A bit of maths and physics

Tag: locality

  • Quantum entanglement: the EPR paradox and Bell’s Theorem

    Quantum entanglement: the EPR paradox and Bell’s Theorem

    When the state of a subatomic particle cannot be described by a wave function without taking the state of another subatomic particle into account, we speak of quantum entanglement. It’s the special case where both particles can only be described by one and the same wave function. No longer are they separate entities nor do they have separate wave functions. The astonishing consequence is that performing a measurement on one particle has an immediate effect on the measurement of the other particle, no matter how far apart they are from each other. In this article, the second part of our mini-series on quantum entanglement, we will discuss the EPR paradox which Einstein and colleagues put forward. After that, we will discuss Bell’s Theorem which allowed physicists to test Einstein’s proposal. Was Einstein correct?

    A representation of an electron’s spin – do note that this is not what an electron actually looks like nor is it what its spin looks like. The quantum world is simply too strange to depict accurately using ‘classical’ notions as done here. Here we drew a vague ball-like thing which seemingly spins around, which it isn’t and it doesn’t. But it’s the best we’ve got. Although, the best we’ve got is actually something else: a mathematical expression, the wave function.

    Quick summary

    Firstly, let me give a quick summary of the previous post:

    1. we used the property of spin as a way of distinguishing between the two entangled electrons;

    2. the orientation of an electron’s spin is expressed as spin up (anticlockwise) or spin down (clockwise) along the axis of measurement;

    3. you can arbitrarily choose along which axis you want to measure its spin, in three dimensions;

    4. no matter which axis you choose, the result is always going to be a spin up or spin down (there is no spin-a-bit-to-the-right, for instance);

    5. we are able to entangle particles in such a way that they will either always yield opposite spin or they always yield identical spin; once prepared this way, they will never deviate from this correlation when measured;

    6. we used the opposite-spin entanglement in our example and we will do so again here;

    7. quantum mechanics states that before measurement neither electrons have a specific spin: the wave function contains all possible measurement outcomes, in this case pertaining to both spin up and spin down (which can be characterised as having no definite spin yet)(beginfootnote)Analogously, the double-slit experiment showed that before measurement, particles don’t have a specific location yet.(endfootnote);

    8. as soon as you measure one electron’s spin along a certain axis, the other electron’s spin immediately snaps to the opposite orientation along that same axis, regardless of spatial distance between the two entangled particles(beginfootnote)Or, if their entanglement were prepared in such a way that they always have identical spin, the other electron would then immediately snap to the identical spin orientation along the same axis of measurement.(endfootnote).

    EPR paradox

    Even though Einstein understood quantum mechanics like few others, and while accepting these predictions and results, he didn’t quite like the non-local implications brought forth by quantum entanglement. He didn’t like point 8 of the previous section. There seems to be zero time delay between influencing a particle in Amsterdam (through measuring its spin) and influencing its entangled particle in Boston. It violates a pivotal consequence of Einstein’s theory of special relativity: no signal or piece of information – anything within this universe, really – can exceed the speed light(beginfootnote)In a vacuum.(endfootnote) or else causality would not exist. In other words, if information or signals were able to travel faster than light, an effect could occur before its cause had taken place. To put it mildly, this doesn’t seem to be the universe you and I are living in.

    So, Einstein, Podolsky, and Rosen (EPR) hypothesised that something else, something secretive was going on in nature – well out of sight for theoretical and experimental physicists. Quantum mechanics as it was known then had to be incomplete. Obviously, they acknowledged its successes, but when it came to quantum entanglement, they asserted something was missing in the theory of describing nature through wave functions.

    To solve for the seemingly faster-than-light signal, they proposed that what really was going on was that the particles have always been in a specific state. When the electron pair were separated from each other, they have always had either spin up or spin down from the start from the moment of their creation.

    Suppose, a pair of gloves were made. Like all pairs of gloves, they always were each other’s opposite with respect to ‘handedness’(beginfootnote)‘Handedness’ in this context is a form of the more generalised term chirality.(endfootnote). One has always been left-handed, the other has always been right-handed. And if the first one happened to be right-handed, then the other was left-handed. (Or else you’re holding a glove from another pair.)

