According to the Earth-based observer, the spaceship will arrive at the star in 5 years. However, because of time dilation, the spaceship’s clock will only read 3 years of elapsed time on arrival. To an astronaut on the spaceship, the distance to the star appears to be just 2.4 light years because it took him just 3 years to get there while traveling at 80% light speed.

This situation is sometimes explained as a consequence of length contraction. But what is it that’s contracting? Some authors put it down to space itself contracting, or just distance contracting (which it seems to me amounts to the same thing) and others say that’s nonsense because you could posit two spaceships heading in the same direction momentarily side by side and traveling at different speeds, so how can there be two different distances?

So what is the correct way to understand the situation from the astronaut’s perspective?

**Physicist**: Space and time don’t react to how you move around. They don’t contract or slow down just because you move fast relative to someone somewhere. What changes is how you perceive space and time.

When you measure the length of something in space (in other words, “normally”), the total length isn’t just the length in the x or y directions, it’s a particular combination of both that works out exactly the way you’d think it should. When you measure the length of something in *spacetime*, the total length isn’t just the length in the space or time directions, it’s a particular combination of both that works out in more or less the opposite of how you’d think it should.

We don’t talk about the three dimensions of space individually, because they’re not really distinct. The forward, right, and up directions are a good way to describe the three different dimensions of space, but of course they vary from perspective to perspective. Just call someone from the opposite side of the planet and ask them “What’s up?” and you’ll find yourself instantly embroiled in irreconcilable conflict. Everyone can agree that it’s easy to pick three mutually perpendicular directions in our three-dimensional universe (try it), but there’s no sense in trying to specify which *specific* three are the “true” directions.

A meter stick is a meter long (hence the name), so if you place it flat on a table and measure its horizontal length (with a… tape measure or something), you’ll find that its horizontal length is 100cm and its vertical length is zero. Given that, you could reasonably divine that it must be 100cm long. But if you tilt it up (or equivalently, tilt your head a bit), then the horizontal and vertical lengths change. There’s nothing profound happening. To handle a universe cruel enough to allow such differing perspectives we use the “Euclidean metric”, , to find the total length of things given their lengths in each of the various directions. The length in any given direction (x, y, z) can change, but the *total* length (d) stays the same.

Einstein’s big contribution (or one of them at least) was “combining” time and space under the umbrella of “spacetime”, so named because Germans love sticking words together in a traditional process called (roughly translated) stickingwordstogethertomakeonereallylongdifficulttoreadandoftunpronounceableword.

The different spatial dimensions are equivalent. To see for yourself, walk north and south, then walk east and west. Unless you’re carrying a compass, you shouldn’t notice any difference. But clearly time is different. To see for yourself, first walk north and south, then walk to tomorrow and back to yesterday. So when someone cleverly volunteers “we live in a 4 dimensional universe!”, they’re being a *little* imprecise. Physicists, who love precision slightly more than being understood, prefer to say “we live in a 3+1 dimensional universe!” to make clear that there are three space dimensions and one time dimension.

But while time and space are different, they’re not completely separate. In very much the same way that the forward direction varies between perspectives, the “future direction” also varies. And in the same way that rotating perspectives exchanges directions, moving at different velocities exchanges the time direction and direction of movement. The total “distance” between points in spacetime is called the “interval”, L. For folk familiar with the Euclidean metric, the “Minkowski metric” should look eerily familiar: . Some folk will flip the sign on this, , because it makes it a little easier to talk about the time experienced on a particular path (in fact, I’m gonna do that in a minute), but the important thing is not the sign of this equation, it’s that it’s *constant* between different perspectives. It should bother you that can be negative, but… don’t worry about it. It’s fine.

If you’re wondering, the spacetime interval is a direct consequence of rule #1 in relativity: the speed of light is the same to everyone. The short way to see this is to notice that if you find the interval between the start and end points of a light beam’s journey, the interval is always zero because . The long way to see why the interval is what it is, is a little long.

