Bell's Theorem

Bell's Theorem shows that no physical theory that depends on precisely existing values for unmeasured quantities (termed local hidden variables) can reproduce all of the predictions of quantum mechanics. However the Many Worlds Interpretation does not depend on hidden variables so how does it produce the correct predictions?

Consider a series of experiments with each experiment involving an entangled photon pair named A and B respectively. In each experiment, the photons will each be measured by polarized filters which can be set to any angle from 0° to 360°. When a measurement is taken, the photon will either align with the filter and therefore be transmitted or it will not align and therefore be absorbed or reflected.^[1] The entangled photons have been prepared so that if they are measured by polarizers set to the same angle then they will either both be transmitted or both absorbed, with a 50% chance of observing either outcome.

The amplitude for configurations where the photons are correlated (i.e., have the same alignment) is the cosine of the angle between the two polarizers. Conversely, the amplitude for configurations where the photons are anti-correlated is the sine of the angle. In our experiments, the polarizers are set to either 0°, 120° or 240°.

For the first experiment, the two polarizers are set to 0°. The configurations (along with amplitudes) that are in superposition prior to measurement are as follows:

0°-1. Photon A is aligned and photon B is aligned (cos 0° = 1)
0°-2. Photon A is aligned and photon B is non-aligned (sin 0° = 0)
0°-3. Photon A is non-aligned and photon B is non-aligned (cos 0° = 1)
0°-4. Photon A is non-aligned and photon B is aligned (sin 0° = 0)

When the photons are measured, different versions of the observer become entangled with each configuration that contains amplitude. The probability that an observer will find themselves entangled with a particular configuration is determined by the squared modulus of the amplitude.^[2] So this is 50% each for configurations 1 and 3 and 0% for the other two configurations. That is, there is a 100% probability that the photon pair will have the same alignment (i.e., be correlated).

The same configurations apply if both polarizers are set to 120° or 240° respectively since the angle between the polarizers is still 0°. However what happens if the polarizer for photon A is set to 0° and the polarizer for photon B is set to 120°? In this case, the configurations in superposition are:

120°-1. Photon A is aligned and photon B is aligned (cos 120° = -0.5)
120°-2. Photon A is aligned and photon B is non-aligned (sin 120° = 0.866)
120°-3. Photon A is non-aligned and photon B is non-aligned (cos 120° = 0.5)
120°-4. Photon A is non-aligned and photon B is aligned (sin 120° = 0.866)

The probabilities for configurations 1 and 3 are 12.5% each and 37.5% each for configurations 2 and 4. That is, there is a 25% probability that the photon pair will have the same alignment and a 75% probability that it will not. The same configurations apply if the polarizer for photon A is set to 120° and the polarizer for photon B is set to 240° since the angle between the polarizers is still 120°. Similarly if the polarizers are set to 240° and 0° respectively.

The same probabilities also occur if the polarization angle difference is set to 240°. However the configurations are slightly different, as follows.

240°-1. Photon A is aligned and photon B is aligned (cos 240° = -0.5)
240°-2. Photon A is aligned and photon B is non-aligned (sin 240° = -0.866)
240°-3. Photon A is non-aligned and photon B is non-aligned (cos 240° = 0.5)
240°-4. Photon A is non-aligned and photon B is aligned (sin 240° = -0.866)

Note that if the polarizers are set randomly, then the average probability that the measurements will correlate is 50%. That is, (100% + 25% + 25%) / 3 = 50%. Similarly, for measurement anti-correlation, (0% + 75% + 75%) / 3 = 50%.

Now suppose that the entangled photons are light years apart and the polarizer angles are set randomly just prior to measurement. Observers can only be entangled with one of the configurations listed above, so measurements will continue to conform to the specified probabilities.

So Many Worlds produces the correct predictions. However the above scenario is impossible to replicate by any single-world theory that depends on local hidden variables. Such theories entail that a photon has a definite polarization for each of the three angles 0°, 120° and 240° prior to measurement. For this to be true, either the photon has the same alignment value for all three angles (two options - AAA and NNN) or it has two alignment values the same and one different (six options - AAN, ANA, NAA, NNA, NAN, ANN).

Now if we could measure any pair of polarization values, they would either correlate 100% of the time for the first two options or 33% of the time for the last six options. For example, with AAN, the pairings are AA, AN and AN, with only AA correlating. So overall the measurements would correlate at least 33% of the time. This is called a Bell inequality.

