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 downtoearth 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.
Entanglement is often
regarded as a uniquely quantummechanical phenomenon, but it is not. In fact,
it is enlightening, though somewhat unconventional, to consider a simple
nonquantum (or “classical”) version of entanglement first. This enables us to
pry the subtlety of entanglement itself apart from the general oddity of
quantum theory.
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 cons. 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 cons as cakes.
Our cons come in two
shapes, square or circular, which we identify as their possible states. Then
the four possible joint states, for two cons, 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.
Olena Shmahalo/Quanta Magazine
We say that the cons 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 con (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 cons are entangled when information about one improves our knowledge
of the other. Our second table demonstrates extreme entanglement. In that case,
whenever the first con is circular, we know the second is circular too. And
when the first con 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 Î¦_{■}, Î¦_{●} for
the wave functions describing system 1 in its square or circular states, and Ïˆ_{■}, Ïˆ_{●} for the
wave functions describing system 2 in its square or circular states, then in
our working example the overall states will be
Independent: Î¦_{■} Ïˆ_{■} + Î¦_{■} Ïˆ_{●} + Î¦_{● }Ïˆ_{■} + Î¦_{●} Ïˆ_{●}
_{}
Entangled: Î¦_{■} Ïˆ_{■} + Î¦_{●} Ïˆ_{●}
_{}
We can also write the
independent version as
(Î¦_{■} + Î¦_{●})(Ïˆ_{■} + Ïˆ_{●})
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 EinsteinPodolskyRosen (EPR)
and GreenbergerHorneZeilinger (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 cons 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 cons 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 qon — the situation is profoundly
different. The fact that a qon 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 qon, but in doing so we lose all information about its color. Or we can
measure the color of our qon, 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:
1.
A property that is not measured need not
exist.
2.
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
qons, 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 qon’s color, we will surely get red. But as we discussed
earlier, when introducing complementarity, if we choose to measure a qon’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 another brilliantly illuminating
example of quantum entanglement. It involves three of our qons, prepared
in a special, entangled state (the GHZ state). We distribute the three qons 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 qons starting out
in the GHZ state.
Each experimenter,
separately, finds maximally random results. When she measures a qon’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 mindexpanding.
IV.
Thus far we have considered
how entanglement can make it impossible to assign unique, independent states to
several qons. Similar considerations apply to the evolution of a single qon
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 qon, which we monitor at two
different times. We can imagine situations where we determine that the shape of
our qon 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 qon 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 qon 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 SchrÃ¶dinger, 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 “SchrÃ¶dinger 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 qons that
are, in a real sense, SchrÃ¶dinger kittens. Their full description requires, at
intermediate times, that we take both of two contradictory
propertytrajectories into account.
The controlled experimental
realization of entangled histories is delicate because it requires we gather
partial information about our qon. 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.

No comments:
Post a Comment