    Suppose, the machine which had made the pair put each glove in a separate box. We can’t see which glove went in which box until we open the box. The boxes were sent to Amsterdam and Boston. The experimental physicists then open the box in Amsterdam: it’s the right-handed one! And so, we now instantly know, the one in Boston is left-handed. No magic, no non-locality, no lightspeed-breaking shenanigans.

    This is what Einstein and friends said was happening in the case of electrons. An electron pair always had specific spins to start with. It’s only in Amsterdam and Boston that we ‘open the box’ aka measure their spin. It’s only logical now that as soon as you know which spin the Amsterdam electron has, you immediately know which spin the Boston electron has.

    So, said Einstein, non-locality is an illusion. It’s all just normal local laws of nature and a bit of logical thinking. For one, spin orientation is merely hidden from us and not principally uncertain. Secondly, there’s no spooky action at a distance[1], as he famously described it(beginfootnote)In German, he wrote ‘spukhafte Fernwirkung'[1].(endfootnote).

    In everyday parlance, physicists call this a local version of the ‘hidden variables’ theory. ‘Hidden variables’ pertain to the stuff that we can’t see yet (such as spin orientation or other variables influencing this) because our quantum mechanical description (the wave function) is incomplete, however, they are there, they do exist – they do not not exist yet, according to the hidden variables theory.

    Bell’s inequalities

    Unfortunately, Albert Einstein passed away in 1955. And Niels Bohr, the other great physicist with whom he used to debate the fundamental nature of quantum mechanics passed away in 1962. In both cases too soon for them to be able to read John Stuart Bell’s 1964 paper called ‘On the Einstein Podolsky Rosen Paradox'[2]. Bell realised that Einstein’s proposal was in principle testable. It yielded a clear prediction, called Bell’s inequality.

    At this point, we must note that over the years, more than one Bell’s inequalities have been put forward by physicists(beginfootnote)Besides his original inequality, there’s the much-used CHSH-inequality, for instance.(endfootnote). To explain Bell’s inequality, we will apply a version of David Mermin’s original version as mentioned in his fantastic Boojums All the Way Through: Communicating Science in a Prosaic Age[3].

    Recall from point 3 before that we can measure an electron’s spin orientation along any axis. We’re going to be measuring along three axes. These axes will be at an angle of 120° relative to each other.

    The first axis will be the spin orientation along the vertical axis, which we will denote with the following symbols for spin up and spin down:

    $$\uparrow \downarrow$$

    The spin orientations up and down will also be measured along this second axis:

    $$\nwarrow \searrow$$

    And the spin orientations along the third axis will be denoted by:

    $$\nearrow \swarrow$$

    So, imagine two entangled electrons being separated in space from each other. The usual quantum-mechanical description of each electron is that they are in a superposition of spins up and spins down for all three axes.

    Except, Einstein says, no, no, not really: hidden behind the ‘veil of superposition’ they are in fact already in definite, specific spin orientations for each of the three axes. We just don’t yet know which until we measure them!

    He says, the electron in Amsterdam may already be in the specific spin states as follows:

    $$\left( \uparrow \searrow \swarrow \right)_A$$

    So, along axis 1 it’s spin up, along axis 2 it’s spin down, and along axis 3 it’s also spin down.

    Einstein continues and says that the entangled electron in Boston has to already be in the opposite states:

    $$\left( \downarrow \nwarrow \nearrow \right)_B$$

    And so, Einstein concludes, as soon as you actually perform a measurement in Amsterdam along the first axis, of course, you get the opposite spin in Boston. Only logical!

    Bell’s insight was that if you would work out this entire argument for all possible combinations, you could actually get a prediction of a ratio of outcomes. Here’s how that goes.

    First of all, if you measure along axis 1 in Amsterdam, that doesn’t mean you have to measure along that same axis in Boston. You could just choose to measure along axis 3. So, with the two examples above, your results would simply be that in Amsterdam you get spin up and in Boston you also get spin up:

    $$\left( \uparrow \right)_A \text{ and } \left( \nearrow \right)_B$$

    Bell then argued, if you would count the number of times you would get the combinations up-up, down-down, and of course up-down and down-up like this, you should get ratios of these combinations which should match experiment. If, however, these ratios don’t appear in the experiments, then Einstein’s hypothesis is incorrect. In that case, something entirely different is going on. The electrons were not already in a specific state, which in turn means that the non-local measurement effect in quantum entanglement does exist!