There are two things to notice about the spacetime interval. First, that “c” is the speed of light and it *basically* provides a unit conversion between meters and seconds (or furlongs and fortnights, or whatever units you prefer for distance and time). So 1 second has an interval of about 300,000 km (one “light-second“), which is most of the distance between here and the Moon. It turns out that the speed at which light travels comes from the “c” in this equation. So the speed of light is dictated by the nature of space and time (as described by the Minkowski metric), not the other way around. Which is good to know.

Second and more important is that negative. That *really* screws things up. It is arguably responsible for damn-near all of the weird, unintuitive stuff that falls out of special relativity: time dilations, length contractions, twin paradoxes, Einstein’s haircut and marriages, everything. In particular (and this is why the exchange between distance and duration is so unintuitive), if is constant, then when d increases, so does t.

This is in stark contrast to regular distance, where if is constant, an increase in x means a decrease in y. Picture *that* in your head and it makes sense. Picture relativity in your head and it doesn’t.

Now brace yourself, because here comes the point. The original question was about a journey that, from the perspective of Earth, was d = 4 light-years long, at a speed of v = 0.8c, and taking t = 5 years. The beauty of using “light units” (light-years, light-seconds, etc.) is that the spacetime interval is really easy to work with. The interval between the launch and landing of the spaceship is:

So the interval is L = 3 light-years.

Like regular distance, the power of the spacetime interval is that it is the same from all perspectives. From the perspective of the spaceship the launch and landing happen in the same place. It’s like a narcissist on a train: they get on and get off in the same place, while the world moves around them. So d = 0 and it’s just a question of how much time passes:

So, t = 3 years because 3 light-years divided by the speed of light is 3 years.

So just like changing your perspective by tilting your head changes the horizontal and vertical lengths of stuff (while leaving the total length the same), changing your perspective by moving at a different speed changes length-in-the-direction-of-motion and duration (while keeping the spacetime interval the same).

That’s time dilation (5 years for Earth, but 3 years for the spaceship). Length contraction is a little more subtle. Normally when you measure something you get out your meter stick (or yard stick, depending on where you live), put it next to the thing in question and boom: measured. But how do you measure the length of stuff when you’re moving past it? With a stopwatch.

So, like the original question pointed out, if it takes you 3 years to get to your destination, which is approaching you at 0.8c, then it must be 3×0.8 = 2.4 light-years away. Notice that in the diagram with the planets above, on the left they’re 5 light-years apart and on the right they’re 2.4ish light-years apart (measure horizontally in the space direction).

It feels like length contraction should be more complicated than this, but it’s really not. You can get yourself tied in knots thinking about this too hard. After all, when you talk about “the distance to that whatever-it-is” you’re talking about a straight line in spacetime between “here right now” and “over there right now”, but “now” is a little slippery when the “future direction” is relative. Luckily, “time multiplied by speed is distance” works fine.

There are a few ways to look at the situation, but they all boil down to the same big idea: perspectives moving relative to each other see all kinds of things different. Reality itself, space and time and the stuff in it doesn’t change, but how we view it and interact with it doesn’t *quite* follow the rules we imagine.

But the Uncertainty Principle is a statement about what things are actually like in reality, and the weird limitations on how well we can do measurements is just a symptom. In a very fundamental, profound, and *physical* sense, everything is a little uncertain.

To be clear, you *can* measure the position and momentum of a single particle (or many particles or whatever quantum system you prefer) very precisely, and there’s nothing stopping you from getting very precise results from both measurements. The problem is that those precise measurements don’t really tell you much about the actual quantum state you’re looking at. You can prepare a series of identical quantum states and measure each of them in exactly the same way, but because quantum states are generally a combination of many different states together (what folk in the quantum biz call a “superposition of states”) you generally don’t get the same result over and over.

An electron in an atomic orbital is in a state with one amount of energy, but many positions. The electron is kinda “smeared out” around the atom. So if you measure the energy you get a definite result, but if you measure the position you could get any result within a range of positions (a small range, what with atoms being small). You can picture this as being like a musical chord and either asking “what chord was played?” or “what note was played?”. For example, the C chord is composed of the C, E, and G notes. If you do a chord measurement (this is not an actual thing, but bear with me), you get a definite answer: C. If you do a note measurement (again, more quantum metaphor than mechanics), then you get one of three results: C, E, or G.

