two detectors). Such a split photon can
entangle two unrelated particles so as to create an EPR pair. For example, two
atoms positioned across its two possible paths will become entangled due to the
correlation between their ground and excited states. Can the more peculiar,
$\Lambda$-shaped case (two sources, one detector) be similarly used to create
an inverse EPR?

**\section
{Hardy’s Hybrid Experiment} **

\label{sec:hardy_exp}

\begin{figure}

\centering

\includegraphics[scale=0.45]{hardy.eps}

\caption{Hardy's
experiment.}

\label{fig:hardy}

\end{figure}

Before
we show how to do that, let us study an experiment due to Hardy
\cite{Hardy92a}, in which he has elegantly integrated the peculiarities of the
EPR experiment, single-particle interference and the interaction-free
measurement -- all in one simple setting.

Let a
single photon traverse a MZI. Let two spin $1 \over 2$ atoms be prepared in the
following way: Each atom is first prepared in an up spin-x state ($x^+$) and
then split by a non-uniform magnetic field $M$ into its spin-z components. The
two components are then carefully put into two boxes $z^+$ and $z^-$ while
keeping their superposition state:

\beq

\Psi=\ket{\gamma}
\cdot \frac{1}{\sqrt{2}}(iz^+_1+z^-_1) \cdot \frac{1}{\sqrt{2}}(iz^+_2+z^-_2).

\eeq

The
boxes are transparent for the photon but opaque for the atoms. Atom 1's (2's)
$z_1^+$ ($z_2^-$) box is positioned across the photon's $v$ ($u$) path in such
a way that the photon can pass through the box and interact with the atom
inside in a 100\% efficiency. Now let the photon be transmitted by $BS_1$:

\beq

\Psi=\frac{1}{\sqrt{2}^3}(i\ket{u}
+ \ket{v}) \cdot (iz^+_1+z^-_1) \cdot
(iz^+_2+z^-_2).

\eeq

After
the photon was allowed to interact with the atoms, we discard the cases in
which absorption occurred (50\%), to get:

\bea

\Psi&=\frac{1}{\sqrt{2}^3}(&-i\ket{u}z^+_1z^+_2
- \ket{u}z^-_1z^+_2 \\

&&\quad
+ i\ket{v}z^-_1z^+_2 + \ket{v}z^-_1z^-_2 ). \nonumber

\eea

Now,
let photon parts $u$ and $v$ pass through $BS_2$, following the evolution:

\[

\ket{v}\stackrel{BS_2}{\longrightarrow}\frac{1}{\sqrt{2}}\cdot
(\ket{d}+i\ket{c}), \qquad

\ket{u}\stackrel{BS_2}{\longrightarrow}\frac{1}{\sqrt{2}}\cdot
(\ket{c}+i\ket{d}),

\]

giving:

\bea

\Psi&=\frac{1}{4}(&\ket{d}z^+_1z^+_2
+ \ket{d}z^-_1z^-_2 \\

&&+
i\ket{c}z^-_1z^-_2 - i\ket{c}z^+_1z^+_2 -2 \ket{c}z^-_1z^+_2 ). \nonumber

\eea

If we
now post-select only the experiments in which the photon was surely disrupted
along its way, thereby hitting detector $D$, we get:

\beq

\Psi=\frac{1}{4}\ket{d}(z^+_1z^+_2
+ z^-_1z^-_2 ).

\eeq

Consequently,
the atoms, which have never met in the past, become entangled in an EPR-like
relation. Unlike the ordinary EPR, where the two particles have interacted
earlier, here the only common event in the past is the single photon that has
``visited" both of them.

In the
next section we shall show how to achieve this result even without any common
past. Then, the measurement’s effect on past evolution will become even more
striking.

**\section
{Inverse EPR (``RPE’’)} **

\begin{figure}

\centering

\includegraphics[scale=0.55]{entangle2.eps}

\caption{Entangling two atoms.}

\label{fig:entangle}

\end{figure}

Let two coherent photon beams be emitted from two distant sources as in Fig.~\ref{fig:entangle}. Let the sources be of sufficiently low intensity such that, on average, one photon is emitted during a given time interval. Let the beams be directed towards an equidistant BS. Two detectors are positioned next to the BS:

\bea

\phi_{\gamma
u} &=& p\ket{1}_u+q\ket{0}_u, \\

\phi_{\gamma
v} &=& p\ket{1}_v+q\ket{0}_v, \\

\psi_{A
1} &=& {1\over \sqrt 2}(i z^-_1 + z^+_1), \\

\psi_{A
2} &=& {1\over \sqrt 2}(i z^-_2 + z^+_2),

\eea

where
$\ket{1}$ denotes a photon state (with probability $p^2$), $\ket{0}$ denotes a
state of no photon (with probability $q^2$), $p\ll 1$, and $p^2+q^2=1$.