It so happens that this inequality can be tested by measuring a pair of photon values via entanglement. To do this, the first polarizer angle is set to 0° and the second polarizer angle is set to 120°. Since photon A and B have identical attributes (i.e., they always correlate when measured at the same angle) then the value measured for photon B at 120° is the value that would have been the value measured for photon A at 120° if it had been measured. As was shown above, the pair of values for photon A at 0° and 120° should correlate at least 33% of the time and so the actual measured values for photon A and photon B should also correlate at least 33% of the time. However, as shown earlier with the 120° configurations, Quantum Mechanics predicts that they will correlate only 25% of the time (cos² 120° = 0.25). So Quantum Mechanics violates the Bell inequality. Therefore photons cannot have precisely-defined values for angles that have not been measured.

One possible loophole remains and that is to ensure that the entangled photons do not somehow communicate with each other. This loophole can be closed by ensuring that the photons are space-like separated when the polarizer angles are set and the measurements performed (i.e., at a distance that would require superluminal communication between the photons). The results are that experiments support the predictions of quantum mechanics and definitively rule out local hidden variable theories.

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[1] This is how polarized sunglasses work. Sunlight normally radiates in all directions. However sunlight that reflects off flat horizontal surfaces such as lakes and roads will be polarized horizontally resulting in a strong glare that vertically polarized sunglasses can block.

[2] This is the Born rule. Note: In order to keep the tables simple, the configuration amplitudes have not been normalized (i.e., the modulus squared probabilities do not add to 1). To normalize, multiply each amplitude by 1/√2.

Three sides to every story

Have you ever imagined yourself as the hero in a story where, against all odds, you take on the villains to save the world? Feels good doesn't it?

Politics provides an opportunity to take part in such a story. The villains consist of those that conspire to thwart your team's noble cause, either by their actions or their indifference.

For conservatives, the battle is between civilization and barbarism. Conservative themes that fit this narrative include terrorism, socialism and moral issues like abortion.

For progressives, the struggle is for oppressed or marginalized groups and include concerns such as inequality, racism and corporate power.

For libertarians, the conflict is coercion against liberty, particularly with respect to the role of government in the lives of individuals and communities, such as the extent of trade regulations, taxes and legislated personal morality.

Any political issue can be described in terms of these axes.[1] For example, illegal immigrants can be regarded as experiencing oppression, importing barbarism, or seeking freedom depending on the chosen framing narrative. In a particular election cycle, the party that tells the most persuasive story around the political issues of the day will win the election. It is not the facts and policies themselves that matter for winning the vote but, instead, how those facts and policies are framed.

In the upcoming US election, Donald Trump's slogan is "Make America Great Again". This fits particularly well with the civilization/barbarism narrative and taps into people's sense that things aren't so great right now. It also makes it easy for anyone to create their own vision for what America's greatness means. It's a difficult slogan for Democrats to attack. So they instead need to challenge Trump's credibility for delivering on it.

Hillary Clinton's slogan is "Stronger Together". This strikes a solidarity chord with progressives. But it doesn't speak to people's aspirations or struggles. So Clinton's challenge is to find a message that emotionally resonates with people.[2]

Libertarian candidate Gary Johnson's slogan is "Live Free". This fits the libertarian narrative. But, as with Clinton's slogan, it will miss the mark with most voters. Libertarians need to inspire independents who are not enamored by either of the major party candidates. So how would freedom lead to a more civilized, safe and just society or improve their day-to-day lives?

Terrorist threats and economic hardship feed into Trump's story that there are systemic problems that only he, the uncompromising Washington outsider, can solve. Unless his opponents create a persuasive alternative story, Trump wins in November.

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[1] See Arnold Kling's Tribal Politics in the 21st Century.

[2] See Scott Adam's Battle of the Campaign Slogans.

Heisenberg's Uncertainty Principle

Diagram 1 - Hydrogen atom

In everyday life we are used to the idea of things like cars and trees having a precise velocity and position. For example, a car could have a velocity of 10km/h southwards and be currently leaving my driveway. And a tree is next to my driveway and not moving.

However it turns out that in our quantum universe the precision of the car's position is inversely related to the precision of the car's momentum (which is mass times velocity).^[1] Same for the tree's position and momentum. This imprecision is miniscule at human scales which is why we normally don't notice it. The inverse relationship between position and momentum is captured by Heisenberg's uncertainty principle.^[2]

The word "uncertainty" suggests that we can't accurately determine both values at the same time. However the principle is stronger than that. It means that the particle does not have an exact position and an exact momentum at the same time. This is somewhat analogous to how a bucket of water can't be completely solid and completely liquid at the same time.

