Quantum quandary #5: Klyachko’s theorem
...and the manifestation of the world (to us)
For any quantum system whose state space has three or more dimensions — which means that measurements performed on the system can have three or more outcomes — one can prove a theorem to the effect that the system’s measurable quantities cannot be in possession of pre-existing values: their values are not simply revealed by being measured; they only exist if they are measured. The proof, moreover, is state-independent: it doesn’t depend on which measurements were previously performed on the system or which outcomes were previously obtained. The first theorems of this kind were proved by Simon Kochen and Ernst Paul Specker1 and separately by John Bell.2
To get a feel for what’s at stake, consider a measurement M₁ with three possible outcomes. In quantum mechanics, these outcomes are represented by three mutually perpendicular axes 1, 2, 3 in a three-dimensional state space. Next consider a second measurement M₂ with outcomes represented by three mutually perpendicular axes 1, 2’, 3’. Because axes 2’ and 3’ are neither identical with nor orthogonal to axes 2 and 3, the two measurements are incompatible: they cannot be simultaneously made.
One peculiar thing about quantum mechanics is that every triplet of mutually perpendicular axes in a system’s state space represents an in principle possible measurement. For illustration, let’s imagine a sphere whose center coincides with the tails of the arrows in the above figure, and whose surface is touched by the arrows’ tips. And let’s assume what is disproved by the Kochen–Specker–Bell theorem, namely that measurements reveal pre-existing values. It will then be possible to color the arrows (or their tips on the surface of the sphere): say, green if the probability assigned to the arrow (or to the outcome it represents) equals 1, and red if it equals 0. If this is done, one third of the points on the sphere will be green, while two thirds of the points will be red. What’s worse, if one point of the sphere is green (let’s say, the north pole), then all points on the equator — coincident with the tips of arrows which are perpendicular to the arrow that points due north — will be red.
It is one thing to feel intuitively that coloring all points on the sphere in this particular way is impossible, and quite another to prove it. The original Kochen–Specker proof involved 117 directions, or 117 points on the surface of the sphere, associated with 43 triplets of mutually perpendicular directions (with some directions belonging to more than one triplet). What Kochen and Specker proved, in effect, is that it’s impossible to color the 117 arrows or points red and green in such a way that every triplet of mutually perpendicular arrows contains one green and two red arrows. It’s impossible because some arrows belong to more than one triplet, and because there will be at least one arrow that, as member of one triplet. would have to be green while, as member of another triplet, would have to be red.
These two triplets correspond to different measurements or measurement contexts. If measurements merely revealed pre-existing values, these values would have to be contextual in general. In the context of measurement M₁, arrow 1 might have to be green, while in the context of measurement M₂, it might have to be red. But this would defeat the whole rationale for positing pre-existing values, which is to rid quantum mechanics of its dependence on measurements, or to replace the language of finding (this or that outcome) with the language of possessing (this or that value), or to talk about beables rather than observables.
By now there are several proofs involving 33 directions, and John Conway and Simon Kochen have produced one with a “mere” 31 directions. The polyhedron of three intersecting cubes on top of the left tower in M.C. Escher’s engraving “Waterfall” contains a representation of the 33 directions in a version of the theorem by Asher Peres.3 (The 33 directions of Peres’ proof are defined by the lines connecting the center of the three cubes to their vertices and to the centers of their faces and edges.) Needless to say, one would like to eventually see an experimental test of the Kochen–Specker–Bell theorem, but even the 31 directions required by Conway and Kochen are far too many for such a test to become feasible.
Imagine, then, the surprise when Alexander Klyachko and coworkers4 published a proof that uses a mere five directions. The relatively small price to be paid for this superlative economy is that the proof is not state-independent: it makes use of a quantum system that has been prepared in a specific way.
