The first half of the twentieth century witnessed the birth of a theory that changed our perception of the universe at an extraordinarily tiny scale. Yes, I’m talking about the birth of quantum mechanics. And with the advent of this theory, there originated one of the most incredible scientific debates between two intellectual wizards: Neils Bohr and Albert Einstein. So what was the debate actually about, and how is it connected to the title of this article, i.e., Bell’s inequalities? Let’s find out!

How One Man Challenged Einstein's Theory of Hidden Variables In Quantum Mechanics. 2
Niels Bohr with Albert Einstein in the late 1920s (Image: Emilio Segre Visual Archives/AIP/SPL)

Copenhagen v/s Einstein’s interpretation of quantum mechanics

Quantum mechanics has always been a theory of weirdness, rivalries, and criticism. And the very first debate in the field was between two drastically different interpretations of the theory: the Copenhagen interpretation propounded by Neils Bohr and Werner Heisenberg, and the other one was Einstein’s interpretation of quantum mechanics.

The Copenhagen interpretation of quantum mechanics claimed that physical systems only have probabilities. Neils Bohr proposed that the reality of a physical system is not only unknown but also unknowable until a measurement has been made. This means that rather than having specific and fixed properties, everything is unknown and doesn’t exist until a measurement has been performed.

However, Einstein boldly criticized this perception. He believed that probability doesn’t define everything. Instead, there exists an objective reality that is independent of the observations. For example, an electron has an intrinsic spin and moment, whether measured or not. Einstein even made a dramatic statement to support his claim. He once asked that “Do you believe that the Moon exists only when you look at it?” However, he never doubted the correctness of the theory. He just believed that the theory is incomplete and involves some hidden variables.

The EPR paradox and quantum entanglement

By 1935, it was already recognized that quantum mechanics doesn’t point to exact results. Rather its predictions are probabilistic.  In the same year, Albert Einstein, Boris Podolsky, and Nathan Rosen published a paper known as the EPR paper, according to which the particles can interact in such a way that it is possible to measure both their position and momentum more accurately.

The trio came up with a thought experiment to argue the incompleteness of the theory of quantum physics. They started with an assumption that the wave function does not give a complete description of physical reality and hence arrived at the conclusion that the two physical quantities with non-commuting operators can have simultaneous reality and finally it was concluded that the quantum mechanical description of physical reality given by the wave function is not complete and needed to be extended with hidden variables.

The main scenario of the EPR paper involved a pair of widely separated physical objects that were prepared so that the quantum state of the pair was entangled. Now, what does it mean to be entangled in quantum physics?

Entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described concerning each other, even though the individual objects may be spatially separated. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin up, the other one will always be spin down. So if you measure one of the entangled particles, you will automatically know what the results would be for the other particles.

It’s like putting each glove from a pair in two different boxes and then shuffling the boxes and taking them far away from each other. Once you open a box and find a right-handed glove there, you will automatically know that the other box contains a left-handed glove. It is just a simplified example. Quantum measurements are not as simple as looking out for a pair of gloves!

The information must travel between the two entangled particles at speeds greater than light for quantum entanglement to exist. However, this is not possible as per the laws of the theory of relativity. So this is where the trio stated that there exist some hidden variables that stimulate the transfer of information between the two entangled species.

Some mind-boggling paradoxes in physics:

The entry of the Bell’s inequalities:

The EPR paradox argued that since nothing can travel faster than light, this should invalidate the Copenhagen interpretation, and a theory supporting hidden variables should exist. This was fundamentally accepted for 30 years until John bell came up with an equation to determine who was right in 1964.

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John Bell (Image: Queens University Belfast)

Bell took forward the analysis of quantum entanglement to an advanced level. He deduced that if measurements are performed independently on the two separated halves of an entangled pair, then the assumption that the outcomes depend upon hidden variables within each half implies a constraint on how the outcomes on the two halves are correlated. This constraint is what we know as the Bell’s inequalities. Bell’s inequalities primarily concern the measurements made on pairs of particles that have previously interacted and then got separated.

