An Exposition: Quantum Entanglement
An aura of indefinite mystery accompanies quantum entanglement. Yet in the end, it’s nothing more than a scientific idea. We expect an intuitive meaning and a concrete implication but quantum mechanics goes beyond that. Quantum entanglement explains, to a great deal, the limitations of our understanding. Things do travel faster than light. And here’s how. This is my attempt at exploring the very fundamentals of quantum entanglement.
Entanglement is often regarded as a uniquely quantum-mechanical phenomenon, but it is not. In fact, it is enlightening, though somewhat unconventional, to consider a simple non-quantum (or “classical”) version of entanglement first. This allows us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.
Entanglement arises in situations where we have partial knowledge of the state of two systems. Consider a system of squares and circles. It’s an odd example but I believe it’d do the trick. Our “objects” of interest, in particular, come in two shapes, square or circular. We identify this as their possible states.
The possible states are:
- square, square
- square, circle
- circle, square
- circle, circle.
We say that the objects are “independent” if knowledge of the state of one of them does not give useful information about the state of the other. Our first table has this property. If the first object is a square, we’re still in the dark about the shape of the second. Similarly, the shape of the second does not reveal anything useful about the shape of the first.
On the other hand, we say our two objects are entangled when information about one improves our knowledge of the other. Our second table demonstrates extreme entanglement. In that case, whenever the first object is circular, we know the second is circular too. And when the first object is square, so is the second. Knowing the shape of one, we can infer the shape of the other with certainty.
The quantum version of entanglement is essentially the same phenomenon — that is, lack of independence. In quantum theory, states are described by mathematical objects called wave functions. These wave functions consist of all the information we know about our quantum system.
To demonstrate entanglement, we’ll first consider spin. Particularly, the spin of an electron. Now, for the uninitiated, spin is a relatively counterintuitive idea. If you’d like an insight into it, read up:
Spin, explained very briefly, is a mathematical understanding of particle behavior. Some particles behave like they have an inherent angular momentum. They don’t actually have this momentum, they just behave as they do. This “behavior” is quantified by spin.
Now back to electrons. Electrons have half-integer spins. In the sense that they take half and negative half. But what does that mean? It means that they can take two possible states: spin up and spin down:
By the way, the line and angled bracket is just a notation thing. It’s a way to represent these quantum states. But we aren’t necessarily concerned with that here. If you are, however, interested in understanding the notation, here’s a primer:
The Language of Quantum Physics
An introduction to kets, wavefunctions, and quantum systems
medium.com
Anyway, if we were to now measure the spins of these two electrons, we’d get one of four possible permutations:
- |↑A⟩ and |↑B⟩,
- or |↑A⟩ and |↓B⟩,
- or |↓A⟩ and |↑B⟩,
- or |↓A⟩ and |↓B⟩
I understand that this is possibly difficult to read but quantum mechanics never had straightforward notation to begin with. If it helps, however, you could consider the spin-up and spin-down states as 0 and 1 respectively. Since these states are discrete, we get the following:
- 1, 1
- 1, 0
- 0, 1
- 0, 0
Now as we’ve established, finding the spin of an electron will yield one of those four possibilities. But when there is no external influence on our two-electron system, our electrons are in a superposition of all possible states.
Presupposing you know little to nothing about superposition, it is the ability of a quantum system to be in multiple states at the same time until it is measured. Take me on blind faith here.
Well, we know that an electron can take one of the two states:
|↑⟩ = spin up
|↓⟩ = spin down
But here’s the caveat. If our electron — our quantum system — is just left alone then it is said to be in a superposition of both of these states, In other words, the electron isn’t |↑⟩ or |↓⟩, it’s |↑⟩ and |↓⟩.
All we need to know is that our quantum system can be in a superposition of states and supposedly in more than one state at one time. However, that only holds for as long as it’s left alone and nothing is interacting with it. If you somehow manage to measure its spin, you’ll find that it’s in either one of the two states.
So now, back to our two-electron system. When it’s left alone, we can write its state as the following:
We’ve shown that it’s a superposition by adding all possible states together. And coincidentally, the numbers in front of each of the combinations (a, b, c, and d) are directly related to the probability of finding our system in those specific states.
Another thing worth noting is that when something external interacts with our system — say we make a measurement — then the system collapses into one of the four possible states. We’ll never measure these electrons to be in a superposition. They’ll always be discrete, specific values. And well squaring the numbers in front of those states tells us the probability we’ll find our system in that state. So, a² + b² + c² + d² is equal to 1.
