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.

Are you smarter than a chimp?

I watched a great talk called, "How not to be ignorant about the world" by Hans Rosling. He demonstrates that people, regardless of education level, are often less knowledgeable about the world than they think. Hans jokes in his talk that monkeys randomly selecting the answers to his quiz do better than the people answering them!

To get the idea, try answering the four questions below to see how you go.

1. In 1950 there were fewer than one billion children (aged 0-14) in the world. By 2000 there were almost two billion. How many do UN experts think there will be in 2100?
    
   A. Four billion
   B. Three billion
   C. Two billion
    
2. What percentage of adults in the world today are literate – can read and write?
    
   A. 80%
   B. 60%
   C. 40%
    
3. On average, in the world as a whole today, men aged 25-34 have spent 8 years in school. How many years on average have women in the same age group spent in school?
    
   A. 7 years
   B. 5 years
   C. 3 years
    
4. In the last 20 years the proportion of the world population living in extreme poverty has...
    
   A. Almost doubled
   B. Remained more or less the same
   C. Almost halved

You can find the answers to the quiz at the end of this post along with the percentage of people who selected the correct answer. The questions were part of the Ignorance Survey that was conducted in the US in 2013.

So how did you go? If you didn't do very well then you're in good company with the vast majority of respondents! As you can see by the percentages, the respondents did worse than monkeys randomly choosing the right answers 33% of the time.

One of the issues Hans raised was whether the problem was due to people not reading and listening to the media. It turns out that after surveying the media with these questions, they fared no better than the general population. The problem was that even the media themselves didn't know basic facts about the world!

In the second part of the talk, Han's son Ola provided some tips for beating the chimps. He pointed out that the reason that we often get these questions wrong is due to:

1. Personal bias - we tend to generalize from our own experience which is not representative of the broader population
2. Teachers often teach outdated information based both on what they learnt during their schooling and on outdated text books
3. News bias - sensational and unusual events are more likely to make headlines and these are not representative of world events

These biases lead to the misconceptions that:

1. Everything is getting worse
2. The gap is increasing between rich and poor
3. People need wealth before social development can occur
4. Sharks are dangerous

Ola suggests that people's scores on the world fact survey will immediately and dramatically improve if they generalize in the opposite direction, recognizing instead that:

1. Most things improve
2. Global wealth can be represented as a normal bell curve with most people in the middle
3. Most people are already socially developed before they have wealth
4. Sharks are not actually very dangerous to us since they kill very few people - so recognize that fear exaggerates danger

--
Answers to the quiz (along with the percentage of respondents who selected the correct answer):

1. C (7%)
2. A (22%)
3. A (24%)
4. C (5%)

 

The Blue Eyes Puzzle (Part 2)

This follow-up post gives the solution to the Blue Eyes puzzle described here.

The answer is that the 100 blue-eyed people will leave the island on the 100th day.

The way to understand this answer is to first work through a much simpler version of the problem where there is only one-blue eyed person and one brown-eyed person on the island. Let's first try to solve it before the outsider arrives. The blue-eyed person doesn't see anyone with blue eyes, while the brown-eyed person sees one person with blue eyes. Since the blue-eyed person doesn't know whether there are any blue-eyed people on the island, he doesn't know whether he himself has blue eyes. So he will not leave the island.

However once the outsider informs the people that there is at least one blue-eyed person, the single blue-eyed person immediately knows that it must be him. Thus he leaves the island that night. Conversely, when the outsider makes her statement, the brown-eyed person doesn't yet know the color of his own eyes because he can already see someone with blue eyes. But he knows that if he does have brown eyes, then the other person will deduce their own blue eyes and leave the island that night. Or if he himself has blue eyes, then the other person will similarly not yet know the color of his eyes and will stay on the island. Once he observes the blue-eyed person leave the island that night, he then immediately knows that he himself has brown eyes.

So in this simple case, at least, the outsider's statement conveyed crucial information to the blue-eyed person that he didn't previously know and that made the difference to whether he left the island or not. Also, on observing the blue-eyed person leave the island, the brown-eyed person deduces that he himself has brown eyes.

