realism – Interpretations of Quantum Mechanics

Thu 11 January 2024Wed 21 February 2024

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.

Fri 17 July 2020Wed 6 October 2021

Einstein, Podolsky, Rosen, and Bell

Notes by Benjamin W.

In this lecture two theorems were treated. One was the EPR theorem, due to Einstein, Podolsky and Rosen (1935), the other was the Bell theorem (1964), as a logical continuation of the EPR theorem. The EPR theorem states that if all predictions of quantum mechanics are correct and if physical reality can be described in a local framework, then quantum mechanics is necessarily incomplete: there are elements of reality in nature that are not represented in this theory.

EPR (Einstein, Podolsky and Rosen, 1935)

Consider a source that emits pairs of particles. Two assumptions are made by EPR, namely separability and locality.

The first assumption of separability is that at the time of separation, each system (particle) has its own reality. EPR assume that each system retains its own identity, which is characterized by a real physical state, although each system is also strictly correlated with the other. In the 1935 paper, the correlations deal with the momentum and position variables of two particles. Following Bohm, the theorem was also formulated in terms of spin variables: the total spin of two particles is assumed in a singlet, $S = 0$ state.

The second assumption is that of locality. This assumes that no real change can take place in one system as a direct result of a measurement on the other system. EPR justify this assumption: at the time of measurement the two systems no longer interact. It should be noted that locality does not assume that nothing at all in one system can be directly disturbed by a remote measurement on the other system. (As an example: the information about the other system changes instantly after a measurement is made on one system.) Locality simply excludes the possibility that a remote measurement may directly disturb or alter what is considered real in relation to one system, a reality that guarantees separability. Based on these two assumptions, EPR conclude that each system can have definite values (elements of reality) for both position and momentum simultaneously (or for any spin components, in Bohm’s formulation).

The key idea here is that an experimental result that is known from the outset (with 100% probability) can only be the consequence of a physical quantity that already exists. The fundamental conviction of EPR is therefore that the regions of space contain their own elements of reality. These elements would evolve in time, after the emission of the particles, in a local manner. It follows from the EPR theorem that measurement results are well-defined functions of these variables. Thus nowhere would a random, non-deterministic process take place.

Another characteristic of EPR is completeness. If the description of systems by state vectors were complete, values of quantities that can be predicted with certainty, should be derivable from a state vector for the system or from a state vector for a composite system of which the system is a part.
It follows from the EPR theorem that separate systems have definite position and momentum values at the same time. The quantum mechanical description of the system by means of state vectors is therefore incomplete.

Bell (1964)

Some 30 years later, Bell again looked at the elements of reality and relied on locally realistic considerations to show that quantum mechanics cannot be supplemented in any possible way without altering experimental predictions, at least in certain cases. Bell considered correlations between measurement results for systems of two particles in separate laboratories, where the measurements of the particles differ by locally defined angles of spin projection axes. In this Gedanken experiment, he showed that correlations measured in different runs of an EPRB experiment satisfy a set of conditions.

These considerations are encoded in the Bell inequalities, which apply to all measurements that provide random results, whatever the mechanism that generates correlations between spins. Thus, any theoretical model that remains within the framework of local realism must lead to predictions that satisfy the Bell inequality. Here, realism is a necessary assumption, since the concept of element of reality introduced by EPR has been used in the proof. The proof is based on locality because it excludes the possibility that the measured result A depends on the measurement setting b in the other laboratory and that, conversely, B depends on a. Because of these inequalities, it was expected that any reasonable physical theory would produce predictions that were consistent with the Bell inequalities. However, the opposite was the case and violations of the Bell inequalities were even experimentally confirmed. A number of measurements provided valid violations of the Bell inequalities with a high degree of accuracy. This has led to the conclusion that the predictions of quantum mechanics are correct, even if the Bell inequalities are violated.

Example

Bell’s inequality for an ensemble of objects with the following set of properties:

female or male (w or m)
drives a car or not (a or -a)
speak german or not (d or -d)

Assignment of properties:

$n(w,a)$ is a woman driving a car
$n(a,-d)$ is a person who drives a car and does not speak German

Then the following inequality applies:

(1) $n(w,a) \le n(w,d) + n(a,-d)$

Meaning: The number of women driving a car is less than or equal to the number of women who speak German added to the number of persons driving a car who do not speak German.

We now consider a pair of photons and consider the polarizer settings:

– $\alpha$ and $\beta$ can take values among 0°, 30° and 60°

Number of measurements where the orientation combination $(\alpha,\beta)$ was present and where both photons passed through, according to one of the entangled Bell states

(2) $n( \alpha, \beta) = n_0 \cos^2( \alpha - \beta)$

Note the perfect correlation when $\alpha = \beta$ . Number of measurements where the left photon passed and the right photon was absorbed:

(3) $n( \alpha, \bar{\beta}) = n_0 \sin^2( \alpha - \beta)$

(to avoid confusion, the event “the photon did not pass the polarizer with setting $\beta$ is denoted $\bar{\beta}$ rather than $-\beta$ ).

The key assumption is now: the properties “first photon passes polarizer with setting $\alpha$ ” and “second photon passes polarizer with setting $\beta$ ” can be considered in the same way as the properties “is woman” or “speaks German” as above. We suppose they are “elements of reality” in the sense of EPR. The application of Bell’s inequality (1) then gives

$n(\alpha,\beta ) \leq n(\alpha,\gamma) + n(\beta,\bar{\gamma})$

Insert equations (2) and (3) and divide by $n_0$ :

$\cos^2(\alpha - \beta) \leq \cos^2(\alpha-\gamma) + \sin^2(\beta-\gamma)$

For $\alpha = 0^\circ$ , $\beta = 30^\circ$ and $\gamma = 60^\circ$ , it follows:

$\cos^2(30^\circ) \le \cos^2(60^\circ) + \sin^2(30^\circ) \qquad \text{or} \qquad \frac{3}{4} \leq\frac{1}{4}+\frac{1}{4}$

Since this is wrong, Bell’s inequality is violated by the predictions of quantum mechanics.

Literature

Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë (2020), Quantum Mechanics III: Fermions, Bosons, Photons, Correlations, and Entanglement, chap. XXI: Quantum entanglement, measurements, Bell’s inequalities, Wiley-VCH.

A. Einstein, B. Podolsky, and N. Rosen (1935), “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, Physical Review 47, 777–80.

Franz Embacher (2000), EPR-Paradoxon und Bellsche Ungleichung, Universität Wien, online at https://homepage.univie.ac.at/franz.embacher/Quantentheorie/EPR.

Sat 22 July 2017Fri 6 December 2024

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.