Popper's experiment
Encyclopedia
Popper's experiment is an experiment proposed by the 20th century philosopher of science Karl Popper
, an advocate of an objective interpretation of quantum mechanics. He wanted to test the Copenhagen interpretation
, a popular subjectivist interpretation of quantum mechanics
. Popper's experiment is similar in spirit to the subsequent thought experiment of Einstein, Podolsky and Rosen (The EPR paradox
) although not as well known.
is an astoundingly successful hypothesis when it comes to explaining or predicting physical
phenomena
. There are various interpretations of quantum mechanics
that do not agree with each other. Despite their differences, they are experimentally nearly indistinguishable from each other. The most widely accepted interpretation of quantum mechanics is the Copenhagen interpretation put forward by Niels Bohr
. The spirit of the Copenhagen interpretation is that the wavefunction
of a system
is treated as a composite whole, so disturbing any part of it disturbs the whole wavefunction. This leads to the counter-intuitive result that two well separated, non-interacting systems show a mysterious dependence on each other. Einstein called this spooky action at a distance. Einstein's discomfort with this kind of spooky action is summarized in the famous EPR
argument.
Karl Popper shared Einstein's discomfort with quantum theory. While the EPR argument involved a thought experiment, Popper proposed a physical experiment to test the Copenhagen interpretation of quantum mechanics.
of particles traveling to the left and to the right along the x-axis. The
momentum along the y-direction of the two particles is entangled in such a
way so as to conserve the initial momentum at the source,
which is zero. Quantum mechanics allows this kind of entanglement
. There are two slits, one each in the paths of the two particles.
Behind the slits are semicircular arrays of detectors which can detect the
particles after they pass through the slits (see Fig. 1).
Popper argued that because the slits localize the particles to a narrow
region along the y-axis, from the uncertainty principle
they experience large uncertainties in the y-components of their momenta.
This larger spread in the momentum will show up as particles being
detected even at positions that lie outside the regions where particles
would normally reach based on their initial momentum spread.
Popper suggests that we count the particles in coincidence, i.e., we count only those particles behind slit B, whose other member of the pair registers on a counter behind slit A. This would make sure that we count only those particles behind slit B, whose partner has gone through slit A. Particles which are not able to pass through slit A are ignored.
We first test the Heisenberg scatter for both the beams of particles going to the right and to the left, by making the two slits A and B wider or narrower. If the slits are narrower, then counters should come into play which are higher up and lower down, seen from the slits. The coming into play of these counters is indicative of the wider scattering angles which go with narrower slit, according to the Heisenberg relations.
Now we make the slit at A very small and the slit at B very wide. According to the EPR
argument, we have measured position "y" for both particles (the one passing through A and the one passing through B) with the precision
, and not just for particle passing through slit A. This is because from the initial entangled EPR state we can calculate the position of the particle 2, once the position of particle 1 is known, with approximately the same precision. We can do this, argues Popper, even though slit B is wide open.
We thus obtain fairly precise "knowledge" about the y position of particle 2 – we have "measured" its y position indirectly. And since it is, according to the Copenhagen interpretation, our knowledge which is described by the theory – and especially by the Heisenberg relations — we should expect that the momentum
of particle 2 scatters as much as that of particle 1, even though the slit A is much narrower than the widely opened slit at B.
Now the scatter can, in principle, be tested with the help of the counters. If the Copenhagen interpretation is correct, then such counters on the far side of slit B that are indicative of a wide scatter (and of a narrow slit) should now count coincidences: counters that did not count any particles before the slit A was narrowed.
To sum up: if the Copenhagen interpretation is correct, then any increase in the precision in the measurement of our mere knowledge of the particles going through slit B should increase their scatter.
Popper was inclined to believe that the test would decide against the Copenhagen interpretation, as it is applied to Heisenberg's uncertainty principle.
If the test decided in favor of the Copenhagen interpretation, Popper argued, it could be interpreted as indicative of action at a distance.
Popper's experiment was realized in 1999 by Kim and Shih using a SPDC photon source. Interestingly, they did not observe an extra spread in the
momentum of particle 2 due to particle 1 passing through a narrow slit. Rather, the momentum spread of particle 2 (observed in coincidence with particle 1 passing through slit A) was narrower than its momentum spread in the initial state. This led to a renewed heated debate, with some even going to the extent of claiming that Kim and Shih's experiment had demonstrated that there is no non-locality in quantum mechanics.
The ideal EPR
state is written as
, where the two labels in the "ket" state represent the positions or momenta of the two particle. This implies perfect correlation, meaning, detecting particle 1 at position 
will also lead to particle 2 being detected at 
. If particle 1 is measured to have a momentum 
, particle 2 will be detected to have a momentum 
. The particles in this state have infinite momentum spread, and are infinitely delocalized. However, in the real world, correlations are always imperfect. Consider the following entangled state
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
, an advocate of an objective interpretation of quantum mechanics. He wanted to test the Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...