To understand what this means physically, observe Diagram 2.

Diagram 2 - Wave superposition

The top four matter waves are sine waves from which a precise particle momentum can be derived from the wavelength of each wave.^[3] However amplitude for the particle position is concentrated in equal amounts along the entire wave. So the particle position for each wave is undefined (or, put differently, a particle is in a superposition of all possible positions when it has a precise momentum).

The bottom wave is formed by combining the top four waves in a superposition. In this case, the wave is more pronounced in the middle where the peaks and troughs are larger. While amplitude for the particle position is still spread out along the wave, it is no longer present in equal concentrations. It is more likely that a position measurement will locate a particle at the center of the wave (with the probability equal to the squared modulus of the amplitude).^[4] Note that since the bottom wave does not have a single wavelength, it does not have a precisely-defined particle momentum. A momentum measurement will return the momentum of one of the top four waves (resulting in decoherence and, consequently, indeterminacy in the particle's position).

Diagram 3 - Pulse wave

In the extreme case where all possible momenta are added, the result is a pulse wave as in Diagram 3. The particle has a precisely-defined position (since the amplitude of the horizontal lines is zero) but the momentum is undefined (i.e., the particle is in a superposition of all possible momenta).^[5]

In summary, the Heisenberg Uncertainty Principle states that a particle's position and momentum cannot both be sharply defined at the same time. This is because the range of momenta are derived from the wave's component wavelengths while the range of positions are derived from the concentrations of amplitude. A matter wave cannot be both a sine wave (precisely defining momentum) and a pulse wave (precisely defining position) at the same time.

For my first post in this series, see Visualizing Quantum Mechanics.

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Diagram 4 - Fourier transform

[1] Specifically, position and momentum are conjugate variables. This means that a matter wave can be represented in terms of position or equivalently, via a Fourier transform, in terms of momentum.

Another example of conjugate variables are frequency and time. Consider Diagram 4, where a sound signal over time has been recorded in the red box. A Fourier transform decomposes the signal into three uniform waves and produces a new signal in the blue box. An inverse Fourier transform can be used to recreate the original time-based signal.

Our ears perform a similar function to this when they receive sound waves as air vibrations and transform them into separate frequencies that are sent to the brain.

[2] The equation for Heisenberg's Uncertainty Principle is ΔxΔp ≥ h/4π where Δx and Δp are the standard deviations of position and momentum, and h is Planck's constant (10^-34). The indeterminacy in the position of a thrown baseball is 10^-30 millimeters. This indeterminacy becomes significant at tiny scales such as shown in Diagram 1 where the position of the electron in a Hydrogen atom is represented as a cloud.

[3] The de Broglie wavelength: λ = h/p where p is momentum.

[4] This is the Born rule. Note that the probability must be normalized so that the sum of the squared moduli of the probability amplitudes of all the possible positions is equal to one. For a sense of how the bottom wave in Diagram 2 would be represented in three dimensions, imagine the wave as a corkscrew winding along and around the x-axis with a larger bulge where the higher amplitude is concentrated.

[5] This is nicely demonstrated by Walter Lewin in this single-slit experiment video. As the slit becomes narrower, the position of the photons in the slit become more precisely defined which results in a wider spread of photons on the back screen (since direction is a component of momentum). See also Diagram 5 below. The reason why this wider spread of photons occurs is because amplitude flows from all positions in the slit to all positions on the back screen but a wider angle from the narrow slit is needed for destructive interference to occur.

Diagram 5 - Single-slit experiment

Is color real? (Part 2)

Are tiles A and B the same color?

In my previous post I raised the question of whether color is a property of things in the world (such as paint, strawberries and fire-engines) or whether it is a property of the way that humans see things.

In everyday usage, we generally associate color with things in the world. We suppose that the fire-engine is red not merely when we look at it but also when we are not.

That has raised some scientific and philosophical issues. One issue is that sometimes people name colors differently because of vague boundaries. That seems to be the case with my disagreement with Jason. The color of the stool was on the boundary of red and orange. Since we learn about objects and their colors by looking at them and categorizing them, it is possible that people categorize colors slightly differently.

Another issue is that physical differences in our eyes can cause perceptual differences. This is the case with color blind people who can't differentiate between red and green.