Since
the two sources’ radiation is with equal wavelength, a static interference
pattern will be manifested by different detection probabilities in each
detector. Adjusting the lengths of the photons' paths $v$ and $u$ will modify
these probabilities, allowing a state where one detector, $D$, is always silent
due to destructive interference, while all the clicks occur at the other
detector, $C$, due to constructive interference.

Notice
that each single photon obeys these detection probabilities only if both paths
$u$ and $v$, coming from the two distant sources, are open. We shall also
presume that the time during which the two sources remain coherent is long
enough compared to the experiment’s duration, hence we can assume the above
phase relation to be fixed.

Next,
let two spin-$1 \over 2$ atoms be prepared as in Hardy’s experiment
(Sec.~\ref{sec:hardy_exp} above) and let each ``half atom’’ be placed in one of
the possible paths. After the photon was allowed to interact with the atoms, we
discard the cases in which absorption occurred (50\%), to get:

\bea

\Psi&&=\frac{1}{\sqrt{2}^3}(-i\ket{u}z^+_1z^+_2
- \ket{u}z^-_1z^+_2 \\

&&\quad\qq\qq
+ i\ket{v}z^-_1z^+_2 + \ket{v}z^-_1z^-_2 ). \nonumber

\eea

If we
now post-select only the cases in which a single photon reached detector $D$,
which means that one of its paths was surely disrupted, we get:

\beq

\Psi=\frac{1}{4}\ket{d}(z^+_1z^+_2
+ z^-_1z^-_2 ),

\eeq

which
entangles the two atoms into a full-blown EPR state:

\[

z^+_1z^+_2
+ z^-_1z^-_2.

\]

In
other words, tests of Bell’s inequality performed on the two atoms will show
the same violations observed in the EPR case, indicating that the spin value of
each atom depends on the choice of spin direction measured on the other atom,
no matter how distant.

The two
photon sources, though unrelated, must still be coherent in order to
demonstrate interference. Dropping the coherency requirement would make the EPR
inversion even more prominent. This has been accomplished by Cabrillo {\it et.
al.} \cite{Cabrillo99} in a different setup, devised
for generating pairs of entangled atoms. Their setup involves atoms with three
energy levels: two, mutually close ``ground’’ states, $\ket{0}$ and $\ket{1}$,
and one excited state $\ket{2}$. Two distant such atoms in $\ket{0}$ state are
shone by a weak laser beam tuned to the $\ket{0}\rightarrow\ket{2}$ transition
energy. If a detector then detects a single photon of the
$\ket{2}\rightarrow\ket{1}$ energy, the entangled state
$\ket{1}\ket{2}+\ket{2}\ket{1}$ ensues.

Here,
in the absence of coherency, one cannot talk about interference. Still, since
only one photon is detected, the uncertainty about the photon’s origin suffices
to make the two atoms entangled, leading eventually to an EPR state.

Unlike
the ordinary EPR generation, where the two particles have interacted earlier,
here the only common event lies in the particles’ future. These two versions,
one involving coherent light and the other with incoherent light, highlight
different peculiarities of the inverse EPR, henceforth termed ``RPE.’’ We shall
discuss their implications below.

**\section
{Histories for Choice} **

The
``RPE’’ experiment offers several options for studying the way in which
measurement determines a history. Consider, first, its delayed-choice aspect,
which can be best demonstrated in the incoherent setup of Cabrillio {\it et.
al.}:

\begin{itemize}

\item
If the experimenter chooses at the last moment to pull out the BS, then the
photon’s two possible histories, i.e., ``it originated from the right atom’’
and ``it originated from the left atom,’’ become distinguishable. Consequently,
the photon’s ``footprints’’ become distinguishable too and no entanglement
between the atoms will be observed.

\item
Conversely, inserting the BS will entangle the two atoms, even though their
interaction with the photon has taken place earlier. In other words, {\it what
seems to be the generation of uncertainty only in the observer’s mind, gives
rise to a testable entanglement in reality.} Unlike the delayed-choice
experiment, here the history ``chosen’’ leaves observable footprints.

