Quantum Entanglement

Quantum entanglement is one of the most striking differences between the behaviour of the universe described by quantum theory, and that given by classical physics. If two physical systems interact then, even if they later separate, their future evolutions can no longer be considered purely in isolation. Any attempt to describe the systems with classical logic leads inevitably to an apparent link between them, where simply observing one instantaneously impacts the state of the other. This effect remains, regardless of how far apart the systems become.

As it is a very famous quantum phenomenon, a lot has been written about entanglement in both the scientific and popular literature. However, it does still seem to be frequently misunderstood, with many surrounding misconceptions. I will attempt to explain the effects of entanglement in as straightforward a way as possible, with some very basic thought experiments. These can be followed without any understanding of what physical processes may be going on underneath. They only involve pressing a button on a box and observing the colour of a light bulb mounted on it. In fact, this is one of the features of quantum entanglement. It does not matter how you describe the physical world, whether you think of things as particles, waves, or whatever. Entanglement is an observable property independently of how, or even if, we try to describe the physical processes.

As this entry is rather long, I include a list of contents:

The EPR Experiment
Black Boxes
Experiment 1: GHZ Boxes
Can the Boxes Transfer Information?
Constructing the Boxes Classically
Experiment 2: Mermin’s Boxes
Experiment 3: Alice and Bob’s Impossible Boxes
Experiment 3 Redux: Bell’s Boxes
Experiment 4: Magic Squares
Summary
The Uncertainty Principle
Blender
References

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The EPR Experiment

One of the earliest descriptions of quantum entanglement in literature was written in 1935 by the physicists Einsten, Podolsky and Rosen, and is known as the EPR paper¹¹. This was quickly followed by another paper in the same year, by Erwin Schrödinger²¹, coining the term ‘entanglement’, which he called the characteristic trait of quantum mechanics. These physicists were all sceptical of the idea, with Einstein in particular, famously deriding it as spooky action at a distance. At the time, entanglement was a feature of ‘orthodox’ quantum mechanics, but had not been explicitly observed in nature, nor had experiments been proposed to resolve this issue. So, the reservations of some physicists in considering it to be real would seem justified. Instead, it was suggested that quantum mechanics is incomplete, and that so-called hidden variables models may be able do away with entanglement as a real phenomenon. Responses to the EPR paper, led by Niels Bohr⁸, went some way to resolving the questions raised as far as most working physicists were concerned. They did not settle the issue though, and did not definitively answer whether entanglement is a real property of the universe. It was not until 1964 when John Bell published his seminal paper, On the Einstein Podolsky Rosen Paradox⁴, that things changed. This showed that quantum entanglement has directly observable physical consequences. These have been seen in practical experiments known as Bell tests²⁴, such as those performed by Alain Aspect in 1980-82^2,22, so establish quantum theory as something entirely new and distinct from the classical understanding of the universe.

I now mention the issues raised by the EPR paper. This is just for the historical interest, before describing the much more convincing thought experiments that were promised, and can be skipped entirely if preferred. Although it originally looked at measurements of the position and momentum of a pair of particles, the EPR thought experiment was reformulated in 1951 by David Bohm⁷ in terms of their spin. As an intrinsic component of its total angular momentum, the spin of a particle can be measured with respect to any fixed axis. For example, as in a Stern-Gerlach experiment³², it can be measured by observing how much the particle is deflected as it passes through a non-uniform magnetic field. For spin 1/2 particles, which have the smallest possible nonzero spin, the component measured in any one direction can only give one of two values. This includes electrons, positrons, protons, and many atoms. See figure 1 above, which shows the simultaneous measurement of the spin of a pair of particles. According to the direction in which they are deflected, they are measured to be in either a spin up or spin down state. By rotating the magnets, we can choose whether to measure the spin along a vertical or horizontal axis. Now, by conservation of angular momentum, it is possible for particle pairs to created in a state with total spin equal to zero. Whichever axis we choose for the measurement, if one is in a spin up state, then the other will be discovered to be spin down. The box in figure 1 inscribed ‘EPRB’ is creating such particle pairs, and ejecting them along opposite directions. How they are created is not important to the argument. It could, for example, be done by the spontaneous creation of electron-positron pairs, or by the dissociation of a molecule into a pair of atoms. What matters is that the spin of each particle is completely random. It will always have exactly equal chance to be measured spin up as spin down.

Quantum theory describes the spin of the particles as a superposition of up and down states. It is only when measuring that it is forced to collapse to one or the other. This is where entanglement starts to enter the picture. Look again at the experiment in figure 1. Both particles are originally in a superposition. When we measure the particle on the left to be spin up, it collapses into this state. However, to conserve angular momentum, the particle on the right must simultaneously collapse into the spin down state. It is as if there is some instantaneous link between the two particles, even though they could be far apart by the time measurement is made.

The entanglement so far described for the EPR-Bohm experiment is nothing special. If this was all there is, then it would not be interesting or impressive at all. Consider taking a pair of shoes, putting each in a separate parcel, and shuffling so that it is not possible to tell which parcel contains which shoe. Then mail one parcel to Alice, who resides in Vladivostok, and the other to Bob in Timbuktu. When Alice receives her parcel, she has no idea whether it contains the left or right shoe. Similarly, the best she can say about Bob’s parcel, is that it has 50% probability of containing a left shoe and 50% of containing the right. However, the instant she opens her parcel and sees the left shoe, she immediately knows that Bob’s parcel, across the other side of the world, contains a right shoe. Did Alice opening her parcel cause a sudden influence to rush around the world to Timbuktu and collapse the probability distribution of Bob’s parcel to be 100% that it contains a right shoe? Of course not! Bob’s parcel always contained the right shoe, all that happened is that Alice gained this knowledge. There is nothing interesting happening here. I only mention it in case you were thinking that quantum entanglement is really just this kind of thing, and are not impressed. It is not! In fact John Bell also gave a similar description of what quantum entanglement isn’t, involving the socks of his colleague Reinhold Bertlmann which, apparently, are always different colours⁵.