Now consider the next simplest case where there are only two blue-eyed people and two brown-eyed people. Again, let's try to solve it before the outsider arrives. Each blue-eyed person can see one other blue-eyed person, while the brown-eyed people can see two people with blue eyes. Therefore everyone knows that there is at least one blue-eyed person on the island.

The first blue-eyed person realizes that if he himself has brown eyes then his scenario is identical to the single blue-eyed person scenario that we just analyzed. However, without the outsider's information, the other blue-eyed person cannot deduce that he has blue eyes and will therefore stay on the island. He similarly realizes that if he has blue eyes, he has no way of knowing this, so will stay on the island.

However when the outsider states that there is at least one blue-eyed person, the first blue-eyed person will now reason differently. He again realizes that if he has brown eyes then his scenario is identical to the single blue-eyed person scenario that we just analyzed. But he now knows that the other blue-eyed person would deduce that he himself has blue eyes and leave the island on the first night. So the first blue-eyed person now just has to wait until that night to see if the other blue-eyed person leaves the island. If he does, then he can deduce that he himself has brown eyes. If he does not leave, then he can deduce that he himself has blue eyes, and will leave on the second night. The other blue-eyed person reasons in the same way and also leaves on the second night. The brown-eyed person also reasons in a similar way. However, because he can see two blue-eyed people, he will wait until the second night to see what happens. When he observes the blue-eyed people leave the island on the second night, he then immediately knows that he himself has brown eyes.

So the outsider's statement, which only seemed crucial in the one blue-eyed person scenario was also crucial in the two blue-eyed people scenario. That's because it allowed each blue-eyed person to rule out the one-blue-eyed person scenario by observing what happened after the first night. It also allowed each brown-eyed person to confirm the two blue-eyed people scenario after the second night.

The same line of reasoning applies to the three blue-eyed people scenario. Without the outsider's statement, no-one will leave the island. With the outsider's statement the three blue-eyed people can rule out the two blue-eyed people scenario after the second night, deduce that they themselves have blue eyes and leave on the third night.

How do they conclude this? All three blue-eyed people hypothesize that if they themselves have brown eyes, then there are only two blue-eyed people on the island who, in turn, hypothesize that if they themselves have brown eyes, then there is only one blue-eyed person on the island. That hypothesized single blue-eyed person will deduce that he has blue eyes and leave on the first night as a result of the outsider's statement. But if no-one leaves on the first night, then that falsifies the hypothesized two blue-eyed people's hypothesis, so they would leave on the second night. But if no-one leaves on the second night, that falsifies the hypothesis of the three blue-eyed people, so they will all leave on the third night.

Conversely, the brown-eyed people will observe the three blue-eyed people leave on the third night and deduce that they themselves have brown eyes. The same style of reasoning applies to the 100 blue-eyed people scenario, with the blue-eyed people leaving on the 100th night and the remaining islanders deducing that they themselves have brown eyes.

So the outsider's statement did enable everyone to deduce their eye colors. But what new and useful information did it convey? In the single blue-eyed person scenario, it informed the blue-eyed person that someone had blue eyes. In the two blue-eyed people scenario, it informed the two blue-eyed people that the other blue-eyed person knew that someone had blue eyes. In the three blue-eyed people scenario, it informed the three blue-eyed people that the other blue-eyed people knew that the other blue-eyed people knew that someone had blue eyes. And so on.

In other words, the information that at least one person had blue eyes had become common knowledge. This meant that they all knew it, they all knew that they knew it, they all knew that they all knew that they knew it, and so on ad infinitum.