Like Bell’s demonstration of the inconsistency of quantum theory with pre-existing values, the proof of Klyachko’s team features an inequality that is violated by the theory. Unlike Bell’s theorem, or the one by Greenberger, Horne, and Zeilinger, Klyachko’s doesn’t involve measurements on more than one system. (For simultaneous measurements on two or more systems, you need at least 2 possible outcomes for each individual system, and thus a minimum of 4.) When you have component systems, the statistical correlations predicted by quantum mechanics could be inconsistent either with pre-existent values or with the speed limit imposed by the special theory of relativity on the transmission of information. In the absence of component systems, you do not need to worry about the possibility of superluminal (“spooky”) action at a distance. Three years after the publication of the paper by Klyachko’s team, the violation of Klyachko’s inequality by quantum mechanics was experimentally confirmed.5
To set up the proof of this inequality (and of its violation by quantum mechanics), we’ll inscribe a pentagram with vertices labeled 1 through 5 in the equator of a sphere of radius 1, as shown in the below figure. Next we’ll pull the circle containing the pentagram towards the north pole, until the angle δ subtended at the center O of the sphere by the endpoints of a line of the pentagram equals 90°. Now we have five measurement contexts, each represented by three mutually perpendicular lines or arrows starting at O. The first context contains the arrows pointing to vertices 1 and 2 as well as a third arrow perpendicular to these two. The second contains the arrows pointing to vertices 2 and 3 plus a third arrow perpendicular to those two, and so on.
The next step is to determine the maximum probability of finding that an arrow — randomly selected from the five that point to a vertex — is associated with (or corresponds to, or represents) a pre-existent value, assuming that observables have pre-existing values. All we need to know to be able to determine this probability is that both endpoints of any line of the pentagram (or the arrows pointing to them) represent different outcomes of the same measurement.
It may be that none of the arrows pointing to a vertex corresponds to a pre-existing value, in which case the probability of obtaining the outcome represented by such an arrow is 0. It’s also possible that exactly one such arrow corresponds to a pre-existent value. In this case the probability of obtaining, by a randomly selected measurement, the outcome represented by such an arrow equals 1/5. And it’s possible that exactly two such arrows correspond to pre-existent values.
What is not possible is that more than two such arrows correspond to pre-existent values. This is because the endpoints of each line of the pentagram (or the arrows pointing to them) represent different possible outcomes of the same measurement, and of these at most one can be the actual outcome. Thus if (the arrow pointing to) vertex 1 corresponds to a pre-existent value, neither vertex 2 nor vertex 5 can correspond to a pre-existent value: a measurement that is certain to yield the outcome associated with vertex 1 will never yield the outcome associated with either vertex 2 or vertex 5. This leaves room for only one other pre-existing value, which can correspond to either vertex 3 or vertex 4. The maximum probability that a randomly selected vertex corresponds to a pre-existent value therefore is 2/5.
To compare this result with the quantum-mechanical prediction, we need to sum these maximum probabilities over the five vertices. The obvious result is five times 2/5, which is 2. This is Klyachko’s inequality. As said before, Klyachko’s proof of its being violated by quantum mechanics requires the system to be prepared in a particular way. The system must have been previously observed to possess the property or value which corresponds to the arrow that points to the north pole of the sphere. In this case the sum of the probabilities assigned to the five vertices (or to the arrows pointing to them) equals √5 ≈ 2.236, which is greater than 2.
This once again shows that quantum mechanics is inconsistent with pre-existing values. A quantum-mechanical observable has a value only if its value is indicated by a measurement. And this, once again, implies that neither macroscopic objects (including those by means of which measurements are made) nor microscopic objects (of which macroscopic objects are commonly thought to be composed) can be substances in the sense of being independently existing property carriers.
Because there are different kinds of properties and various possibilities of dependence, let’s consider the more important ones. The properties of an object may be intrinsic (or necessary, or essential) in the sense of belonging to the type (or species, or kind) to which the object belongs, or in the sense of being independent of the relations that the object entertains with other objects. Properties may be extrinsic in the sense of being accidental or contingent (they may or may not be possessed by an object of a given type) or in the sense of being possessed in virtue of an object’s relations to other objects. To these the philosopher Duns Scotus has famously added the property of haecceity or “thisness,” in virtue of which an individual object is the individual that it is. Haecceity is not only strikingly absent from quantum physics but also superfluous in classical physics, where individuality (in the form of transtemporal identity) is secured by determinism.