A further issue is that background lighting influences the color we perceive. This is primarily the case with "the dress" phenomenon where some people report the dress as being blue-and-black and others that it is white-and-gold. It turns out that people see the actual dress as blue-and-black so the phenomenon only occurs when viewing an image of the dress. The colors people see generally depends on how much background is shown in the image and also the external lighting when viewing the image on a computer screen.

Then we get the issues of light reflection. If a paint can is closed (or it is open at night with the lights off), then paint would not appear red since color perception requires light to reflect off the surface of the paint onto our eyes. And, of course, a blind person never sees the paint as red.

Finally, there are the wavelengths of light. Visible light covers a small range of wavelengths in the electromagnetic spectrum and constitutes most of the light that the sun radiates. So color can be associated directly with the wavelength as well.

So there are a lot of things to consider. From a pragmatic point-of-view, we don't want to have to deal with all these issues (and others we haven't thought of) every time we refer to color in our everyday lives. We just want to say that the fire-engine is red regardless of the background lighting or the perceptual capabilities of different people.

The way we have achieved this with our everyday concept of color is by abstracting away all the complicating details. We simply associate color with the object [1]. If there are differences in perception due to lighting or a person's perceptual capabilities, then we use the word "appears" to make that distinction. Thus, the fire engine is red, but it doesn't appear red at night or to a blind person. The paint in the closed can is also red even though there is no light reflecting off the paint inside the can.

The benefit to abstracting in this way is that not only is the concept simple to learn and intuitive to use, it is also independent of any particular physical explanation for why color appears as it does. If future scientific discoveries are made that replace our best current theories of light, the reflective properties of surfaces, our eyes, or how our brain implements perception, then our everyday language around color would not need to change. We would just know more about color than we did before. [2]

So this is why color is a property of things and not a property of how we perceive things.

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[1] We learn concepts ostensively by paradigmatic examples. In this case, we learn colors by pointing at objects like fire-engines under normal lighting and comparing and contrasting them with other colored objects. Color is real (it is about objects in the world, not our perception of objects) and abstract (it need not carve nature at its joints).

[2] A further consideration is that other creatures perceive colors differently to humans as a consequence of their different brain and eye structures. So the scope and context of color concepts needs to be recognized.

Yes, they are the same color - they just appear different

Is color real?

I had a metaphysical discussion with my two-year old son this morning. I put a stool back in the room where it belonged (the chair kind of stool, not the other kind).

My son says, "Want orange stool."

So I retrieve the stool and say, "Jason, it's not an orange stool, it's a red stool."

He says, "Orange."

I go and get an obviously red drum stick and an obviously orange toy building. Showing him the drum stick, I say, "Is this red?"

"Yes."

Pointing to the toy building, I say, "Is this orange?".

"Yes."

Pointing to the stool, I say, "So Jason, see how the color of the stool is like the color of the red drum stick? So the stool is red."

"No, orange."

So my son and I are clearly having a serious metaphysical disagreement. Surely the stool has to be one color or the other. Can the stool be orange for him merely because he thinks it is orange?

Before you take the objective or subjective side of the dispute, here is a further puzzle about color. When we buy a can of paint, it has the name of a color on the label, such as "Red". However what is the color of the paint inside the can when the lid is on? (Note: there is no light inside the closed can). Also, when the can is opened under normal lighting, what is the color of the paint for a blind person?

For my solution, see Part 2.

Quantum entanglement

Consider a box that contains two marbles, one red and one blue. Alice and Bob each take a marble from the box without looking at it. They then walk to separate rooms and Alice looks at her marble. If Alice sees a red marble, then she immediately knows that Bob has a blue marble. Hopefully you agree there is no great mystery here. Her inference depends merely on the correlation of the marbles. Whatever colored marble Alice has, Bob must have the opposite-colored marble.

Quantum entanglement is very much like that, except for one crucial difference. The two marbles are similarly correlated but they exist in a superposition of "the first marble is red and the second marble is blue" and "the first marble is blue and the second marble is red". To help visualize this, imagine a double exposure photograph that exhibits features of two different landscapes at the same time.

Without looking, Alice takes a marble from the box (which I will refer to as the first marble), Bob takes the second marble and they walk to different rooms.^[1] When Alice looks at her marble, the superposition disappears and there is a 50% chance of her seeing either a red or a blue marble. If she sees a red marble then, due to the correlation, she immediately knows that Bob has a blue marble. Indeed when Bob looks, his marble is blue. It does not matter if Bob is in the next room, or on the opposite side of the world, or in a galaxy millions of light years away, the marbles will always be the opposite color.

