Popper's experiment

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Popper's experiment is an experiment proposed by the philosopher Karl Popper. As early as 1934 he was suspicious of, and was proposing experiments to test, the Copenhagen interpretation, a popular subjectivist interpretation of quantum mechanics.^[1]^[2] Popper's experiment is a realization of an argument similar in spirit to the thought experiment of Einstein, Podolsky and Rosen (the EPR paradox) although not as well known.

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 known interpretation of quantum mechanics is the Copenhagen interpretation put forward by Niels Bohr. It says that observations lead to a wavefunction collapse, thereby suggesting the counter-intuitive result that two well separated, non-interacting systems require action-at-a-distance. Popper argued that such non-locality conflicts with common sense, and also with what was known at the time from astronomy and the "technical success of physics." "[T]hey all suggest the reality of time and the exclusion of action at a distance."^[3] While Einstein's EPR argument involved a thought experiment, Popper proposed a physical experiment to test for such action-at-a-distance.

Popper's proposed experiment

Popper first proposed an experiment that would test indeterminacy in quantum mechanics in two works of 1934.^[4]^[5] However, Einstein wrote a letter to Popper about the experiment in which he raised some crucial objections,^[6] causing Popper to admit that his initial idea was "based on a mistake".^[7] In the 1950s he returned to the subject and formulated this later experiment, which was finally published in 1982.^[8]^[9]

Popper wrote:

I wish to suggest a crucial experiment to test whether knowledge alone is sufficient to create 'uncertainty' and, with it, scatter (as is contended under the Copenhagen interpretation), or whether it is the physical situation that is responsible for the scatter.^[10]

Popper's proposed experiment consists of a low-intensity source of particles that can generate pairs of particles traveling to the left and to the right along the x-axis. The beam's low intensity is "so that the probability is high that two particles recorded at the same time on the left and on the right are those which have actually interacted before emission."^[10]

There are two slits, one each in the paths of the two particles. Behind the slits are semicircular arrays of counters which can detect the particles after they pass through the slits (see Fig. 1). "These counters are coincident counters [so] that they only detect particles that have passed at the same time through A and B."^[11]

Fig.1 Experiment with both slits equally wide. Both the particles should show equal scatter in their momenta.

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 partner has gone through slit A. Particles which are not able to pass through slit A are ignored.

The Heisenberg scatter for both the beams of particles going to the right and to the left, is tested "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 a narrower slit, according to the Heisenberg relations."^[11]

Fig.2 Experiment with slit A narrowed, and slit B wide open. Should the two particle show equal scatter in their momenta? If they do not, Popper says, the Copenhagen interpretation is wrong. If they do, it indicates action at a distance, says Popper.

Now the slit at A is made very small and the slit at B very wide. Popper wrote that, 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 $\Delta y$ , 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.^[11]

Therefore, Popper states that "fairly precise "knowledge"" about the y position of particle 2 is made; its y position is measured indirectly. And since it is, according to the Copenhagen interpretation, our knowledge which is described by the theory – and especially by the Heisenberg relations — it should be expected that the momentum $p_{y}$ 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 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.^[12]

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 $\psi (y_{1},y_{2})=\int _{{-\infty }}^{\infty }e^{{iky_{1}}}e^{{-iky_{2}}}\,dk$ , already contained an infinite spread in momenta (tacit in the integral over k), so no further spread could be seen by localizing one particle.^[13]^[14] 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.^[15]

In 1987 there came a major objection to Popper's proposal from Collet and Loudon.^[16] 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.

Furthermore, Redhead analyzed Popper's experiment with a broad source and concluded that it could not yield the effect that Popper was seeking.^[17]

Realization of Popper's experiment

Fig.3 Schematic diagram of Kim and Shih's experiment based on a BBO crystal which generates entangled photons. The lens LS helps create a sharp image of slit A on the location of slit B.