There is a further sense in which the properties of microscopic or quantum objects in particular are extrinsic. They are extrinsic in that they depend for their existence on being indicated by a measurement. This brings into play the experiencing subject, not only because nothing can be indicated in the absence of an experiencing subject, but also because it is now necessary to distinguish between objects that are directly accessible to human sensory experience and objects that are empirically accessible only indirectly, with the help of a macroscopic apparatus. This means is that quantum physics is incompatible with the elision of the human subject, which classical physics seemed able to achieve.
Like the intrinsic properties of macroscopic or “classical” objects, the intrinsic properties of microscopic or quantum object are those that make it the kind of object that it is. The intrinsic properties of an electron are the ones necessary for its being an electron. But here it must be stressed that what holds for the extrinsic properties of a quantum object also holds for its intrinsic ones: they exist (as properties actually possessed by an actually existing individual) only if the object is individuated by an experimental apparatus. While its extrinsic properties depend for their existence on the property-defining feature of an apparatus (like the position-defining region of a detector), its intrinsic properties depend for their existence on the macroscopic properties of the quantum phenomenon from which they can be inferred: “The carrier of these properties is nothing but the quantum phenomenon itself, say, a particle track on a bubble chamber photograph or the interference pattern of the double slit experiment”.6
This brings up a question that has already been touched upon: if a molecule of a given type is individuated by an experimental apparatus or a quantum phenomenon, what about the atoms of which it is common thought to be composed? Or the electrons, protons, and neutrons of which its atoms are commonly thought to be composed? Are they individuated as well? The answer is that they aren’t.
When we think of quantum objects in terms of interacting components parts, we are projecting the traditional concepts of substance and causality — which allow us to attribute properties as well as causal powers to individual property carriers — into the quantum domain, where they lack meaningful application. As stressed by Brigitte Falkenburg,7 “our classical construal of physical reality necessarily gives a distorted picture of subatomic structure. It simply makes us look into the atom through the wrong glasses. Unfortunately, we do not have any better tools.” Our language isn’t designed for this. (See Bohr’s remark on the prospect of one day replacing classical concepts by quantum-theoretical ones.)
What is known about the internal structure of an individuated quantum object pertains to its kind — the type to which it belongs — and this information is encapsulated in an S-matrix. The S-matrix is a mathematical tool that serves to calculate the probability with which, in a scattering experiment, any given set of incoming particles gets transformed into any given set of outgoing particles. Before and after the scattering event we have a set of individuated quantum objects, but the scattering event itself is a black box, and to think of the outgoing particles as the component parts or fragments of some object would be a serious mistake.
The energy-dependence of the internal structure of a “composite” quantum object provides an interesting case in point. At relatively low energies, an atomic nucleus bears some resemblance to a composite of protons and neutrons interacting via the exchange of mesons, while at higher energies it rather resembles a composite of quarks interacting via the exchange of gluons. What, furthermore, speaks against thinking of quarks in particular as components of quantum objects is their confinement: it’s impossible to isolate an individual quark and thus to individuate it. If, for example, we tried to separate the two quarks that are commonly said to make up a meson, the required energy would produce a second meson instead.
Let us re-consider in this light the idea that the multiplicity of the world experienced by us (or manifested to us) rests on reflexive spatial relations that are entertained by a single Ultimate Constituent (see here or here). The way I see it, the framework of thought that makes most sense of all available data has its roots in the original Vedanta of the Upanishads (as expounded by Sri Aurobindo). There we have a single ultimately reality (Brahman or Sachchidananda), which relates to the world as a supramental consciousness that contains the world, as a substance that constitutes it, and as an infinite quality and/or delight (ānanda) that experiences and expresses itself in it.
“In a sense,” Sri Aurobindo wrote, “the whole of creation may be said to be a movement between two involutions, Spirit in which all is involved and out of which all evolves downward to the other pole of Matter, Matter in which also all is involved and out of which all evolves upward to the other pole of Spirit” [LD 137]. The first casualty in the downward evolution is the consciousness which “sees the universe and its contents as itself in a single indivisible act of knowledge” [LD 147]. The result is a consciousness exclusively identified with some part of the universe and therefore unaware not only of the one all-containing consciousness and of the one all-constituting substance but also of the identity of that consciousness with this substance. This divided and dividing consciousness is what Sri Aurobindo refers to as “mind.”
