Using their reasoning, he said, a choice of measurement setting here should not affect the outcome of a measurement there and vice versa.

## "Quantum Chance: Nonlocality, Teleportation and Other Quantum Marvels" [GAP-Optique]

After providing a mathematical formulation of locality and realism based on this, he showed specific cases where this would be inconsistent with the predictions of quantum mechanics theory. In experimental tests following Bell's example, now using quantum entanglement of photons instead of electrons, John Clauser and Stuart Freedman and Alain Aspect et al.

In October , Hensen and co-workers [8] reported that they performed a loophole-free Bell test which might force one to reject at least one of the principles of locality, realism, or freedom-of-choice the last "could" lead to alternative superdeterministic theories. Conclusive experimental evidence of the violation of Bell's inequality would drastically reduce the class of acceptable deterministic theories but would not falsify absolute determinism, which was described by Bell himself as "not just inanimate nature running on behind-the-scenes clockwork, but with our behaviour, including our belief that we are free to choose to do one experiment rather than another, absolutely predetermined".

However, Bell himself considered absolute determinism an implausible solution. In July , physicists reported, for the first time, capturing an image of a strong form of quantum entanglement , called Bell entanglement. Bell's theorem states that any physical theory that incorporates local realism cannot reproduce all the predictions of quantum mechanical theory. For a hidden variable theory, if Bell's conditions are correct, the results that agree with quantum mechanical theory appear to indicate superluminal faster-than-light effects, in contradiction to the principle of locality.

The theorem is usually proved by consideration of a quantum system of two entangled qubits with the original tests as stated above done on photons. The most common examples concern systems of particles that are entangled in spin or polarization. Quantum mechanics allows predictions of correlations that would be observed if these two particles have their spin or polarization measured in different directions. Bell showed that if a local hidden variable theory holds, then these correlations would have to satisfy certain constraints, called Bell inequalities. Following the argument in the Einstein—Podolsky—Rosen EPR paradox paper but using the example of spin, as in David Bohm 's version of the EPR argument [6] [12] , Bell considered a Gedankenexperiment or thought experiment in which there are "a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions.

Since total angular momentum is conserved, and since the total spin is zero in the singlet state, the probability of the same result with parallel antiparallel alignment is 0 1. This last prediction is true classically as well as quantum mechanically. Bell's theorem is concerned with correlations defined in terms of averages taken over very many trials of the experiment. The correlation of two binary variables is usually defined in quantum physics as the average of the products of the pairs of measurements.

### References

Note that this is different from the usual definition of correlation in statistics. The quantum physicist's "correlation" is the statistician's "raw uncentered, unnormalized product moment ". The correlation is related in a simple way to the probability of equal outcomes, namely it is equal to twice the probability of equal outcomes, minus one.

Measuring the spin of these entangled particles along anti-parallel directions i. On the other hand, if measurements are performed along parallel directions i. This is in accord with the above stated probabilities of measuring the same result in these two cases.

These basic cases are illustrated in the table below. Columns should be read as examples of pairs of values that could be recorded by Alice and Bob with time increasing going to the right. Experimental results match the curve predicted by quantum mechanics. Over the years, Bell's theorem has undergone a wide variety of experimental tests.

However, various common deficiencies in the testing of the theorem have been identified, including the detection loophole [13] and the communication loophole. In , the first experiment to simultaneously address all of the loopholes was performed. To date, Bell's theorem is generally regarded as supported by a substantial body of evidence and there are few supporters of local hidden variables, though the theorem is continually the subject of study, criticism, and refinement, and the popularity of non-local hidden variable theories such as Many Worlds Theory have been on the rise.

Bell's theorem, derived in his seminal paper titled On the Einstein Podolsky Rosen paradox , [6] has been called, on the assumption that the theory is correct, "the most profound in science". David Mermin has described the appraisals of the importance of Bell's theorem in the physics community as ranging from "indifference" to "wild extravagance".

The title of Bell's seminal article refers to the paper by Einstein, Podolsky and Rosen [22] that challenged the completeness of quantum mechanics. In his paper, Bell started from the same two assumptions as did EPR, namely i reality that microscopic objects have real properties determining the outcomes of quantum mechanical measurements , and ii locality that reality in one location is not influenced by measurements performed simultaneously at a distant location.

