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Bertlmann's Socks and the Nature of Reality

Celebrated around the world for his theories of special and general relativity, the young Albert Einstein was hailed as a bold, daring, revolutionary scientist. But as Einstein grew older, he became increasingly conservative and reactionary, especially when it came to quantum mechanics.

Einstein was a determinist. He believed in a certain world and disliked the inherently random, stochastic character of quantum mechanics. The idea that a cat could be in a superposition of being both dead and alive seemed totally absurd to Einstein. Clearly, something was missing from quantum mechanics, something which would take away this uncertainty and fuzziness. Notice that Einstein was not claiming that quantum mechanics was incorrect. He merely believed quantum theory offered an incomplete picture of the nature of reality.

Einstein and Bohr

Niels Bohr (left) and Albert Einstein (right).

His greatest opponent at that time was the Copenhagen scientist Niels Bohr, who had no such difficulties with quantum mechanics. The two giants entered into one of the fiercest public debates in the history of science. Thought experiment after thought experiment, Einstein tried to reveal the fatal flaws in the formalism of quantum mechanics, but Bohr remained unimpressed by Einstein's arguments. When Einstein told him that "God does not play dice with the universe", Bohr bitterly responded: "Stop telling God what to do."

 

The EPR paradox

But in 1935, Einstein believed he had finally found the Achilles' heel of quantum theory. Engraved in the formalism of quantum mechanics was a concept so bizarre and counter to our classically oriented minds, Einstein was convinced it held the key to proving quantum mechanics incomplete. The concept came to be called entanglement (or Verschränkung) by Erwin Schrödinger who referred to it as "the characteristic trait of quantum mechanics that enforces its entire departure from classical lines of thought."

Basically, whenever two particles interact, they become entangled. That is, their properties become inextricably connected, and remain so even when the particles separate at a later time by flying off in opposite directions. At first hearing, this idea of entanglement might sound rather innocent, but its consequences are so strange and ridiculous they defy common sense. Here was something so baffling it just had to be wrong.

Entangled Atoms

Artistic depiction of two entangled atoms.

So in a last heroic attempt to prove quantum mechanics incomplete, Einstein teamed up with two of his postdoctoral research associates at the Institute for Advanced Study — Boris Podolsky and Nathan Rosen. Together, they wrote a paper titled "Can Quantum Mechanical Description of Physical Reality be Considered Complete?" in which they introduced an intriguing Gedankenexperiment, which came to be known as the EPR paradox (for Einstein, Podolsky and Rosen).

The aim of this post is to look in some detail at the EPR paradox, which was intended to highlight the intrinsic weirdness of quantum entanglement in order to substantiate their claim that the formalism of quantum theory was only sketchy and incomplete. Regrettably, the crux of the EPR paradox has often been misunderstood in the past. So in order to prevent this from happening again, I believe it will be best to proceed in small steps.

One way of doing this is by telling you the story of Dr. Bertlmann — a bright yet somewhat eccentric scientist who was known far and wide for his unusual taste in clothing. There are two versions of the story: the classical version and the quantum version. Now, since there's nothing deeply mysterious about the classical version, we'll start with that one. The quantum version, on the other hand, is just the EPR paradox in disguise and will lead us straight into the heart of the problem. Here we go!

 

The curious case of Dr. Bertlmann's socks

Dr. Bertlmann loved socks, particularly flashy coloured socks. In his late teens, "as a protest against the establishment", Bertlmann got in the rather bizarre habit of never wearing matching pairs again, and he kept on doing so for the rest of his life. To be specific, whenever you saw him walking into the room with a purple striped sock on one foot, you could be absolutely certain he was wearing a yellow dotted sock on the other.

Mismatched Socks

What's more, since Bertlmann put on his pair of mismatched socks at random, there always was a 50% chance for the purple sock to end up on his right foot, with the yellow sock on his left, and another 50% chance for the exact opposite. The first situation can be represented by the state

\begin{equation}\left|purple\right\rangle_R\left|yellow\right\rangle_L,\label{B}\end{equation}

and the second by the sate

\begin{equation}\left|yellow\right\rangle_R\left|purple\right\rangle_L.\label{C}\end{equation}

Now, since no one could predict what the configuration would be on a given day before actually seeing the socks, everyone started off in the following entangled state:

\begin{equation}\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left|purple\right\rangle_R\left|yellow\right\rangle_L+\frac{1}{\sqrt{2}}\left|yellow\right\rangle_R\left|purple\right\rangle_L,\label{A}\end{equation}

where the \frac{1}{\sqrt{2}} in front of both terms refers to the fact that both terms occur with equal probability \mathfrak{P}=\left|\frac{1}{\sqrt{2}}\right|^2=\frac{1}{2}. If you're having any troubles understanding this equation, don't worry ! Go back to my first post on Schrödinger's cat and (re-)read the sections on quantum entanglement and the collapse postulate. That should clarify a lot!

