Alhassanain(p) Network for Heritage and Islamic Thought

3. How quantum theory limits description of physical reality

In their famous EPR ([1935]), Einstein, Podolsky and Rosen assumed that quantum theory has descriptive resources but argued that these do not permit acomplete description of physical reality. Specifically, their argument set out to show that the wave-function fails to give a complete description of the state of an individual system. But they took it for granted that the wave-function’s role was indeed to describe physical reality, however incompletely: Einstein ([1949], pp.82-7, p.682) suggested that the wave-function should be taken incompletely to describe a statistical ensemble of similarly prepared systems.

       On the pragmatist approach I am presenting here, the quantum state does not itself purport to describe physical reality at all - not even incompletely. But in addition to its role in generating quantum probabilitiesvia the Born rule, it has an important secondary role in licensing limited claims about physical reality by an agent applying quantum theory.

       One can appreciate the need for such a role only after one has abandoned the idea that quantum theory itself makes available new descriptive or representational resources, either in the form of quantum state ascriptions or in some other way (perhaps by allowing dynamical variables to take on operator-values -q -numbers- instead of, or as well as, real numbers -c -numbers). Rather than thinking of quantum theory as providing distinctively new ways of describing or representing physical reality, focus instead on its effect onnon-quantum descriptions and representations.

       It is tempting to refer to such non-quantum descriptions and representations asclassical , following Bohr and others. But there are at least two reasons not to yield to this temptation. First, it encourages the mistaken thought that any use of such a description carries with it the full content of classical physics, including dynamical laws such as those of Newton and Maxwell. More importantly, it tends unduly to limit the scope of non-quantum descriptions to exclude what Bell called ‘the familiar language of everyday affairs, including laboratory procedures’ as well as descriptions made available by further advances in non-quantum physics (for example, possible modifications of classical relativity to secure empirical adequacy in light of new observations attributed to high-energy cosmic rays or so-called dark matter). The scope of non-quantum descriptions is very wide: indeed, if quantum theory itself provides no resources for describing or representing physical reality, thenall present and future ways of describing or representing it will be non-quantum.

       It is critical for the present approach to have available non-quantum descriptions of outcomes of quantum measurements. Call a claim expressed by a sentence of the formS : ‘The value ofA ons lies in Δ’ a non-quantum magnitude claim (NQMC).  If one could not express the result of a measurement in a NQMC, then the Born rule could acquire no empirical support from measurements and we should have little or no reason to believe quantum theory. Quantum theory itself does not imply sentences of the formS on the present approach, even in a case in which the Born rule assignsS probability 1. But physicists make claims using such sentences (or their equivalents) all the time, when describing the results of quantum measurements and in many other circumstances (e.g. in describing the operation of particle accelerators and nuclear reactors, as well as the position of ions in a crystal and the velocity and polarization of photons propagating through an optical fiber delay). How can one reconcile this practice with their acceptance of quantum theory, in light of the no-go results mentioned in section 2 (see footnote 6)?

       Answering this question will require excursions into pragmatist philosophy as well as the quantum physics of decoherence. Claims of environmentally-induced decoherence, to solve the quantum measurement problem and explain the emergence of classical behavior of macroscopic objects, are now widely (and wisely) regarded with suspicion.[13]But it is hard to dismiss the thought that decoherence hassome important role to play in resolving interpretational problems of quantum theory. As Bacciagaluppi ([2003/7]) and Schlosshauer ([2007]) explain, decoherence plays different roles within different attempted interpretations of the theory. So after a sketch of relevant quantum physics of decoherence, my main task here will be carefully to explain how a pragmatist can use this to explain when and how quantum theory itself can license the kind of non-quantum descriptive claims physicists do, and must, make in order successfully to apply quantum theory.