Fig.4 Results of the photon experiment by Kim and Shih, aimed at realizing Popper's proposal. The diffraction pattern in the absence of slit B (red symbols) is much narrower than that in the presence of a real slit (blue symbols).

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. They write:

"Indeed, it is astonishing to see that the experimental results agree with Popper’s prediction. Through quantum entanglement one may learn the precise knowledge of a photon’s position and would therefore expect a greater uncertainty in its momentum under the usual Copenhagen interpretation of the uncertainty relations. However, the measurement shows that the momentum does not experience a corresponding increase of uncertainty. Is this a violation of the uncertainty principle?"^[18]

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.

They concluded that:

"Popper and EPR were correct in the prediction of the physical outcomes of their experiments. However, Popper and EPR made the same error by applying the results of two-particle physics to the explanation of the behavior of an individual particle. The two-particle entangled state is not the state of two individual particles. Our experimental result is emphatically NOT a violation of the uncertainty principle which governs the behavior of an individual quantum."^[18]

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.^[19]

Unnikrishnan (2001), discussing Kim and Shih's result, wrote that the result:

is a solid proof that there is no state-reduction-at-a-distance. ... Popper's experiment and its analysis forces us to radically change the current held view on quantum non-locality.^[20]

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.^[21] 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.^[22] 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.^[23]^[24]

The ideal EPR state is written as $|\psi \rangle =\int _{{-\infty }}^{\infty }|y,y\rangle \,dy=\int _{{-\infty }}^{\infty }|p,-p\rangle \,dp$ , 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 $x_{0}$ will also lead to particle 2 being detected at $x_{0}$ . If particle 1 is measured to have a momentum $p_{0}$ , particle 2 will be detected to have a momentum $-p_{0}$ . 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

\psi (y_{1},y_{2})=A\!\int _{{-\infty }}^{\infty }dpe^{{-p^{2}/4\sigma ^{2}}}e^{{-ipy_{2}/\hbar }}e^{{ipy_{1}/\hbar }}\exp[-{(y_{1}+y_{2})^{2}/16\Omega ^{2}}]

where $\sigma$ represents a finite momentum spread, and $\Omega$ is a measure of the position spread of the particles. The uncertainties in position and momentum, for the two particles can be written as

\Delta p_{{2}}=\Delta p_{{1}}={\sqrt {\sigma ^{2}+{\hbar ^{2}/16\Omega ^{2}}}},~~~~\Delta y_{1}=\Delta y_{2}={\sqrt {\Omega ^{2}+\hbar ^{2}/16\sigma ^{2}}}.

The action of a narrow slit on particle 1 can be thought of as reducing it to a narrow Gaussian state: $\phi _{1}(y_{1})={\frac {1}{(\epsilon ^{2}2\pi )^{{1/4}}}}e^{{-y_{1}^{2}/4\epsilon ^{2}}}$ . This will reduce the state of particle 2 to $\phi _{2}(y_{2})=\!\int _{{-\infty }}^{\infty }\psi (y_{1},y_{2})\phi _{1}^{*}(y_{1})dy_{1}$ . The momentum uncertainty of particle 2 can now be calculated, and is given by

\Delta p_{{2}}={\sqrt {{\frac {\sigma ^{2}(1+\epsilon ^{2}/\Omega ^{2})+\hbar ^{2}/16\Omega ^{2}}{1+4\epsilon ^{2}(\sigma ^{2}/\hbar ^{2}+1/16\Omega ^{2})}}}}.

If we go to the extreme limit of slit A being infinitesimally narrow ( $\epsilon\to 0$ ), the momentum uncertainty of particle 2 is $\lim _{{\epsilon \to 0}}\Delta p_{{2}}={\sqrt {\sigma ^{2}+\hbar ^{2}/16\Omega ^{2}}}$ , 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 ${\sqrt {\sigma ^{2}+\hbar ^{2}/16\Omega ^{2}}}$ (the initial spread), for any value of $\epsilon ,\sigma$ , and $\Omega$ . Thus, particle 2 does not acquire any extra momentum spread than what it already had. This is the prediction of standard quantum mechanics. So, the momentum spread of particle 2 will always be smaller than what was contained in the original beam. This is what was actually seen in the experiment of Kim and Shih. Popper's proposed experiment, if carried out in this way, is incapable of testing the Copenhagen interpretation of 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.