    Bell’s theorem

    So, let’s put them all together. Let’s first take our example above:

    $$\left( \uparrow \searrow \swarrow \right)_A \text{ and } \left( \downarrow \nwarrow \nearrow \right)_B$$

    If you measure along axis 1 in Amsterdam and along axis 1 in Boston you get spin up, spin down. If you measure along axis 1 in Amsterdam and along 2 in Boston, you get spin up, spin up. And so on, and so forth! We’ve put it in a little table:

    Here you can see all the possible combinations of measurement outcomes along the three possible axes of the electrons in Amsterdam (A) and Boston (B). We used U for spin up and D for spin down.

    Bell then says that if Einstein was correct, and the states of the spin orientations along these three axes were already there, then these are the expected outcomes.

    Let’s focus on the number of UD or DU combinations, in other words, let’s focus on the number of times we find the opposite spin orientations, irrespective of the axes along which they are measured. We’ve marked them yellow.

    Exactly five out nine times you will find the opposite spin directions.

    Let’s check for other spin combinations. Suppose, the electron in Amsterdam is secretly in the following spin states, $\left( \downarrow \nwarrow \swarrow \right)_A$, and the electron in Boston is then the opposite, $\left( \uparrow \searrow \nearrow \right)_B$. If we count again the number of times the measurement outcome of opposite spins, we get, again, five out of nine.

    Okay, I think you can imagine where this is going. We’re not going to go by all the tables, but I do want to do one more, just for fun. Suppose, the one in Amsterdam is all spin down, $\left( \downarrow \searrow \swarrow \right)_A$, and, obviously, the Boston one is its opposite, $\left( \uparrow \nwarrow \nearrow \right)_B$. In that case, we would get opposite spins in nine out of nine times.

    And so, this particular Bell inequality states that the probability (P) of finding opposite spins along all three axes is at least $\frac{5}{9}$ or 55% (and at most 1 or 100%). In other words, $P(\text{opposite}) \geq \frac{5}{9}$. If this inequality were violated by experiment, the underlying theory will have been proven to be incorrect.

    Experimental outcomes

    Over the past thirty years, many experiments were carried out to test multiple versions of Bell’s inequality. Usually, these tests involved photons rather than electrons and pertained to measurement of polarisation rather than spin.

    Freedman and Clauser did the first Bell test. They used a version of the so-called CH74 inequality[4].

    The most well-known test was performed by Alain Aspect and colleagues. As Bell had originally suggested, they were able to have the two measurement devices randomly select the method of measurement before the entangled photons had arrived[5].

    In all tests, all versions of Bell’s inequalities were violated. Instead, the statistical outcome was congruent with the predictions of quantum mechanics. The conclusion has to be that Einstein’s local hidden variable theory was incorrect. There is nothing local about measuring entangled particles.

    In our particular inequality, the result was that the occurrence of opposite spins turned out to be exactly 50%, not 55%.

    Conclusions

    Let’s summarise what we have established over the course of the last two posts, including this one.

    In quantum mechanics, particles which have not been measured yet don’t have a definite, specific state. Instead, they are best described by a wave function which incorporates all the possible future states it can snap into once measured.

    When a particle can only be described in tandem with another particle, i.e. both particles can only be described by one and the same wave function, they are maximally quantum entangled(beginfootnote)In practice, in the real world, particles aren’t maximally entangled like the way we can prepare them in the laboratory. The world is too messy for those ‘pure states of entanglement’ to exist for any significant amount of time. There are simply too many particles around to not interact with any other particle. Every particle will invariable interact with thousands of trillions of other particles and so any previous entanglement will quickly decohere into either a very weak version of the original entanglement or simply to zero entanglement. Every interaction represents a measurement. Since our brains are too large and consist of thousands of trillions of particles, they will never be in a pure state of superposition nor entanglement. Not to mention our much larger body, which will never be in any sort of quantum state. It is statistically so unlikely that you’d have to become as old as $(10^{100})^{100}$ times the age of our current universe to witness such an event. And that number was a metaphorical one. It’s much larger.(endfootnote).

