National Geographic’s Video on Spooky Action at a Distance

An aura of glamorous mystery attaches to the concept of quantum entanglement, and also to the (somehow) related claim that quantum theory requires “many worlds.” Yet in the end those are, or should be, scientific ideas, with down-to-earth meanings and concrete implications. Here I’d like to explain the concepts of entanglement and many worlds as simply and clearly as I know how.

I.

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 enables us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.

Quantized

A monthly column in which top researchers explore the process of discovery. This month’s columnist, Frank Wilczek, is a Nobel Prize-winning physicist at the Massachusetts Institute of Technology.

Entanglement arises in situations where we have partial knowledge of the state of two systems. For example, our systems can be two objects that we’ll call c-ons. The “c” is meant to suggest “classical,” but if you’d prefer to have something specific and pleasant in mind, you can think of our c-ons as cakes.

Our c-ons come in two shapes, square or circular, which we identify as their possible states. Then the four possible joint states, for two c-ons, are (square, square), (square, circle), (circle, square), (circle, circle). The following tables show two examples of what the probabilities could be for finding the system in each of those four states.

We say that the c-ons 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 c-on (or cake) is 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 c-ons are entangled when information about one improves our knowledge of the other. Our second table demonstrates extreme entanglement. In that case, whenever the first c-on is circular, we know the second is circular too. And when the first c-on 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. The rules connecting wave functions to physical probabilities introduce very interesting complications, as we will discuss, but the central concept of entangled knowledge, which we have seen already for classical probabilities, carries over.

Cakes don’t count as quantum systems, of course, but entanglement between quantum systems arises naturally — for example, in the aftermath of particle collisions. In practice, unentangled (independent) states are rare exceptions, for whenever systems interact, the interaction creates correlations between them.

Consider, for example, molecules. They are composites of subsystems, namely electrons and nuclei. A molecule’s lowest energy state, in which it is most usually found, is a highly entangled state of its electrons and nuclei, for the positions of those constituent particles are by no means independent. As the nuclei move, the electrons move with them.

Returning to our example: If we write Φ_{■}, Φ_{●} for the wave functions describing system 1 in its square or circular states, and ψ_{■}, ψ_{●} for the wave functions describing system 2 in its square or circular states, then in our working example the overall states will be

Independent: Φ_{■} ψ_{■} + Φ_{■} ψ_{●} + Φ_{● }ψ_{■} + Φ_{●} ψ_{●}

Entangled: Φ_{■} ψ_{■} + Φ_{●} ψ_{●}

We can also write the independent version as

(Φ_{■} + Φ_{●})(ψ_{■} + ψ_{●})

Note how in this formulation the parentheses clearly separate systems 1 and 2 into independent units.

There are many ways to create entangled states. One way is to make a measurement of your (composite) system that gives you partial information. We can learn, for example, that the two systems have conspired to have the same shape, without learning exactly what shape they have. This concept will become important later.

The more distinctive consequences of quantum entanglement, such as the Einstein-Podolsky-Rosen (EPR) and Greenberger-Horne-Zeilinger (GHZ) effects, arise through its interplay with another aspect of quantum theory called “complementarity.” To pave the way for discussion of EPR and GHZ, let me now introduce complementarity.

Previously, we imagined that our c-ons could exhibit two shapes (square and circle). Now we imagine that it can also exhibit two colors — red and blue. If we were speaking of classical systems, like cakes, this added property would imply that our c-ons could be in any of four possible states: a red square, a red circle, a blue square or a blue circle.

Yet for a quantum cake — a quake, perhaps, or (with more dignity) a q-on — the situation is profoundly different. The fact that a q-on can exhibit, in different situations, different shapes or different colors does not necessarily mean that it possesses both a shape and a color simultaneously. In fact, that “common sense” inference, which Einstein insisted should be part of any acceptable notion of physical reality, is inconsistent with experimental facts, as we’ll see shortly.

We can measure the shape of our q-on, but in doing so we lose all information about its color. Or we can measure the color of our q-on, but in doing so we lose all information about its shape. What we cannot do, according to quantum theory, is measure both its shape and its color simultaneously. No one view of physical reality captures all its aspects; one must take into account many different, mutually exclusive views, each offering valid but partial insight. This is the heart of complementarity, as Niels Bohr formulated it.

