Note: This post is (somewhat) a continuation of my previous post about making secret codes using quantum superposition. Check it out to better understand the analogy to pennies and quarters I make in this post!
To get us started, we are going to spend a little time getting comfortable with something that made Einstein very uncomfortable. First, check out this Veritasium video, which does a fantastic job of explaining the fundamental features of quantum entanglement. You should watch the whole thing (’cause it’s awesome), but you only really need the first 3 or 4 minutes for most of this post.
As described above, all fundamental particles have an intrinsic property called spin. They aren’t actually spinning (if electrons were actually small spinning spheres, they would be spinning so fast that they moved faster than the speed of light!). But it’s a good analogy, because they have a measurable angular momentum and orientation, just like anything that does spin, like a top. Just don’t use an electron to test whether you’re in the dreamworld though, because they never stop having spin.
Thanks to good ol’ quantum superposition, whenever you try to measure the spin of a particle, it fundamentally changes the spin to either spin ‘up’ or spin ‘down’ relative to the direction we measured it in (‘up’ and ‘down’ correspond to our 1s and 0s for polarizers, Penny and Quarter Machines, or whichever quantum measurement device you prefer) .
There are situations where two particles must have equal and opposite spin to conserve angular momentum, such as when an electron-positron pair form spontaneously out of thin air, or maybe a spin zero particle decays into two spin 1/2 particles. As described in the video, because of quantum mechanics, these particles don’t have a well-defined spin direction (yet), but because of conservation of angular momentum, their spin must be opposite that of the other particle. This phenomenon is known as quantum entanglement.
Spukhafte trist Fest
Now here’s why its spooky. If we measure the spin direction of one particle, we immediately (like, immediately) know what direction the spin of the other particle must be.
Now, if we tried to Skype with the New Horizons probe when it was getting all paparazzi with Pluto, we’d be waiting for 10 hours before realizing the screen was lagging. If the sun spontaneously ceased to exist, we wouldn’t even know it for a solid eight minutes. Information just can’t travel faster than the speed of light. Those are the rules. Yet, somehow, if our friend and us have two entangled particles, and we measure the spin of our particle (let’s call him Andy), we immediately know the spin of the other particle (Ollie), even if it’s lightyears away! Spooky.
However, it is important to realize that this entanglement is fragile. When you measure Andy’s spin, Andy and Ollie are no longer superpositions of possible states. They now have a well-defined spin direction, and no longer rely on each other to ensure conservation of angular momentum.
Now let’s shift into pennies and quarters from last post. Two entangled coins shoot out of a magic Black Box® (it’s magic, black, a box, and spits out entangled coins. That’s all we need to know). One gets to you immediately and one begins the journey to your friend who lives on Mars. You made a deal that whoever gets a heads penny gets to have a party.
The coins are not ‘well-defined’ until your coin goes through your Penny Maker®, which measures a heads up penny. Not only do you get to have a party, but you can take pleasure in knowing immediately that your friend doesn’t get to have a party! It doesn’t even matter if the coin has made it to him yet, it is inevitable that he will spend his evening sad and alone, questioning his choice of friends.
If your head hurts, that’s good, because this is the only situation we know of where information is legitimately traveling faster that the speed of light! But don’t get out the champagne yet. The only ‘information’ that breaks the rules is basically the result of a completely random coin toss, and there’s no way to use the information to do anything clever or interesting. Odds are that the next time around, you will be the one alone, questioning your gambling habits.
The EPR Paradox, Hidden Variable Theory, and Bell’s Theorem
Now, Einstein and his buddies Podolsky and Rosen had a problem with the murkiness of saying the entangled particles don’t have a “well-defined” spin direction until one of them is measured. What does that even mean? In 1935 they questioned the completeness of our understanding of quantum entanglement with a thought experiment that came to be known as the EPR Paradox (awesome band name btw).
Basically, Einstein thought that what is really happening is simple: the coins coming out of the black box are predetermined, just in a way that is hidden to us. So if the left coin ‘knows’ to always give heads no matter how it’s measured , the right coin knows to always give tails. Equivalently, if the left coin changes its identity over time (say, a tails up penny one second, a heads up quarter the next), the right one would change in a complimentary way simultaneously (in this case a heads up penny, then a tails up quarter) without any of this spooky action at a distance nonsense. Whatever the measured results, each coin was just following the orders it got from the black box. If that’s the case, then they are correlated by design, but not because of any tricky communication between them. This is called (local) hidden variable theory.
As an example, imagine that you see two people on either side of a wall, unable to see each other, but doing exactly the same dance routine to a funky beat.
You wouldn’t assume that one of them is coming up with a move, then communicating it quickly to their friend through a headset, would you? The simpler explanation is that they both learned the choreography, timed to the music, ahead of time. It looks like they are performing in relation to each other, but really, they’re just following the choreography, the variable that is hidden to us. If we knew the choreography, then we could predict what move they would each do next. Simple.
