The EPR Paradox

The Einstein–Podolsky–Rosen paradox (EPR paradox) is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) that they interpreted as indicating how the explanation of physical reality provided by quantum mechanics was incomplete. The reason why the EPR thought experiment ends up as a paradox is the reliance of its authors on hypotheses that appear to be universally valid and truthful, yet turn out to be misleading and incomplete. That set of hypotheses is that 1) any physics theory must be rooted in the notion of “objectively existing reality” (any observed particle should be having fixated values that remain such irrespective of any measuring done) and 2) that of locality (the result from the act of measuring performed on a given particle should not be influencing the result from the act of measuring performed on another). The EPR authors were convinced that the quantum-mechanical formulation of the world was incomplete and there must have been some missing element to it that would transform it into a fuller (deterministic) theory.  In that sense, the EPR paradox represents a more fundamental shift from the deterministic to probabilistic view of the world that would go on to make its way beyond the scientific domain (but let’s stick physics for now – putting the puzzle together one piece at a time).

The state in which particles are linked among one another on a quantum level is called quantum entanglement. Thus, the entangled particles are forming a quantum system between them, and the different particles within it – let’s take only a single pair of particles to make matters easy – those can be referred to as subsystem A and B respectively. And within the state of quantum entanglement, the act of measuring subsystem A has an immediate non-local effect on sub-system B. As postulated by quantum mechanics, measuring of one part of the quantum system leads to the collapse of the wave function to this part. However, what appeared as truly paradoxical to the EPR trio was how ‘’the news about the measuring’’ to one of the parts gets carried immediately to the other part of the same quantum system. This effect of immediate non-local connectivity is in direct contradiction with special relativity. The EPR trio was convinced that this state of non-local connectivity – or as Einstein called it “spooky action at a distance” – was not revealing the entire truth about the probable nature of quantum dynamics. Instead, there must have been some “hidden variable” to complete the picture.

Measuring entangled states

Although the original EPR thought experiment involved position and momentum measurements, David Bohm reformulated the EPR paradox into a more practical experiment utilizing spin or polarization measurements. Let’s imagine two quantum entangled electron particles moving apart in opposite directions.

At some distance from their common origin, Alice measures (observes) the spin of one of the particles (electron A), while Bob (observes) measures the spin of the other one (electron B). According to quantum mechanics, two electrons can be “set” while being entangled. That is, they are not initially defined, but instead, form quantum superposition of two states that we can name as State I and State II. In quantum State I, the electron A’s spin is rotating parallel to the z axis in one direction (let’s name this spin as z+), whereas the electron B’s spin is rotating in the other direction (two opposingly rotating spins). In quantum State II, the electron A’s spin would be having a value of z-, whereas electron B’s spin would be having that of z+. It turns out that it is impossible to ascribe a definite spin rotation to either of the two electrons, as they are instead mutually conditional. Such way of pairing is called “singlet state”, which is pretty much equivalent to “quantum entanglement”.

So, if Alice gets to measure the spin of the electron by having the z axis as a frame of reference, she would either obtain z+ or z- as a result. The wave function would then be collapsing into State I. And if Alice obtains z+, Bob would surely be obtaining z-, given that he uses the same frame of reference for his measuring.

But the z axis is not some privileged frame of reference. The same relationship can be illustrated through measurements along different axis of spin. Let’s take the x spinning axis. And let’s say that in a state which we might call Ia Alice obtains x+, while Bob obtains x-. Now let’s reverse the scenario – Alice obtains x-, while Bobby obtains x+. That would be our IIa state. Just as in the case of the z axis, the entangled state is expressed by the superposition formed between the Ia and IIa states. If Alice obtains x+, the system’s wave function would collapse into Ia state – i.e. if Alice obtains x+, Bobby would get x-, and vice versa.

In quantum mechanics, the spin along the x and z axes are “incompatible observables” in relation to one another. This means that the values deriving from rotations around both axes could not possibly be simultaneously obtained within the same quantum condition. If Alice gets to measure along the z axis, she would obtain z+. The wave function would then collapse into I state. But if Bobby wants to actually measure along the x axis, and since the system is being in I state, this means that Bobby would get either x+ or x- at 50% probability each. And nothing more can be said before Bobby has actually measured.