The puzzle for the various interpretations of Quantum Mechanics is to explain how the marbles are always correlated even though the color of Alice's marble is randomly determined at the precise moment that she looks (and not merely unknown to her as in the classical scenario). This is the substance of the EPR Paradox that Einstein (along with colleagues Podolsky and Rosen) devised to challenge the Copenhagen Interpretation. If Alice happens to see a red marble, it seems that Bob's marble somehow "knows" it has to be blue, either by communicating faster than the speed of light or via local rules that determine its color. Einstein presumed the latter mechanism, but this was subsequently proven to be impossible by John Bell some years after Einstein's death.^[2]

So how does the Many-World's Interpretation resolve this puzzle? The key insight is that it is not merely the pair of marbles that are in a superposition before Alice looks at her marble. Everything in the environment, including Alice and Bob, are in a superposition along with the marbles. The scenario configurations can be described as follows:

1. Alice picks the red marble and Bob picks the blue marble
2. Alice picks the blue marble and Bob picks the red marble
3. Alice has a marble and Bob has a marble (a superposition)^[3]
4. Alice sees a red marble and Alice has a red marble and Bob has a blue marble
5. Alice sees a blue marble and Alice has a blue marble and Bob has a red marble

When Alice and Bob each pick a marble that is in a superposition, amplitude flows to both configuration 1 and configuration 2. The total amplitude flowing to configuration 3 is the sum of the amplitudes from configurations 1 and 2. This is analogous to the double-slit experiment where photon amplitude flows through both slits and causes an interference pattern on the back screen.

Alice then looks and sees a red marble (configuration 4). She is now entangled with all the things on that amplitude path which includes the red marble and Bob who has the blue marble. On the other amplitude path, her twin is entangled with the blue marble and also Bob's twin who has the red marble. There are no amplitude paths back to configuration 3 (or over to configuration 5) from Alice's world branch. This is decoherence - an irreversible transition for Alice from independence to entanglement (correlation) with a single world branch. If she hides the marble and then looks at it again, it will continue to be red.^[4]

On this interpretation, there is no mystery to how the marbles correlate. Their correlation is simply the logical consequence of having a red and a blue marble in each world branch just as it is the logical consequence of having a red and a blue marble in the classical scenario. The 50% probability of seeing a red marble merely reflects Alice's knowledge of which of the two world branches that she will find herself in when she looks at her marble.^[5]

For my earlier posts in this series, see Visualizing Quantum Mechanics and Interpreting Quantum Mechanics.

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[1] In practice, decoherence effects would prevent actual marbles being put in a superposition, much less picked up and moved by people. However, as a thought experiment, it is analogous to actual experiments of two entangled photons with correlated spins that can be separated by great distances while remaining in a superposition.

[2] See Bell's Theorem which rules out local hidden variables.

[3] The superposition described in configuration 3 can be conceptualized in either of two equivalent forms:

• Alice has the first marble in a superposition of "red marble" and "blue marble" and Bob has the second marble in a superposition of "red marble" and "blue marble"
• A superposition of "Alice has a red marble and Bob has a blue marble" and "Alice has a blue marble and Bob has a red marble"

Mathematically, this can be represented as: A.B.(r1.b2 + b1.r2) = A.r1.B.b2 + A.b1.B.r2

In this expression, Alice (A), Bob (B) and the marbles (where 1 and 2 represent the first and second marbles while r and b represent red and blue) are each configurations with probability amplitudes (which are complex numbers). In the left-hand-side expression, the full system configuration is the complex product of Alice, Bob and the marble superposition. In turn, the marble superposition is the complex addition of the two distinct marble configurations (one of which is the complex product of the first red marble and the second blue marble while the other is the complex product of the first blue marble and the second red marble).

Notice that the left-hand-side expression is the factorization of the right-hand-side expression. In the left-hand-side expression, Alice and Bob are factored independently of the marble superposition. In the right-hand-side expression, Alice and Bob are, equivalently, part of a more general superposition where their amplitude is found in both branches. In both expressions, a red marble (r1 or r2) is always correlated with a blue marble (b1 or b2).

It's always useful to remember that a complex number can be represented as an arrow (phase vector) with its tail at the origin. To add two arrows, move the second arrow so its tail is at the head of the first arrow. Then draw a new arrow from the origin to the head of the second arrow. To multiply two arrows, sum the two angles and multiply the two lengths to draw a new arrow.