As a consequence, quantum theory forces us to be circumspect in assigning physical reality to individual properties. To avoid contradictions, we must admit that:

- A property that is not measured need not exist.
- Measurement is an active process that alters the system being measured.

II.

Now I will describe two classic — though far from classical! — illustrations of quantum theory’s strangeness. Both have been checked in rigorous experiments. (In the actual experiments, people measure properties like the angular momentum of electrons rather than shapes or colors of cakes.)

Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) described a startling effect that can arise when two quantum systems are entangled. The EPR effect marries a specific, experimentally realizable form of quantum entanglement with complementarity.

An EPR pair consists of two q-ons, each of which can be measured either for its shape or for its color (but not for both). We assume that we have access to many such pairs, all identical, and that we can choose which measurements to make of their components. If we measure the shape of one member of an EPR pair, we find it is equally likely to be square or circular. If we measure the color, we find it is equally likely to be red or blue.

The interesting effects, which EPR considered paradoxical, arise when we make measurements of both members of the pair. When we measure both members for color, or both members for shape, we find that the results always agree. Thus if we find that one is red, and later measure the color of the other, we will discover that it too is red, and so forth. On the other hand, if we measure the shape of one, and then the color of the other, there is no correlation. Thus if the first is square, the second is equally likely to be red or to be blue.

We will, according to quantum theory, get those results even if great distances separate the two systems, and the measurements are performed nearly simultaneously. The choice of measurement in one location appears to be affecting the state of the system in the other location. This “spooky action at a distance,” as Einstein called it, might seem to require transmission of information — in this case, information about what measurement was performed — at a rate faster than the speed of light.

But does it? Until I *know* the result you obtained, I don’t know what to expect. I gain useful information when I learn the result you’ve measured, not at the moment you measure it. And any message revealing the result you measured must be transmitted in some concrete physical way, slower (presumably) than the speed of light.

Upon deeper reflection, the paradox dissolves further. Indeed, let us consider again the state of the second system, given that the first has been measured to be red. If we choose to measure the second q-on’s color, we will surely get red. But as we discussed earlier, when introducing complementarity, if we choose to measure a q-on’s shape, when it is in the “red” state, we will have equal probability to find a square or a circle. Thus, far from introducing a paradox, the EPR outcome is logically forced. It is, in essence, simply a repackaging of complementarity.

Nor is it paradoxical to find that distant events are correlated. After all, if I put each member of a pair of gloves in boxes, and mail them to opposite sides of the earth, I should not be surprised that by looking inside one box I can determine the handedness of the glove in the other. Similarly, in all known cases the correlations between an EPR pair must be imprinted when its members are close together, though of course they can survive subsequent separation, as though they had memories. Again, the peculiarity of EPR is not correlation as such, but its possible embodiment in complementary forms.

III.

Daniel Greenberger, Michael Horne and Anton Zeilinger discovered anotherbrilliantly illuminating example of quantum entanglement. It involves three of our q-ons, prepared in a special, entangled state (the GHZ state). We distribute the three q-ons to three distant experimenters. Each experimenter chooses, independently and at random, whether to measure shape or color, and records the result. The experiment gets repeated many times, always with the three q-ons starting out in the GHZ state.

Each experimenter, separately, finds maximally random results. When she measures a q-on’s shape, she is equally likely to find a square or a circle; when she measures its color, red or blue are equally likely. So far, so mundane.

But later, when the experimenters come together and compare their measurements, a bit of analysis reveals a stunning result. Let us call square shapes and red colors “good,” and circular shapes and blue colors “evil.” The experimenters discover that whenever two of them chose to measure shape but the third measured color, they found that exactly 0 or 2 results were “evil” (that is, circular or blue). But when all three chose to measure color, they found that exactly 1 or 3 measurements were evil. That is what quantum mechanics predicts, and that is what is observed.

So: Is the quantity of evil even or odd? Both possibilities are realized, with certainty, in different sorts of measurements. We are forced to reject the question. It makes no sense to speak of the quantity of evil in our system, independent of how it is measured. Indeed, it leads to contradictions.

The GHZ effect is, in the physicist Sidney Coleman’s words, “quantum mechanics in your face.” It demolishes a deeply embedded prejudice, rooted in everyday experience, that physical systems have definite properties, independent of whether those properties are measured. For if they did, then the balance between good and evil would be unaffected by measurement choices. Once internalized, the message of the GHZ effect is unforgettable and mind-expanding.

