The idea of quantum entanglement and the (somehow related) assertion that quantum theory necessitates "many worlds" are shrouded in a glamorous mystery. However, those are, or ought to be, scientific concepts with practical applications and real-world significance. Here, I will attempt to make the ideas of entanglement and many worlds as understandable and straightforward as possible.
Contrary to popular belief, entanglement is not exclusively
a quantum-mechanical phenomenon. In fact, starting with a basic non-quantum (or
"classical") version of entanglement is instructive if a little
unconventional. This allows us to separate the general oddity of quantum theory
from the subtlety of entanglement itself.
When we only have a partial understanding of the states of
two systems, entanglement occurs. Our systems, for instance, can consist of two
items that we'll refer to as c-ons. The letter "c" is intended to
imply "classical," but if you would rather have a more precise and
agreeable image in mind, consider our c-ons as cakes.
We distinguish between the two possible states of our c-ons,
which are square and circular in shape. Consequently, for a pair of c-ons, the
four possible joint states are (square, square), (square, circle), (circle,
square), and (circle, circle). Two examples of the probabilities for finding
the system in each of those four states are provided in the following tables.
If knowledge of one of the c-ons' states does not provide
meaningful information about the other's state, we refer to the c-ons as
"independent." This is a feature of our first table. We still don't
know the shape of the second c-on (or cake) if the first one is square.
Likewise, nothing meaningful about the shape of the first can be inferred from
the shape of the second.
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.