The randomness of the Uncertainty Principle has the same root cause: a single quantum state being composed of many different states at the same time. Like the chord example, a position state is made up of a range of many momentum states. Unlike the chord example, the reverse is also true; a momentum state is composed of many position states. Unintuitively, the fewer position states something is in (the more specific the position) the more momentum states it is also in. Unfortunately, because of this fact (that position is describable in terms of momentum and vice versa) you can directly derive the uncertainty principle mathematically. In other words, assuming that every experiment ever designed to refute the basic physics didn’t all fail accidentally, the Uncertainty Principle is built into the universe and no cleverness or engineering will overcome it.

When you prepare many particles (or any other quantum system) in identical quantum states, measure them one after another, and write down the results, you’ll find that there’s some spread in where they show up as well as their momenta. You can prepare states with very little spread in their position or with very little spread in their momenta (so called “squeeze states”), so the Uncertainty Principle isn’t as simple as “everything is random and unpredictable”, it’s about *pairs* of measurements applied to “conjugate variables” (position and momentum are the classic example).

The Uncertainty Principle says that if you look at the spread (standard deviation) of those two measurements for many copies of any given state and multiply those spreads together, their product is always greater than some minimum amount. Explicitly, if and are how spread out the position and momentum are, then . This, it’s worth noting, is a *really* tiny lower bound. If you’re certain that a brick is in a box, then (were you so compelled) you could nail down its velocity to within around 0.000000000000000000000000000000002 meters per second, which is arguably fairly certain.

Entanglement doesn’t do much to change the picture. With entangled pairs of particles, a measurement on one mirrors a measurement on the other. You can entangle any property (energy, polarization, delectability, hell even *existence*), including position and momentum. In a dangerously succinct nutshell, entanglement basically/sorta gives you two chances to make a measurement on a quantum state. Assume particles A and B are position-entangled. If you measure the position of A you’ll be able to say “ah, it’s over here” and if you measure the position of B you’ll be able to say “it sure is”. The two measurements, although otherwise random, agree.

But what you’d really like to see is a precise position measurement on A and a precise momentum measurement on B. It turns out: that’s fine. Once again, that spread of results shows up. If A shows up in some region, there is a corresponding set of momentum states that are compatible with that (if a certain “chord is played” there is a particular “set of notes” involved) and when you measure B, you’ll see one of those.

So entanglement does give you another chance to measure a quantum state with as much precision as you might desire, but… it doesn’t really change anything. The Uncertainty Principle doesn’t say “you can’t simultaneously measure position and momentum with nigh perfect precision!”, it says “it doesn’t matter if you do!”.

The giant microscope picture is from here.

]]>Clouds are just a bunch of moisture in the air (contain your shock). What’s a little surprising is that the transparent, non-cloudy air around them typically has almost the same amount of moisture. So when you see clouds in the sky, you’re not seeing a few wet blobs surrounded by dry air, you’re seeing a *lot* of humid air, some fraction of which has tipped past the dew point and begun to condense into visible vapor.

A patch of air can only hold so much water vapor. Hotter or denser air can hold more, and colder or thinner air can hold less. The “dew point” is where the air can no longer hold water vapor, which instead begins to condense out and become visible.

Evaporated water condenses when the humidity of the air it’s in increases too much, or when the temperature drops too much. That’s why you can see your breath when it’s cold outside. The air in your lungs is warm (because you’re a mammal) and it has plenty of opportunity to pick up moisture (because lungs are soggy and gross). Mixing with cold air means that the temperature of your exhalation crashes and drops through the dew point, making it visible. Mixing with lots of nearby drier air dilutes your breath, dropping the humidity in any particular parcel of air, and raises the breath/outside-air mixture back above the dew point.

Clouds are governed by the same physics. If you spend some time staring at clouds (and why wouldn’t you?), you’ll find that they continuously grow and shrink, appearing and disappearing. A given cloud maintains its rough shape and position because it takes time for conditions in air to change; if the air in some region is on one side of the dew point, then it’ll probably stay there for the next few minutes. But if you speed up time, you find that clouds don’t hold their shape any better than steam coming out of a kettle.