IV.

Thus far we have considered how entanglement can make it impossible to assign unique, independent states to several q-ons. Similar considerations apply to the evolution of a single q-on in time.

We say we have “entangled histories” when it is impossible to assign a definite state to our system at each moment in time. Similarly to how we got conventional entanglement by eliminating some possibilities, we can create entangled histories by making measurements that gather partial information about what happened. In the simplest entangled histories, we have just one q-on, which we monitor at two different times. We can imagine situations where we determine that the shape of our q-on was either square at both times or that it was circular at both times, but that our observations leave both alternatives in play. This is a quantum temporal analogue of the simplest entanglement situations illustrated above.

Using a slightly more elaborate protocol we can add the wrinkle of complementarity to this system, and define situations that bring out the “many worlds” aspect of quantum theory. Thus our q-on might be prepared in the red state at an earlier time, and measured to be in the blue state at a subsequent time. As in the simple examples above, we cannot consistently assign our q-on the property of color at intermediate times; nor does it have a determinate shape. Histories of this sort realize, in a limited but controlled and precise way, the intuition that underlies the many worlds picture of quantum mechanics. A definite state can branch into mutually contradictory historical trajectories that later come together.

Erwin Schrödinger, a founder of quantum theory who was deeply skeptical of its correctness, emphasized that the evolution of quantum systems naturally leads to states that might be measured to have grossly different properties. His “Schrödinger cat” states, famously, scale up quantum uncertainty into questions about feline mortality. Prior to measurement, as we’ve seen in our examples, one cannot assign the property of life (or death) to the cat. Both — or neither — coexist within a netherworld of possibility.

Everyday language is ill suited to describe quantum complementarity, in part because everyday experience does not encounter it. Practical cats interact with surrounding air molecules, among other things, in very different ways depending on whether they are alive or dead, so in practice the measurement gets made automatically, and the cat gets on with its life (or death). But entangled histories describe q-ons that are, in a real sense, Schrödinger kittens. Their full description requires, at intermediate times, that we take both of two contradictory property-trajectories into account.

The controlled experimental realization of entangled histories is delicate because it requires we gather *partial* information about our q-on. Conventional quantum measurements generally gather complete information at one time — for example, they determine a definite shape, or a definite color — rather than partial information spanning several times. But it can be done — indeed, without great technical difficulty. In this way we can give definite mathematical and experimental meaning to the proliferation of “many worlds” in quantum theory, and demonstrate its substantiality.

*This article was reprinted on Wired.com.*

### More evidence to support quantum theory’s ‘spooky action at a distance’

By Adrian Cho

## Some Quantum History

THE UNCERTAINTY PRINCIPLE says that you can’t know certain properties of a quantum system at the same time. For example, you can’t simultaneously know the position of a particle and its momentum. But what does that imply about reality? If we could peer behind the curtains of quantum theory, would we find that objects really do have well defined positions and momentums? Or does the uncertainty principle mean that, at a fundamental level, objects just can’t have a clear position and momentum at the same time. In other words, is the blurriness in our theory or is it in reality itself?

### Case 1: Blurred glasses, clear reality

The first possibility is that using quantum mechanics is like wearing blurred glasses. If we could somehow lift off these glasses, and peek behind the scenes at the fundamental reality, then of course a particle must have some definite position and momentum. After all, it’s a thing in our universe, and the universe must know where the thing is and which way it’s going, even if we don’t know it. According to this point of view, quantum mechanics isn’t a complete description of reality – we’re probing the fineness of nature with a blunt tool, and so we’re bound to miss out on some of the details.

This fits with how everything else in our world works. When I take off my shoes and you see that I’m wearing red socks, you don’t assume that my socks were in a state of undetermined color until we observed them, with some chance that they could have been blue, green, yellow, or pink. That’s crazy talk. Instead, you (correctly) assume that my socks have always been red. So why should a particle be any different? Surely, the properties of things in nature must exist independent of whether we measure them, right?

### Case 2: Clear glasses, blurred reality

On the other hand, it could be that our glasses are perfectly clear, but reality is blurry. According to this point of view, quantum mechanics is a complete description of reality at this level, and things in the universe just don’t have a definite position and momentum. This is the view that most quantum physicists adhere to. It’s not that the tools are blunt, but that reality is inherently nebulous. Unlike the case of my red socks, when you measure where a particle is, it didn’t have a definite position until the moment you measured it. The act of measuring its position forced it into having a definite position.