Where the EPR thought experiment appears to either conflict with quantum theory, or predict Einstein’s spooky action at a distance, is when we consider it in conjunction with the Heisenberg uncertainty principle. This states that the spin of a particle can never be known simultaneously along the vertical and horizontal axes. If we know that, along the vertical axis it is either spin up or spin down then, along the horizontal axis, it will be completely random. In fact, the spin states along the vertical axis are the same thing as certain superpositions of states with respect to the horizontal axis. We might consider that the situation with EPR is the same as with the shoes, and the particles were always in a definite spin state from the start, just that we do not which. The measurement simply reveals this knowledge. However, if that was the case for spin along the vertical axis, we could simply rotate by 90 degrees, and the uncertainty principle says that they are then in a true superposition, so that measurement of the horizontal spin component of one particle again instantaneously collapses the state for them both. We can even mix this up, and measure the spin of the left particle along the vertical axis, and the right one horizontally. As the measurement of the particle on the right also determines the horizontal spin of the one on the left, since they are opposite, it would seem to simultaneously determine both the horizontal and vertical spin components of the particle on the left. This would be in contradiction with the uncertainty principle which, again, can be restored by assuming that the measurement of the right hand particle collapses the state of the left hand one.

The argument given by the EPR experiment does not conclusively demonstrate that there is anything weird happening. We could simply suppose that the uncertainty principle is wrong, and that the horizontal and vertical components of spin are determined from the outset but, other than each particle having the opposite values, are completely random. This would in fact be consistent with the predicted results, even though the theoretical description is different from quantum theory. It is not surprising that some people, such as Einstein, considered quantum mechanics to be incomplete, and that the strange entanglement phenomena could be avoided with a different theory.

In fact, it turns out that we only need to turn the two spin measurement apparatus in figure 1 to a 45 degree angle to each other, rather than 90 degrees as in the EPR-Bohm experiment, to obtain results inconsistent with any classical explanation. That is, unless Einstein’s spooky action at a distance is real.

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Black Boxes

The main piece of equipment used in the experiments to be described below, is a black box. In fact, this is the only piece of equipment. These are nothing other than a fully enclosed box, on which there are mounted some buttons and one or more light bulb. When a button is pressed, the lights each glow either green or red. The simplest black box has just two buttons, labelled L and R, and a single light. This is as pictured below.

The contents of the box are hidden from view, and how they work is not important right now. Maybe the buttons are simply each wired to make the light glow a fixed colour. Maybe it is a bit more complex, and they contain a computer which takes some measurements of its environment, such as the temperature, pressure, time and, when a button is pressed, performs a calculation to decide what colour to light up. It could hold some particles trapped in its interior, and measure the spin of these to decide. Or, maybe a small gremlin living inside the box turns on the lights however he likes. It really does not matter. The aim of the experiments is to demonstrate how quantum entanglement is a fundamental property of the universe, and not just some characteristic of a specific physical system.

With these boxes, we have one, and only one, chance to choose a button. Once we have pressed a button then, we could always try pressing the other one if we like. However, the action of pressing the first button will have already interfered with the state of the box, and caused it to reset. So, what we see may not be the same as if we had pressed the other button in the first place. Any properties that the box is supposed to have will no longer hold for subsequent button presses. If we do want to repeat any of the experiments, then we would have to start over, and re-initialize all the boxes back to their starting states.

I now describe four thought experiments making use of these black boxes, each of which, in their own way, could be argued to be the best possible demonstration of quantum entanglement. Being thought experiments, it is not suggested that we would actually perform them as described here. Real tests of entanglement are difficult to perform, requiring very precise and fast measurements of things such as photons, travelling at the speed of light. Trapping them inside black boxes would not be practical with existing technology although, maybe, with the advent of quantum computing, one day they could be performed in reality. All experiments are consistent with the predictions of quantum mechanics, except where explicitly stated otherwise, and are possible, in theory. I will not give the actual quantum states used in the experiments, but do include references to the original papers explaining these, and include the mathematical descriptions of the states in a later post.

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Experiment 1: GHZ Boxes

Consider three black boxes, each of which has two buttons, labelled L and R, and a single light. When a button is pressed, the light instantly glows either green or red. One box is handed to Alice, one to Bob, and one to Charlie. These three participants are then allowed to leave with their box, and they each press one, and only one, of the buttons. The boxes do not communicate with each other. To ensure this, the participants can each travel to distant locations if they like, and press their buttons closely enough to an agreed time so that light, electromagnetic radiation, or any other kind of signal, would not be able to travel between them quickly enough to affect the outcomes of each other’s observations. They could also sit inside Faraday cages to block any possible communication between boxes, if preferred. They agree to meet up at a future specified time and compare results. The boxes come with guaranteed properties, or parity constraints:

1. If Alice, Bob and Charlie all press L, then an even number of them (none or two) will see their box light up red.
2. If precisely one of Alice, Bob and Charlie press L, with the other two pressing R, then an odd number of them (one or three) will see their box light up red.

Thinking about these properties for a few moments, you should see that this is rather weird behaviour, maybe even self-contradictory. Start with the case where each button of each box is simply wired up to turn the light a fixed colour, either green or red. Across the 3 boxes, with two buttons each, there are 6 buttons in total with 64 possible combinations of ways that they can be wired up. None of these satisfy the required properties. We can see this without resorting to counting through all 64 combinations. By the second stated property, there are an odd number of red lights in each of the following three situations.

Alice presses L, Bob presses R, Charlie presses R

Alice presses R, Bob presses L, Charlie presses R

Alice presses R, Bob presses R, Charlie presses L

Now add together the total number of red lights in these scenarios. As it is the sum of three odd numbers, we still obtain an odd number. However, the result of Alice pressing R has been included twice, which can contribute a count of zero or two and does not affect whether the result is odd or even. Similarly, we have counted the effect of Bob pressing R twice, and also for Charlie. After removing these cases, we are left with an odd number of red lights for the scenario,

Alice presses L, Bob presses L, Charlie presses L.