All of that poses the question: How come Bobby’s electron “knows” when Alice will decide to measure her electron? Also, how does it know what the value derived from her measuring is? The Copenhagen interpretation tells us that with the act of measuring and the collapse of the wave function, an immediate effect is being triggered. This means that either the first part of the quantum system transmits an immediate message to the other part of the system, or that the second electrons knows the outcome in advance. Classical causality is being violated in either case.

The electron’s spin was used in the examples above, but the same principles would work with any other physical values – or “observables” as referred to by quantum mechanics – that can be used for setting entangled states.

Realism and completeness

The argumentation of Einstein, Podolsky and Rosen rests on the premises of two fundamental notions: a) element of physical reality and b) theoretical completeness.

The EPR authors do not openly initiate the philosophical debate regarding the meaning behind the concept of “element of physical reality”. Instead, they bring forth the suggestion that if a given physical object has a value independent of any acts of measuring or other interventions, it then follows that this object is an element of physical reality. And even if any elements of physical reality display alternative ways of being, the EPR’s postulation would remain unchanged. In other words, the underlying understanding of the EPR authors is rooted in their conviction that “the independent existence of an objective physical reality” is a statement that is truthful and complete in itself. Therefore, in order for any theory to be complete as well, it should be relating and describing elements of this physical reality.  In short, the notion of “objective physical reality” is fundamentally truthful and unchangeable, and any theory failing to reinforce and reaffirm such notion follows to be incomplete.

 

Let’s see how that set of ideas is being applied to the actual thought experiment and what the outcomes are. Assume that Alice decides to measure the spin along the z axis (let’s call it z-spin for shorter). This act of measuring predetermines the value that Bob shall obtain when measuring the same z-spin – i.e. Bob’s z-spin turns into an element of physical reality only after Alice has done her measuring. Similarly, if Alice decides to measure the x-spin, this would automatically turn Bobby’s measuring of the x-spin into an element of physical reality.

 

As it was previously outlined, the values of the x and z spins cannot be simultaneously identified within the same quantum state. And if quantum mechanics is accepted as a complete theory by the EPR authors, this would mean that the x and z spins cannot simultaneously be elements of physical reality. This means that the Alice’s decision to measure has an immediate effect on Bob’s elements of physical reality. Such outcome violates another principle of essential importance as far as objective physical reality is concerned: locality.

The principle of locality

According to the principle of locality, it is not possible for physical processes taking place within a given section of space to have an immediate effect on elements located within another section in space. This is a sensible postulation as far as the theory of special relativity is concerned. According to the latter, information could not be transferred at a speed greater than the speed of light without this violating the principle of causality. The commonly accepted assumption is that any theory to be violating the principle of causality should be internally contradicting and therefore – unsatisfactory. But as it turns out, when the conventional set of rules gets applied to quantum mechanics, it is the principle of locality that is being violated, whereas the principle of causality remains intact.

Indefinite order of causality

A recent experiment by Austrian researchers (Giulia Rubino et al. 2016) aiming to identify the order of causality in quantum mechanics revealed quantum phenomena is fundamentally defined by an indefinite order of causality.

A brief summary of the experiment: Let’s assume that Alice is sending Bob a present. Or no, let’s make it the other way around – Bob is sending Alice a present. Or actually, they are sending each other presents at the same time. The experiment shows how identifying a definite causality order becomes a confusing task when quantum mechanics are involved. Rubino remarks:

 “When trying to incorporate classical causality into quantum mechanics, this results in a situation where there is no definite order of cause and effect. And this contradicts all logic.’’

 The experiment itself involved shooting a photon through a pair of optical devices marked as Alice and Bob. Those devices would reconfigure the photon’s quantum state in such a way as if it first passes through Alice and then Bob to produce different results, then if it passes through Bob first.

To make things clearer, let’s imagine this photon to be a present intended for a third party. Assume that Alice likes to wrap her presents in paper, while Bob prefers simply putting a ribbon on his. If Alice gets the present first, she would wrap it up, and when it arrives to Bob, he would just tie a ribbon over the readily wrapped-up paper. But if Bob is first, he would then tie his ribbon on, and Alice will then cover the ribbon with her wrapping paper. The final appearance of the present would be different from the first scenario.