Now, you might think that this is one of those ‘if-a-tree-falls-in-the-forest’ types of metaphysical questions that can never have a definite answer. However, unlike most philosophical questions, there’s an actual experiment that you can do to settle this debate. What’s more, the experiment has been done, many times. In my view, this is one of the most underappreciated ideas in our popular understanding of physics. The experiment is fairly simple and tremendously profound, because it tells us something deep and surprising about the nature of reality.

**Here’s the setup.** There’s a source of light in the middle of the room. Every minute, on the minute, it sends out two photons, in opposite directions. These pairs of photons are created in a special state known as quantum entanglement. This means that they’re both connected in a quantum way – so that if you make a measurement on one photon, you don’t just alter the quantum state of that photon, but also immediately alter the quantum state of the other one as well.

With me so far?

On the left and the right of this room are two identical boxes designed to receive the photons. Each box has a light on it. Every minute, as the photon hits the box, the light flashes one of two colors, either red or green. From minute to minute, the color of the light seems quite random – sometimes it’s red, and other times it’s green, with no clear pattern one way or another. If you stick your hand in the path of the photon, the light bulb doesn’t flash. It seems that this box is detecting some property of the photon.

So when you look at any one box, it flashes a red or a green light, completely at random. It’s anyone’s guess as to which color it will flash next. But here’s the really strange thing: Whenever one box flashes a certain color, the other box will always flash the same color. No matter how far apart you try to move the boxes from the detector, they could even be in opposite ends of our solar system, they’ll flash the same color without fail.

It’s almost as if these boxes are conspiring to give the same result. How is this possible? (If you have your own pet theory about how these boxes work, hold on to it, and in a bit you’ll be able to test your idea against an experiment.)

“Aha!” says the quantum enthusiast. “I can explain what’s happening here. Every time a photon hits one of the boxes, the box measures its quantum state, which it reports by flashing either a red or a green light. But the two photons are tied together by quantum entanglement, so when we measure that one photon is in the red state (say), we’ve forced the other photon into the same state as well! That’s why the two boxes always flash the same color.”

“Hold up,” says the prosaic classical physicist. “Particles are like billiard balls, not voodoo dolls. It’s absurd that a measurement in one corner of space can instantaneously affect something in a totally different place. When I observe that one of my socks is red, it doesn’t immediately change the state of my other sock, forcing it to be red as well. The simpler explanation is that the photons in this experiment, like socks, are created in pairs. Sometimes they’re both in the red state, other times they’re both in the green state. These boxes are just measuring this ‘hidden state’ of the photons.”

The experiment and reasoning spelt out here is a version of a thought experiment first articulated by Einstein, Podolsky and Rosen, known as the EPR experiment. The crux of their argument is that it seems absurd that a measurement at one place can immediately influence a measurement at totally different place. The more logical explanation is that the boxes are detecting some hidden property that both the photons share. From the moment of their creation, these photons might carry some hidden stamp, like a passport, that identifies them as being either in the red state or the green state. The boxes must then be detecting this stamp. Einstein, Podolsky and Rosen argued that the randomness we observe in these experiments is a property of our incomplete theory of nature. According to them, it’s our glasses that are blurry. In the jargon of the field, this idea is known as a hidden variables theory of reality.

It would seem the classical physicist has won this round, with an explanation that’s simpler and makes more sense.

The next day, a new pair of boxes arrives in the mail. The new version of the box has three doors build into it. You can only open one door at a time. Behind every door is a light, and like before, each light can glow red or green.

The two physicists play around with these new boxes, catching photons and watching what happens when they open the doors. After a few hours of fiddling around, here’s what they find:

1. If they open the same door on both boxes, the lights always flashes the same color.

2. If they open the doors of the two boxes at random, then the lights flash the same color exactly half the time.

After some thought, the classical physicist comes up with a simple explanation for this experiment. “Basically, this is not very different from yesterday’s boxes. Here’s a way to think about it. Instead of just having a single stamp, let’s say that each pair of photons now has three stamps, sort of like holding multiple passports. Each door of the box reads a different one of these three stamps. So, for example, the three stamps could be red, green, and red meaning the first door would flash red, the second door would flash green, and the third door would flash red.”