This flatly contradicts the first stated property. So, it is not possible to build such boxes by simply wiring each button to light up the box a specific colour. In other words, it is not possible to fill in the light colours for the table below such that each person’s colour depends only on whether they press L or R, and such that the first three rows each contain an odd number of red lights but the fourth row contains an even number of red lights.

Button			Light
Alice	Bob	Charlie	Alice	Bob	Charlie
L	R	R	R	R	R
R	L	R	G	G	R
R	R	L	G	R	G
L	L	L	R	G	G

How about if we are a bit more ingenious? We could try randomizing the behaviour. Maybe by having a computer inside each box, which samples from a random number generator, measures the temperature or the precise time, and decides what colour to light up based on these — and other — factors. Well, if we are still restricted by having no communication between the boxes then, no, it is not possible to construct boxes with the required properties. At least, not according to classical physics. The considerations above still apply. Whatever data our computer program uses to decide which colour to light up then, conditional on the values of this data, there will always be some combination of button presses for which the three participants will see a result contradicting the guaranteed properties. In fact, the second property alone is sufficient to conclude that the boxes cannot have any intrinsic randomness, and that their reponse to any button press is predetermined by the state of the other two boxes. Regardless of which button Alice presses, for example, Bob and Charlie can always choose their buttons such that, between the three of them, exactly one presses L. The colour of Alice’s light will then be determined by the other two boxes. If exactly one of Bob and Charlie’s boxes lights red, then Alice’s is green, otherwise it is red.

The argument given can also be expressed succinctly with a small amount of algebra, which may be clearer for some people. I will write A_L for the outcome of Alice’s box, supposing that she presses button L. Set A_L = 1 to mean that the light glows red and A_L = -1 for green. Note that this need not be a fixed deterministic value, but could depend on other random data, such as the temperature, the time, or the outcomes of a random number generator installed in the box. All that we can say is that it should not depend on the actions of Bob and Charlie. We similarly write A_R for the outcome supposing that she presses button R. In the same way, we define B_L and B_R to represent the outcome of Bob’s box depending on which button he presses, and C_L and C_R for Charlie’s box. The two guaranteed properties of the boxes can be written algebraically,

ALBLCL = -1,

ALBRCR = ARBLCR = ARBRCL = 1.

This is using the fact that the product of a set of numbers, each equal to 1 or -1, is 1 whenever there is an even number of -1’s and -1 when there is an odd number. Multiplying the three expressions on the second row gives A_LB_LC_L(A_RB_RC_R)² = 1. As 1 and -1 both have square equal to 1, this is equivalent to A_LB_LC_L = 1, which directly contradicts the first equality!

We can be more precise and give some quantitative predictions. Suppose that Alice, Bob and Charlie each independently decide which button to press, and choose either button with probability one half. They could, for example, each toss a coin. Out of the eight possible combinations of button presses, which all occur with equal probability, there must be at least one for which the boxes’ guaranteed properties fail. This means that, with probability 1/8, the boxes should fail to live up to the guarantee. Equivalently , they can only satisfy the requirements with probability 7/8. Alice, Bob and Charlie reason that if they repeat the experiment one hundred times, then the probability of satisfying the guaranteed properties every time cannot be more than (7/8)¹⁰⁰, or about 0.00016%. This is highly unlikely. So they repeat the experiment one hundred times and compare results. To their surprise, the guaranteed properties held every time!

Of course, what Alice, Bob and Charlie were not accounting for, is that their boxes are quantum entangled. In fact, pressing the button caused a measurement of a single quantum bit of information held within the box and, at the start of each experiment, these qubits had been put into an entangled state. More precisely, they had been put into the Greenberger–Horne–Zeilinger, or GHZ, state²⁶. The experiment described here is based on the 1990 paper Quantum Mysteries Revisited¹⁶ by David Mermin and the 1989 paper Going Beyond Bell’s Theorem¹³ by Greenberger, Horne and Zeilinger. The internal workings of these boxes could be something along the following lines. Contained inside is a spin 1/2 particle, such as an electron, proton, or atom. It could be held in place by an ion trap²⁸. Pressing a button triggers a measurement of the spin of this particle, with a ‘spin-up’ result causing the light to glow green and ‘spin-down’ resulting in it glowing red. The only difference between the two buttons, is that they trigger the spin measurement along distinct axes, at right-angles to each other.

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Can the Boxes Transfer Information?

Having confirmed that their boxes are either breaking basic logical rules of classical physics, or are somehow communicating between each other, Alice, Bob and Charlie think that their boxes must be communicating. Surely then, they have stumbled upon a magical communication device capable of transferring information across time and space! So, they decide to try and use these boxes to actually communicate between each other.

Is this possible? Well, there is a quick way to tell. As they have already run the experiment many times, they have the data. All that needs to be done is to check the results and see how the distributions of the various outcomes depends on the choice of which button is pressed. If, for example, the distribution of Bob and Charlie’s observations depends in any way on whether Alice presses L or R, then this could be used by Alice to transmit information to the other two. After analysing the data, up to statistical error, they discover the following.

• If precisely two of Alice, Bob and Charlie press L, or none of them do, then the guaranteed properties are not applicable. Each box lights up red or green with probability one half, and the boxes are independent of each other. That is, each of the 8 possible combinations of results occur with probability 1/8.
• If precisely one of Alice, Bob and Charlie press L, or if they all do, then the guaranteed properties apply. In this case, each of the 4 possible combinations of results obeying the required parity constraint occurs with probability 1/4.

Upon discovering this, they give up. Although, reasoning with classical probability, they were not able to explain the behaviour of the boxes without assuming that they are communicating, there is no way to actually use the behaviour to communicate any information. Look at the behaviour of Bob and Charlie’s boxes, for example. They always see both boxes light up red and green with probability one half, and independently of each other. This does not depend in any way on which button Alice presses. It is not possible for Alice to send any information to Bob and Charlie through pressing either button. The situation is symmetric so, similarly, neither Bob nor Charlie can transmit information to the others in this way.

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Constructing the Boxes Classically

As discussed above, according to classical logic, it is not possible to build boxes with the stated properties. However, this was assuming that the boxes do not communicate with each other. As soon as we allow some communication between the boxes, then it becomes straightforward. Start with the following construction.