, a popular subjectivist interpretation of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. Popper's experiment is similar in spirit to the subsequent thought experiment of Einstein, Podolsky and Rosen (The EPR paradox
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
) although not as well known.
Background
Quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is an astoundingly successful hypothesis when it comes to explaining or predicting physical
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
phenomena
Phenomenon
A phenomenon , plural phenomena, is any observable occurrence. Phenomena are often, but not always, understood as 'appearances' or 'experiences'...
. There are various interpretations of quantum mechanics
Interpretation of quantum mechanics
An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...
that do not agree with each other. Despite their differences, they are experimentally nearly indistinguishable from each other. The most widely accepted interpretation of quantum mechanics is the Copenhagen interpretation put forward by Niels Bohr
Niels Bohr
Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in...
. The spirit of the Copenhagen interpretation is that the wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
of a system
System
System is a set of interacting or interdependent components forming an integrated whole....
is treated as a composite whole, so disturbing any part of it disturbs the whole wavefunction. This leads to the counter-intuitive result that two well separated, non-interacting systems show a mysterious dependence on each other. Einstein called this spooky action at a distance. Einstein's discomfort with this kind of spooky action is summarized in the famous EPR
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
argument.
Karl Popper shared Einstein's discomfort with quantum theory. While the EPR argument involved a thought experiment, Popper proposed a physical experiment to test the Copenhagen interpretation of quantum mechanics.
Popper's proposed experiment
Popper's proposed experiment consists of a source of particles that can generate pairsof particles traveling to the left and to the right along the x-axis. The
momentum along the y-direction of the two particles is entangled in such a
way so as to conserve the initial momentum at the source,
which is zero. Quantum mechanics allows this kind of entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
. There are two slits, one each in the paths of the two particles.
Behind the slits are semicircular arrays of detectors which can detect the
particles after they pass through the slits (see Fig. 1).
region along the y-axis, from the uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
they experience large uncertainties in the y-components of their momenta.
This larger spread in the momentum will show up as particles being
detected even at positions that lie outside the regions where particles
would normally reach based on their initial momentum spread.
Popper suggests that we count the particles in coincidence, i.e., we count only those particles behind slit B, whose other member of the pair registers on a counter behind slit A. This would make sure that we count only those particles behind slit B, whose partner has gone through slit A. Particles which are not able to pass through slit A are ignored.
We first test the Heisenberg scatter for both the beams of particles going to the right and to the left, by making the two slits A and B wider or narrower. If the slits are narrower, then counters should come into play which are higher up and lower down, seen from the slits. The coming into play of these counters is indicative of the wider scattering angles which go with narrower slit, according to the Heisenberg relations.
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
argument, we have measured position "y" for both particles (the one passing through A and the one passing through B) with the precision
, and not just for particle passing through slit A. This is because from the initial entangled EPR state we can calculate the position of the particle 2, once the position of particle 1 is known, with approximately the same precision. We can do this, argues Popper, even though slit B is wide open.We thus obtain fairly precise "knowledge" about the y position of particle 2 – we have "measured" its y position indirectly. And since it is, according to the Copenhagen interpretation, our knowledge which is described by the theory – and especially by the Heisenberg relations — we should expect that the momentum
of particle 2 scatters as much as that of particle 1, even though the slit A is much narrower than the widely opened slit at B.Now the scatter can, in principle, be tested with the help of the counters. If the Copenhagen interpretation is correct, then such counters on the far side of slit B that are indicative of a wide scatter (and of a narrow slit) should now count coincidences: counters that did not count any particles before the slit A was narrowed.
To sum up: if the Copenhagen interpretation is correct, then any increase in the precision in the measurement of our mere knowledge of the particles going through slit B should increase their scatter.
Popper was inclined to believe that the test would decide against the Copenhagen interpretation, as it is applied to Heisenberg's uncertainty principle.
If the test decided in favor of the Copenhagen interpretation, Popper argued, it could be interpreted as indicative of action at a distance.
The debate
Many viewed Popper's experiment as a crucial test of quantum mechanics, and there was a debate on what result an actual realization of the experiment would yield.- In 1985, Sudbery pointed out that the EPR state, which could be written as
, already contained an infinite spread in momenta (tacit in the integral over k), so no further spread could be seen by localizing one particle. Although it pointed to a crucial flaw in Popper's argument, its full implication was not understood.
- Kripps theoretically analyzed Popper's experiment and predicted that narrowing slit A would lead to momentum spread increasing at slit B. Kripps also argued that his result was based just on the formalism of quantum mechanics, without any interpretational problem. Thus, if Popper was challenging anything, he was challenging the central formalism of quantum mechanics.
- In 1987 there came a major objection to Popper's proposal from Collet and Loudon. They pointed out that because the particle pairs originating from the source had a zero total momentum, the source could not have a sharply defined position. They showed that once the uncertainty in the position of the source is taken into account, the blurring introduced washes out the Popper effect.