To understand the physical meaning of Bell’s inequalities, let’s consider a pair of entangled particles originated from a system having a total angular momentum equal to zero. Now, we can have measurement devices at two distant locations that have settings to measure the spin of these entangled particles in some direction.

The experimenters (Alice and Bob, as named by Bell) choose the directions to make spin measurements for each particle separately. We suppose that the measurement outcome is binary (e.g., spin up, spin down) and the two particles are perfectly anti-correlated. This means that whenever both are measured in the same direction, one will get identically opposite outcomes, while they will always give the same outcomes when measured in opposite directions.

Now, let’s have three directions of measurement. The Z-axis, the X-axis, which is orthogonal to the Z-axis, and the third direction being something that lies in between the X and Z-axis, say Q, as shown below. In a universe where local hidden variables are true when the two particles are emitted, they will always know what their state is going to be in all three directions from their birth itself. And there are only 8 possibilities of spins that each particle could have.

Measurements that contribute  to Bell's inequalities
Measurements that contribute to Bell’s inequalities

Now talking about Alice, there are 4 events where the spin measured in Z is positive. So, the number of events when Alice has a positive measurement in Z and Bob has a positive measurement in X would exactly be 2, i,e. event 3 and event 4. This is because for Bob to have a positive spin in the X direction, Alice must have it negative there. So to get the probability, we have to divide the sum by the total number of events, i.e., 8 in our case. This corresponds to our measurement of P(Z+; X +).

Now, what would be the probability that when Alice measures in the Z direction, she gets a positive spin and Bob measures positive in the Q direction? Again, it would correspond to two events, i,e., event 2 and event 4. Again we can have the probability by dividing E2+E4  by 8.

Similarly, event 3 and event 7 correspond to the scenario when Alice measures positive in the Q direction, and Bob measures positive in the X direction? So overall, the probabilities of these three cases can be summarized as:

  • P( Z+; X+) = (E3 + E4)/8
  • P(Z+;Q+) = (E2 + E4)/8
  • P(Q+;X+) = (E3 + E7)/8

These are the three probabilities that satisfy the hidden variables theory. Now, as per Bell, if we take the total number of events and multiply that by the probability that Alice measures Z positive and Bob measures X positive, this has to be less than or equal to the total number of events times the probability that Alice measures Z positive, and bob measures Q positive, plus the probability that Alice measures Q positive, and bob measures X positive. In mathematical form,

8*P(Z+;X+) ≤ 8*[P(Z+;Q+) + P(Q+;X+)]

This further gives,

P(Z+;X+) ≤ [P(Z+;Q+) + P(Q+;X+)]

This is what we call the Bell’s inequalities. The above equation can easily be proved by doing simple mathematics. So the Bell’s inequalities have to be true for any hidden variables theory to be true. But in quantum physics, things are not as simple as they appear to be!

In a scenario where the laws of quantum mechanics are correct, and we don’t have any hidden variables, the probability of Bob measuring Q to be positive after Alice has measured Z to be positive is given by : P(Z+;Q+) = [sin2(45o)]/2. So, yes, the equations of quantum physics provide a formula for probability as a function of the angles between the measurement axes.

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Linear and sinusoidal probability (Image: Richard Gill, creative commons)

This is an important difference between quantum mechanics and the hidden variables theory. The probability here is not linear but looks like a sine wave. When a graph is plotted for the same, the results for the two systems agree at 0°, and multiples of 90o as shown in the above graph. But in between, there is a visible variation. Moreover, the probabilities calculated using this formula don’t obey Bell’s inequalities. This means that Bell’s inequalities are violated in the theory of quantum physics.

Over the years, physicists have made increasingly precise experimental tests of Bell’s inequalities, and the sine function correlation has been confirmed. Unfortunately, this establishes that quantum particles do not behave linearly, which means the hidden variables theory cannot be correct.

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