September Blog

Bell’s inequality is both incredibly simple and incredibly powerful, and speaks to the fundamental nature of reality; it will be the focus of this blog.

So what is Bell’s inequality? It was derived by John Stewart Bell in the 1960s, and has now been tested by experiments. The following derivation is based on work by Eugene Wigner, Bernard d’Espagnat and Anton Zeilinger, who helped in providing a more digestible understanding of Bell’s original work.

To understand Bell’s inequality, we start by thinking about identical twins. We choose 100 twins, where the twins are selected with the following requirement: a twin must have either blue or brown eyes, brunette or blond hair, and is either tall or short; twins with hair colour different from brunette or blond, or eye colour different from blue or brown, or heights different from tall or short, are not selected.

We define, \( F_{tall} \) as the fraction of tall twins and \( F_{short} \) as the fraction of short twins, and therefore \( F_{tall} + F_{short} = 1 \); similarly, \( F_{blue} + F_{brown} = 1\) and \( F_{brunette} + F_{blond} = 1\).

We define, \( F_{tall ~\&~ blue} \), as the fraction of twins that are tall and have blue eyes, and \( F_{tall ~\&~ blue ~\&~ brunette} \) as the fraction of twins that are tall and have blue eyes and have brunette hair.

Using this terminology, \( F_{tall ~\&~ blue} = F_{tall ~\&~ blue ~\&~ brunette} + F_{tall ~\&~blue ~\&~blond} \), because \( F_{brunette} + F_{blond} = 1\).

The two terms ( \(F_{tall ~\&~ blue ~\&~ brunette} \) and \( F_{tall ~\&~blue ~\&~blond} \)) on the right hand side of the above equation can be expressed as follows:

\( F_{tall~\&~blue~\&~brunette} + F_{tall~\&~brown~\&~brunette} = F_{tall~\&~brunette} \)

and therefore: \( F_{tall~\&~blue~\&~brunette} = F_{tall~\&~brunette} – F_{tall~\&~brown~\&~brunette} \)

Similarly,

\( F_{tall~\&~blue~\&~blond} + F_{short~\&~blue~\&~blond}= F_{blue~\&~blond} \)

and therefore, \( F_{tall~\&~blue~\&~blond} = F_{blue~\&~blond} – F_{short~\&~blue~\&~blond}\)

Using the above, it is clear that the following is true:

\( F_{tall ~\&~ blue} = F_{tall~\&~brunette}- F_{tall~\&~brown~\&~brunette} + F_{blue~\&~blond} – F_{short~\&~blue~\&~blond}\)

Removing the negative terms, leaves the following inequality:

\( F_{tall ~\&~ blue} \le F_{tall~\&~brunette} + F_{blue~\&~blond} \)

This inequality says that the number of twins that are tall and have blue eyes will always be less than or equal to the number of twins that are tall and have brunette hair + the number of twins that have blue eyes and blond hair.

Now, if we associate the properties of twins with the properties of photons (the quantum particles of light), we arrive at Bell’s inequality.

The property of a photon we will use is its polarisation, which is the property that Polaroid sunglasses use to reduce the glare of the sun. The polarisation of a photon can be thought of as an arrow attached to the photon that represents the direction of the oscillating electromagnetic field; we assume that this arrow always point in the same direction for a specific photon prior to impacting the sunglasses, and that a group of typical un-polarised photons have their arrows randomly pointing in every different direction. Polaroid sunglasses work by allowing a fraction of photons to pass through, with the fraction dependent upon the direction of the polarisation arrow. If the polarisation arrow is aligned with the direction of the molecules in the sunglasses, then the photon passes through. If the polarisation arrow is perpendicular to the molecules in the sunglasses, then the photon does not pass through. If the polarisation arrow is somewhere between aligned with and perpendicular with the molecules in the sunglasses, then a fraction of the photons pass through; the fraction, from Malus’ Law, is found to be proportional to the angle \(\theta \) between the polarisation arrow and the direction of the molecules in the sunglasses: \( \propto \cos^2 \theta \).

In the experiment that will be performed, two entangled photons (the twins) will each be passed through a polariser (sunglasses) which can be set to transmit a photon with a polarisation arrow set to one of three angles (\( 0^{\circ}, 30^{\circ}, 60^{\circ}\)); equivalent to height, eye colour and hair colour. The resulting photon will then be passed through a beam splitter, which according to Malus’ Law, will result in either a horizontally polarised photon (H) or a vertically polarised photon (V); equivalent to getting either tall or short if height is selected, blue or brown if eye colour is selected or brunette or blond if hair colour is selected.

Here is the required association, where \( \theta \) represents the angle of the polariser (sunglasses), and H or V represents the resulting photon from the beam splitter:

\( tall \rightarrow H_{0^{\circ}} ~,~~short \rightarrow V_{0^{\circ}} ~,~~\)\(blue \rightarrow H_{30^{\circ}} ~,~~brown \rightarrow V_{30^{\circ}}\)\( brunette \rightarrow H_{60^{\circ}} ~,~~blond \rightarrow V_{60^{\circ}} \)

Which gives Bell’s Inequality for these photons:

\( F_{H_{0^{\circ}} ~\&~ H_{30^{\circ}}} \le F_{H_{0^{\circ}}~\&~H_{60^{\circ}}} + F_{H_{30^{\circ}}~\&~V_{60^{\circ}}} \)

The experiment, when performed, gives the following impossible result:

\( 0.75 \le 0.25 + 0.25 \)

i.e. \( 0.75 \le 0.50\)

Which is truly shocking! What does this mean?

Well, when an inequality is violated, it means an assumption(s) leading to the derivation of the inequality must be wrong.

Looking back at the relatively simple derivation of the inequality above, the main assumption is local realism: i.e. the twins and photons are assumed to have definite properties of height, hair colour and eye colour that persist whether they are being observed or not, and that these properties can only be influenced by local surroundings.

For the twins, this is true – local realism is true, and the properties of the twins exist whether being observed or not..

But for entangled photons, as we have seen in previous blogs, local realism is not true, and we must give up the cherished belief that quantum particles have definite properties before observation – they do not. Bell’s inequality and the experiments performed with polarised photons show that local realism is not true in the quantum realm!

Interestingly, with a knowledge of entanglement and use of Malus’ Law, this violation of the inequality was expected: \(\cos^2 (30^{\circ} ) = 0.75\) and \( \cos^2 (60^{\circ}) = 0.25\), which are the values observed from the experiment discussed above.

Future blogs will continue to explore what it means for quantum particles (of which we are all made) not to obey local realism; local realism is something that we as a collection of non local realistic quantum particles, take for granted!

This blog is dedicated to an amazing mum, who though no longer with us, is entangled in all our thoughts.