    If their entanglement entails their spins will always correlate in a certain way – be it identical spins or opposite spins – a measurement on one particle, causing it to snap into one of the possible, specific, definite states, has immediate effect on the state of the other particle: it instantly snaps out of its wave function haze into a correlating, specific, definite state.

    Einstein didn’t like this as this would imply some kind of information was somehow transported beyond the speed of light from one particle to the other.

    He postulated that particles have always been in a specific, definite state to begin with. The only reason we don’t know which is because we haven’t measured it yet. There is no ‘snapping out of the haze’ going on.

    John Bell showed that Einstein’s hypothesis can be tested. If you would perform many, many measurements of many, many maximally entangled particles, eventually, the occurrences of the variety of correlated states should show up in a certain ratio, an inequality, as it happens.

    Experiments showed they do not. Instead, the ratio is exactly according to the predictions of quantum mechanics.

    This demonstrated that particles indeed snap out of their haze upon measurement and not that particles had always been in a hidden but definite state.

    And if that is true, then non-locality has to be true – there is no other way the other particle snaps into the correct, correlated state.

    Nobody knows how this happens. Certain non-local but still hidden-variables hypotheses have been proposed. One of the more famous versions is called the ER=EPR conjecture by Juan Maldacena and Leonard Susskind. Perhaps we’ll dive into that later on.

    Einstein’s aversion to this ‘particles have no definite state until measured upon’ made him utter his famous complaint, ‘God does not play dice’.

    Unfortunately, he was wrong here on two occasions. God(beginfootnote)We are using the word ‘God’ in a purely metaphorical way. This does not pertain to any specific religious entity as revered by many in a variety of societies in human culture.(endfootnote) does play dice. Moreover, He throws them where we can’t see them. Even God seems to be bound by Heisenberg’s Uncertainty Principle. But that’s a subject for another bit of maths and physics.

    [1] Einstein, A., Podolsky, B. and Rosen, N. (1935) “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?,” Physical Review, 47(10), pp. 777–780. doi: 10.1103/PhysRev.47.777.

    [2] Bell, J. S. (1964) “On the Einstein Podolsky Rosen Paradox,” Physics Physique Fizika, 1(3), pp. 195–200. doi: 10.1103/PhysicsPhysiqueFizika.1.195.

    [3] Mermin, N. D. (1990) Boojums all the way through : communicating science in a prosaic age. Cambridge England: Cambridge University Press.

    [4] Fry, E. S. and Thompson, R. C. (1976) “Experimental Test of Local Hidden-Variable Theories,” Physical Review Letters, 37(8), pp. 465–468. doi: 10.1103/PhysRevLett.37.465.

    [5] Aspect, A., Dalibard, J. and Roger Gérard (1982) “Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers,” Physical Review Letters, 49(25), pp. 1804–1807. doi: 10.1103/PhysRevLett.49.1804.

    Featured image: Portrait of theoretical physicist John Bell at CERN, June 1982 (CERN, CC BY 4.0)

  • Quantum entanglement: non-locality and the state of a two-particle system

    Quantum entanglement: non-locality and the state of a two-particle system

    To this day, quantum entanglement and its effects are phenomena which still leave physicists scratching their heads when trying to get a deeper understanding of what is actually happening. This series on quantum entanglement is going to be a two-parter. In this post, we will discuss what is meant by locality and non-locality and what quantum entanglement is. The term quantum entanglement has been used in many instances of popular culture pertaining to spirituality, healing, and a flurry of new age approaches to human consciousness. This is not the kind of ‘quantum entanglement’ we will discuss here. We will purely look at the physics of it, its original and proper meaning. We will study the state of a two-particle system. In the next post, we will discuss what Einstein and his friends proposed, what Bell wrote, and whether Einstein was right. And then there are also exciting caveats which we will explore.

    The basics

    Let’s go over the basics one more time. ‘Particles’ aren’t particles in the classical sense at all – they’re absolutely not like tiny balls or pellets. They are best described by the wave function, a mathematical expression containing all possible states the particle can be in. This pertains to its energy levels, its positions or a number of other properties it can have.

    As long as no measurements have been performed on it, the particle has no definite state or states. It displays wave-like behaviour like being caught in a haze of all possible states. However, as soon as you measure it, the particle will snap out of its haze and it will appear to be a particle, an actual particle in the classical sense, with a definite state.