1. Wire up the R buttons however we like.
2. Next, wire up the L button on each box so that, if it is pressed and the R button is pressed on the other two boxes, then an odd number of them will light up red.

Unfortunately, if the boxes are wired up like this, then there will also be an odd number of red lights if each of Alice, Bob and Charlie press L. All that is required to remedy this is to install a single communication channel from any one of the boxes to any one of the others. For example, suppose that when Alice presses her button, it transmits her choice to Bob’s box. We can add a switch to Bob’s box in the wiring from the L button. This is set up to wait to see what button Alice presses. If she presses L, then Bob’s box will switch its behaviour for the L button and, if she presses R, then it keeps the original behaviour. Bob may have to wait for his box to receive the signal from Alice, so it will not be instant but, once it responds, the behaviour will be as guaranteed.

With this construction, then the only way that Bob’s box can act instantly without waiting for Alice, is for information to travel backwards in time. This is easily remedied by having a bidirectional communication channel. If Bob presses L first, then it is Alice’s box which switches its behaviour. This last point is often confused in popular accounts of entanglement. In such descriptions, entanglement is described using classical logic, but with an immediate effect from one system on another distant system. Then, when the second system is measured before the first, it is made to seem like the effect occurs before the cause. In fact, it is always possible to explain such entangled systems classically with information travelling in either direction, so it is never necessary to talk about effects going backwards in time. We should keep in mind that, just as in the example above, it is not actually possible to transmit any information simply by making local measurements of entangled systems. Descriptions involving instantaneous effects at a distance are a result of trying to give a classical explanation of inherently quantum phenomena.

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Experiment 2: Mermin’s Boxes

The discussion above concerned three boxes with strange ‘entanglement’ properties, which could not be explained by classical probability without assuming some instantaneous communication between them. You could ask why we looked at an example with three observers, rather than two. Is entanglement possible between just two observers? Yes it is. Consider the following scenario.

There are two boxes, again each with two buttons labelled L and R, and a single light. As before, when a button is pressed, the light instantly glows either green or red. One box is handed to Alice and one to Bob. These two participants are then allowed to leave with their respective boxes and, at some point, press one of the buttons. They agree to meet up at a future time to compare results. As in the previous experiment, they can travel far apart and press the buttons at the same time, so that there is no time for the boxes to communicate in any way, even using signals travelling at the speed of light. The boxes have the following guaranteed properties.

1. If Alice and Bob both press L, then at least one of the boxes lights up green.
2. If one of Alice and Bob presses L and the other presses R, then at least one of the boxes lights up red.
3. If they both press R, then the boxes sometimes both light up green.

The final rule here is bit more vague than the others, as it concerns a property which holds only some of the time. We can be more precise though and say that, if they both press R, then the probability of both lights glowing green is 1/12, or about 8.3% of the time. The precise probability does not really matter. Consider the situation where they both press R and see their box lighting green. What could have happened if Alice had pressed L instead? As this cannot affect the fact that Bob pressed R and saw green, the second property guarantees that Alice would have seen the light glow red. In a similar way, we can conclude that Bob would have seen his box glow red if he had pressed L. Hence, if they both pressed L, then both their boxes must have lit up red, contradicting the first guaranteed property! Equivalently, it is not possible to fill in the colours in the following table, such that the R column is all green, both diagonals contain at least one red, but the L column contains a green.

	L	R
Alice	R	G
Bob	R	G

Using classical logic, the third property is inconsistent with the first two, unless there is some communication, so that the behaviour of Alice or Bob’s box depends on the choice made by the other participant. However, such behaviour is allowed by quantum probability, even without any communication channel. As in the first experiment, we can ask whether the boxes can be used by Alice and Bob to instantaneously communicate between themselves. To answer this, we require a more complete statistical description. Supposing that they were to repeat the experiment many times, trying all different combinations of pressing L and R, then the probability of each combination of G (green) and R (red) observed by the pair of participants is as in the table below.

Alice	Bob	G	G	G	R	R	G	R	R
L	L	1/3		1/3		1/3		0
L	R	0		2/3		1/6		1/6
R	L	0		1/6		2/3		1/6
R	R	1/12		1/12		1/12		3/4

This shows that Alice’s box satisfies the following statistics, regardless of which button Bob presses.

1. If she presses L, then it glows green with probability 2/3 and red with probability 1/3.
2. If she presses R, then it glows green with probability 1/6 and red with probability 5/6.

Hence, it is not possible for Bob to have any effect on the statistical behaviour of Alice’s box through his choice of button. In the same way, Alice is not able to affect the behaviour of Bob’s box so, although we cannot explain the behaviour of the boxes with classical logic unless there is some communication between them, there is no way to employ them to actually communicate.

If we do allow a communication channel, then there is no problem with building boxes with the above properties. Simply construct Alice’s box to always light up green, but to also transmit which button she presses to Bob’s box. Then, wire up Bob’s box to light red unless both Alice and Bob press R, in which case it lights green. It is easily seen that this will satisfy the three required properties. However, this set-up does not match the statistical properties described above, and does allow Alice to communicate to Bob. If Bob presses R, then he can tell which button Alice presses by observing the colour which his box lights. In fact, the entire statistical properties described above can be replicated with a slightly more complicated setup, where we install random number generators in the boxes to give the correct probabilities, and have a single communication channel which transmits Alice’s button press and light colour to Bob’s box. His box will light red or green with the correct conditional probabilities, given which buttons each participant pressed and given the colour of Alice’s light.

This experiment gives a striking demonstration of quantum entanglement between two participants using a simple two-button black box. The boxes themselves could be exactly the same as the ones used in the first experiment above, with the only difference being that their qubits start off entangled in a different state. It is based on the entanglement scenario described in the 1995 paper The Best Version of Bell’s Theorem¹⁸ by David Mermin, and is closely related to Hardy’s paradox²⁷, which gives a similar situation for a specific physical setup described in the 1992 paper Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories¹⁴. However, it is not quite as strong as we might hope. The third property is statistical, only specifying a constraint on the output some of the time, and not with every run of the experiment. This suggests a further question. Is it possible to set up the boxes with the properties above such that if Alice and Bob both press R then their boxes always glow green? In fact, without a communication channel between the boxes, this is not possible, even if they are quantum entangled.

