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Lecture 1
Systems and Experiments
Lenny and Art wander into Hilbert’s Place.
Art: What is this, the Twilight Zone? Or some kind of fun
house? I can’t get my bearings.
Lenny: Take a breath. You’ll get used to it.
Art: Which way is up?
1.1 Quantum Mechanics Is
Difffferent
What is so special about quantum mechanics? Why is it so
hard to understand? It would be easy to blame the “hard
mathematics,” and there may be some truth in that idea.
But that can’t be the whole story. Lots of nonphysicists are
able to master classical mechanics and fifield theory, which
also require hard mathematics.
Quantum mechanics deals with the behavior of objects
so small that we humans are ill equipped to visualize them
at all. Individual atoms are near the upper end of this scale
in terms of size. Electrons are frequently used as objects of
study. Our sensory organs are simply not built to perceive
the motion of an electron. The best we can do is to try
to understand electrons and their motion as mathematical
abstractions.
“So what?” says the skeptic. “Classical mechanics is fifilled
to the brim with mathematical abstractions—point masses,
rigid bodies, inertial reference frames, positions, momenta,
fifields, waves—the list goes on and on. There’s nothing new
about mathematical abstractions.” This is actually a fair
point, and indeed the classical and quantum worlds have
some important things in common. Quantum mechanics,
however, is difffferent in two ways:
1.
Difffferent Abstractions. Quantum abstractions are fun
damentally difffferent from classical ones. For example,
we’ll see that the idea of a state in quantum mechanics
is conceptually very difffferent from its classical counter
part. States are represented by difffferent mathematical
objects and have a difffferent logical structure.
2.
States and Measurements. In the classical world, the
relationship between the state of a system and the re
sult of a measurement on that system is very straight
forward. In fact, it’s trivial. The labels that describe
a state (the position and momentum of a particle, for
example) are the same labels that characterize mea
surements of that state. To put it another way, one
can perform an experiment to determine the state of a
system. In the quantum world, this is not true. States
and measurements are two difffferent things, and the
relationship between them is subtle and nonintuitive.
These ideas are crucial, and we’ll come back to them again
and again.
1.2 Spins and Qubits
The concept of spin is derived from particle physics. Par
ticles have properties in addition to their location in space.
For example, they may or may not have electric charge, or
mass. An electron is not the same as a quark or a neutrino.
But even a specifific type of particle, such as an electron, is
not completely specifified by its location. Attached to the elec
tron is an extra degree of freedom called its spin. Naively,
the spin can be pictured as a little arrow that points in some
direction, but that naive picture is too classical to accurately
represent the real situation. The spin of an electron is about
as quantum mechanical as a system can be, and any attempt
to visualize it classically will badly miss the point.
We can and will abstract the idea of a spin, and for
get that it is attached to an electron. The quantum spin
is a system that can be studied in its own right. In fact,
the quantum spin, isolated from the electron that carries it
through space, is both the simplest and the most quantum
of systems.
The isolated quantum spin is an example of the gen
eral class of simple systems we call qubits—quantum bits—
that play the same role in the quantum world as logical
bits play in defifining the state of your computer. Many
systems—maybe even all systems—can be built up by com
bining qubits. Thus in learning about them, we are learning
about a great deal more.
1.3 An Experiment
Let’s make these ideas concrete, using the simplest example
we can fifind. In the fifirst lecture of Volume I, we began by
discussing a very simple deterministic system: a coin that
can show either heads (H) or tails (T). We can call this a
two-state system, or a bit, with the two states being H and
T. More formally we invent a “degree of freedom” called σ
that can take on two values, namely +1 and −1. The state
H is replaced by
σ = +1
and the state T by
σ = −1.
Classically, that’s all there is to the space of states. The
system is either in state σ = +1 or σ = −1 and there is
nothing in between. In quantum mechanics, we’ll think of
this system as a qubit.
Volume I also discussed simple evolution laws that tell
us how to update the state from instant to instant. The
simplest law is just that nothing happens. In that case, if
we go from one discrete instant (n) to the next (n + 1), the
law of evolution is
σ(n + 1) = σ(n).
(1.1)
Let’s expose a hidden assumption that we were careless
about in Volume I. An experiment involves more than just
a system to study. It also involves an apparatus A to make
measurements and record the results of the measurements.
In the case of the two-state system, the apparatus interacts
with the system (the spin) and records the value of σ. Think
of the apparatus as a black box1 with a window that displays
the result of a measurement. There is also a “this end up”
arrow on the apparatus. The up-arrow is important because
it shows how the apparatus is oriented in space, and its di
rection will affffect the outcomes of our measurements. We
begin by pointing it along the z
axis (Fig. 1.1). Initially,
we have no knowledge of whether
σ = +1 or σ = −1. Our
purpose is to do an experiment to fifind out the value of σ.
Before the apparatus interacts with the spin, the window
is blank (labeled with a question mark in our diagrams).
After it measures σ, the window shows a +1 or a −1. By
looking at the apparatus, we determine the value of σ. That
whole process constitutes a very simple experiment designed
to measure σ.
Now that we’ve measured σ, let’s reset the apparatus to
neutral and, without disturbing the spin, measure σ again.
