Category: Philosophy

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Can you prove you do not know something?

A question raised by Narges T. S.

We were sitting in early September 2023 in a restaurant, and Narges raised the question:

“How can we prove that we do not know something?”

No, this was not asking for a strategy in a police interview.

I don’t know any more what we discussed and what kind of quick-shot answers were given in that moment. But two weeks later, I was able to formulate an idea that quantum physics may provide a setting and an answer. Here is what I wrote to Narges:

Dear Narges,

… I had the impression that experimental settings where Bell inequalities are violated, can be used to “prove” that “we do not know something”. The challenge is to formulate precise “what we do not know”.

You probably know the typical setting where one deals with an entangled state of two particles, say

|\psi\rangle = |u, d\rangle + |d, u\rangle

where |u\rangle means “spin up” (or right-handed polarisation) and |d\rangle “spin down” (left-handed). The two entries u, d in the first ket describe the state of the first and the second particle – we may them picture spatially separated in two laboratories (“Alice and Bob”).

Space-time diagram of particles (photons, say) sent in pairs from a common source (lowest corner) to laboratories A and B for measurements. At the point C, correlations between the two data sets are calculated. Adapted from D. Pullman & al, “Characterization of a remote optical element with bi-photons”, Phys. Scr. 91 (2016) 023006.

When Alice and Bob have made a series of experimental checks and are sure that their source and detectors work reliably (I spare you these details, although they are interesting in themselves), they “know” the state |\psi> of the two-particle system.

What Alice does not know is in which state “her” particle is (either |u\rangle or |d\rangle, from the first or second term), and therefore when she does a spin measurement, she does not know what result she will get. This is perhaps not surprising since we are accustomed with random outcomes of single-photon counters. 

The funny feature (where I have the feeling that one can “prove” this “non-knowledge”) is that by using different measurement tricks (technically: checking that a Bell inequality is violated), one can prove that the measurement data of Alice and Bob cannot be described by “local hidden variables”. I have the impression that this “local hidden variable” statement can be re-formulated in the sense of your question.

In a local hidden variable context, one imagines that the particles are in definite states with “sufficient information” to produce measurement results even in settings (imagine the angle of a polariser) that the particles do not “know” beforehand. (And the experimentalists even try hard to make these choices after the particles are emitted, this is implicit in the word “local”.) Broadly speaking, one may say, in a “local hidden variable” description, somebody (perhaps some great world demon) knows the “real state” of the two particles. (The word “real” means: physical properties with definite values rather than some non-intuitive superpositions.) So in principle, also other people may get this knowledge. (That’s the job of “spies” or “eavesdroppers”, typically called Eve, who try to intercept the particles sent to Alice or Bob.)

The key point is now: if Alice and Bob manage to operate their setup in such a way that the Bell inequality is violated, then Bell’s theorem shows: this result cannot be understood / reproduced by any “local hidden variable” model / theory / knowledge. I have the impression that this statement is similar to what you ask for: checking for entanglement by a Bell inequality is equivalent / closely related to a proof that nobody (not Eve nor the world demon) had knowledge about the “real state” of either Alice’s or Bob’s particles.

Of course, you see that this is closely tied to a very special definition of “knowledge” (and about physical reality). 

Best regards and sorry for writing so long,
Carsten H.

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Lecture 1: A Little of Quantum Philosophy

20 Oct 2016 [notes by Johannes P. H. and Martin S., expanded by C. Henkel]

— Goals of the Course —

  • introduction to interpretations
  • and their key ideas,
  • provide an overall picture.

How to understand “Interpretation”:

A physical theory is providing a connection between a mathematical model (formalism) and the physical world. An interpretation tries to formulate in ordinary words what the theory (the formalism) “means” and what it tells us about “reality”. In the lecture, we shall try to sketch the corresponding world view (Weltbild). We shall encounter a strange aspect of quantum physics:
there are things that are difficult even to formulate (J. Bell uses the word “unspeakable”), and things that are not even “thinkable” (we shall see what this may mean).
Many interpretations of quantum mechanics come with an elaborate mathematical formalism: we shall try to illustrate their world view and connect the “elements of reality” (a concept taken from A. Einstein’s work) to the mathematical concepts. This may also require to create new ways of speaking about physical reality.

Background and Methods

The lecture should be understood in many places as amateur philosophy. Let us mention the old philosophical problem whether “the world exists” independently of our sensory impressions or whether all our impressions are just the product of our own mind. A scientist who is amateur philosopher will probably recognise herself in the viewpoint of “intersubjective positivism” (or better “realism”):

  • Through the communication between persons about impressions related to an object, the object enters our reality. This does not necessarily mean that the object does not exist without talking (which would be an extreme point of view), but if we talk about it, we recognise it in our reality. The word “intersubjective” is meant to emphasize the role of communication for this approach.
  • An independent reality may not exist or not be directly accessible to us, but as we exchange information about our observations, we are led to conclude that there are things in reality that we can agree upon — we may infer that they “exist”.
  • It may be that two interpretations are equivalent to another with respect to experimental predictions. But that does not mean that the world views behind these interpretations are the same.

Mathematics as a language:

  • The lecture tries not to develop too much formalism.
  • But it turns out that a careful understanding of the mathematical concepts reveals also a meaningful physical understanding.
  • In this way, one can connect the elements of the formalism to “physical objects” of the “outside world”.

Notes

The “Quantum Technologies” flagship is a strategic investment of the European Union with 1 billion (1 Milliarde) euros over the next 5–10 years. The promises of the flagship are related to communication, computing, precision measurements, and simulations, which can be improved by using quantum technologies.

We shall often hear that the results of certain measurements in quantum physics are random and can be predicted only in a statistical sense. But it seems difficult to certify that a sequence of results is random in quantum theory. The mathematical theory of random numbers can help us to understand how this may be done. (Topic not covered in the course.) A server of quantum random numbers can be found at the web site qrng.anu.edu.au. If you want to buy a quantum device that generates random numbers, try www.qutools.com.

Quotation

“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.” (“God made the integers, everything else is the work of man.”)

Leopold Kronecker, 7 December 1823 — 29 December 1891, was a German mathematician who worked on number theory, algebra and logic.