This particular aspect can be experimentally tested.

Popper's experiment and ghost diffraction

It has been shown that this effect has actually been demonstrated experimentally in the so-called two-particle ghost interference experiment.^[25] This experiment was not carried out with the purpose of testing Popper's ideas, but ended up giving a conclusive result about Popper's test. In this experiment two entangled photons travel in different directions. Photon 1 goes through a slit, but there is no slit in the path of photon 2. However, Photon 2, if detected in coincidence with a fixed detector behind the slit detecting photon 1, shows a diffraction pattern. The width of the diffraction pattern for photon 2 increases when the slit in the path of photon 1 is narrowed. Thus, increase in the precision of knowledge about photon 2, by detecting photon 1 behind the slit, leads to increase in the scatter of photons 2.

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 mechanics. Indeed, some authors have criticized Popper's experiment based on this impossibility of superluminal communication in quantum mechanics.^[26]^[27] Use of quantum correlations for faster-than-light communication is thought to be flawed because of the no-communication theorem 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 theorem 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).

References

↑ Popper, Karl (1982). Quantum Theory and the Schism in Physics. London: Hutchinson (from 1992 published by Routledge). pp. 27–29. ISBN 0-8476-7019-8.
↑ Popper, Karl (1985). "Realism in quantum mechanics and a new version of the EPR experiment". In Tarozzi, G.; van der Merwe, A. Open Questions in Quantum Physics. Dordrecht: Reidel. pp. 3–25. doi:10.1007/978-94-009-5245-4_1.
↑ Popper (1982), p. 26.
↑ K. Popper (1934). "Zur Kritik der Ungenauigkeitsrelationen". Naturwissenschaften. 22: 807–808. doi:10.1007/BF01496543.
↑ K. Popper (1934). Logik der Forschung [The Logic of Scientific Discovery, 1959] (in German). Vienna: Springer.
↑ K. Popper (1959). The Logic of Scientific Discovery. London: Hutchinson. appendix *xii. ISBN 0-415-27844-9.
↑ Popper (1959), p. 236, note.
↑ M.H. Hacohen (2002). Karl Popper: The Formative Years, 1902-1945: Politics and Philosophy in Interwar Vienna. Cambridge: Cambridge University Press. p. 259.
↑ William M. Shields (2012). "A Historical Survey of Sir Karl Popper's Contribution to Quantum Mechanics". Quanta. 1 (1): 1–12. doi:10.12743/quanta.v1i1.4.
1 2 Popper (1982), p. 27.
1 2 3 Popper (1982), p. 28.
↑ Popper (1982), p.29.
↑ A. Sudbery (1985). "Popper's variant of the EPR experiment does not test the Copenhagen interpretation". Philosophy of Science. 52: 470–476. doi:10.1086/289261.
↑ A. Sudbery (1988). "Testing interpretations of quantum mechanics". In Tarozzi, G.; van der Merwe, A. Microphysical Reality and Quantum Formalism. Dordrecht: Kluwer. pp. 470–476.
↑ H. Krips (1984). "Popper, propensities, and the quantum theory". British Journal for the Philosophy of Science. 35 (3): 253–274. doi:10.1093/bjps/35.3.253.
↑ M. J. Collet; R. Loudon (1987). "Analysis of a proposed crucial test of quantum mechanics". Nature. 326 (6114): 671–672. doi:10.1038/326671a0.
↑ M. Redhead (1996). "Popper and the quantum theory". In O'Hear, A. Karl Popper: Philosophy and Problems. Cambridge: Cambridge University Press. pp. 163–176.
1 2 Y.-H. Kim & Y. Shih (1999). "Experimental realization of Popper's experiment: violation of the uncertainty principle?". Foundations of Physics. 29 (12): 1849–1861. doi:10.1023/A:1018890316979.
↑ C.S. Unnikrishnan (2002). "Is the quantum mechanical description of physical reality complete? Proposed resolution of the EPR puzzle". Foundations of Physics Letters. 15: 1–25. doi:10.1023/A:1015823125892.
↑ C.S. Unnikrishnan (2001). "Resolution of the Einstein-Podolsky-Rosen non-locality puzzle". In Sidharth, B.G.; Altaisky, M.V. Frontiers of Fundamental Physics 4. New York: Springer. pp. 145–160.
↑ A. J. Short (2001). "Popper's experiment and conditional uncertainty relations". Foundation of Physics Letters. 14 (3): 275–284. doi:10.1023/A:1012238227977.
↑ P. Sancho (2002). "Popper's Experiment Revisited". Foundations of Physics. 32 (5): 789–805. doi:10.1023/A:1016009127074.
↑ Tabish Qureshi (2005). "Understanding Popper's Experiment". American Journal of Physics. 73 (6): 541–544. doi:10.1119/1.1866098.
↑ Tabish Qureshi (2012). "Popper's Experiment: A Modern Perspective". Quanta. 1 (1): 19–32. arXiv:1206.1432. doi:10.12743/quanta.v1i1.8.
↑ Tabish Qureshi (2012). "Analysis of Popper's Experiment and Its Realization". Progress in Theoretical Physics. 127: 645–656. doi:10.1143/PTP.127.645.
↑ E. Gerjuoy; A.M. Sessler (2006). "Popper's experiment and communication". American Journal of Physics. 74 (7): 643–648. arXiv:quant-ph/0507121. doi:10.1119/1.2190684.
↑ G. Ghirardi, L. Marinatto, F. de Stefano (2007). "A critical analysis of Popper's experiment". arXiv:quant-ph/0702242.