If the boxes obey the properties above, and always both light green when Alice and Bob both press R, then it would be possible to use them to transmit information in contradiction with the properties of quantum entanglement. Suppose Bob presses R. If he saw his light glow red then he would know that Alice had pressed L. The only way that this information can be avoided is if Bob’s box always glows green when he presses R. In the same way, by pressing R, Alice would sometimes gain information about Bob’s choice unless her box also always glows green in this situation. Then, if either Alice or Bob press L and see their box light green, they would know that the other participant has also pressed L. This is by the second guaranteed property, which would have to be broken if the other participant had pressed R and also seen green. However, the first property says that if they both press L, at least one of the participants would see their light glow green, so they would know that the other had also pressed L. By this argument, it would be possible for one of Alice and Bob to be able to be certain of the other’s button press by observing the colour of their own light, at least a positive proportion of the time. This is not possible for boxes with no communication between them, even with quantum entanglement.

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Experiment 3: Alice and Bob’s Impossible Boxes

So far, we have seen two experiments, both involving simple black boxes with two buttons and a single light which glows either green or red in response to one of the buttons being pressed. The first experiment, with three participants, involved boxes which satisfied some guaranteed properties on every run of the experiment, and could not be explained by classical logic without assuming that the boxes are able to instantaneously communicate between themselves. The second experiment above is similar, but only involved two participants. However, it was a little less striking as one of the stated properties of the box could only be guaranteed to hold some of the time, and not with every run. You may ask if there are examples showing entanglement between a single pair of participants, involving boxes with similarly strange properties which are guaranteed to hold on every run of the experiment? Consider the following. We again have two participants, Alice and Bob, who are each handed a simple black box with two buttons labelled L and R and a single light which will immediately glow either green or red when a button is pressed. As before, Alice and Bob are both allowed to leave with their box. They each press one of the buttons when they like, and agree to meet up at a later date to compare results. These boxes come with the following guaranteed properties.

1. If at least one of Alice or Bob presses L, then they see the same result. Either, both their boxes light red or both light green.
2. If both of Alice and Bob press R, then they see different results. One of their boxes lights red and one lights green.

Are such boxes possible? As before, if we assume classical probability and no communication between them then, no, it is not possible. The situation is even simpler than that above. Suppose Alice and Bob both press L. Whatever colour Bob sees, according to the guaranteed properties, Alice will see the same. However, she would also see the same colour as Bob if she had pressed R instead. Since there is no communication between the boxes, the colour that her box lights up should not depend on which button Bob pressed, and we conclude that the colour is independent of which button she chooses. Exactly the same argument applies for Bob. So, the colour that the boxes light up cannot depend at all on which button they press. By the first guaranteed property, they should both see the same colour regardless. This directly contradicts the second property. In other words, it is not possible to fill in the colours in the table below such that the L column, and both of the diagonals, each consist of two entries of the same colour, while the R column contains entries of distinct colours.

	L	R
Alice	G	G
Bob	G	G

As in the first experiment, this argument can also be expressed neatly with a small amount of algebra. As before, write A_L for the outcome of Alice’s box supposing that she presses L, with A_L = 1 meaning that the light glows red and A_L = -1 for green. In a similar way, use A_R for the outcome if she presses R, and B_L and B_R for the possible outcomes for Bob. As the behaviour of the boxes need not be deterministic, these values can be random and could also be functions of any measurable quantities in the environment of the respective boxes (pressure, temperature, time, etc). The guaranteed properties can be written algebraically.

ALBL = ARBL = ALBR = 1.

ARBR = -1,

Multiplying together the terms in the first row gives A_RB_R(A_LB_L)² = 1. As 1 and -1 both have square equal to 1, this is equivalent to A_RB_R = 1, contradicting the second equality.

According to classical logic, it is not possible for Alice and Bob’s boxes to exist with the stated properties, unless there is some instant communication between them. So, is it possible for Alice and Bob to use these boxes to actually communicate some information between themselves? To answer this, we need a more complete statistical description of the boxes’ behaviour. For any specific choice by Alice and Bob on whether to press L or R, according to the stated properties there are just two possible outcomes. Suppose that these each occurs with probability one half.

1. If at least one of Alice or Bob presses L, then they both see green with probability 1/2, or both see red with probability 1/2.
2. If both of Alice and Bob press R, then Alice sees green and Bob sees red with probability 1/2, or Alice sees red and Bob sees green with probability 1/2.

We see that Bob sees either green or red, each with probability one half, and that this is the case regardless of which button Alice presses. So, Alice cannot influence the behaviour of Bob’s box by her choice of which button to press. Similarly, Bob cannot influence Alice’s box. There is no way for the two protagonists to use these boxes to transmit information between themselves.

As previously, we can be a bit more quantitative and describe a statistical test that Alice and Bob can perform to determine if they really follow the guaranteed properties. Suppose that they each choose which button to press independently of each other, and choose either L or R with probability one half. For example, they can both toss a coin to decide. According to classical logic, if there was really no communication between the boxes, then out of the four possible choices that can be made between the two of them, at least one must be inconsistent with the guaranteed properties. So, the properties should have probability at least 1/4 of being broken. Equivalently, the properties should hold with probability no more than 3/4, or no more than 75% of the time. This is known as Bell’s theorem²³, or its slight generalization, the CHSH inequality²⁵ — although stated in a slightly different form than usual. Alice and Bob can repeat the experiment some large number of times. Let’s say, 100 times. The probability of the guaranteed properties holding on every single trial is no more than (3/4)¹⁰⁰, which is an absolutely minuscule number. This would be sufficient to rule out the boxes behaving according to classical probability without there being some communication between them.

