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, 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:

- is a woman driving a car
- is a person who drives a car and does not speak German

Then the following inequality applies:

(1)

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:

– and can take values among 0°, 30° and 60°

Number of measurements where the orientation combination was present and where both photons passed through, according to one of the entangled Bell states

(2)

Note the perfect correlation when . Number of measurements where the left photon passed and the right photon was absorbed:

(3)

(to avoid confusion, the event “the photon did not pass the polarizer with setting is denoted rather than ).

The key assumption is now: the properties “first photon passes polarizer with setting ” and “second photon passes polarizer with setting ” 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

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

For , and , it follows:

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.