Wlodzislaw Duch,
Institute of Physics, Nicolaus Copernicus University, Grudziadzka 5, Torun, Poland
Complementarity, Superluminal Telegraph and the Einstein-Podolsky-Rosen Paradox.
Submitted to: American J. of Physics (1989)
ABSTRACT
A new version of the Einstein-Podolsky-Rosen paradox in a double
Mach-Zehnder interferometer is described. Particles created in
multiparticle processes are in the spatially extended nonfactorizable
multiparticle states. It is shown that if the complementarity principle
is valid the existence of such extended quantum states, leading to
correlations violating Bell's inequality, is necessary to avoid
superluminal signaling.
PACS 03.65
1. Introduction.
Quantum mechanics is a marvellous theory: it gives numbers matching experimental results with astonishing precision and yet, as Feynman [1] writes "it is safe to say that no one understands quantum mechanics". This statement is more true now than ever before because, thanks to modern technology, many theoretical predictions, quite alien to our classical intuitions, were verified recently. There were at least as many surprises on the theoretical as on the experimental side of physics. To mention just a ams of theoreticians that turned true thanks to development of quantum optics, neutron interferometry and other techniques: interference of single neutrons and photons, delayed-choice interference experiments, quantum beats, spin superposition and 4-spinor symmetry, Bose-Einstein condensation, Josephson effect, macroscopic quantum tunneling, super and sub-radiation, non-classical properties of light, Aharonov-Bohm effect, correlations violating Bell's inequality. The last of these phenomena is the subject of the present paper.
Over 60 years ago Einstein, Podolsky and Rosen [2] (EPR) have published a famous paper that motivated generations of physicists to think about foundations of quantum mechanics. For many years Einstein tried to show that quantum mechanics is not a complete theory [3]. In particular, Heisenberg uncertainty relations were frequently his target.
EPR paper was an ingenious attempt to find both the position and the momentum of a particle: considering two correlated particles one may find the position or the mom f the first particle, without disturbing it in any way, by making measurements on the second particle. It means that the choice of measurement on the second particle changes the wave
function of the first one - from the position eigenfunction \Psi(x) to the plane wave momentum eigenfunction - although the two particles are well separated and do not interact. Einstein called it "telepathy" and strongly rejected. Bohr based his answer to the EPR challenge firmly on the complementarity principle: "... during the last stage of the measuring procedure [...] there is essentially the question of an influence of the very conditions which define the possible type of predictions regarding the future behaviour of the system" [4]. The space-time description, involving measurements of time and position, and the casual description, involving measurements of energy and momenta,
are "complementary but exclusive features of the description", as Bohr [5] pointed out already during his Como lecture in 1927. Quantum mechanics forces us t o think in terms of "unbroken wholness", where, strictly speaking, no objects exist because it is not possible to separate anything from the rest of the world [6]. Whenever we speak, we are invoking approximations and different approximations frequently are mutually exclusive. During experiments our decision how to interact with Nature brings out certain responses of physical systems. Complementarity is the basis of Bohr, and in general of the Copenhagen, understanding of quantum mechanics.
EPR argument was formulated as a Gedankenexperiment and it was not until David Bohm's book [7] appeared that a new formulation in a more practical terms, for measurements of the spin components of two particles in a singlet state, was given. The idea was again to learn about the first particle by making measurements on the second particle. The real interest in such experiments started however with the paper of J.S. Bell [8] showing the incompatibility of quantum mechanical predictions with those based on ealism. Correlations between measurements of polarization components of two spatially separated photons were determined in a series of experiments, indicating that even in cases
when the two measurements could not be connected by light signals correlation between the particles is still greater then the Bell inequality (for most physicists synonymous with local realism) allows [9]. Since the concepts of locality and realism are so basic to all of science one should not abandon them lightly. A number of authors [10-11] criticized the conclusions drawn from these experiments urging to look for loopholes, auxiliary assumptions and alternative explanations before rejecting local realism.
Below I will show that local realism is incompatible with complementarity: if both are true than it should be possible to built a superluminal telegraph. Since this last possibility leads to such paradoxes as closed time loops [12] it is a rather strong argument against local realism, assuming that complementarity holds. However, it seems
that a direct experimental evidence against complementarity exists: measurements [13] of the spread of the momentum in the beam of Auger electrons emitted by a single la argon atoms lead to the precision of an order of magnitude greater than allowed by the uncertainty principle!