The big difference between clouds-in-the-sky and breath-on-a-cold-day is what causes the air to pass through the dew point. The dew point depends on both the humidity and temperature. For breath, the local humidity fluctuates a lot (as anyone with close-talking friends can tell you). For clouds, the humidity stays relatively constant, but as the air changes temperature (mostly through the expansion or contraction from changing altitude) different regions pass through the dew point and become clouds. If you’re in the middle of the ocean or Kansas (or any other flat, featureless landscape), there’s no particular reason for any given location in the sky to have a cloud. It’s just the luck of the draw (humidly speaking).

But when the conditions change predictably, the clouds appear and disappear predictably and you get “lenticular clouds”. Here are some beautiful examples. Lenticular clouds are a great way to see that clouds aren’t “objects” that move through the sky, they’re regions that, for whatever reason, are below the dew point.

Normally air moves across the land or sea in a “laminar”, smooth way, with nary a tumble or nor turbulence. Across grasslands clouds tend to vary slowly and randomly, but when there’s an obstruction, like mountains, suddenly the air is forced to flow in a particular pattern. If that pattern involves abrupt changes of altitude, then the air experiences abrupt changes in pressure and temperature which leads to abrupt changes in cloudness. The same amount of moisture is present in the air at the foot of the mountain as there is at the top (give or take), but the top of the mountain is where you’ll see clouds. In fact, if the conditions are just right, this is a clever/cheap way to get some insight into what the (normally) invisible air currents are.

So clouds *roughly* hold their shape (for a little while) because it takes time for the humidity or temperature to change, or for the cloud to be twisted up by local air currents. But since they’re not “blobs-of-water” so much as “patches-of-air-with-slightly-different-conditions” they change continuously and are free to pop in and out as they cross the dew point.

As for how they manage to be pretty and invariably end up looking like something familiar (cotton, marshmallow men, other clouds, negative Rorschach blots, etc.), that’s more psychological than climatological.

The clouds passing by overhead video is from here.

The lenticular cloud video is from here.

]]>And that’s exactly what’s happened! You can hear the episode here.

Before you ask, the shirt says “∃x : Ix” which is mathspeak for “there exists x such that I love x”; a statement which is demonstrably true.

]]>If our choices are expressions of the activity of nerves cells, which are soggy bags of molecules clacking together (to paraphrase Gray’s Anatomy), which are governed entirely by fundamental universal laws, then everything we do is dictated by physical mechanics. Even the feeling that we have free will would just a bunch of atoms all following the same set of simple rules every time they meet another atom, repeated super ad nauseam.

So if physical laws are all deterministic, then everything anyone does is just as “fated” as a rock rolling down a hill or a clock chiming. Now, you may be lacking free will in some idealized sense, but if there’s no way to tell, then what are you really missing? Whether your actions are predictable *in theory* is not quite as important as whether your actions are predictable *in practice*. The tiny, individual interactions that make up everything we do are easy (well… kinda) to predict, but big systems are a lot more complicated than the sum of their parts.

Conway’s “Game of Life” (different from the one where the goal is to die rich) provides a beautiful example of this. The Game of Life, which really should have been called “Staring at Pixels”, is a very short list of rules applied to pixels on a grid that describes which will be “alive” in the next “generation”. Systems of tiny, simple things (like pixels with basic interaction rules) are called “cellular automata”.

If you’re not familiar with the Game of Life, it’s well worth taking a moment to play around with it. Despite baby-simple rules that only govern individual pixels and their immediate neighbors, the ultimate behavior of the Game can not, in general, be predicted from the initial conditions without actually running through the generations. In fact, the Game of Life is capable of the same level of complexity as the computer that runs it (if allowed to run on a large enough grid); so a computer can simulate the Game of Life, but at the same time the Game of Life can simulate a computer.

The Game of Life is to free will, as a misplaced red wig full of bearer bonds is to clowns. If you spend enough time with the former, you can’t help but ask probing questions about the latter. Whether you’re talking about neurons or molecules or fundamental particles, people are also made up of lots of tiny “cellular automata”. The behavior of every little piece follows a set of known rules, so *in theory* you should be able to determine the outcome of even large systems (like people or worlds or whateveryougot) so long as you know everything about the initial conditions. But in practice: nope.