The situation reflects the first two experiments above. Namely, they each have a box which responds to them pressing one of its two buttons. These boxes come with guaranteed properties which cannot be explained by classical logic, other than by assuming that there is some instantaneous communication between the boxes. In addition, their statistical behaviour is such that it is not possible for either of the protagonists to influence the behaviour of the other boxes through their choice of which button to press, so the boxes cannot be used to transmit information.

Using quantum entanglement, how could such boxes be created? In fact, it is not possible! This behaviour is too weird, even for quantum theory! These boxes are really impossible. Why this is the case is not clear by simply looking at the physical consequences of the properties of such boxes. It just seems to be a deep consequence of quantum theory, and is ruled out by Tsirelson’s theorem³³. If both participants choose independently to press either L or R with probability one half, then this theorem states that the properties above cannot hold more than a proportion (2+√2)/4 of the time on average, which is a probability of about 85%. It is not possible to get any closer to 100%.

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Experiment 3 Redux: Bell’s Boxes

According to quantum theory, it is impossible for the properties of Alice and Bob’s boxes, described in experiment 3, to be guaranteed all of the time. However, the following modification of the experiment can be achieved in reality. Once again, Alice and Bob are each handed a simple black box which has two buttons labelled L and R, and a single light which instantly glows either green or red when a button is pressed.

• Regardless of which buttons Alice and Bob press, the outcome satisfies the ‘guaranteed’ properties above with probability (2+√2)/4 ≈ 85%.

As explained above using classical logic, out of the four possible combinations of choices of button presses for the two participants, there is always at least one for which the stated properties fail. Consequently, if Alice and Bob both choose which button to press completely at random, such as by flipping a coin, then there is always at least a one in four chance that the stated properties fail to hold. This means that the maximum probability that we would hope for the properties to hold is 75%. So this modified experiment is still impossible to explain by classical logic, assuming that the boxes cannot communicate, and would irrefutably demonstrate the effects of quantum entanglement. In fact, even Mermin’s boxes from the previous experiment will break the 75% bound, so long as we switch the behaviour of Bob’s box so that it lights green instead of red, and vice-versa. Using the statistics from figure 5, we see that they would obey the stated properties here with probability 19/24, or about 79%.

This entanglement situation is essentially a version of that described by John Bell in his famous 1964 paper On the Einstein Podolsky Rosen paradox⁴, and by Clauser, Horne, Shimony and Holt in the 1969 paper Proposed Experiment to Test Local Hidden-Variable Theories¹⁰. It is what CHSH Bell tests²⁴ are designed to check. The black boxes used here can be the same ones as used in the previous two experiments, with the only difference being that their internal qubits are initially entangled in a ‘Bell state’.

We can again ask whether the boxes used in this experiment can be used to actually transmit information. Could Alice change the behaviour of Bob’s box in any noticeable way simply through her choice of button? The following fact is sufficient to answer this: whatever buttons Alice and Bob press, the probability of each specific outcome is the same as with the colours of the lights inverted. That is, replacing red lights by green, and vice-versa, does not affect the probability of each outcome. This means that Bob sees his light glow green with probability one half, and red with probability one half. This is true regardless of which button Alice presses, so she cannot influence Bob’s box in an observable way. Similarly, Bob is not able to noticeably influence Alice’s box.

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Experiment 4: Magic Squares

For a really striking example of quantum entanglement, it would be good to see a case involving just a single pair of black boxes satisfying some guaranteed properties for every run of the experiment. These properties should be impossible to explain classically unless the boxes are instantaneously communicating between themselves. The first experiment above showed such a situation for three boxes. Mermin’s boxes of the second experiment got very close but, as one of the properties was only guaranteed to hold sometimes, it is not quite as strong as we are looking for. The proposed boxes in the third experiment would fit the bill perfectly but, unfortunately, they were not physically possible according to quantum theory. The closest we could get was for the properties to hold 85% of the time. This is still a convincing demonstration of quantum entanglement, but we would like to do better. So, is it possible? In fact, yes, it is, as we will see shortly. We will, however, have to go beyond the simple two-button black boxes used above. First, a short diversion on magic squares²⁹.

A magic square is a square grid of numbers, with the sums of the rows, columns, and sometimes the diagonals, satisfying certain special properties. Finding examples whose entries are all distinct whole numbers and whose rows, columns and diagonals all sum to the same value, has been a part of recreational mathematics for thousands of years. We will not discuss those here, and only look at a seemingly very simple case. Try constructing a 3×3 grid of whole numbers where each row has an odd sum and each column has an even sum. How can this be done? It is actually impossible. Consider the sum of all nine entries of the square. As this equals the total of the row sums, it must be odd and, as it also equals the total of the column sums, it should also be even. This type of magic square cannot exist. If we try constructing one then, once we get to the final entry, we always find ourselves in the situation below. The value would need to be filled in differently depending on whether we want its row to sum to an odd number or its column to have an even sum.

0	1	0
1	1	1
1	0	?

I now describe an experiment involving two participants, Alice and Bob. They are handed an identical looking black box each, on which there are three buttons in a row, labelled 1, 2 and 3, and also three lights in a row, one above each button. These are as pictured below.

Alice and Bob are now free to leave with their boxes, and press one of the buttons at a time and place of their choosing. The boxes do not communicate and, to be sure of this, the two participants can travel to distant locations if they like. They can then do this such that there is insufficient time for any signal travelling at the speed of light to pass between the boxes in between their respective button presses. This should ensure that the boxes are unable to communicate in any way as to affect the possible outcome. They agree to meet up at a future time to compare results. These three-button black boxes come with the following guaranteed properties.

1. When one of the buttons on a box is pressed, its lights instantly turn on, each either green or red, such that the number of red lights is even (zero or two).
2. If Alice presses button i and Bob presses button j, then Alice’s j’th light and Bob’s i’th light are different colours.

These properties are rather strange and seemingly self-contradictory. How can Alice’s box respond to her pressing each individual button? Consider filling in a 3×3 square grid, where each row corresponds to the colours that Alice will see if she presses the respective button on her box. That is, the first row of the grid will contain the three colours that she sees if she presses button 1, in the same order. Similarly, for the second and third rows. Filling in each entry of the grid with a ‘0’ if the corresponding light colour is red, and ‘1’ if it is green, then we obtain something like the following square.