Should the results of these experiments be confirmed one should reconsider the status of the complementarity principle itself, and this would require a major revision of the interpretation and may be even a change of the formalism of quantum mechanics. Until there is an undisputable need for such revolutionary changes we may assume that
complementarity is well proven.
2. EPR correlations and superluminal signaling.
A number of experiments connected with the EPR paradox were proposed, most of them involving measurements of correlation of the spin projections or of photon polarization, i.e. they are based on the conservation of the angular momentum. In particular all experiments performed so far were of this type. Very recently a practical proposal that is essentially a realisation of Einstein's Gedankenexperiment with positions and linear momenta was proposed by Klyshko [14]. The same author has proposed an exper nvolving time and energy in the EPR type of situation [15]. Although these propositions are very interesting there is not much doubt what will be the outcome of such experiments. Moreover, it would be rather hard to formulate an analogue of Bell's inequality in the original EPR situation. Another class of experiments, for which one can easily formulate Bell's inequality, was described recently [16], very close to the original EPR argument, showing in an even more striking way the peculiarity of the quantum mecha nical predictions. The strangeness of the EPR situation results from the fact that, according to quantum mechanics, two (or more) particles in nonfactorizable state do not have a separate existence, even if they are separated by a large distance. The wave-particle duality is usually presented in context of a single-particle experiments which do not force us to abandon concepts of Reality and Locality (RL). Multi-particle processes are qualitatively different. Existence o f the spatially extended two-particle nonfactorizable states would, as pointed out by Piccioni and Mehlhop [10], force us to abandon RL concept independently of the inequality of Bell. Below it is proven that if such states do not exist and complementarity holds then a faster-than-light telegraph should work.
Consider an interference experiment with one source or two independent sources of particles. According to the usual understanding of quantum mechanics (cf. Feynman [17]) based on the complementarity principle as long as we do not know which route the particle takes it contributes to the interference pattern with the intensity proportional to
(1) I = |1+2|2
where 1 and 2 are the wavefunctions (plain waves) of the particle
going through one of the two possible paths. On the other hand particles
for which the trajectory is known do contribute to the intensity but
not to the interference, i.e.
(2) I = |1|2 + |2|2
The important thing is that the difference between the case with
interference and with no interference is not connected with the energy
necessary for different distribution of the pattern on the screen but
only with the knowledge of the particle's paths. This knowledge comes
from measurements of particle's momenta and is complementary with the
knowledge of particle's positions. The simplest example of this
complementarity is seen in the interferometer. Detailed calculations
were performed for the n interferometer18 where one can trace the loss
of coherence in the neutron beam, or reduction of the contrast of an
interference pattern, the contrast C defined by:
(3) C = (Imax-Imin)/(Imax+Imin)
If we know in PN cases out of N which route in the interferometer a
particle takes the contrast is:
(4) C = (1-P)/(1+P)
i.e. it goes to zero with P going to one. This is a quantitative
expression of complementarity between momentum and position or
interference and knowledge of the paths in an interferometer. Although
usually two-slit experiments are discussed in this context19 it is more
practical and instructive to use Mach-Zehnder interferometer instead20.
The two beams are in this case very well separated. As is easily
verified if the maxima of the interference fringes separated by x are
visible than the uncertainty of m omentum is of the order of the total
momentum of particles in the beam, so that there is no knowledge which
route the particles take. Can one gain this knowledge without disturbing
the particles that form the pattern on the screen of an interferometer?
In analogy to the EPR reasoning let us introduce the source of
pairs of particles, sending each time two particles in opposite
directions vertically or horizontally (Fig.1). The particles do not have
to be correlated in any special way, they may even have different
energy, the only requirement is that detection of a particle in the
upper region should correspond to detection of a particle in the lower
region (mixing mirrors Ml, Mr in Fig.1a should be removed to check
this). Under these circumstances ould not expect interference since one
knows which route the particle takes. As long as such information exists
in nature there should be no interference. However, mixing mirrors Ml,
Mr destroy all information about the particle's paths. It seems that in
such double Mach-Zehnder interferometer one should see the interference
effects. Manipulating with the semi-transparent mirror Ml without in any
way disturbing the experiment in the lower region one obtains at xr, yr
interfe rence effects or their absence. Moreover, it seems that one can
change from one situation to the other almost instantaneously: the time
needed to recognize if there is interference may be very short comparing
with the time necessary for the light signal to travel from the
mirror/detectors in one region of space to the mixing mirror/detectors
in the other region. Worse than that, making the upper paths longer than
the lower ones we can decide a posteriori whether to detect particles or
to let them int erfere thus sending signals backwards i n time.