Now imagine Mr. Bertlmann entering the room, with his right foot appearing first from behind the corner. As soon as you catch a glimpse of his foot, observing (i.e. measuring) the colour of his right sock, the wavefunction \left|\psi\right\rangle collapses (with equal probability 1/2) onto one of the two terms in equation \eqref{A}.

Suppose, for instance, you observe his right sock to be purple. In that case, the wavefunction has collapsed onto the first term:

\begin{equation}\left|\psi\right\rangle\rightarrow\left|purple\right\rangle_R\left|yellow\right\rangle_L,\end{equation}

and since \left|purple\right\rangle_R is perfectly correlated with \left|yellow\right\rangle_L you instantly know that his left foot is wearing the yellow sock, even though that foot is still hidden behind the corner.

 

Intergalactic socks

Rocket ShipSo far, so good. Now, let's spice things up a little, and imagine a wilder thought experiment. Suppose Bertlmann comes home and decides to store each sock in a different box. One box stays here on Earth, while the other is placed inside a rocket ship and fired to another galaxy 3 billion light years away. Since there's an equal chance for both socks to end up in either of the two boxes, we end up with the following state:

\begin{equation}\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left|purple\right\rangle_{\text{box}\;1}\left|yellow\right\rangle_{\text{box}\;2}+\frac{1}{\sqrt{2}}\left|yellow\right\rangle_{\text{box}\;1}\left|purple\right\rangle_{\text{box}\;2}.\label{D}\end{equation}

Now, let's say that after peeking inside the box on Earth (box 1), we find a yellow sock. At that point, we instantaneously know that the box in the other galaxy (box 2) must contain the purple sock, despite the humongous distance separating us from that sock and the fact that no one has actually looked inside it. Once again, there's nothing deeply mysterious about this.

 

Enter quantum mechanics

But here comes the twist! There's a fundamental difference between the classical story I just told, and its analogous quantum version. In the classical version of Bertlmann's socks, the socks always have a determinate colour; they are either purple or yellow. In our last example, for instance, the sock on Earth was yellow, and the other one was purple. This particular configuration was already set when Bertlmann first packed the socks and sent them away. Of course, we didn't know which sock was in which box until we actually looked in one of the boxes, but the act of looking obviously did not affect either of the two socks. Their colour was already fully determined long before our measurement.

Not so in quantum theory ! In the quantum version of Bertlmann's story, the colour of a sock, like most quantum properties, would generally be undetermined as long as we don't measure it. So a pair of entangled quantum socks in the state \eqref{D} would not have any definite colour; each sock would be in a superposition of being both purple and yellow (just as Schrödinger's cat can be in a superposition of being both dead and alive). Actually, when dealing with states such as \eqref{D}, any talk about the colour of the socks would be meaningless mumbo-jumbo. The socks only settle on a definite colour when a measurement is performed and the wavefunction \left|\psi\right\rangle randomly collapses onto one of the two colour-definite terms.

Intergallactic Sock 2

The colour of the sock in the faraway galaxy is undetermined until the moment we measure the colour of the sock on Earth.

Now, where does all this lead us? Well, in this version of the story, each box contains a sock whose colour is completely fuzzy and uncertain until we measure it. What is more, since the socks are entangled, we know that whenever we measure a yellow sock here on Earth, the other sock is guaranteed to settle on purple, and vice versa. That is, whenever we measure the colour of a quantum sock on Earth, we not only affect that sock, but also its entangled partner in the other galaxy.

 

Spooky action at a distance and quantum nonlocality

This however raises a puzzling question. How does the second sock in the faraway galaxy "know" that we performed a measurement on the first sock here on Earth? As David Keiser of MIT observed, there are no forces, no pulleys, no telephone wires or transmitters connecting both socks. So how could they be communicating with one another? Suppose that upon measurement, the sock on Earth would emit a radio signal to the other sock with the message "my colour is yellow." Given that the distance between both socks is 3 billion light years, it would take exactly 3 billion years for the message to reach the other galaxy, and only at that point would the second sock "know" it has to turn purple.

Yet, quantum mechanics tells us quite the opposite. According to quantum theory, it doesn't take 3 billion years at all, not even a flash second; it happens instantaneously ! The only way for this to work would be for a signal to travel much faster than the speed of light. But superluminal communication was something Einstein himself had proven to be utterly impossible within his theory of relativity. In the words of Alain Aspect, even though both socks "are separated enough so that there is no signal able to allow them to communicate, they still seem to be talking to each other." In short, how can our actions over "here" affect the state of affairs over "there", no matter how distant "there" might be?

Spooky ActionsThe idea that socks could be linked across space, and affect one another instantly as if the space between them did not even exist, ran counter to Einstein's cherished ideas about local realism according to which objects can only be directly influenced by their immediate surroundings. The violation of this principle of locality seemed to imply in other words that quantum mechanics was nonlocal. Einstein was so deeply bothered by these weird, mysterious long-range connections between socks and other quantum objects, he came to call them spooky actions at a distance.