Another way to think about this puzzle is that the blue-eyed people know that they are either in a 99 or a 100 blue-eyed people scenario, but they don't know which one. Whereas the brown-eyed people know that they are either in a 100 or a 101 blue-eyed people scenario, but they also don't know which one. So the blue-eyed people will wait to see if the other blue-eyed people leave the island on the 99th night. If they do not leave, then the blue-eyed people deduce that they are in the 100 blue-eyed people scenario, and leave on the 100th night. The brown-eyed people observe the blue-eyed people leave the island on the 100th night, deduce that they are in the 100 blue-eyed people scenario, and thus know that they themselves have brown eyes. But remember that no-one will leave the island without the outsider's statement since her statement is necessary to falsify or confirm each persons' deeply-nested hypothetical about what the others know.

The key to solving this puzzle is to recognize that it has a recursive structure, observe the pattern that emerges with the simpler scenarios, and then apply that pattern to the more complex 100 blue-eyed people scenario. It's like the pattern that emerges when solving the factorial of four (or 4!). 4! = 4 * 3!, 3! = 3 * 2!, 2! = 2 * 1!, 1! = 1. Once the base case of 1! is solved, and the factorial pattern is understood, it's then easy to solve for higher-numbered factorials.

For some other expositions of the blue eyes puzzle, see xkcd and Terence Tao.

[Added Sep 24]
Bonus question: In a different version of the puzzle, the outsider comes back the day after making her public announcement, calls together all the people on the island, and makes a new public announcement:

"I'm terribly sorry, but I retract my statement from yesterday. I had a migraine that caused me to not see colors correctly so I don't actually know that I saw someone with blue eyes."

What effect (if any) will the outsider's new statement have?

The Blue Eyes Puzzle (Part 1)

I came across a fascinating logic puzzle that goes like this:

On an island, there are 100 people who have blue eyes and 100 people who have brown eyes. No-one on the island knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island that night. On the island, each person knows every other person's eye color, there are no reflective surfaces, and there is no discussion of eye color. Also it is public knowledge that the islanders are perfect logicians - if a conclusion can be deduced then they will immediately do so.

One day, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement:

"I can see someone with blue eyes."

Who leaves the island and on what night do they leave?

If you haven't seen this puzzle before, I encourage you to think about it before continuing.

...

All done?

Is your answer that no-one will leave the island? If so then I'm sorry to say that you are incorrect! Despite the fact that all the islanders can see many blue-eyed people, the outsider's statement does convey new information to the islanders and that information enables them to determine the color of their own eyes.

I will give the actual answer and explanation in a few days time. In the meantime, here is a suggestion to help you find the correct solution.

First try to solve a much simpler version of the puzzle. The simplest version is the scenario where there is one blue-eyed person and one brown-eyed person. Consider what the islanders can deduce before the outsider arrives. Then consider what they can deduce after the outsider makes his statement.

Now solve the next simplest version where there are two blue-eyed people and two brown-eyed people. Again consider what the islanders can deduce both before and after the outsider makes his statement. This will help you notice what the islanders learned from the outsider's statement and how it is necessary for solving the subsequent versions with three blue-eyed people and so on.

[Added Sep 22: The Blue Eyes Puzzle solution]

A simple solution to the Gettier problem

Long ago, Plato proposed that knowledge is justified, true belief. That is, your beliefs count as knowledge if you have good evidence for them and they are actually true. Otherwise they are either mere opinions or false beliefs.

This definition seemed to reflect the common use of the term through history. However, in more recent times, Edmund Gettier provided counterexamples showing that if a belief relies on a false premise, then it won't commonly be regarded as knowledge. For example, you may believe the correct time by looking at the clock on your wall. But your belief doesn't count as knowledge if your tacit assumption that the clock is working is mistaken.

My amendment to Plato's definition is to recognize that for a conclusion to be known, any essential premises also need to be known. This is a slightly stronger formulation than William Lycan's excellent no-essential-false-assumptions analysis.

My proposal for the definition of knowledge: For S to know that P is true means that:
(1) P is justified
(2) P is true
(3) P is believed
(4) S knows that the essential premises of P (if any) are true

One intuition for this definition is that knowledge is constructed hierarchically like building blocks. If a low-level building block fails, then building blocks that sit on it are not supported and so also fail.