Let us first note that time independent formalism is quite
sufficient to discuss the situation described above. Usually the
collapse of the wavefunction due to a measurement is considered to be
instantenous. There are at least two kinds of experiments that prove
that quantum correlations are really instantenous: the last of Aspect's
experiments on Bell inequality [9], where correlations did not change when
the position of polarizers was changed in the last moment before
detection, and the delayed-choice experiments [20].
Few years ago N. Herbert has proposed an experimental set-up for
faster-then-light communication but it was soon proven that the
linearity of quantum mechanics does not allow a "photon xeroxing
device", crucial for his experiment, to exist [21]. I have a more general
argument against Herbert's set-up: if one assumes realism, i.e.
existence of all three spin components prior to measurement, the EPR
correlations should not violate Bell's inequality. It is the
undetermined character of non-commuting obse that leaves room for
non-trivial correlations. One cannot expect that a photon for which no
value of spin component is determined may be copied with its state
unchanged. Superluminal signaling was also discussed by Klyshko [14] and
other authors and the conclusions were always negative. However, the
experiment proposed above is of quite different type. Is there a way to
escape the absurd consequences of such experiment? Can locality be
saved?
The answer may be based on Schrodinger's analysis of the EPR
paradox, in particular his statement [22]: "Best possible knowledge of a
whole does not include best possible knowledge of its parts". Whenever
there is an interaction individual wavefunctions become 'entangled' (as
he calls it) and loose their separate existence. Non-trivial
correlations (i.e. violating Bell's inequality) are then present but the
price is that individual particles may no longer be in superposition of
pure states, for example h arms of an interferometer. They behave in a
more 'classical' way until further interaction with two slits or
semi-transparent mirror breaks the 'entanglement'. This behavior will be
illustrated in details on the double Mach-Zehnder interferometer (a
realistic proposition for an experiment using such interferometer is
described in [23]). There is no law of nature forbidding non-trivial
correlations and whatever is not forbidden is sooner or later found.
Moreover, these correlations mus t exist to save locality . Another
point illustrated by the double interferometer is how knowledge or
rather the possibility of its gain changes the description from pure to
mixed states or from interference to no interference.
3. Calculations.
In the usual Mach-Zehnder interferometer semi-transparent mirror
splits the incoming beam into two. Here instead of the initial mixing
mirror a source of particle pairs is placed (Fig.1). The wave travelling
to the right or in the x direction is represented by the vector |x> and
the one moving down as |y>. Consider first a single interferometer with
a screen instead of the final mixing mirror Mr. The wavefunction at the
screen is
_ i
(5) |> = / (|x>+e |y>)
Choosing the origin of coordinates x=y=0 at the screen the plain waves
are there |x>=|y>=1 and the intensity
i 2
(6) I = |1+e | = 1+cos = 2cos2/2
It is more convenient to use the mixing mirror Mr instead of a screen
and place detectors at xr and yr. Reflection at the optically denser
medium changes the sign of the incoming wave. Suppose that when |y> is
reflected it changes to -|x> and reflection of |x> changes it simply to
|y>. Intensities measured by the detectors xr, yr are then
i 2
(7) I(xr) = |1-e | = 2sin2/2
i 2
I(yr) = |1+e | = 2cos2/2
Thus different intensities measured by xr, yr are a signature of
interference. Consider now a source of particle pairs. Removing Ml
mirror and l phase shifter the approximate function describing the
two-particle states is
_ ir
(8) |> = / (|x>|-x>+e |y>|-y>)
and the wavefunctions at xr, yr detectors are
_ ir
(9) |xr> = / |x>(|-y>-e |-x>)
_ ir
|yr> = / |y>(|-y>+e |-x>)
ir
|(1,2)>={|-x>e (-|x>+|y>)+|-y>(|x>+|y>)}
These formulas clearly show that there is no correlation between firing
of xr and yr detectors and detection of particles in -x or -y
directions. for example, finding a particle in -y direction is
associated in half cases with detection at xr and in half cases with
detection at yr. Arrangement described above (Ml mirror removed) allows
to find the route the particle takes in the interferometer to reach xr
or yr by making measurements on the spatially separated particles in -x
or -y regions, therefore it is incompatible with the possibility of
interference. Technically the zero value of |-x>|-y> function outside
the particle's source or vanishing of the overlap integral <-x|-y>
results in the independence of and on the phase r. What
happens if the information about particle's paths is destroyed by
putting a screen in place of the Ml mirror? If the interference at xr,
yr appears one could use this effect to send signals faster than light.
