Quantum quandary #4: bomb testing

The conundrum of "interaction-free" interactions

To begin with, let’s revise quantum theory’s two most fundamental rules. Suppose that we have made a measurement and obtained a particular outcome O₁. And suppose that we want to calculate the probability of a particular outcome O₂ of another measurement that will be made at a later time. Here is how we may proceed: we choose a sequence of measurements that may (or may not) be made in the meantime, and we apply the appropriate rule:

• [Rule 1] If the intermediate measurements are made (or if it is possible to find out what their outcomes would have been), we square the magnitudes of the amplitudes associated with all possible sequences of intermediate outcomes, and we add the results.

• [Rule 2] If the intermediate measurements are not made (and it is not possible to find out what their outcomes would have been), we add the amplitudes associated with all possible sequences of intermediate outcomes, and we square the magnitude of the result.

Once again we shall consider a setup in which there is just one intermediate measurement, which may or may not be made, and which has just two possible outcomes. Thus there are two alternatives, to each of which we have to assign a complex number called “amplitude.” The setup involves a beam splitter (S₁), two mirrors (M₁ and M₂) and two photodetectors (D₁ and D₂).

A photon enters S₁ from the left, and the next thing we know is that either of the two photodetectors clicks, indicating that it has absorbed a photon. (Note that I do not straight away say that it has absorbed the photon. It is always worth keeping in mind that particles, including photons, lack intrinsic sameness. We can speak of he same photon only if the experimental context permits it, as the present context does.) So if D₂ clicks, we can conclude that the photon went straight through S₁ and was then reflected upward by M₂, and if D₁ clicks, we can conclude that the photon was reflected first upward by S₁ and then to the right by M₁.

Beam splitters are designed so that the two possibilities are equally likely. (In the context of classical physics, a beam splitter produces two light beams each having half the intensity of the incoming beam.) It would, however, be wrong to think that a photon either goes straight through the beam splitter or is reflected upward. These alternatives exist only in experimental contexts like the one presently considered, in which one of two detectors clicks, indicating either that the incoming photon went straight through or that it was reflected upward. Now let’s add a second beam splitter (S₂):

Now there are two alternatives that lead to the photon’s detection by D₁ (via M₁ or via M₂), and there is one possible intermediate measurement, designed to determine the route taken by the photon. If this measurement is not made, Rule 2 applies, and this requires that we begin by adding the amplitudes associated with the two alternatives. Because the two paths from S₁ to S₂ are equal in length, the corresponding amplitudes (A₁ and A₂) are equal except that each reflection causes a phase shift by π/2 or 90°. Since each path involves two reflections, each amplitude is shifted by the same phase, so A₁ = A₂. Thus we may drop the indices and write 2A for the sum of the two amplitudes. The probability with which D₁ will click is therefore |2A|².

There are also two alternatives that lead to the photon’s detection by D₂. But now one path (via M₁) involves three reflections, while the other path (via M₂) only involves one reflection. The two amplitudes therefore differ by a phase shift of π or 180°, and this is the same as saying that they differ by a factor of –1. The sum of the two amplitudes is therefore zero, as is the probability with which D₂ will click. So D₂ never clicks, while D₁ is certain to click every time. This is a classic example of quantum-mechanical interference. In one case we speak of constructive interference, in the other we speak of destructive interference.

Now imagine a bomb — I hope that your spam filter can handle the word — so sensitive that a single photon (or any other type of particle) will set it off. Obviously, this bomb will have to be kept in complete darkness. Next imagine a scenario in which such a bomb may be present but we are not sure whether it is actually present. Is it possible to find out whether the bomb is there without looking — and thus without setting it off? Let’s try this experiment:

Bomb testing scheme designed by Avshalom Elitzur and Lev Vaidman

If the bomb explodes, which happens in 50% of all trials, we have failed to detect its presence without setting it off. If D₁ clicks, which happens in 25% of all trials, we haven’t caused it to explode, but we have failed to detect it since the possibility that D₁ clicks also exists in the absence of a bomb. But if D₂ clicks, which happens in the remaining 25% of all trials, we have succeeded. Because D₂ never clicks in the absence of a bomb, we can infer the presence of a bomb, and this without having set it off. So the answer is: yes, it is possible to find out whether a bomb is present without detonating it — but only with a success rate of 25%.