    Note that ‘the state of an electron’ can refer to a particle with no definite set of states when no measurement was performed. The state of an electron is then best described by the wave function, which contains all possible definite states upon measurement.

    Hereafter, ‘wave function’ and ‘state’ are used interchangeably.

    In This is not an atom, the wave function is discussed. In The double-slit experiment, the wave-like and the particle-like behaviours are showcased.

    Locality vs non-locality

    Isaac Newton knew he had a problem when he formulated his theory of gravity. While it beautifully described the extent to which two masses exert gravitational forces upon each other, his theory didn’t explain how they did that. He didn’t like the conclusion that the gravitational influence between Earth and the Moon seemed to spookily operate at a distance through the vacuum. He wrote it was ‘so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it’. He famously stated to leave this unsolved mystery to ‘the Consideration of my readers’[1].

    In other words, Newton wasn’t big on non-locality. And yet, his own theory did entail an invisible force operating over vast distances through the vacuum. Moreover, it seemed to be an instantaneous effect: if the Sun were to suddenly disappear, then Earth would be flung off its trajectory immediately. Of course, today, we know that nothing can travel faster than light, so the gravitational changes of the Sun would take about eight minutes to ‘reach’ Earth.

    The following years, physical phenomena such as magnetism and electricity proved, in fact, to be very local indeed. It became clear there is always an indirect way through which one object is able to influence another object at a distance. What is meant with locality? Here’s the mechanism: an object interacts with its immediate environment, a field embedded within the three-dimensional space we live in, i.e. the electromagnetic field, which then passes on that ripple of disturbance onto the other object. In terms of ‘fields’, one could say that at one particular location the field’s value is changed by some object. That value change then changes the values of the field in the direct vicinity, which then change the values in their vicinity, and so on. It’s a bit like ‘the wave’ done by thousands of sports fans in a stadium. Or like falling dominoes. Every change is ever local and the propagation of that change through space is limited to the speed of light.

    Tumbling telephone boxes are definitely a ‘local phenomenon’. The sculpture Out of Order by David Mach is situated in Kingston upon Thames (UK). Photo by 272447.

    Many years later, Einstein replaced Newton’s theory with his own theory of gravity, General Relativity (GR). It showed that Newton’s intuition was correct. Gravity couldn’t be non-local and Einstein showed it isn’t. In GR, space and time itself are the stretchy substance through which gravitational disturbances propagate at the speed of light towards the other object. When a mass curves or disturbs spacetime around it, that curvature or disturbance then ripples through the universe, on its way to influence other objects. In fact, on 11 February 2016, a large collaboration of incredibly talented scientists physically measured these gravitational ripples in spacetime as predicted by Einstein in 1916. It won three key figures the Nobel Prize.

    And so, it seems there is no spooky influence at a distance in physics. Even still to this day, in modern quantum physics, our best understanding and most successful theory is that quantum fields pervade our universe, forming the mediums through which forces are propagated, limited by the speed of light.

    Non-locality entails a change in one patch of space instantaneously influencing another patch of space irrespective of their distance. Locality entails the propagation of change through space by influencing only neighbouring patches of space at a maximum of the speed of light.

    Spin

    Electrons have several properties. One of the more obvious is (negative) charge. The Stern-Gerlach experiments showed that they possess another property which was given the name spin angular momentum or simply spin for short, for lack of a better term as electrons aren’t exactly like spinning balls.

    Nevertheless, as it stands, electrons have an intrinsic spin, which cannot in any sensible way be described like a classical-mechanical rotation. Like with any object in three-dimensional space, you can measure its spin along any angle within 360 degrees in three dimensions. With respect to whichever axis you choose, they can only ever spin clockwise or anticlockwise(beginfootnote)Yes, this does sound like there is an actual rotation around an axis in the classical sense. And maybe, in some deep sense, there is after all, however, this deserves a post of its own, so suffice to say for now, our language is simply too limited to avoid using classical terms for quantum mechanical phenomena, misleadingly.(endfootnote). The latter is called spin up and the former spin down, according to the right-hand rule.