Fixing the coordinate origin at this screen the wavefunc tions at the
lower detectors are
_ ir
(10) |xr> = / |x>(1-e )
_ ir
|yr> = / |y>(1+e )
Thus the conditional probability or correlation between the detection of
particles at (0,0) point of the screen at Ml and detectors at xr, yr has
to be of (1cos) type although there is no sign of interference in the
upper or the lower interferometer. Correlations are a substitute for
interference patterns that cannot form without violation of locality.
Multiparticle states always give (at least in principle) a possibility
of determining the trajectories without disturbing particles themselves,
so the very existence of such states forbids all interference effects.
Total function for the double interferometer (Fig.1) with both
mixing mirrors Ml, Mr in place is
il ir
(11) |(1,2)>={(|xl>|xr>+|yl>|yr>)(e +e )
il ir
+ (|xl>|yr>+|yl>|xr>)(e -e )}
It gives conditional probabilities (1cos(l-r)), of the kind that
violate Bell's inequality, for coincidence measurements between xl and
xr detectors, but no interference in the individual interferometers:
(12) l=Trr|><|= (|xl><|= (|xr> not vanish, as
in the case of the CP noninvariant processes26 tha above considerations
do not hold and it is not cle ar why superluminal signaling should not
be possible.
4. Discussion.
Although the above calculation shows that superluminal telegraph will not work double Mach-Zehnder interferometer is a very interesting system well worth an experimental investigation. First, it is a new
system of non-spin type in which correlations violating Bell's inequality should be observed. Second, it proves that spatially extended pure states have to exist unless superluminal signaling is allowed or complementarity does not hold. These two points rule out local realism and are immune against ections11 raised in case of experiments where polarization is measured. Third, this system illustrates how particles
created in multiparticle processes lose their individuality and do not have individual wave functions, thus not being able to interfere. This point is a matter of some controversy [26] and deserves a comment. What
does it mean in practical terms that a particle is not in a pure state?
It means that there is no way to prepare a coherent beam of such particles because they c annot interfere. Even if we destroy the second particle correlated with the first so that absolutely no information
about the first particle's path could be gained still coherence will not appear since it is the conditions at the source that prevent it.
Coherence can be regained only if a division of the wavefront is made, for example by splitting the beam with a semitransparent mirror. So far all interference experiments are made splitting a single beam. A suitable source of pairs of particles for interference experiments is described in [14,23].
Finally I would like to make two suggestions. First, it seems that the fact that coherence is lost whenever an interaction creates 'entalgement' is not explored in the measurement problem, where it can be an obvious route from quantum (pure states) to classical (mixed states) world. Second, in the cascade emission the two photons are not emitted simultaneously but there is a slight time delay so that the first photon may be already detected while the second one is still not emitted. How can the fir know that the second will be emitted in the future (more technically, how does the two-particle state (8) depend on the time delay)? What if, after detection of the first photon, we decide to destroy the atom, for example ionizing it? There should be no reason then why the photons should not build the interference pattern. However, the photons are detected before the decision is made whether the second photon would be emitted or not. Therefore it should be interesting to analyse the whole process using time-dependent formalism and find out how this time delay and the interference patterns are connected.
Acknowledgements: Discussions with J-P. Vigier, A. Kyprianidis and Z. Maric gave me a lot of pleasure and inspired me to write this paper. Grant from the Institute for Low Temperature and Structure Research,
Polish Academy of Science, Program CPBP 01.12 is gratefully acknowledged.
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Fig 1. a) Double Mach-Zehnder interferometer with a single source.
b) Experiment with two independent sources. M are mirrors, Ml, Mr semi-transparent mirrors, x, y are detectors, l, r are phase shifters.