## "Quantum Chance: Nonlocality, Teleportation and Other Quantum Marvels"

Bell was able to derive from those two assumptions an important result, namely Bell's inequality. The theoretical and later experimental violation of this inequality implies that at least one of the two assumptions must be false. In two respects Bell's paper was a step forward compared to the EPR paper: firstly, it considered more hidden variables than merely the element of physical reality in the EPR paper; and Bell's inequality was, in part, experimentally testable, thus raising the possibility of testing the local realism hypothesis.

Limitations on such tests to date are noted below. Whereas Bell's paper deals only with deterministic hidden variable theories, Bell's theorem was later generalized to stochastic theories [23] as well, and it was also realised [24] that the theorem is not so much about hidden variables, as about the outcomes of measurements that could have been taken instead of the one actually taken. Existence of these variables is called the assumption of realism, or the assumption of counterfactual definiteness.

After the EPR paper, quantum mechanics was in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of a finite propagation speed of physical effects. In a modified version of the EPR thought experiment, two hypothetical observers , now commonly referred to as Alice and Bob , perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state.

It is the conclusion of EPR that once Alice measures spin in one direction e. In QM, predictions are formulated in terms of probabilities — for example, the probability that an electron will be detected in a particular place, or the probability that its spin is up or down. The idea persisted, however, that the electron in fact has a definite position and spin, and that QM's weakness is its inability to predict those values precisely.

The possibility existed that some unknown theory, such as a hidden variables theory , might be able to predict those quantities exactly, while at the same time also being in complete agreement with the probabilities predicted by QM. If such a hidden variables theory exists, then because the hidden variables are not described by QM the latter would be an incomplete theory.

The concept of local realism is formalized to state, and prove, Bell's theorem and generalizations. A common approach is the following:. The hidden parameter is often thought of as being associated with the source but it can just as well also contain components associated with the two measurement devices.

## Quantum Chance: Nonlocality, Teleportation and Other Quantum Marvels

Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. Assuming local realism, certain constraints must hold on the relationships between the correlations between subsequent measurements of the particles under various possible measurement settings.

Let A and B be as above. Define for the present purposes three correlation functions:. The two-particle spin space is the tensor product of the two-dimensional spin Hilbert spaces of the individual particles. Each individual space is an irreducible representation space of the rotation group SO 3. The product space decomposes as a direct sum of irreducible representations with definite total spins 0 and 1 of dimensions 1 and 3 respectively.

Full details may be found in Clebsch—Gordan decomposition. The total spin zero subspace is spanned by the singlet state in the product space, a vector explicitly given by. One has, by definition of the Pauli matrices ,. The inequality that Bell derived can then be written as: [6]. This inequality is however restricted in its application to the rather special case in which the outcomes on both sides of the experiment are always exactly anticorrelated whenever the analysers are parallel.

The advantage of restricting attention to this special case is the resulting simplicity of the derivation. In experimental work, the inequality is not very useful because it is hard, if not impossible, to create perfect anti-correlation.

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This simple form has an intuitive explanation, however. It is equivalent to the following elementary result from probability theory. Consider three highly correlated, and possibly biased coin-flips X, Y , and Z , with the property that:. Imagine a pair of particles that can be measured at distant locations. Suppose that the measurement devices have settings, which are angles—e.

### Nonlocality, Teleportation and Other Quantum Marvels

The experimenter chooses the directions, one for each particle, separately. Suppose the measurement outcome is binary e. Suppose the two particles are perfectly anti-correlated—in the sense that whenever both measured in the same direction, one gets identically opposite outcomes, when both measured in opposite directions they always give the same outcome.

The only way to imagine how this works is that both particles leave their common source with, somehow, the outcomes they will deliver when measured in any possible direction. How else could particle 1 know how to deliver the same answer as particle 2 when measured in the same direction? They don't know in advance how they are going to be measured The measurement on particle 2 after switching its sign can be thought of as telling us what the same measurement on particle 1 would have given. Start with one setting exactly opposite to the other.