The first experiment of this kind was proposed by Avshalom Elitzur and Lev Vaidman,¹ who gave the following description: “Consider a stock of bombs with a sensor of a new type: if a single photon hits the sensor, the bomb explodes. Suppose further that some of the bombs in the stock are out of order: a small part of their sensor is missing so that photons pass through the sensor’s hole without being affected in any way, and the bomb does not explode. Is it possible to find out which bombs are still in order?” The first experiment of this kind to be actually made (with non-lethal bombs to be sure) was performed by Anton Zeilinger and collaborators.²

When a simulation of the experiment was presented at a science fair in Groningen, the Netherlands, in 1995,³ the reactions of non-physicists differed markedly from those of physicists. Everyone was perplexed, for the detection of the photon by D₂ seemed to have contradictory implications: the bomb was present, yet the photon never came near it. If the photon never came near the bomb, how was it possible to learn that the bomb was present? While most ordinary folks thought that some physicists will eventually solve this puzzle, the physicists themselves were less hopeful.

In a previously quoted passage, Bryce DeWitt and Neill Graham remarked that

physicists are, at bottom, a naïve breed, forever trying to come to terms with the ‘world out there’ by methods which, however imaginative and refined, involve in essence the same element of direct contact as a well-placed kick.

In each of the three publications referenced above the experimental test for the presence of a bomb has been described as “interaction-free.” What their authors mean by “interaction-free” is an interaction that involves no direct contact.

That the notion of a kick-like interaction is a non-starter should be clear from the fact that the shapes of things resolve themselves into spatial relations between entities that lack internal structure. Whether we think of these entities as pointlike or realize that they are formless makes no difference here. To be kickable, things need surfaces, and neither pointlike nor formless things have surfaces — nor do things whose shapes resolve themselves into spatial relations between pointlike or formless entities.

Every interaction, quantum-mechanically described, involves some action at a distance, spooky or otherwise. If Einstein referred to the interactions between EPR-correlated particles as “spooky actions at a distance,” it was because he could not conceive of interactions that did not ultimately involve some form of direct contact. To account for EPR-correlations non-spookily, either one needs something that travels from one particle to the other, or one needs something that travels from a common cause to each particle. In other words, one needs interactions that satisfy the principle of local action, which requires a sequence of actions each of which takes place locally (as against at a distance) — first the action by a particle or a common cause on that which travels, then the continuous action by that which travels on itself (that infinitesimal bucket brigade), and finally the action by that which travels on another particle. Strangely enough, these “viscerally unproblematic” kick-like actions hardly ever receive the attention they deserve.

Visceral suggestiveness aside, how do physicists come to believe (i) that interactions between particles need to be mediated by travelling particles (which, as we just saw, implies a direct, not further mediated action by and on the mediating particle), and (ii) that in order to be able to interact without further mediation, particles need to be in the same place?

What forms the evidentiary basis of the quantum-mechanical interaction laws is scattering experiments. These laws ensure that the total energy-momentum of the incoming particles equals the total energy-momentum of the outgoing particles, and that the same holds for all conserved charges. Whatever happens between the creation of the incoming particles and the detection of the outgoing particles is treated as a black box, which is known as the S-matrix. The S-Matrix is a calculational tool with two inputs and one output. If you insert a particular set of incoming particles and a particular set of outgoing particles, it yields the probability with which the incoming set will transform into the outgoing set. A typical calculation of this probability involves a series of Feynman diagrams.

For illustration, consider the following Feynman diagram. It is the first in a series of diagrams that are used to calculate the probability with which a neutron (consisting of one u quark and two d quarks) transforms into a proton (consisting of one d quark and two u quarks) plus an electron and an antineutrino. The wiggly line represents a so-called virtual particle, i.e., a particle that only exists inside the black box, as part of a mathematical procedure.

The cardinal error committed by most particle physicists is to mistake a mathematical procedure for an actual physical process. One of the more egregious examples can be found in Antony Zee’s introduction to quantum field theory,⁴ where he writes that

Feynman diagrams are literally pictures of what happened.... Feynman diagrams can be thought of simply as pictures in spacetime of the antics of particles, coming together, colliding and producing other particles, and so on.