But what does it mean for a system to be entangled? How do we know that it is entangled? Well, we established that the mathematical expression of our quantum state — when left alone — is a quantum superposition of all the conceivable states.
Now, if we can separate this needlessly long expression into two halves where one considers electron A and the other electron B, then we’d have something of that looked like this:
And if the wavefunction can indeed be broken down into these chunks, then it would be considered not entangled. Since this is a separable state, any information about electron A tells us virtually nothing about electron B. The mathematics behind separable states is brilliantly straightforward. But I suppose including it here won’t do justice.
Anyway, if we cannot separate this mathematical expression into just one part talking about A and the other about B, then it would be considered entangled.
But first, what exactly do we mean when we call a system entangled? Well for the purposes of this article, we’ll say the following:
The probability of a measurement on one particle in a certain state, changes after we interact with the other particle.
So if you measured the spin on particle A, you'd find that although unrelated, you’ve affected the spin of particle B. So essentially what we can say is this:
The influence of a measurement changes not just one particle, but the entire system itself
Bell States
Well, to illustrate quantum entanglement, here’s an idea. Consider a very specific example where the wave function of the system is as shown below:
Although this may look remarkably confusing, the amount of information we can derive from this function is beyond brilliant. First, notice that as opposed to the superposition of four states, we only have two here. Meaning that the possibilities of these two electrons being the same spin state is 0. So we can render those components void.
Before measurement, we know that we have a superposition of two states. One, where electron A is spin up and B is spin down, and two, where electron A is spin down and B is spin up. So as far as that goes, we can say that the chances of finding electron A in a spin-up state is 50%.
Now, hypothetically, let’s assert that we’ve made a measurement on our quantum system. And we find that the second electron — electron B — is in a spin-down state. So after measuring B, we understanding that A must be in a spin-up state since those are the only two possibilities.
And well, vice versa. Suppose electron B is in a spin-up state, we understand that A must be spin-down. Since again, that’s the only other possibility. But why’s this important?
Well before we did any measuring to our system, there was an equal probability of being presented with two outcomes: spin-up, spin-down, or spin-down, spin-up. Once we did make the measurement, however, the system collapsed to just one of these two outcomes and from the measurement of just one electron, we found the spin state of the other’s. In other words, this system was quantumly entangled.
The Good Stuff
Well the math checks out and we’ve established that wavefunctions that can be “seperated” are not entangled. But one caveat that goes over the heads of many is that in the specific Bell State I’ve proposed, measuring particle A first will yield some probability of B and measuring B first will yield some probability of A. These probabilities are equal. However, this is true only for a very specific scenario.
In most cases — almost all — measuring A first will yield a completely different probability for B as opposed to measuring B first and finding the probability on A. As it turns out, A affects B and B affects A. Einstein famously called this “spooky action at a distance”.
A discourse with my Physics teacher was particularly insightful and I hope it makes sense to you too. It went like this:
Two particles, A and B, can be in a quantum system together. But suppose you somehow have managed to separate these two particles. You have A in one hand and B in another. Now, take them and throw them off in opposite directions. A goes left and B, to the right. Now before any measurement, we know that A is in a superposition of both a spin-up and a spin-down state. But the instant you do make a measurement — as we’ve established here — you affect the entire system itself. Which would mean you also affect particle B. And this must mean that information from A travels to B. And the speed, to our surprise, is about four order of magnitude greater than that of light. Quantum entanglement transfers information at around 3-trillion meters per second. This is a lower speed limit, meaning as we collect more precise data, you can expect that number to get larger. Now at a speed greater than light, physics breaks down. Newtonian and classical mechanics, as far as we’re concerned, has always explained a cause-and-effect relationship. That’s most of how we understand physical phenomena. But here, with the advent of entanglement, it appears that it’s an effect-and-cause relationship.
Something is the result of something happening before something happens.
Now if that didn't make sense, which I suppose it didn’t, consider this: the electron A “communicates” with B. Since A and B are both in a superposition, once A is measured, B is too. That’s what it means for a system to be entangled. But now, if you consider the idea that A has been measured and B’s state is apparent due to A’s measurement, then you’d hit walls of theoretical nonsense. Because if information between A and B travels faster then light, then it must mean that the quantum information traveled back in time. And if it travels back in time, we understand that the state of A must be the effect of a measurement on B that never happened.
Here we have a remarkably precise theory that tells us something we can’t even begin to wrap our heads around. We just don’t know. Maybe scientists are seriously stupid. it’s got to be God after all.
And with that, I hope I made sense. Thank you for reading!