	1	2	3
Button 1	0	1	0
Button 2	1	1	1
Button 3	1	0	0

By the first guaranteed condition, each row contains an even number of red squares and an odd number of greens. So, the values of each row sum to an odd number. Now, consider the colours that Bob will see on his box when he presses any one of the buttons. As the boxes cannot communicate, it should not depend on which button Alice actually pressed, but must be consistent with the guaranteed properties regardless. By the second property, when Bob presses a button, the colours of the lights should correspond to the respective column of the 3×3 square, but with the colours inverted. That is, ‘0’ corresponds to green and ‘1’ to red. For example, according to the square above, if he presses the second button then, reading from the second column, he will see the colours red, red and green in that order. By the first condition above, the columns of the 3×3 square must sum to even numbers. We have already seen that such ‘magic squares’ with odd row sums and even column sums are impossible. In the square above, the third column sums to one, signifying that Bob would see precisely one red light if he had pressed the third button, in contradiction with the first guaranteed property. According to classical logic, such boxes are impossible unless they are allowed to communicate, so that Bob’s box changes behaviour depending on which button Alice presses, or vice-versa.

As these boxes appear to be transmitting information between themselves, we can ask if it is possible for Alice and Bob to use them as an instantaneous communication channel. Having already asked the same thing for each of the previous three experiments, it should come as no surprise that the answer here is the same. It is not possible. To understand this, we need a more complete statistical description of the box. In fact, for any choice of buttons by Alice and Bob, the boxes light up completely at random, subject to the two guaranteed properties. There are a total of eight possible combinations of ways in which the lights can glow, each of which will occur with probability 1/8. This means that Bob’s box lights up completely at random. Each of the four possible outcomes of his box, where the number of red lights is even, occur with probability 1/4, and this does not depend on which button Alice presses.

The properties of these boxes would be easy to explain if only there was some communication between them. Suppose that when Alice presses a button, her lights glow at random, so that each of the four possibilities with an even number of red lights, occurs with probability 1/4. If her box also transmits her choice of button and the light colours to Bob’s box, then it is straightforward for his box to satisfy the required properties. The second property fixes the colour of one of his lights, and the first will either imply that the remaining two lights glow with the same colour, or that they glow different colours. In both cases, there are exactly two possibilities. If Bob’s box selects each of these with probability 1/2, then the boxes would not only satisfy the required properties, but would also have the statistics just described. Quantum entanglement enables the properties described here to be satisfied, even if the boxes are not able to communicate. However, these black boxes are slightly more complex than in the previous experiments. Pressing any one of the three buttons causes measurements to be made on a pair of qubits held inside. Across the two boxes, this is four qubits in total, which have been entangled in a specific way from the start. It is based on the ideas of the 1990 papers Incompatible results of quantum measurements²⁰ by Asher Peres and Simple Unified Form for the Major No-Hidden-Variables Theorems¹⁷ by David Mermin, and formulated as a thought experiment along similar lines as in the present post in the 2004 paper Quantum mysteries revisited again³ by P.K. Aravind. The experiment can also be expressed in terms of the Mermin–Peres magic square game³¹.

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Summary

Hopefully the situation is clear. The setup in each experiment is as follows.

1. A collection of two or more black boxes, each of which has two or more buttons, and some lights.
2. Each box is handed to a participant in the experiment, who are then allowed to leave with their box.
3. Each participant presses one, and only one, of the buttons on their box. When they do this, the lights on their boxes instantly glow with certain colours.
4. The participants meet up at a future time and compare results.

It is assumed that the boxes cannot communicate while they are pressing the buttons. To ensure this, the participants are free to travel to distant locations, so that there is not enough time for any signal, travelling at the speed of light, to pass between them while they press the buttons.

So far, so straightforward. Entanglement comes in when we assign properties to the boxes which cannot be explained with classical logic. For the strongest scenarios, as in experiments 1 and 4:

• The boxes come with guaranteed properties. For certain combinations of button presses amongst the participants, these place restrictions on the possible combinations of results that they see.
• There is no possible way to assign a specific response to each button, for each box, which simultaneously satisfies all of the guaranteed properties.

According to classical probability, if each participant presses a button at random, and independently of each other, then they must see that the guaranteed properties fail at least some of the time. Somehow, the response of the boxes depends also on the buttons which are pressed on the other boxes. This would require an instantaneous effect occuring between them.

We also saw the, slightly weaker, statistical entanglement in experiments 2 and 3. Although this might not seem quite so striking, for all practical purposes, it is just as convincing a demonstration of entanglement as above.

• The boxes come with some statistical properties. For certain combinations of button presses among the participants, this assigns probabilities to the different combinations of results that they can see.
• There is no way to assign probabilities for the outputs that will be seen simultaneously for each button for each participant, which satisfies with the probability constraints above.

In figure 5, we saw the probabilities for each outcome for Mermin’s boxes conditional on which buttons were pressed. For example, if Alice pressed L and Bob pressed R, then the probability of them seeing green and red respectively is 2/3. We did not give probabilities that Alice would see green if she presses L and red if she presses R, and that Bob would see red regardless of which button he presses. This is not even an observable probability, since each participant only presses one button. In any case, once they have pressed a button, the state of the box changes, so pressing the other button may not show the same behaviour as if they had pressed that first. However, according to classical logic, such probabilities must exist and, if the boxes do not transmit information between themselves, it is not possible for a button press on one box to change the probabilities of the other ones. The fact that it is impossible to assign such probabilities demonstrates the non-classical behaviour of these boxes.

Despite the entanglement that was identified in the experiments, we saw that it was not possible to actually transmit information using the boxes. For any choice of buttons by any of the participants, the resulting probability of seeing any particular outcome does not depend on the choice made by the other participants.