       The basic idea of environmentally-induced delocalization of coherence is well known.[14]

It may be illustrated by this toy model. Given an arbitrary superposed pure quantum state of a systems interacting with a systems′ in an appropriate initial quantum state |b 0,, there are Hamiltonians on the tensor product Hilbert space Hs⊗ Hs′  that will induce the following unitary evolution of the total quantum state ofs +s ʹ:

∑i c i |a i ,|b 0 ,→ ∑i c i |a i ,|b i ,               (2)

for some complete orthonormal basis |a i,of Hs. The resulting quantum state ofs is then given by partial tracing over Hs′  as ρs = ∑i|c i|2 |ai,+ai|, which contains no terms diagonal in the preferred |a i,basis defined by the Hamiltonian for this interaction. Thinking ofs′ as the environment ofs , such an interaction with its environment has delocalized the coherence ofs ’s initial state into the more inclusive systems +s′ (which in this case remains pure): every Born probability for an observable ons alone equals the weighted average (with weights  |ci|2 ) of Born probabilities of all states |a i,. Following such an interaction,s will display none of the interference characteristic of quantum mechanical superpositions. To observe any interference it would be necessary to perform an appropriatejoint measurement involvingboth ofs ands′ .

       Environmentally induced decoherence has only relatively recently become the subject of experimental investigation. One particularly revealing set of experiments studies interference phenomena involving large molecules including fullerenes (C60 and C70 molecules). Hackermüller et al. ([2004]) investigated the effects of increased temperature in matter wave interferometer experiments in which C70 molecules lose their quantum behavior by thermal emission of radiation. They prepared a beam of C70 molecules of well-defined velocity, passed them through two gratings of a Talbot-Laue interferometer in a high vacuum, and detected those that passed through a third movable grating set at the appropriate Talbot distance and used as a scanning mask, by ionizing them and collecting the ions at a detector. Each molecule is sufficiently large and complex to be assigned a temperature as it stores a considerable amount of energy in its internal degrees of freedom. Interaction with the electromagnetic vacuum may result in emission of photons with an intensity and frequency that increases as the internal temperature is raised. These photons may be considered the environment of the molecule. Entanglement between such photon states and the state of the emitting molecule tends to induce environmental decoherence.

       Hackermülleret al . ([2004]) present a theoretical model of this decoherence that fits their observations quite well, as the observed interference dies away when the molecules’ temperature is raised from 1000°K to 3000°K. This model bears an interesting correspondence to more informal discussions of how the possibility of observing through which slit a particle passed will prevent observation of any consequent interference pattern.[15]Such discussions often focus on particular methods for trying to observe through which slit each particle passes, and proceed to argue that quantum features of the required apparatus necessitate a trade-off between success in this attempt and success in obtaining any resulting interference pattern. Following Heisenberg ([1930]), one often considers shining light on the particles and collecting reflected light in a microscope focused on them as they pass the slits. In order to tell through which slit a particle goes one would need to use a microscope capable of resolving distances at least as small as the slit separation. Now the resolving power of a microscope is limited by the wavelength of light used: better resolving power requires light of shorter wavelength. However, photons of light of short enough wavelength would have such a large momentum as to disturb the particle and effectively to destroy the interference pattern. Even though no observation of the positions of C70 molecules as they pass through the apparatus is contemplated in these experiments, and no light is shone on them, the theoretical model of decoherence shows that the possibility of photonemission of short enough wavelength to make itpossible to determine through which slit each molecule goes is enough effectively to destroy the interference pattern. Moreover, the detailed form of the quantum state of the fullerene (expressed in the off-diagonal elements of the fullerene center-of-mass position density operator) describes the diffraction limitation of a hypothetical microscope used to obtain which-path information on the molecules.

       The phrase “which-path information” (or „welcher-Weg-Information”) that occurs repeatedly in Hackermülleret al . ([2004]) and many other experimental as well as theoretical treatments of quantum interference is puzzling but highly suggestive. I shall pursue the suggestion after discussing another related experiment recently conducted in the same laboratory in Vienna.

         Juffmanet al. ([2009]) prepared a beam of C60 molecules with well-defined velocity, passed them through two gratings of a Talbot-Laue interferometer in a high vacuum, and collected them on a carefully prepared silicon surface placed at the Talbot distance. They then moved the silicon about a meter into a second high vacuum chamber and scanned the surface with a scanning tunneling electron microscope (STEM) capable of imaging individual atoms on the surface of the silicon. After running the microscope over a square area of approximately 2μm2 they were able to produce an image of some one to two thousand C60 molecules forming an interference pattern.[16]They reported that the surface binding of the fullerenes was so strong that they could not observe any clustering, even over two weeks. Clearly they felt no compunction in attributing very well defined, stable, positions to the molecules on the silicon surface, and even recommended developing this experiment into a technique for controlled deposition for nano-technological applications.