First, you’re not going to figure out the exact position of every particle in any system anywhere near as large as a person, and even if you could, the universe is right there to pile on more stuff to keep track of. Look around. See anything at all? Then it’s happening already.

Second, like the Game of Life, it’s unlikely that there’s a “computational short cut” for physical systems. That is to say, if you wanted to *precisely* predict what a person will do and think and whatever else, you’d have to simulate all of their bits and pieces, run the simulation forward, and see what happens. But that’s exactly how you handle a being with free will: you just… see what they do. Maybe this wild train we call life is on tracks, but if there’s no way to know where that train is going without actually riding it to the destination, what’s the difference? That’s at least free-will-adjacent.

That all said, the universe is not entirely deterministic. A little over a decade ago Conway and his buddy Kochen, defined something to be “free” when its actions are not explicitly determined by the past, and then proved the “Free Will Theorem” which is both profoundly bonkers and deeply frustrating:

If people have free will, then so do individual particles.

The difference between scientists and philosophers, is that scientists force each other to nail down *exactly* what they’re talking about and philosophers tend to be a little more loosey goosey (also, scientists are big fans of empirical evidence). So while you may disagree with how they defined the “free” in “free will”, that just means we need more words to work with.

In their paper, they spend buckets of time establishing a known physical principle, “fundamental randomness“, and just a little establishing what in the hell they’re talking about. To wit:

Here’s the idea:

For most of history, scientists labored under a rather modest assumption; that all possible experiments have a result, regardless of whether you bother to do those experiments. For example, if your experiment is “is this card the queen of spades?”, then there is an answer whether or not you actually look. By looking you gain a little information, but the result is there whether you bother to do the experiment or not. The card is whatever it is.

Even better, if you have a bunch of cards, it doesn’t matter which you choose to look at; all of them are what they are. If you assume that things are in definite states just waiting around to be uncovered, then there are a variety of mathematical statements you can make that are always true. Statements like “if there are three cards total, then there are a different number of red and black cards”. No matter how clever you are with actual playing cards, this statement (and a hell of a lot besides) *must* be true.

But it turns out that a lot of quantum phenomena, entanglement in particular, are incompatible with some of the equations that come from the assumption that “each card (or particle in this case) already is what it is”. In their paper, Koch ‘n Con provide an explicit example of an incompatibility based on particle spin and this old post goes into a lot more detail on another example, the math behind it, and what it means. The point is: seriously, as weird as it sounds, for many quantum systems it is literally impossible for the results of *all* possible experiments to exist (and therefore to be predetermined) because those results would be logically inconsistent with each other.

Now, if you assume that *everything* that happens is completely predetermined, then this isn’t actually a problem. The weirdness of quantum mechanics, the experimenters, and the experiments they choose to do, are all “on tracks”. The person doing the experiment had no choice about which experiment to do, which means that the result of only that one experiment needs to exist. The others won’t be done, so they can be safely swept under the existential carpet.

On the other hand, if the person doing the experiment has free will (in the sense that their actions are not determined by the past), then suddenly there’s an issue. If they’re free to choose *any* experiment and the result of that experiment is predetermined, then *all* of the results have to be predetermined. But that’s impossible.

Of course when we do these fancy quantum experiments we always get a result. No big deal. But if we have free will in the sense that our behavior is not entirely determined by the past, then the quantum systems we’re playing around with have free will in exactly the same sense.

The universe would need to be dead-set on conspiring against quantum theorists on a massive scale, and in a very specific way, in order to create the experimental results we’ve seen so far. In order to back up the assumption that the particles involved, the experimenters, and the machines used to do the measurements aren’t all subject to some all-encompassing, atom-by-atom, perfectly executed, and needlessly nefarious systematic bias, the methods used to make the “choices” in the experiments have become a little over-the-top. For example, by measuring entangled particles fast enough and far enough apart that light can’t travel between them, and by using cosmic microwave background radiation from opposite sides of the universe to randomly orient the measurement devices after the entangled particles are created and but before they’re measured. That way, either you really are randomly choosing which experiment you want to do, or the entire universe has been conspiring against you and this particular experiment since the beginning of time. It’s important to be sure about things, but this level of caution can blur the line between due diligence and paranoia.