 Things get a bit more complicated in the case of the photon, since the optical devices “Alice” and “Bob” can perform various actions with different probabilities, thus there are more than two possible outcomes.

Who's first?

In the experiment, the quantum switch arranges Alice and Bob’s order of action by determining the path of the photon. In order to fuse the causalities of both scenarios, the switch was set into a state of superposition. In this way the researchers tried to put both Alice and Bob at the starting position. Of course, this is not exactly what happened in practice, but words cannot do justice for the weirdness of the quantum world underlying our visible reality.

At the scale of Plank’s length (1.10-33), the vacuum fluctuations are so intense that space is literally boiling and it produces quantum foam. Space appears to be all plain up until the scale of 10-12 , but it begins to get really rugged when approaching the scale of  10-30. The smaller space gets, the more uncertain things become.

 Mateus Araújo – a member of the research team – further remarks: ‘’Time itself appears to be undefined in such situations… The whole confusion with quantum mechanics is due to lack of knowledge and understanding – it is a paradigm which simply does not correspond with out classical macroscopic experience.’’

Matty Hoban of Oxford University adds: ‘’We pushed the real quantum mysteries and messes to the absolute limit…And we have no good idea about what they actually are.’’

Conclusions and takeaways from the EPR paradox

[I can’t accept quantum mechanics because] “I like to think the moon is there even if I am not looking at it.”– Albert Einstein

According to the Copenhagen interpretation of quantum mechanics, until a measurement is made, neither particle has a definite state –the particles are really in a superposition of all possible states. They each exist as a probability cloud. The state of both particles is determined only after one of them has been measured. In its turn, the second particle was configuring its state in accordance with its measured counterpart.

According to the EPR authors, there is an objective reality, which means that the measurements done on a given particle should be independently existing of any other measurements being done. And moreover, further according to them, the result of that measurement should not be affecting any other measurement of any other particle. Einstein’s expression to his intransigency of the way quantum mechanics work lead him to the following version for resolving his own paradox – he supported an alternative approach called the hidden-variables theory, which suggested that quantum mechanics was incomplete. In this viewpoint, there had to be some aspect of quantum mechanics that wasn’t immediately obvious but which needed to be added into the theory to explain the non-local entanglement of particles. The assumption for this hidden variable represents a wider scientific discourse at the time for having definitions of “physical reality” or “matter” as steady and credible points of examination. This means rejecting the notion of immateriality – the ether, the aether, the nothingness. At the same time, the negation of matter has been classified as “anti-matter”. In the EPR trio’s view, one theory is completed only when it is completely and fully articulated solely through elements of physical reality. Such paradigm allows for the elimination and suppression of discoveries made by the likes of David Bohm, Nikola Tesla, Louis de Broglie and others.

The EPR paradox is quite illustrative of how uncertainty in quantum mechanics doesn’t just represent a lack of knowledge but a fundamental lack of definite and objective material reality. As the EPR paradox shows us, having the notion of objective reality as a starting point going into a thought experiment involving quantum entanglement, brings the objectively existing reality as such to a dead-end. And this paradox may be reconciled either through accepting objective reality as non-existent, or through dismissing quantum mechanics as incomplete (which was the EPR’s way out). In more philosophical terms, the steadiness and certainty of physical reality is primarily incompatible with the elusiveness and uncertainty of quantum mechanics. However, the actual reconciliation of the EPR paradox has resulted in the alternative excluding the existence of objective reality. Unlike the nature of the quantum phenomena, the actual experimentation work done for supporting the math behind quantum mechanics was unambiguous. The major nail in the coffin of the hidden-variables theory came from the physicist John Stewart Bell, in what is known as Bell’s Theorem. He developed a series of inequalities (called Bell inequalities), which represent how measurements of the spin of Particle A and Particle B would distribute if they weren’t entangled. In experiment after experiment, the Bell inequalities are violated, meaning that quantum entanglement does seem to take place.

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