[4] Note that the effects of Alice's observation can't reach Bob faster than the speed of light. So if Bob doesn't look at his own marble, Bob remains independent of either world branch until, for example, Alice walks over and shows Bob her red marble (in accordance with the no-communication theorem). Decoherence is a physical process where Alice and, later, Bob are transformed from independence to correlation with one of the world branches.

[5] When Alice finds herself in configuration 4, it may seem that configuration 2 was not part of her history. However if it were not, then Alice's history would just be the classical scenario where she had picked the red marble from the box at the beginning but just didn't know it yet. However, in our quantum scenario, the marble was in a superposition before she looked which could only be caused by amplitude for Alice flowing through both configurations 1 and 2.

Interpreting Quantum Mechanics

In my last post, I discussed the mathematical intuition underlying Quantum Mechanics, which is the idea that particle configurations with probability amplitudes can cancel out when combined. That is because amplitudes can be positive, negative or complex numbers, not just positive numbers as classical probabilities are.

I also noted that when a photon arrives at the back screen in the double-slit experiment, we only see one of the slit detectors activated, which corresponds to one of the configurations. But how do we account for the configuration where the other detector was activated?

There are several interpretations, all disturbing, so therefore best described by reference to superheroes.

Copenhagen Interpretation: You, the conscious observer, create reality. If you look at the back screen, the photon will be a wave. If you look at the slits, the photon will instead be a particle. You don't feel like you're in a superposition, so you're an exception to the laws that those little particles follow.




De Broglie–Bohm (or pilot-wave) Interpretation: The photon surfs on a wave which carries it through one of the slits to a location on the back screen where destructive interference doesn't occur. Added bonus: Entangled particles can communicate faster than the speed of light in violation of special relativity.

Many-Worlds Interpretation: You observe one of the photons going through one slit while your twin observes the other photon going through the other slit. You are either observing a superposition on the back screen or participating in a branch of one when you become entangled with one of the photons. Also the entire universe is in a superposition. Things just look classical because you have to be standing in one configuration or another.





Instrumentalist Interpretation: Yada yada yada ... who cares? You can build really amazing stuff using quantum mechanics! Also affectionately called, "shut up and calculate!".

So there we have it. The Copenhagen Interpretation posits an observer-dependent reality and a mysterious wave function collapse. The Bohmian Interpretation requires information to travel faster than light.^[1] The Many-Worlds Interpretation causes incredulous stares. And, finally, the Instrumentalist Interpretation isn't an interpretation at all.^[2]

Since each interpretation uses the same mathematical formalism, is there any reason to prefer one to another on philosophical grounds? I think there is.

The famous Schrodinger's Cat thought experiment vividly demonstrates the logic of quantum behavior in terms of familiar, everyday things. The upshot is that the cat is in a superposition of being both dead and alive until we look in the box. But what does that mean? And why don't we normally observe such things?

To take the second question first, we actually can observe superpositions involving objects that are (just barely) visible to the naked eye. For example, a recent experiment demonstrated interference effects for the superposition of a tiny tuning fork vibrating and not vibrating. This is analogous to the double-slit experiment where photon amplitude flows through both slits and we observe an interference pattern on the back screen.

So what does this mean? It means that we have observed the effects of a single amplitude, and it is the sum of the amplitude for a vibrating tuning fork and the amplitude for a non-vibrating tuning fork.

How can this observation be explained in a coherent way? The idea that there is a single tuning fork that is both vibrating and not vibrating is a contradiction, so that fails. The idea that there is a single tuning fork that is either vibrating or not vibrating also fails, since possibilities can't cause interference effects. That leaves the idea that there is one tuning fork that vibrates and a second one that does not and that we are observing the combined effects of both.

If there are two tuning forks, what would explain the observation of a single tuning fork (that is either vibrating or not) when you try to detect the vibration? There are paths that entangle you with the vibrating tuning fork and paths that entangle you with the non-vibrating tuning fork and amplitude flows along both paths. Becoming entangled with one of the tuning forks destroys the interference pattern from your point-of-view (that is, you can no longer observe the effects of the other tuning fork because there are no amplitude paths from here to there).^[3]

The Many-World's Interpretation, despite its startling implications, seems to me to be the most coherent interpretation of both our everyday and quantum observations. That, of course, doesn't mean that it is true - that is ultimately an empirical question^[4]. However it is an intuitive and natural framework for conceptualizing our observed experience.

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[1] See Bells Theorem which rules out local hidden variables.