    If electrons were like tiny, fluffy balls such as displayed here, you could picture their spin as an anticlockwise or clockwise rotation about the axis of measurement. Using the right-hand rule, we can designate this spin-up or spin-down. Of course, in three dimensions, any axis of measurement at any angle can be chosen with respect to which it will be found spinning. Disclaimer: this classical-mechanical illustration does not portray actual electrons nor actual quantum mechanical spins. But it’s perhaps useful as a simile. (Illustration by KJ Runia)

    Symbols

    As we take our readers seriously, we’ll take this opportunity to introduce a few mathematical symbols which will prove to come in handy at later stages of this series.

    Let’s use the symbol $\lvert A \rangle$ to denote the state of the electron in Amsterdam with respect to its spin. As long as we haven’t performed any measurements on the electron, it has no definite state. However, upon measurement, its spin with respect to the vertical axis of measurement is ever either spin up or down. Let’s write these two possible measurement outcomes as $\lvert\uparrow\rangle_A$ or $\lvert\downarrow\rangle_A$.

    Likewise, if the state of an electron in Boston $\lvert B \rangle$ is spin up or spin down, we write $\lvert\uparrow\rangle_B$ or $\lvert\downarrow\rangle_B$.

    Assuming the state of the electron in Amsterdam hasn’t been measured yet, we can express this (with respect to spin) as a combination of both spin states:

    $$\lvert A \rangle = \alpha \lvert\uparrow\rangle_A + \beta \lvert\downarrow\rangle_A .$$

    This is why physicists often poetically say that the unmeasured particle is in a state of both spins at the same time while it’s more accurate to say it has no definite state. Mathematically, its state is an amalgam of all possible, linearly superposed (added together), algebraic solutions to the Schrödinger equation, hence, it’s said to be in quantum superposition.

    What’s that $\alpha$ and $\beta$, you ask? Well, they’re numbers of probability we need to find in order to complete our expression. The Born rule states that if we square the (modulus of the) wave function (the state), we will get the probability (density) of either possible outcome after measurement. Now, experiments have shown that either outcome, spin up or spin down, $\lvert\uparrow\rangle_A$ or $\lvert\downarrow\rangle_A$, appears in 50% of the total number of measurements. In other words, the probability of measuring either spin state is exactly $\frac{1}{2}$. So, if we put $\alpha=\beta=\frac{1}{\sqrt{2}}$, then $\lvert\alpha\rvert^2 = \lvert\beta\rvert^2 = \frac{1}{2}$. After all, $(\frac{1}{\sqrt{2}})^2 = \frac{1}{2}$, which is exactly what we want. So, the state (wave function) of our Amsterdam electron with respect to spin can be represented by

    $$\lvert A \rangle = \frac{1}{\sqrt{2}} \lvert \uparrow\rangle_A +\frac{1}{\sqrt{2}} \lvert \downarrow\rangle_A .$$

    Similarly, the state of the electron in Boston with respect to spin is then represented by

    $$\lvert B \rangle = \frac{1}{\sqrt{2}} \lvert \uparrow\rangle_B +\frac{1}{\sqrt{2}} \lvert \downarrow\rangle_B .$$

    What you need to take from this is the following: the state of an electron before measurement is the sum of all possible states (multiplied by a probability factor, in this case $\frac{1}{\sqrt{2}}$).

    In the case of spin as measured along the vertical axis, the state of the electron is the sum of two possible states, spin up $\lvert \uparrow \rangle$ or spin down $\lvert \downarrow \rangle$.

    Note that there are other possibilities: we could measure the spin along a horizontal axis. We could represent this with spin left $\lvert \leftarrow \rangle$ or spin right $\lvert \rightarrow \rangle$. Or we could measure the spin at angles of +120 or -120 degrees from the vertical axis, which we might represent as $\lvert \nwarrow \rangle$ and $\lvert \searrow \rangle$ or $\lvert \nearrow \rangle$ and $\lvert \swarrow \rangle$. We will get to that in the discussion of Bell’s Theorem in the next post.

    Quantum entanglement

    So, what is quantum entanglement? Recall that the most complete description of a particle is the wave function. This has always been about a free, single particle, not interacting with anything. In the case of quantum entanglement, however, this doesn’t fly anymore.

    When the state of a particle can no longer be described without a description of the state of another particle, those two particles are said to be quantum entangled. No longer can we describe either particle by one wave function each. They can only be described as a two-particle system by one and the same wave function.