- Redhead analyzed Popper's experiment with a broad source and concluded that it could not yield the effect that Popper was seeking.
Realization of Popper's experiment
momentum of particle 2 due to particle 1 passing through a narrow slit. Rather, the momentum spread of particle 2 (observed in coincidence with particle 1 passing through slit A) was narrower than its momentum spread in the initial state. This led to a renewed heated debate, with some even going to the extent of claiming that Kim and Shih's experiment had demonstrated that there is no non-locality in quantum mechanics.
- Short criticized Kim and Shih's experiment, arguing that because of the finite size of the source, the localization of particle 2 is imperfect, which leads to a smaller momentum spread than expected. However, Short's argument implies that if the source were improved, we should see a spread in the momentum of particle 2.
- Sancho carried out a theoretical analysis of Popper's experiment, using the path-integral approach, and found a similar kind of narrowing in the momentum spread of particle 2, as was observed by Kim and Shih. Although this calculation did not give them any deep insight, it indicated that the experimental result of Kim-Shih agreed with quantum mechanics. It did not say anything about what bearing it has on the Copenhagen interpretation, if any.
Criticism of Popper's proposal
Tabish Qureshi has published the following analysis of Popper's argument.The ideal EPR
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
state is written as
, where the two labels in the "ket" state represent the positions or momenta of the two particle. This implies perfect correlation, meaning, detecting particle 1 at position will also lead to particle 2 being detected at . If particle 1 is measured to have a momentum , particle 2 will be detected to have a momentum . The particles in this state have infinite momentum spread, and are infinitely delocalized. However, in the real world, correlations are always imperfect. Consider the following entangled state-
where
represents a finite momentum spread, and 
is a measure of the position spread of the particles. The uncertainties in position and momentum, for the two particles can be written as
The action of a narrow slit on particle 1 can be thought of as reducing it to a narrow Gaussian state:
. This will reduce the state of particle 2 to 
.
The momentum uncertainty of particle 2 can now be calculated, and is given by
-
If we go to the extreme limit of slit A being infinitesimally narrow (
), the momentum uncertainty of particle 2 is 
, which is exactly what the momentum spread was to begin with. In fact, one can show that the momentum spread of particle 2, conditioned on particle 1 going through slit A, is always less
than or equal to
(the initial spread), for any value of 
, and 
. Thus, particle 2 does not acquire any extra momentum spread than what it already had. This is the prediction of standard quantum mechanics.
On the other hand, if slit A is gradually narrowed, the momentum spread of particle 2 (conditioned on the detection of particle 1 behind slit A) will show a gradual increase (never beyond the initial spread, of course). This is what quantum mechanics predicts. Popper had said
...if the Copenhagen interpretation is correct, then any increase in the precision in the measurement of our mere knowledge of the particles going through slit B should increase their scatter.
Popper's experiment and faster-than-light signalling
The expected additional momentum scatter which Popper wrongly attributed to the Copenhagen interpretation can be interpreted as allowing faster-than-light communication, which is thought to be impossible, even in quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. Indeed some authors have criticized Popper's experiment based on this impossibility of superluminal communication in quantum mechanics. Use of quantum correlationsQuantum entanglementQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
for faster-than-light communication is thought to be flawed because of the no-communication theoremNo-communication theoremIn quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics such as the EPR paradox or...
in quantum mechanics. However the theorem is not applicable to this experiment. In this experiment, the "sender" tries to signal 0 and 1 by narrowing the slit, or widening it, thus changing the probability distribution among the "receiver's" detectors. If the no-communication theoremNo-communication theoremIn quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics such as the EPR paradox or...
were applicable, then no matter if the sender widens the slit or narrows it, the receiver should see the same probability distribution among his detectors. This is true, regardless of whether the device was used for communication (i.e. sans coincidence circuit), or not (i.e. in coincidence). This is clearly not the case with this experiment. So if superluminal communication is impossible for this device, then it does not come from the so-called "no-communication theoremNo-communication theoremIn quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics such as the EPR paradox or...
."
Some will argue that this is impossible on account of the no cloning theoremNo cloning theoremThe no-cloning theorem is a result of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields.The state of one system can be...
However, cloning of a single quantum state is unnecessary, you just run the experiment like you normally would; i.e. prepare multiple states by down-conversion and collect data on the receiver end from the large number of particles. The only difference, as alluded to above, is that you cannot use a coincidence circuit in using the device for communication. So noise will have to be filtered out somehow. One could conceivably have the receiver collect data in coincidence (or "semi-coincidence") if a three-particle Greenberger–Horne–Zeilinger state is used. The third particle could be sent to the receiver, and particles there collected only in coincidence. Then the only noise will not be from singles, but rather receiver-only doubles.
-