It’s an open question whether the quantum randomness of particles scales up to produce free will, or at least “not-predetermined-ness”, in living critters. Forced to guess, I’d say… maybe/probably? Brains can turn on a dime and unless they’re actively suppressing it, eventually that quantum randomness should “butterfly effect” its way into our actions. At least sometimes. But if you really want to ensure that your will is as free as an atom’s, you can always carry around a Geiger counter and base all of your decisions on what it reads. You’d be the free-will-est person on your block!

]]>**One particle**: Quantum stuff can be (and generally is) in a “superposition”, multiple states at the same time. For example, an electron belonging to Alice (or anyone else), can be both “spin up” and “spin down” in equal parts; a superposition that’s written . Of course, you never look at something and notice that it’s in more than one state (superpositions need to be *inferred* using tricks like the double slit experiment). Before measuring its spin, the electron can be in the superpostion , but after measuring, its state is definite, either or (depending on what the result was). Which of these results you see is fundamentally random.

Now here’s the thing. Quantum randomness is relative. One person’s superposition is another person’s definite state. For example, that superposition from earlier is also a definite state . The spin right/left states are literally a superposition of the spin up/down states (this was as surprising and weird when it was first figured it out as it should be for you right now). If you measure in the vertical direction it’s a superposition of spin up and down, but if you measure it sideways it’s a definite state.

If some devious ne’er-do-well sent you a never-ending stream of or states, flipping coins to determine which every time, it would look the same as a stream of states so long as you measured both vertically. But if you measured sideways, then the stream of states would always yield the same predictable result (), and the stream of random up/down states would still yield random results. In exactly the same way that the left/right spin states are superpositions of the up/down states, the up/down states are superpositions of the left/right states (so they produce random results when measured horizontally).

Here’s the point. If you’re always looking at the same, definite state, then you should be able to find some measurement that always produces a definite result. But if you’re being fed a random set of states, there’s no way to do a measurement that produces consistent results. I suppose you could turn the detector off and walk away, but that’s dirty pool.

*Quick aside: I’m dropping the coefficients in front of the states that would normally dictate their probability of being measured (the state where both results show up 50% of the time, should have been written ). I figure if you notice they’re missing, then you probably already know how to put them back, and if you don’t notice they’re missing, then you probably don’t want to be distracted.
*

**Two Particles**: When you need to worry about multiple systems (particles, quantum computers, needlessly diverse collections of scarves, or whatever else), you don’t worry about their *individual* superpositions of states, you worry about superpositions of their *collective* states. So if you have a second electron, under the custodianship of Bob, you wouldn’t first talk about the states of Alice’s electron ( and ) and then, having done that, talk about the states of Bob’s electron ( and ). Instead, you’d talk about superpositions of their collective states (, , , and ).

For example, if both electrons are in the same equal superposition of spin up and down from earlier, then their collective state is

If Alice measures her electron and finds that it’s spin up, then the state of the pair is now

and if she finds that her electron is spin down, then the overall state is

In other words, in this situation it doesn’t matter what Alice sees; Bob’s state is independent of it. This particular collective state can be described first by considering Alice’s electron and then considering Bob’s (or vice versa), because knowing the result of a measurement on one part tells you nothing about what you’ll see when you measure the other. A state like this is called “separable”, because it can be separated mathematically. Physicists are very clever namers of things, which is why they often refer to themselves as namerators.

Non-seperable states are entangled; a measurement on one part gives you information about what the result of a measurement on another part will be. For example,

is definitely not separable. If Alice were to measure her electron the state of the system as a whole would be either or

In other words, by looking at her own electron, Alice knows what Bob will measure, whenever he gets around to it. This state is “maximally entangled” because knowing one measurement allows you to perfectly predict the other.

But once Alice has measured her electron, and the system as a whole is now either or , then as far as Bob is concerned, his state is *randomly* either or . Unlike the single-particle case with , there’s no “correct” way to measure these randomly selected states to make the result definite.