       Together, these experiments illustrate three different scenarios in which one may contemplate making a claim about the position of an individual fullerene molecule involved in a quantum interference experiment. By reflecting on the inferential commitments entered into by one who makes such a claim, we shall be able to gain a better appreciation of the significance of judgments expressed in NQMC’s of the formS : ‘The value ofA ons lies in Δ’, beginning with the case in whichs is an individual fullerene molecule,A is the horizontal distancex (in nanometers) of its center of mass from a reference point in the plane of the vertically oriented gratings and Δ is an interval of real numbers. To repeat, while quantum theory itself does not imply sentences of the formS , they play an essential role in any application of quantum theory.

       After C60 molecules has been deposited on the silicon substrate in the experiment of Juffmanet al. ([2009]) and imaged by the STEM, their figures 2 and 3 (together with the surrounding discussion) illustrate that some claim of the formS x: ‘The positionx ofs isx s±ε’ for some value of ε<5nm is warranted. The warrant derives substantially from the reliability of the image-forming process, importantly including the (quantum!) theory and practice underlying the successful operation of the STEM used to produce it. But there is a prior issue: given that a C60 molecule may itself be treated as a quantum system, how and why is one entitled to attribute tos a definite, stable position in the first place?

       It is in answering this question that it is appropriate to appeal to environmental decoherence. While it may be difficult to formulate and  solve the Schrödinger equation for a realistic many-body quantum interaction that bindss to the silicon surface, it is clear that this will rapidly and strongly couples to an environment of an exponentially increasing number of degrees of freedom, involving the entire silicon crystal and light reflection from its surface, thermal radiation interacting with phonons in the crystal, vibrations and thermal motion of the supporting structure of the crystal, and eventually the entire laboratory and beyond. Examination of the properties of analytically and computationally solvable models of decoherence in simpler systems justifies one in concluding with a high degree of confidence that the center-of-mass state ofs alone will extremely quickly become, and remain indefinitely in the absence of external disturbances, very close to diagonal in a preferred “position basis” of states, each close to a delta function of position.

       It does not follow that some statement of the formS x is true. On the present approach, no analysis of a decoherence interaction to show the (approximately) diagonal form of the quantum state of a decohering system ever itself thereby implies any such non-quantum statement. The import of the quantum analysis is more subtle. What decoherence shows in this example is that what an agent may legitimately infer from a claim about the position ofs of the formS x is, as it relates to any conceivable goal of that agent, exactly what would follow from the simple truth ofS x.[17]This is how the quantum theory of decoherencelicenses the experimenters in Juffmanet al. ([2009]), anyone reading their paper, and indeed any suitably physically situated agent, human, conscious, or neither, to make some such claim. While quantum theory in this waylicenses many incompatible claims of this form, each ascribing a different valuex s1,x s2 ,x s3 ,... tox , by itself the theorywarrants an agent in claiming none of them: that requires additional, reliable empirical information of a kind acquired by the skillful use of the STEM used by Juffmanet al. ([2009]) to produce data like that displayed in figures 2 and 3. Quantum licensing takes the following form: a quantum state of a system and its environment may be such as to grant an agent permission to issue a judgment of a certain kind concerning that system. Equipped with the necessary permission, the agent may be warranted by its “experience” to issue one rather than another judgment of that kind.

       Feynman ([1963], vol. III, 1.9) said this about the position of an electron as it passes through an analogous 2-hole interference experiment:

if one has a piece of apparatus which is capable of determining whether the electrons go through hole 1 or hole 2, then onecan say it goes through either hole 1 or hole 2. [otherwise] one maynot say that an electron goes through either hole 1 or hole 2. If onedoes say that, and starts to make any deductions from the statement, he will make errors in the analysis. This is the logical tightrope on which we must walk if we wish to describe nature successfully.

Consider instead the status of claims of the formS x about a fullerenes as it passes through the interferometer gratings in either of the two experiments just described. In each diffraction grating in these experiments the slits were regularly spaced at a distance of some hundreds of nanometers. So if one could sayS or:Sx s1 orSx s2 orSx s3 or ... (wherex si marks the center of thei th slit and ε now corresponds to the width of each slit) then one could say the fullerene goes through slit 1 or slit 2 or slit 3 or . Butcan one sayS or?