Karl Popper

Philosophy	Bold hypothesis Critical rationalism Falsifiability Growth of knowledge Open society Popper's experiment Popper's three worlds

Works	The Logic of Scientific Discovery (1934) The Poverty of Historicism (1936) The Open Society and Its Enemies (1945) Conjectures and Refutations (1963)

Quantum mechanics

Background	Introduction History timeline Classical mechanics Old quantum theory Glossary

Fundamentals	Bra–ket notation Complementarity Density matrix Energy level ground state excited state degenerate levels zero-point energy Entanglement Hamiltonian Interference Decoherence Measurement Nonlocality Quantum state Superposition Tunnelling Scattering theory Symmetry in quantum mechanics Uncertainty Wave function collapse wave–particle duality

Formulations	Formulations Heisenberg Interaction Matrix mechanics Schrödinger Path integral formulation Phase space

Equations	Dirac Klein–Gordon Pauli Rydberg Schrödinger

Interpretations	Interpretations Bayesian Consistent histories Copenhagen de Broglie–Bohm Ensemble Hidden variables Many-worlds Objective collapse Quantum logic Relational Stochastic Transactional

Experiments	Afshar Bell's inequality Cold Atom Laboratory Davisson–Germer Delayed choice quantum eraser Double-slit Franck–Hertz experiment Mach-Zehnder inter. Elitzur–Vaidman Popper Quantum eraser Schrödinger's cat Stern–Gerlach Wheeler's delayed choice

Applications	Quantum biology Quantum chemistry Quantum chaos Quantum computing timeline Quantum cryptography Quantum electronics Quantum machine Quantum mind Quantum optics Quantum information science Quantum technology

Extensions	Quantum statistical mechanics Relativistic quantum mechanics Quantum field theory history Quantum gravity Fractional quantum mechanics

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