This randomness isn’t a symptom of Alice having done a measurement, it’s a feature of entangled states in general. After all, Bob and his pet electron could be anywhere. Despite the nearly universal, but categorically false, belief that entangled particles affect each other, nothing that Alice does or doesn’t do with her electron will ever have any impact on Bob’s electron.

If Alice measures her particle, Bob gets a random result. But since the particles can’t communicate and it doesn’t actually matter what Alice does, Bob gets a random result anyway.

Entangled particles follow the same rule that single particles follow: wrong measurement = random result, correct measurement = definite result. But in order to do the “correct measurement” you always need to have access to the entire state. If you have both entangled particles in front of you, then you can easily do a measurement to determine which maximally entangled state you have (for example and are both maximally entangled, but very different states). These “correct” joint measurements always involve a step where the two particles are forced to interact, which is something you can’t do if they’re on opposite sides of the universe.

This leads to a rather profound fact: entangled states always appear random when you only measure part of them. That is, they never seem to be in a superposition of states, they always appear to be a randomly selected state, since there’s no measurement that makes them predictable (see the one particle case).

There are still some clever things you can do with entangled particles that take advantage of their quantum correlations, like quantum teleportation, but two facts remain immutable: 1) when you measure one particle alone the result is random and 2) the two particles never ever influence each other in any way.

**How entanglement is defined**: When you don’t have access to both entangled particles and are forced to measure them one at a time, you get random results. You can describe how random those results are in terms of how much information it takes to describe them: two possible results (up or down) takes 1 bit of information (0 or 1). When you do have access to both entangled particles, you can do a measurement that will always produce the same, predictable result. *One* possible result takes *zero* bits of information. After all, if you had flipped a two-headed coin, would you really need to write down the results at all? Zero bits.

This difference between the randomness of the particles when they’re apart minus the randomness when they’re together is how entanglement is defined. A maximally entangled pair of particles has “1 ebit” of entanglement and a separable pair of particles has 0 ebits.

And yes, there are states in between. For example, is *partially* entangled, because if Alice looks at her electron, she’ll learn a little about Bob’s electron (and vice versa), but not everything. This state has a modest 0.55 ebits of entanglement.

**Three Particles**: If Alice’s electron is in a superposition of spin up and down, its state is . If Alice and Bob each have electrons that are entangled such that they’re spin up and down together, then their state is .

So what’s stopping Carol and her electron from also being both spin up and down in tandem with Alice and Bob’s? Not a damn thing.

Today we can create states like this with not just three, but dozens of particles. No big deal. Any reasonable person (with at least a passing familiarity with quantum notation) would say that state looks pretty entangled.

If Carol measures the spin of her electron, then the state of the system goes from to either or . As far as Alice and Bob are concerned, the state they’re working with is either or .

With the third-act-introduction of Carol, suddenly they’re working with one of two randomly selected states ( or instead of a nice, pure, entangled state ). These states are random when Alice and Bob measure them on their own and when they bring their electrons together, they’re still looking at one of two random states, so even together the result of any measurement is going to end up random.

However! Entanglement is narrowly defined is in terms of how much more random your particles are when they’re apart vs. when they are together. Same amount of randomness means zero entanglement. So because there’s a third (or fourth or fifth…) party involved the state, no two parts can be maximally entangled.

So it’s not that you can’t do fancy quantum states with many particles at once, and it’s not that something terribly profound happens when you move from using two particles to using three, but going by the stringent definition of entanglement: if two parties share a maximally entangled state with each other, they don’t share it with anyone else. One might even go so far as to say that entanglement is monogamous.

Like many seemingly arbitrary mathematical definitions, it turns out that the monogamy of entanglement is a powerful, useful statement. It is the key behind why quantum security is so secure (and quantum). By sharing a string of maximally entangled states with each other, then measuring some and sharing the results, Alice and Bob can check to see if there’s a third party corrupting their state. If there isn’t, they can dive into their private conversation. If there is, then they’ve caught themselves an eavesdropper. Quantum cryptography is Carol-proof!

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