       In the experiment of Juffmanet al. ([2009]) there was no piece of apparatus capable of determining which slit each fullerene goes through. What goes wrong if one saysS or? Feynman’s discussion makes clear the nature of the error he thinks would follow from this claim. Suppose one assumes thej th fullerene goes through sliti . It would have made no difference to its subsequent behavior if all the other slits had been closed, so it would have contributed to a single slit interference pattern centered on sliti . It follows that the total interference pattern will be just a sum of single slit patterns for alli , weighted by the number of fullerenes going through each slit. Since the actual interference pattern is quite different,S or has been empirically falsified.

       The form of the argument isreductio ad absurdum , but as is typical for such arguments it rests on additional premises, any of whose rejection prevents one from drawing the intended conclusion. Bohmians, among others, have principled reasons for denying that the behavior of a fullerene passing through one slit is independent of whether other slits are open or closed: roughly, they take its behavior to be governed by a physically real wave-function that passes throughall the open slits. Bohmian mechanics shows how to draw many more interesting conclusions ofS or consistent with quantum-theoretic predictions, though at the cost of accepting action-at-a-distance.

If one supplementsS or withno additional premises, one will never risk error in making deductions. This is a trivial consequence of two facts: (i)S or is logically consistent, and (ii) no valid deductive argument can lead from logically consistent premises to a contradictory conclusion. But this response misses Feynman’s point, since he is clearly concerned not just with formally valid deductive arguments whose sole premise isS or, but with inferences of the kind anyone with a normal understanding ofS or will naturally make, for example that:

      No particle passes through the material (silicon nitride) in which the slits are cut

      It is possible reliably to observe through which slit each particle passed without altering the interference pattern, or

      If this is not so, then that can only be because any physical mechanism that permitted reliable observation of through which slit each particle passed would inevitably disturb the particle while doing so

       Brandom ([2000]), following Sellars ([1953]), calls inferences such as those fromS or to I, II, IIImaterial inferences. Here and in Brandom ([1994]) he develops what he calls an inferentialist pragmatism about conceptual content. It is a consequence of this kind of pragmatism that thecontent ofS or is a function of the material inferences that connect it to other claims and other actions by a claimant or others in the same linguistic community. Accepting quantum theory in no way undermines the inference fromS or to I: this remains a legitimate material inference even though it is not formally valid. But however natural inferences to II or III may seem, application of quantum theory shows that both II and III lead to conflict with results of experimental (or at leastGedankenexperimental ) findings.[18]So while one cansay S or (pace Feynman), the content of that claim must be understood very differently within a community that has accepted a quantum theoretic analysis of the situation to which the claim applies. Given the possibility of confusion provided by so severely weakening the claim, it may be wise to heed Feynman’s cautionary advice not to sayS or at all in the context of the experiment of Juffmanet al. ([2009]).

       The status ofS or in the experiment of Hackermülleret al. ([2004]) is more complex. In the case of low temperature fullerenes, there is relatively little decoherence of their center of mass motion through the interferometer, so the analysis goes through as for the experiment of  Juffmanet al. ([2009]). While onecan sayS or, it is probably safest not to do so, since anyone who did so would naturally be understood as committed to inferences and other actions they did not intend. Without careful qualification, the weakened content of the claim would make it likely subject to misinterpretation. As the temperature of the fullerenes is increased, the interference contrast decreases. The authors comment

       This is the signature of decoherence due to the enhanced probability for the

       emission of thermal photons that carry ‘which-path’ information. [...] They transmit

(partial) which-path information to the environment, leading to a reduced observability of the fullerene wave nature. [...] Around 3,000°K the molecules have a high probability to emit several visible photons yielding sufficient which-path information to effect a complete loss of fringe visibility in our interferometer.

I believe it would misinterpret their use of the phrase ‘which-path information’ here to take them to presuppose that each fullerene follows a determinate, though unknown, path through the slits, which becomes progressively more open to potential observation as its temperature increases. It is better to regard thecontent of a claim made by a statement likeS or as itself a function of temperature, in the following sense: as the temperature is increased from 1000°K to 3000°K, the inferential power of the claim increases accordingly. This is why it becomes more and more appropriate to think and speak of the fullerenes as having a well-defined path through the interferometer as the degree of thermally induced electromagnetic decoherence into their environment increases. But note that on the present inferentialist view of content, this progressive definition of content has no natural limit such that one could say that when this limit is reached a statement likeS or is simply true because one has finally succeeded in establishing a kind of natural language-world correspondence relation in virtue of which the statement correctly represents some radically mind- and language-independent state of affairs.