    This has an astonishing consequence. Suppose our two electrons become entangled in such a way that they always have opposite spins(beginfootnote)Producing spin-entangled electrons is difficult but clever experimental physicists have their ways.(endfootnote). So, if one has ‘spin up’, $\lvert \uparrow \rangle$, the other always has ‘spin down’, $\lvert \downarrow \rangle$, or vice versa(beginfootnote)It’s also possible to have them correlate such that they have identical spin, but for our example, let’s not.(endfootnote). So, we now have one system with two particles who always have opposite spins, which means that the total spin of our system is 0, zero. Let’s denote the total spin of our system with $\lvert S \rangle$.

    Before our experiment takes place, they are both separated. One is staying in a laboratory in Amsterdam. The other is transported to Boston. Since no measurement has taken place on either particle, they are in a superposition according to the one wave function. They haven’t an exact location (although one is very likely to be somewhere in Amsterdam at the moment of measurement and, likewise, the other in Boston), their energy levels are all over the place, and their spin isn’t either spin up or spin down along this or that axis.

    We can represent this whole situation with respect to spins as follows:

    $$\lvert S \rangle = \dfrac{1}{\sqrt{2}} \left( \lvert \uparrow \rangle_A \lvert \downarrow \rangle_B – \lvert \downarrow \rangle_A \lvert \uparrow \rangle_B \right) .$$

    When you’re looking carefully at the expression above, you can see that the state of the total spin $\lvert S \rangle$ of our two-particle system is a combination of two situations: the electron in Amsterdam is spin up and so the electron in Boston is spin down or the electron in Amsterdam is spin down and the electron in Boston is spin up. They need to be subtracted from each other because the total spin equals 0, remember? Hence, the minus sign. Lastly, both states are multiplied by the fraction $\frac{1}{\sqrt{2}}$ because both states have a 50% chance of occurring (which you get if you square the whole thing).

    And so, what does this mean? As soon as you perform measurements on the one in Amsterdam, and you find it has spin up, the other electron in Boston immediately has spin down along that particular axis upon measurement, even though the probability before measurement was still 50%! How does the electron in Boston ‘know’ what the measurement result in Amsterdam was? En how does it know this so fast? Faster than the speed of light! Besides this, turns out, you’ll always get a definite spin from the other particle opposite to the one you measured first. As soon as the measurement in Amsterdam took place, the measurement outcome in Boston being the opposite result is always 100% all of a sudden! (Or the other way around.) There are never any exceptions!

    In other words, as soon as you do the measurement, the mathematical description changes from

    $$\lvert S \rangle = \dfrac{1}{\sqrt{2}} \left( \lvert \uparrow \rangle_A \lvert \downarrow \rangle_B – \lvert \downarrow \rangle_A \lvert \uparrow \rangle_B \right) ,$$

    to either

    $$\lvert S \rangle = \lvert \uparrow \rangle_A \lvert \downarrow \rangle_B ,$$

    meaning, the state of the total spin equals the one in Amsterdam being spin up and the one in Boston being spin down, or, vice versa:

    $$\lvert S \rangle = \lvert \downarrow \rangle_A \lvert \uparrow \rangle_B .$$

    And here’s the astonishing part: this will always work this way, no matter how great the physical distance between the two particles. Locality out the window. Welcome back, non-locality.

    Einstein accepted this prediction in quantum mechanics as being correct. However, he didn’t like it. How did the other particle instantly ‘know’ which spin to exhibit when Einstein’s fantastically successful theories of relativity relied on the universal law that nothing can exceed the speed of light? He accepted the theory but he concluded it wasn’t complete. There had to be some sort of hidden mechanism which they had overlooked.

    We will discuss Einstein’s attempt at saving the principle of locality and the universal speed limit in the next post. As well as John Bell’s and Alain Aspect’s subsequent work. For now, the question of whether Einstein was right, we will ‘leave up to the Consideration of our readers.’

    [1] Newton, I. (1756) Four Letters from Sir Isaac Newton to Doctor Bentley: Containing Some Arguments in Proof of a Deity [Online]. Available here. (Accessed: 14 May 2020)

    Featured image by KJ Runia