[2] The pragmatic and reasonable version of the Instrumentalist Interpretation is that the math works, whether or not we know what it means. In stronger versions, it asserts that there is no explanation to be had and that the math doesn't mean anything.

[3] This process is called decoherence and is only reversible in practice in microscopic environments where interactions with air and apparatus molecules can be controlled (it's easy to break an egg but difficult to put it back together again). So observing things in a single (or decohered) state is the everyday situation that corresponds to our classical intuitions. The difficult challenge for quantum computing just is how to reliably maintain quantum bits in a superposition of 1 and 0 values.

[4] Scott Aaronson has a great post explaining that while the Many-Worlds Interpretation is the obvious, straightforward reading of quantum mechanics, it is also provisional in a way that heliocentrism (as opposed to geocentrism) isn't. You could, in principle, fly a spaceship above the plane of the solar system and see the Earth and the other planets circling the sun. However you can't similarly travel to another world branch to meet your twin.

Visualizing Quantum Mechanics

Diagram 1: Wave-like interference pattern

In 1803, polymath Thomas Young performed the famous double-slit experiment demonstrating that light exhibits wave-like behavior. When light is shone on a plate pierced by two parallel slits, an interference pattern appears on the screen behind the plate. No light appears at those locations where the wave crests and troughs combine and cancel out, as shown in Diagram 1. Also, the light is most intense at the center of the screen where the waves combine constructively.

In more recent double-slit experiments it has been shown that the same interference pattern emerges even when light is emitted one photon at a time! How can this be? Surely each single photon must go through either one slit or the other, but then why wouldn't that produce the two-clumps pattern^[1] shown in Diagram 2 below? Perhaps an experiment could be done to observe whether the photon really does pass through one slit or the other.


Diagram 2: Particle-like versus wave-like behavior

So when detectors are placed at the slits, each emitted photon is detected passing through just one of the slits, just as we would expect, and not through both slits as a wave would. However the interference pattern then disappears!

Strangely, the photon exhibits particle-like behavior when we detect which slit it passes through but exhibits wave-like behavior when we don't try to detect which slit it passes through. Does the act of observation determine the behavior of the photon?^[2]

These observations and questions highlight the generally-perceived mystery around quantum mechanics. The purpose of this post is to describe a way to visually conceptualize what is going on in the double-slit experiment using the correct mathematical intuition. This can help us to think more clearly about these questions. So let's get started!

Consider a photon that is emitted toward the plate in Diagram 2 above. Let's also consider a single location on the back screen, labeled B0, where destructive interference would occur. There are two paths from the photon emitter to B0, one through each slit, labeled S1 and S2. A distinct state that a photon and experimental apparatus can be in (such as their positions) is called a configuration^[3].

A configuration has a value associated with it called an amplitude^[4]. An amplitude is expressed as a complex number in the form (a + bi) and can be visualized as an arrow that can point in any compass direction. Amplitude flows from prior configurations to subsequent configurations. For each configuration, the incoming amplitudes are summed. The amplitude is also transformed by rules, such as when a photon moves, changes direction through a slit or activates a detector.

Starting with the initial configuration (and amplitude) for our double-slit thought experiment, we can transition to subsequent configurations, follow the rules that transform amplitude, and see where we end up.

The configurations (including the amplitude transformation rules)^[5] are:

1. Initial configuration: (-1 - i) [arrow pointing south-west]
2. A photon goes from the emitter to S1: multiply by -1 = (1 + i) [north-east]
3. A photon goes from S1 to B0: multiply by -i = (1 - i) [south-east]
4. A photon goes from the emitter to S2: multiply by (0.5 + 0.5i) = (0 - i) [south]
5. A photon goes from S2 to B0: multiply by (-1 - i) = (-1 + i) [north-west]
6. A photon arrives at B0: (0 + 0i) [no arrow]

The total amplitude flowing toward B0 is the sum of the individual amplitudes flowing toward B0 (underlined), which is (1 - i) + (-1 + i) = (0 + 0i). The probability of a photon arriving at B0 is the squared modulus of the amplitude (a² + b²)^[6], which is 0² + 0² = 0%. Therefore no photon will arrive at B0, due to the individual amplitudes canceling each other out (i.e., the equal-length arrows pointing in opposite directions). This corresponds to the wave-like behavior that is observed when a series of emitted photons create an interference pattern.^[7]