“Going with this idea, it makes sense that when we open the same door on both boxes, we get the same colored light, because both boxes are reading the same stamp. But when we open different doors, the boxes are reading different stamps, so they can give different results.”

Again, the classical physicist’s explanation is straightforward, and doesn’t invoke any fancy notions like quantum entanglement or the uncertainty principle.

“Not so fast,” says the quantum physicist, who’s just finished scribbling a calculation on her notepad. “When you and I opened the doors at random, we discovered that one half of the time, the lights flash the same color. This number – a half – agrees exactly with the predictions of quantum mechanics. But according to your ‘hidden stamps’ ideas, the lights should flash the same color *more than half* of the time!”

The quantum enthusiast is on to something here.

“According to the hidden stamps idea, there are 8 possible combinations of stamps that the photons could have. Let’s label them by the first letters of the colors, for short, so RRG = red red green.”

RRG

RGR

GRR

GGR

GRG

RGG

RRR

GGG

“Now, when we pick doors at random, a third of the time we will pick the same door by chance, and when we do, we see the same color.”

“The other two-thirds of the time, we pick different doors. Let’s say we encounter photons with the following stamp configuration:”

RRG

“In such a configuration, if we picked door 1 on one box and door 2 on another, the lights flash the same color (red and red). But if we picked doors 1 and 3, or doors 2 and 3, they’d flash different colors (red and green). So in one-third of such cases, the boxes flash the same color.”

“To summarize, a third of the time the boxes flash the same color because we chose the same door. Two-thirds of the time we chose different doors, and in one-third of these instances, the boxes flash the same color.”

“Adding this up,”

⅓ + ⅔ ⅓ = 3/9 + 2/9 = 5/9 = 55.55%

“So 55.55% is the odds that the boxes flash the same color when we pick two doors at random, according to the hidden stamps theory.”

“But wait! We only looked at one possibility – RRG. What about the others? It takes a little thought, but it isn’t too hard to show that the math is exactly the same in all the following cases:”

We just went through the argument of a groundbreaking result in quantum mechanics known as Bell’s theorem. The black boxes don’t really flash red and green lights, but in the details that matter they match real experiments that measure the polarization of entangled photons.

Bell’s theorem draws a line in the sand between the strange quantum world and the familiar classical world that we know and love. It proves that hidden variable theories like the kind that Einstein and his buddies came up with simply aren’t true^{1}. In its place is quantum mechanics, complete with its particles that can be entangled across vast distances. When you perturb the quantum state of one of these entangled particles, you instantaneously also perturb the other one, no matter where in the universe it is.

It’s comforting to think that we could explain away the strangeness of quantum mechanics if we imagined everyday particles with little invisible gears in them, or invisible stamps, or a hidden notebook, or something – some hidden variables that we don’t have access to – and these hidden variables store the “real” position and momentum and other details about the particle. It’s comforting to think that, at a fundamental level, reality behaves classically, and that our incomplete theory doesn’t allow us to peek into this hidden register. But Bell’s theorem robs us of this comfort. Reality is blurry, and we just have to get used to that fact.

### Footnotes

1. Technically, Bell’s theorem and the subsequent experiment rule out a large class of hidden variable theories known as local hidden variable theories. These are theories where the hidden variables don’t travel faster than light. It doesn’t rule out nonlocal hidden variable theories where hidden variables do travel faster than light, and Bohmian mechanicsis the most successful example of such a theory.

I first came across this boxes-with-flashing-lights explanation of Bell’s theorem in Brian Greene’s book Fabric of the Cosmos. This pedagogical version of Bell’s experiment traces back to the physicist David Mermin who came up with it. If you’d like a taste of his unique and brilliant brand of physics exposition, pick up a copy of his book Boojums All the Way Through.

*Homepage Image: NASA/Flickr*

RRG

RGR

GRR

GGR

GRG

RGG

“That leaves only two cases:”

RRR

GGG

“In those cases, we get the same color no matter which doors we pick. So it can only *increase* the overall odds of the two boxes flashing the same color.”

**“The punchline is that according to the hidden stamps idea, the odds of both boxes flashing the same color when we open the doors at random is at least 55.55%. But according to quantum mechanics, the answer is 50%. The data agrees with quantum mechanics, and it rules out the ‘hidden stamps’ theory.”**

If you’ve made it this far, it’s worth pausing to think about what we’ve just shown.