The aim of the EPR paradox, which is basically the quantum version of Bertlmann's story, was to bring out this nonlocal character of quantum mechanics and put it in the spotlights to show how ridiculous it actually was.

 

Hidden variables

The only way out, according to Einstein, was to insist that the socks were behaving according to the first version of Bertlmann's story. That is, the colour-configuration of the socks was already decided when Bertlmann packed them in the two boxes — long before we did any measurement. Here, Einstein was acting as a realist, who believed in a Universe that existed independently of our minds and observations.

But then, if the socks had always been in either of the two colour-definite states \eqref{B} or \eqref{C}, then the wavefunction in \eqref{D} did not represent reality; it merely represented our knowledge, or rather ignorance, about the situation. To put it philosophically, the wavefunction in \eqref{D} had to be interpreted epistemically rather than ontologically. This also implied that the collapse of the wavefunction only corresponded to a change in our degree of ignorance about the true state of the universe, not in a change of reality itself.

From this point of view it was quite clear that quantum mechanics was incomplete. Somehow, there had to be further facts, so-called hidden variables, which determined the exact state but which were not given by quantum mechanics.

 

The clash of titans continues

Bohr's response was as vague as Carroll's Jabberwocky poem, but it was clear he did not agree. Bohr was convinced quantum mechanics offered a complete description of the nature of reality, and he had no problem accepting nonlocality as a necessary consequence.

So who was right? Bohr or Einstein? Well, it turned out that figuring this out was much more challenging than one might expect at first sight. Here's a hypothetical dialogue between the two giants to illustrate the point:

Einstein and Bohr 2Bohr: "Tell me Einstein, what is your opinion about Bertlmann's socks?"
Einstein: "Well, dear Bohr, in my view, Bertlmann's socks already have a definite colour well before you perform any measurement."
Bohr: "OK, suppose you are right. Then how could I check the validity of your claim?"
Einstein: "That's easy! Just look at the socks and you'll notice they have a definite colour."
Bohr: "Ah, but it is the act of looking that brought the socks in a definite state. Before that, there colour was completely fuzzy and uncertain."

As Brian Greene noted: "No one knew how to resolve the problem, so the whole question came to be considered philosophy, rather than science." Einstein died 20 years later, not knowing who was right.

But in the 1960s, this situation changed dramatically when the Irish scientist, John Bell, found a way to turn this philosophical question into an experimental one. What he found, however, will be the topic of my next post!

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Pieter Thyssen

Whereas his left brain was trained as a theoretical scientist, his right brain prefers the piano. At work, Pieter builds time machines (on paper) and loves to dabble in the history and philosophy of science. He often gets stuck in another dimension, contemplating time travel and parallel universes, or thinking about ways to save Schrödinger's cat (maybe). He explores the world on foot, and takes life one cup of (Arabica) coffee at a time. Follow him on Twitter @PieterThyssen or at thelifeofpsi.com. You can reach Pieter via email at pieterthyssen@gmail.com.

  5 comments for “Bertlmann's Socks and the Nature of Reality

  1. November 12, 2013 at 11:09 am

    I am seriously eager to know what John Bell did, well, for a torrent of reasons, but a few of them are here within my limited knowledge in this field: It all sounds to me too very weird and impossible, but more than that, very interesting and intriguing. If science is all about observation, why would we even bother about anything until we observe it? Quantum Physics always uses the language of observables, but not 'observed.' Then, why is this theory included in science? In fact, isn't Quantum Physics all about energy being discrete and discontinuous which brought out the concept of quanta and photons, etc. which eventually helped our understanding of some of the weirdest behavior of some forms of energy, especially light? Why do we have this entanglement and our worry about something out of our observation as part of science? As long as something is not observed/measured, it can always be anything, right? Why did they complicate such a natural wisdom with terms such as superpositions, etc. The cat in the box could be dead or alive only because of Bi's presence there. That means, are we not setting initial boundary conditions just in order to satisfy our thought process? The conditions in this cat experiment, life and death are like binary digits, we cannot have the middle way. But what if we replace these options in the experiment with, say, "cat might have either fainted and awake?" In such a case, doesn't a third probability arise, say, "the cat was crazily jumping when we opened the box," unlike what was expected from the base options of fainted and awake? This is, I guess, the same problem with Perturbation theory of Hydrogen atom; this atom could not be excited beyond a limit. It was straight forward to mathematize this problem. I guess you will eventually post your thoughts on many-body problems of Quantum Physics, I will surely look forward to them. Uff....by the time I framed up my first question and type it here in English (my foreign language), I am losing track of a myriad of other questions. So I am stopping with this long, irritating paragraph, incomplete yet. I am sorry to exploit access to reach you, but it looks like your blog is my only option to delve in this subject, since you seem to use a very friendly layman language. Thank you for sharing this so far!

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