Now consider a second experiment where photon detectors are added at slits S1 and S2, labeled D1 and D2 respectively. They will turn from off to on if they detect a photon passing through their slit. The rules are the same as for the first experiment. The configurations are:

1. Initial configuration: (-1 - i) [arrow pointing south-west]
2. A photon goes from the emitter to S1 and D1 is off and D2 is off: (1 + i) [north-east]
3. A photon goes from S1 to B0 and D1 is on and D2 is off: (1 - i) [south-east]
4. A photon goes from the emitter to S2 and D1 is off and D2 is off: (0 - i) [south]
5. A photon goes from S2 to B0 and D1 is off and D2 is on: (-1 + i) [north-west]
6. A photon arrives at B0 and D1 is on and D2 is off: (1 - i) [south-east]
7. A photon arrives at B0 and D1 is off and D2 is on: (-1 + i) [north-west]

In this experiment, the individual amplitudes flowing toward B0 (underlined) are flowing to two distinct configurations (since there can be no single configuration where D1 is both on and off), so the individual amplitudes are not summed. There is now a positive probability that the photon arrives at B0, with an equal probability of the photon being detected at either slit (the squared modulus of the amplitude for each final configuration is 2, so the ratio is 2:2). This corresponds to the classic particle-like behavior that is observed when the photon is detected going through one of the slits.^[8]

So complex addition of destination configuration amplitudes is the mathematical basis for our observations in the double-slit experiment. The classical intuition is that adding more paths to a destination makes it more likely to reach the destination. The quantum intuition is that adding more paths to a destination can make the destination unreachable since paths can destructively interfere.

This still leaves one more puzzling question. When a photon arrives at B0, we only see one of the detectors activated, which corresponds to one of the configurations. But how do we account for the configuration where the other detector was activated? I will leave this question for a future post.

The ideas presented here were inspired by Eliezer Yudkowsky's post on configurations and amplitude from his series on quantum physics. For my earlier posts on visualizing complex numbers, see Seeing complex numbers and Visualizing Euler's Identity.

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[1] To visualize photons exhibiting particle-like behavior, imagine someone with a gun firing bullets at the plate. For the bullets that pass through the slits, one clump of bullets would accumulate behind the first slit and a second clump of bullets would accumulate behind the second slit.

[2] That is, how should these counter-intuitive observations be interpreted? For one example, theoretical physicist John Archibald Wheeler once commented, "Actually, quantum phenomena are neither waves nor particles but are intrinsically undefined until the moment they are measured. In a sense, the British philosopher Bishop Berkeley was right when he asserted two centuries ago 'to be is to be perceived.'" - Scientific American, July 1992, p. 75

[3] A configuration (quantum state) is a distinct state that the universe is in at a point in time. In reality, it includes all the particles in the universe and all the particles that the emitter, plate, detectors and human observers consist of. What distinguishes one configuration from another is that at least one particle has a different property or position. When two configurations interfere, they combine to form a single configuration - an instance of the superposition principle.


Diagram 3: (a) Rotation to sine wave (b) 180^o phase shift

[4] The amplitude of a wave is the magnitude from rest to crest and is a real number - see Diagram 3. The configuration amplitude, which is the sense used here (and is elsewhere termed a probability amplitude), is a complex number (or phase vector) which additionally encapsulates a phase angle and can be visually represented as a radial arrow that points in any compass direction. For example, (0 + i) represents an anti-clockwise rotation of 90^o from 1 on the real number line and corresponds to an arrow pointing north (the wave peak). Combining two similar waves that are phase-shifted by 180^o (equal-length arrows pointing in opposite directions) results in wave cancellation (destructive interference).

[5] The initial configuration amplitude and the rules in the thought experiment are hypothetical, but serve to demonstrate the key conceptual point of amplitude interference. For the first photon path to B0, (-1 - i) * -1 * -i = (1 - i). For the second photon path to B0, (-1 - i) * (0.5 + 0.5i) * (-1 -i) = (-1 + i). These two final amplitudes have the same magnitude but are 180^o out of phase and therefore cancel out when in a superposition.

[6] This is known as the Born rule.

[7] We can also consider the center of the back screen where the light is most intense. In this case, two incoming configuration amplitudes with the same phase angle are constructively interfering, thus summing their magnitudes. For example, (1 - i) + (1 - i) = (2 - 2i) which corresponds to an arrow sqrt(2² + 2²) units in length pointing south-east.

[8] The detector (with an on or off state) could equally be replaced by a rock (that is or is not perturbed by a photon) and the computational logic would be the same.