       It is important to bear this in mind when reconsidering the role of measurement in quantum theory. I have been careful not to formulate the Born rule narrowly so that it explicitly concerns results of measurements on a quantum system. But it is vital that situations to which the Born rule applies include those in which scientists are warranted in making claims about the values of dynamical variables as a result of performing operations they take to constitute measurements of them. Recall from the introduction how Bell introduced his notion of beables to apply to things

…which can be described in ‘classical terms’, because they are there. The beables must include the settings of switches and knobs on experimental equipment, the current in coils, and the readings of instruments.

emphasizing that by ‘classical terms’ he

refers simply to the familiar language of everyday affairs, including laboratory procedures, in which objective properties -beables - are assigned to objects.

His thought seems to be that at least when it comes to descriptions of experimental equipment and laboratory procedures language must be taken to function in a straightforwardly representational way - as simply saying how things are.

       What “is there” in these fullerene experiments? Because of the massive decoherence between large scale features of the macroscopic laboratory apparatus and its environment, quantum theory licenses claims about the settings of switches and knobs on experimental equipment, the macroscopic current in coils, and the readings of instruments. The content of such claims is almost, but not quite, unchanged by acceptance of quantum theory, since the limits the theory places on their inferential power are of no importance for any practical, or even impractical, purpose. Acceptance of quantum theorydoes significantly modify the content of claims about the microscopic currents produced by electrons tunneling from the fullerenes and silicon surface when scanned by the STEM in the experiment of Juffmanet al. ([2009]). These currents cannot be said to “be there” in the same robust sense, in so far as each results from a tunneling process that is characteristically non-classical. And, as the preceding discussion made clear, acceptance of quantum theory so significantly limits the content of claims about the position of fullerenes as they pass through the interferometer that anyone making them at best courts confusion. But the efficiency of decohering interactions between a fullerene molecule and atoms of the silicon surface in the experiment of Juffmanet al. ([2009]) is such that even though the molecule and atoms are microscopic and the interaction is quantum, a claim that the molecule is there at a specific location on the surface has almost the status of a claim that the entire apparatus is there in the laboratory.

       There is no explicit reference to measurement in the published report of either fullerene experiment. Moreover, in applying quantum theory to account for the features of the interference patterns it is not necessary to interpret the Born rule explicitly to concern probabilities ofmeasured positions of fullerenes in the pattern: one can take it simply to give probabilities for their positions at the detector. But I think it is clear that deposition of a C60 molecule on the silicon substrate in the experiment of Juffmanet al. ([2009]) does count as performance of a quantum measurement of the molecule’s position. Certainly there is no temptation to say that the molecule has no definite position on the surface until and unless a subsequent observation is carried out using the STEM.

       We can now see both why it is natural to formulate the Born rule so that it concerns probabilitiesof measurement outcomes and why the application of that rule is not restricted to measurement contexts. One reason to explicitly mention measurement in a formulation of the Born rule is to stress that the evidence justifying acceptance of quantum theory rests to a large extent on the results of experiments in which observables are measured and the statistical distribution of measurement outcomes compared to those expected on the basis of the Born rule. If the Born rule could not be connected to measurement outcomes in this way, quantum theory would be cut off from its evidential base. But, as we have seen, the link can be preserved by simply assuming that the outcome of a quantum measurement can be expressed in a NQMC, with no mention of any measurement of a kind needed to determine the value of the magnitude in question. This leads to the second, more substantial reason for formulating the Born rule in terms of measurement: the no-go results mentioned in section 2. Not all dynamical variables on a quantum system can consistently be assigned simultaneous real values distributed in accordance with the Born probabilities - not even so as to match just the extremal Born probabilities 0 and 1. But this presents no problems on the present approach, since Born-rule probabilities are well-defined only over claims licensed by quantum theory. According to the quantum theory, interaction of a system with its environment typically induces decoherence in such a way as (approximately) to select a preferred basis of states in the system’s Hilbert space. Quantum theory will fully license claims about the real value only of a dynamical variable represented by an operator that is diagonal in a preferred basis: it will grant a slightly less complete license to claims about approximately diagonal observables. All these dynamical variables can consistently be assigned simultaneous real values distributed in accordance with the Born probabilities. So there is no need to formulate the Born rule so that its probabilities concern only measurement outcomes.