Unfortunately, it turns out Einstein and bros were wrong. As spooky as it sounds, it seems like the useless faster-than-light connection is here to stay, and locality just doesn’t cut it. John Steward Bell came up with a very clever way to prove this. I’d try to explain it, but Veritasium does a far better job in the video at the top than I ever could. Besides, we’ve got to get to the quantum spy stuff!
Quantum Key Distribution: E91
Alright, enough of that, on to making secret codes. Again, the first step in encryption is getting a secret code (or “key”) that both you and the other party know (known as ‘key distribution’ or ‘key exchange’). Artur Eckhart came up with this particular key distribution protocol in 1991, and so of course it is known as E91.
Say hello again to Bob and Alice.
Instead of Alice sending Bob some coins that she flipped, this time a magic Black Box® is going to create a set of entangled coins and send one to each of them. Remember that in order to find out what coin they received, Alice and Bob must first measure it with a Penny or Quarter Maker. A Quarter Maker, for example, lets quarters through unharmed but destroys pennies and mints a new (randomly oriented) quarter.
The coins Alice and Bob receive from the Black Box are equally likely to be measured as a penny or a quarter: it is completely random. This means that whatever she chooses to measure with, there is a 50% chance of it spitting out heads, and a 50% chance of it spitting out tails.
Let’s see what happens. For the first coin, Alice picks a Penny Maker, and Bob a Quarter Maker. First, the Black Box sends them their coins:
Alice gets her coin first, and measures a heads up penny. Because the two coins were entangled, Alice now knows for a fact that the coin headed Bob’s way is a tails up penny.
Bob picked a Quarter Maker, which means there is an equal chance of him measuring tails or heads. He happened to measure a tails up quarter.
Unfortunately, that doesn’t do anyone much good. Alice thinks that Bob has a tails up penny, when in fact he is equally likely to have a heads or tails up quarter.
However, if for the next coin they both happen to choose a Quarter Maker, it gets more interesting. First, the Black Box spits out a new set of coins. Then Alice measures, and gets a tails up quarter. Again, she now knows for a fact that the coin headed Bob’s way is a heads up quarter:
This time though, Bob picked a Quarter Maker, so the heads up quarter makes it through his measuring device unscathed. That means that in this case, Alice believes that Bob has a heads up quarter, and she’s right.
So, thanks to quantum entanglement, Alice knows exactly what Bob will measure, if he uses the same measuring device as her. To take advantage of this to make a key, they get the Black Box to send them a long string of many entangled coins, randomly choosing how to measure each one, and writing down what gets spit out.
Alice and Bob know that they don’t have the same list of coins, so they again have a public yelling match, telling each other whether they received a penny or quarter. Bob publicly yells to Alice whether each coin he received was a penny or a quarter. Again, with two coin types, they chose the same device only 50% of the time.
They throw out the coins where they didn’t choose the same device, and now know that for every heads or tails that Alice kept, Bob has the opposite. They decide to use Alice’s coin orientations (heads or tails) as their secret key.
Just like last time, even though they shouted out loud in public about what kind of coin they measured each time, they never have to say whether they measured heads or tails. That part stays secret the whole time! Now they can finally start sending encrypted messages to each other about how terrible their co-workers are.
Eve the Eavesdropper
In the first post, we talked a lot about Eve the Eavesdropper. This time around, it’s a bit more complicated, but Alice and Bob can still find out whether Eve was listening in.
To verify whether or not Eve was listening in with her own measuring device, Alice and Bob would actually need to have used at least three types of coins (say, nickels and a Nickel Maker). Then, by comparing their results for all the coins not in their key (both coin type and heads or tails), they can perform the same type of Bell inequality test as is explained in the second half of the Veritasium video above.
If Eve was eavesdropping, she would have introduced local realism to the system, and they wouldn’t get the relationship expected by Bell’s theorem. For any given coin, she could pull it off and trick them (as below), but as long as they check enough coins, they’ll catch her, every time.
Let’s clarify this with an example. First, the black box spits out some coins:
The coin gets to Eve first. She measures it, then sends the new coin on to Bob. Now, as soon as Eve peeks at one of the coins (let’s say she gets a tails up penny), she immediately knows what the coin that was sent Alice’s way is (in this case, a heads up penny). However, by measuring it, she has ensured that the coin she sends to Bob and the coin on its way to Alice are no longer entangled.
So when the left coin gets measured by Alice, the right coin doesn’t care anymore. They can measure that their coins are behaving as would classically be expected, instead of how quantum entangled coins have been shown to behave, and surmise that Eve must be the culprit. As in BB84, it doesn’t fix the eavesdropping problem, but it’s still pretty darn spooky.