\end{itemize}

But, in
addition to creating uncertainty at the end of the evolution, the coherent
version (Fig.~\ref{fig:entangle}) gives us the freedom to create uncertainty -- or to
dissolve it -- also at the beginning of that evolution. For even after the
photon was detected at $D$, we can perform two kinds of measurements on the atoms,
measurements that will yield conflicting results:

\begin{itemize}

\item
We can measure the position of each atom in one out of the two boxes. In this
case, one atom must always be found in the intersecting box, while the other
must always reside in the non-intersecting box. Consequently, there is only one
possible history for the photon now: {\it It must have taken the path that was
not blocked by the atom, never the other, blocked path.} As a result, Bell
inequality violations would never be demonstrated by the atoms after this
measurement (recall that Bell-inequality statistics cannot be demonstrated on a
series of same-spin measurements). Hence, the atoms do not demonstrate
non-local correlation.

\item
On the other hand, we can unite the two boxes of each atom using an inverse
magnetic field $-M$, and measure the photon’s spin along the z axis. Here, we
give up the ``which path’’ information about the photon. Consequently,
Bell-inequality violations would be demonstrated in this case, proving that the
photon’s two possible histories cooperated so as to entangle the two distant
atoms.

\end{itemize}

All these variants are, in essence, erasure experiments. When we insert the BS in the ``incoherent RPE’’ or reunites the atoms in the coherent version, we actually erase the still available information about the photon’s two possible histories. Notice, however, that the present erasure experiments (e.g. \cite{Englert91}) demonstrate only the negative result of this information loss, i.e., the disappearance of the interference pattern. The RPE, in contrast, enables erasure to give rise to a positive result, namely, the entanglement of two distant atoms.

{\it ``Nam et ipsa scientia
potestas est} (for knowledge itself is power)’’ was an old maxim of the ancient
Romans, but quantum mechanics rewards one for cases in which {\it ignorance} is
generated.

**\section
{Admit Backward Causation or Abandon Realism?} **

The
time-symmetry of quantum theory’s formalism is well known \cite{ABL64} and has
moreover become the cornerstone of some modern interpretations that render
``affecting the past’’ the main characteristic of quantum interaction \cite{
Cramer86,Reznik95}. As early as in 1983,
Costa de Beauregard \cite{CdB83} gave a CPT-invariant formulation of the EPR
setting that allows a time-reversed EPR. Can we apply such a formulation in our
case and assert that the late entangling
event, i.e., the detection of the photon, really affects backwards the two
histories?

One
might argue that our experiment does not really time-reverse the EPR setting
because, in order to be sure that Bell’s inequality will be violated, the atoms
must be measured only after the detection of the entangling photon. Hence, the
entangling event still remains in the past of the two correlated atoms. The EPR
V shape, so goes the counter-argument, is thus merely flattened rather than
turned upside down into a $\Lambda$ shape.

Notice,
however, that the entangling event can lie outside the past light cones of the
two atoms’ measurements. Here, the argument against backward causation must
take the following form: ``The two atoms begin to violate Bell-inequality only
at the moment the photon was detected at $D$.'' This statement is
relativistically meaningless. By bringing the entangling event itself into
spacelike separation with the entangled particles, we actually render both the
normal and inverse EPRs equally possible.

But
what does ``affecting the past’’ teach us about the nature of time? This
question involves a deeper unresolved issue, that of time’s apparent
``passage.’’ Adherents of the ``Block Universe’’ model \cite{Price96}, argue
that time’s passage is only an illusion. Consequently, all quantum mechanical
experiments that seem to involve a last minute decision involve no free choice
at all. For example, in the EPR, the experimenter’s last-moment decision which
spin direction to measure, or, in the ``delayed choice’’ experiment, the
last-moment decision whether to insert the BS or not, are ``already’’
determined in the four-dimensional spacetime. Within this framework, RPE is
just as possible as EPR.

The
second alternative is that time has an objective ``flow’’ \cite{Prigogine80}.
Then, the retroactive entangling effect would occur in some higher time once
the ``Now’’ has reached the entangling event.

Both views lie at present
outside scientific investigation as both can be neither proved nor disproved.
\footnote{However, we have shown elsewhere
that Hawking’s information erasure conjecture is more consistent with an
objective time ``passage.’’ See \cite{Elitzur99} } Hence, a third and a much
easier answer to the problem would be dismissing
the entire issue by avoiding any reference to objective reality altogether, as
in the Copenhagen Interpretation.

While
two of us (AE and SD) tend to the second interpretation and one (AZ) favors the
third, we prefer to conclude by pointing out that each side can rely on one of
the two giants who have so hotly debated during the first Solvay conferences.

\acknowledgments{We
thank Yakir Aharonov, Terry Rudolph and David Tannor for very helpful comments.
It is a pleasure to thank the participants of the Workshop on Quantum Measurement Theory and Quantum Information at the Schr\"odinger
Institute in Vienna for enlightening discussions.}

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