It should also be clear that the use of black boxes in the experiments above is only for explanatory purposes. By “black box”, we really mean any physical system. Pressing a button refers to making a measurement on this system, and the colours of the lights just refers to observing the result of the measurement. Using black boxes is a handy way to extract out the precise physical setups, or measurements being made, as this is not important for the fact that entanglement is present.

For more information on entanglement, there are of course Wikipedia pages^23,30. The online Stanford Encyclopedia of Philosophy^6,9,12,19 has some detailed and interesting entries on the subject, containing plenty of historical background. See, also Scott Aaronson’s 2015 blog post, Bell inequality violation finally done right¹, concerning a particular Bell test experiment¹⁵, which was designed to close some loopholes.

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The Uncertainty Principle

So, what can we make of entanglement? What is it about classical reasoning that made the experiments above seem impossible, and how does quantum probability resolve this?

The classical descriptions of each of the experiments involved considering, at the same time, the possible responses of a box for pressing each of the buttons. For example, suppose that Alice has a simple two-button black box as in the experiments above and, when she presses a button, it lights up green with probability 1/2 or red with the same probability. Does the box depend at all on which button she presses? We could imagine that it is set up to choose a colour at random, either red or green, and then show this when she presses a button. In this case, it does not matter which button she presses. If she presses L and sees green, then she knows that it would also have been green if she had pressed R instead. Alternatively, the box could have been set up to choose either red or green, and light up this colour if she presses L, or the other colour if she presses R. Then, if she presses L and sees green, she knows that it would have been red if she had pressed R instead.

These two cases just described are the same, as far as can be physically determined. Alice’s box just lights red or green, with equal probability, when she presses a button. Whether the box shows the same colour or not, for the two buttons, is not an observable property. According to quantum theory, it is even a meaningless question! This is Heisenberg’s uncertainty principle. If Alice chooses a button and sees that the green light shows, then it is impossible to ever say what the response would have been if she had made the other choice. The outcome of having pressed the other button is not even a property of the universe. We can never know, at the same time, what the response of the box is for both of the button presses.

The uncertainty principle is frequently stated in terms of the position and momentum of a particle, which cannot both be precisely known at the same time. It does, though, equally apply to the spin of a particle. It is impossible to simultaneously know the component of the spin along two different, orthogonal, axes. It also applies to the outputs of the boxes used in the experiments above. We cannot simultaneously know how the box will respond to each of the button presses. At best, we can know for a single button of our choice, simply by pressing it, but then can never know what would have happened for the other buttons.

Standard probability theory starts with a sample space, which is a set consisting of all possible states of the universe, or of the system under consideration. We assign probabilities to events, which are associated with subsets of the sample space. An example of an event could be that Alice’s box will light up red if she presses L. Another event is that it will light up green if she presses R. Taking the intersection of these gives the event that the light will glow red if she presses L and glow green if she presses R. Classical probability must assign a probability to this. In quantum theory, the two events do not commute, so are not simultaneously observable. There is not even any event corresponding to their intersection. So, the statement that Alice’s box will light red if she presses L and green if she presses R, is not an event represented by the theory, and does not have a probability.

There are two ways of looking at the uncertainty principle. First, in a weak sense, that it is impossible to simultaneously measure certain observable properties, such as the responses of the boxes to pressing each button. The strong sense is that these simultaneous values are not even a property of the universe. The position or momentum of a particle, or its spin, does not exist until it is measured. Practically, it does not make a lot of difference. The existence of something which is fundamentally non-observable may matter from philosophical, interpretational, or theoretical viewpoints, but it has no physical significance.

Going back to the experiments described above, what would be the consequences if we could simultaneously measure how the boxes would respond to each button? We already argued that there is no way that these values can exist unless there is some communication between the boxes. So Alice, for example, would see her boxes properties change when Bob presses a button which would otherwise have caused the guaranteed properties to be broken. This is probably clearest for the fourth experiment. If Alice measured her box to be in the state shown in figure 9, then she knows that Bob cannot have pressed his third button. This is because the third column has an odd sum, inconsistent with Bob seeing an even number of red lights. If Bob has not yet pressed any button then, if he does decide to press button 3, Alice would instantly see her box change to a consistent state. If we were to only accept the weak version of the uncertainty principle, then these instantaneous effects across space would still exist in our model, even if they are not observable.

To be precise, it is the following three aspects of classical reasoning which led to the seemingly contradictory properties of the boxes described in this post.

Reality: Observations made on a system just reveal its underlying properties. These properties exist regardless of whether the observation is actually made. For the experiments above, this means that there is an outcome for each button that the participant could have pressed. When a button is pressed, it reveals the outcome specific for that button press, whereas, if a different choice had been made, it would have revealed the other outcome.

Locality: A measurement made on an isolated physical system cannot simultaneously affect the properties of any other system separated from it in space. Physicists often use a more precise form, making use of special relativity, and using the assumption that information cannot travel faster than the speed of light. Then, the measurement cannot affect the other system faster than it would take light to travel between them. In the experiments above, Alice’s choice of which button to press does not affect the properties of Bob’s box, and vice-versa.

Freedom of Choice: An observer has the freedom to choose which measurement to make. For each of the experiments above, the participants can choose which button to press at random, and independently of each other. Flipping a coin was suggested. Then, by repeating the experiment many times, and recording the results, the statistical behaviour can be described. If freedom of choice was an illusion, and which button they were going to press had been determined at the outset of each experiment in a way correlated with with the state of the boxes, then it could be argued that their results do not give a true statistical description. This form of lack of freedom of choice is called superdeterminism.

The thought experiments show that at least one of these classical properties of the universe must be abandoned. The uncertainty principle, in its strong sense, suggests that we should drop reality. This does seem to be the easiest and most natural way to be consistent with quantum theory. However, there are people who believe that one of the other two properties should be dropped instead, and retain reality. While logically possible, this should be done in a way which is in agreement with the probabilities predicted by quantum theory, at least for all experimental results to date.

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Blender

The images in this post were created using the free 3d computer graphics software, Blender. The following freely available models were used,

• character models from Adobe Mixamo,
• the living room, tropical island and bedroom from free3d,
• beach chairs and umbrella from Blend Swap.

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