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.

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.

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.

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.

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.

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.

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.