To our minds, then, there is a difference between the content of consciousness and a seemingly mind-independent reality that appears to contain the empirically inaccessible causes of our perceptions. Unaware of the single Subject behind the apparent multiplicity of our separate selves, unaware of the single Object behind the apparent multiplicity of the diverse objects we perceive, and unaware that that Subject and this Object are one and the same Reality, we feel that the objects we perceived owe their reality not to a perceiving subject but to their constituent substances. Quantum mechanics, however, has put an end to this notion: because microscopic objects owe their properties to property-defining macroscopic objects, they cannot owe their reality to a multiplicity of constituent substances, nor therefore can macroscopic objects owe their reality to the microscopic objects of which they are commonly thought to be composed.
But if the multiplicity of the world experienced by us (or manifested to us) rests on reflexive spatial relations entertained by a single substance, then the objects we perceive do owe their reality to their constituents substances. It’s just that the number of their constituent substances equals one, and that this all-constituting substance turns out to be identical with a single all-perceiving subject. In the words of Erwin Schrödinger: “we are all really only various aspects of the One.”
So how should we think of the atoms and subatomic particles that mark the stages of the word’s manifestation (to us) — the stages of the transition from the One to the multiplicity of the experiential world? As previously discussed, we should think of them as structures that are instrumental in this manifestation, and that are known to us indirectly through the quantum-mechanical correlations that obtain between possible events in the manifested world. The further we probe the process of manifestation, the less applicable becomes the conceptualization of structure in terms of composition and interacting parts. Conversely, the further we advance from the One to the Many, the more completely multiplicity and distinguishability are realized.
S. Kochen and E.P. Specker, On the problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics 17, 59–87 (1967).
J.S. Bell, On the problem of hidden variables in quantum mechanics, Review of Modern Physics 38, 447–452 (1966). Reprinted in J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1989).
A. Peres, Two simple proofs of the Kochen–Specker theorem, Journal of Physics A: Math. Gen 24, L175–L178 (1991).
A.A. Klyachko, M.A. Can, S. Binicioğlu, and A.S. Shumovsky, Simple test for hidden variables in spin-1 systems, Physical Review Letters 101, 020403 (2008).
R. Lapkiewicz, P. Li, C. Schaeff, N.K. Langford, S. Ramelow, M. Wieśniak, and A. Zeilinger, Experimental non-classicality of an indivisible quantum system, Nature 474, 490–493 (2011).
B. Falkenburg, Particle Metaphysics: A Critical Account of Subatomic Reality, p. 200 (Springer, 2007).
B. Falkenburg, Particle Metaphysics, p. 160.
A bit of a technical comment, but something that also speaks against quarks as components, although also just another aspect of the confinement you mentioned.
Quarks "states" don't actually lie in the QCD Hilbert space, but technically only in a larger Krein space since they have negative norms. They are as such a "trick" to permit the use of local fields, since those give propagators with nice structures for Feynman graph evaluation in perturbation theory.
There's similar technicalities even with electrons where due to infrared issues electrons in QFT don't even have a well defined mass (two-point function has no momentum space pole).
As difficult as materialism/realism is in non-relativistic QM, it becomes even more so in QFT.
Ulrich,
I just read your paper "Bohr, objectivity, and 'our experience'" on the arxiv. Together with your earlier work "Niels Bohr, objectivity, and the irreversibility of measurements" it's a very clear exposition of Bohr's writing.
It's actually both your own writing in those papers and of course Bohr's writings that pinpoint something I never found satisfying in the "decoherence/consistent histories" papers. Namely the attempt to describe the classical world emerging out off the quantum world, by showing various collective coordinate operators for macroscopic bodies (e.g. the position of a mote of dust) are effectively classical variables. This looks fine until one realises that these macro-operators are functions (usually weighted sums) of microscopic operators which themselves represent nothing more than possible quantum phenomena (in Bohr's sense) occurring in the common/shared macroscopic world. So it's really not emergence at all as you say on page six of the new paper.