Richard Mattuck⁵ takes a more considerate stance:

the diagrams are so vividly “physical looking,” that it seems a bit extreme to completely reject any sort of physical interpretation whatsoever.... Therefore, we shall here adopt a compromise attitude, i.e., we will “talk about” the diagrams as if they were physical, but remember that in reality they are only “apparently physical” or “quasi-physical.”

One cannot but wonder what “physical looking” could mean, since we have no way of looking into the black box. We can only calculate the probability with which what goes in turns into what comes out. Brigitte Falkenburg,⁶ restores some needed sanity when she writes that

Feynman diagrams ... have no literal meaning. They are mere iconic representations of the perturbation expansion of a quantum field theory. They make the calculations easier, but they do not represent individual physical processes.

To this she adds the following note:

S-matrix means scattering matrix. The calculation is made within the framework of the quantum theory of scattering. It is usually calculated ... in the lowest order of perturbation theory. For high-precision measurements, higher-order corrections also come into play. The Feynman diagrams are in exact correspondence to the various mathematical terms of the calculations which describe superpositions of quantum theoretical contributions to the interaction.

And if that is not clear enough:

Each Feynman diagram symbolizes one of the virtual processes which contribute to the scattering amplitude. Hence, there are well-defined rules of formal correspondence between the mathematical formalism of the perturbation expansion and the iconic representation of the emission and absorption of virtual field quanta. There is nothing more to Feynman diagrams. The propagators inside the black box only contribute to the perturbation expansion of interacting quantum fields. To talk about them in terms of virtual particles is a mere façon de parler.... this façon de parler provokes conceptual confusion as soon as one insinuates that virtual particles are physical, i.e., on a par with the real field quanta or the incoming and outgoing physical particles of a scattering experiment.

With the notable exception of Roger Boscovich, a Serbo-Croatian polymath who flourished in the 18th Century, it does not seem to have occurred to anyone that action by direct contact is as unintelligible as the ability of material objects to act where they are not, which local action was believed to have explained away, but which thanks to quantum mechanics has re-emerged in an irredeemably spooky form. Fortunately, the conundrum posed by action by direct contact turns out to be spurious, for it only arises if one insists on transmogrifying a mathematical procedure into an actual physical process.

Much the same goes for Einstein’s spooky action at a distance. EPR correlations are experienced — by Alice at the apparatus on the left and by Bob at the apparatus on the right.

These experiences can be objectivized. Alice can convey her apparatus settings and outcome to Bob and the rest of the physics community, and Bob can convey his settings and outcome to Alice and the rest. While these data form part of our shared objective reality, physical interactions responsible for the correlations do not. Whatever is responsible for them is hidden away in a black box that is empirically inaccessible to us. All that is accessible to us are the correlations we are able to predict and observe.

And much the same goes for the correlations that obtain, not between measurements performed on different systems at the same time, but between measurements performed on the same system at different times. The only difference is that in the synchronic (same-time) case, nobody has yet come up with a viable explanation, while in the diachronic (different-time) case, it is widely believed that transmogrifying the wave function — a calculational tool — into an evolving physical state does the trick. In reality, we know as little of a physical process by which an event here and now contributes to determine the probability of a later event here as we know of a physical process by which an event here and now contributes to determine the probability of a distant event now.

1. A.C. Elitzur and L. Vaidman, Quantum mechanical interaction-free measurements, Foundations of Physics 23, 987–997 (1993).

2. P.G. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M.A. Kasevich, Interaction-Free Measurement, Physical Review Letters 74, 4763–4766 (1995).

3. E.H. Du Marchie van Voorthuysen, Realization of an interaction-free measurement of the presence of an object in a light beam, American Journal of Physics 64 1504–1507 (1996).

4. A. Zee, Quantum Field Theory in a Nutshell, pp. 53, 57 (Princeton University Press, 2003).

5. R.D. Mattuck, A Guide to Feynman Diagrams in the Many–Body Problem, p. 88 (McGraw–Hill, 1976).

6. B. Falkenburg, Particle Metaphysics: A Critical Account of Subatomic Reality, pp. 131–132, 235, 237 (Springer, 2007).