Assuming the simple law of Eq. 1.1, we should get the same
answer as we did the fifirst time. The result σ = +1 will be
followed by σ = +1
. Likewise for
σ = −1. The same will be
true for any number of repetitions. This is good because it
allows us to confifirm the result of an experiment. We can also
say this in the following way: The fifirst interaction with the
apparatus A prepares the system in one of the two states.
Subsequent experiments confifirm that state. So far, there is
no difffference between classical and quantum physics.
Now let’s do something new. After preparing the spin
by measuring it with A, we turn the apparatus upside down
and then measure σ again (Fig. 1.2). What we fifind is that if
we originally prepared σ = +1
, the upside down apparatus
records σ = −1. Similarly, if we originally prepared σ = −1,
the upside down apparatus records σ = +1
. In other words,turning the apparatus over interchanges σ = +1 and σ = −1.
From these results, we might conclude that σ is a degree of
freedom that is associated with a sense of direction in space.
For example, if σ were an oriented vector of some sort, then
it would be natural to expect that turning the apparatus over
would reverse the reading. A simple explanation is that the
apparatus measures the component of the vector along an
axis embedded in the apparatus. Is this explanation correct
for all confifigurations?
If we are convinced that the spin is a vector, we would
naturally describe it by three components:
σz, σ
x, and σy.
When the apparatus is upright along the z
axis, it is posi
tioned to measure σz.
So far, there is still no difffference between classical physics
and quantum physics. The difffference only becomes apparent
when we rotate the apparatus through an arbitrary angle,
say π
2 radians (90 degrees). The apparatus begins in the
upright position (with the up-arrow along the z axis). A
spin is prepared with σ = +1
. Next, rotate A so that the
up-arrow points along the x axis (Fig. 1.3), and then make a
measurement of what is presumably the x component of the
spin, σx.
If in fact σ really represents the component of a vector
along the up-arrow, one would expect to get zero. Why?
Initially, we confifirmed that σ was directed along the z axis,
suggesting that its component along x must be zero. But we
get a surprise when we measure σx: Instead of giving σx = 0,
the apparatus gives either σx = +1 or σx = −1. A is very
stubborn—no matter which way it is oriented, it refuses to
give any answer other than σ = ±1. If the spin really is a
vector, it is a very peculiar one indeed.
Nevertheless, we do fifind something interesting. Suppose
we repeat the operation many times, each time following the
same procedure, that is:
•
Beginning with A along the z axis, prepare σ = +1
.
•
Rotate the apparatus so that it is oriented along the x
axis.
•
Measure σ.
The repeated experiment spits out a random series of plus
ones and minus-ones. Determinism has broken down, but
in a particular way. If we do many repetitions, we will find
that the numbers of σ = +1 events and σ = −1 events
are statistically equal. In other words, the average value of
σ is zero. Instead of the classical result—namely, that the
component of σ along the x axis is zero—we fifind that the
average of these repeated measurements
is zero.
Now let’s do the whole thing over again, but instead of
rotating A to lie on the x axis, rotate it to an arbitrary
direction along the unit vector
2
ˆ
n.
Classically, if σ were a
vector, we would expect the result of the experiment to be
the component of σ along the ˆ
n axis. If ˆ
n lies at an angle θ with respect to z, the classical answer would be
σ = cos θ.
But as you might guess, each time we do the experiment we
get σ = +1 or σ = −1. However, the result is statistically
biased so that the average value is cos θ.
The situation is of course more general. We did not have
to start with A oriented along z. Pick any direction ˆm and
start with the up-arrow pointing along ˆm. Prepare a spin
so that the apparatus reads +1. Then, without disturbing
the spin, rotate the apparatus to the direction ˆn, as shown
in Fig. 1.4. A new experiment on the same spin will give
random results ±1, but with an average value equal to the
cosine of the angle between ˆ
n and ˆm. In other words, the
average will be ˆ
n ·
ˆ
m.
The quantum mechanical notation for the statistical av
erage of a quantity Q is Dirac’s bracket notation Q. We
may summarize the results of our experimental investigation
as follows: If we begin with A oriented along ˆm and confifirm
that σ = +1
, then subsequent measurement with A oriented
along ˆ
n gives the statistical result
σ = ˆ
n ·
ˆ
m.
What we are learning is that quantum mechanical systems
are not deterministic—the results of experiments can be sta
tistically random—but if we repeat an experiment many
times, average quantities can follow the expectations of clas
sical physics, at least up to a point.
1.4 Experiments Are Never Gentle
Every experiment involves an outside system—an apparatus—
that must interact with the system in order to record a re
sult. In that sense, every experiment is invasive. This is
true in both classical and quantum physics, but only quan
tum physics makes a big deal out of it. Why is that so?
Classically, an ideal measuring apparatus has a vanishingly
small effffect on the system it is measuring. Classical experi
ments can be arbitrarily gentle and still accurately and repro
ducibly record the results of the experiment. For example,
the direction of an arrow can be determined by reflflecting
light offff the arrow and focusing it to form an image. While
it is true that the light must have a small enough wavelength
to form an image, there is nothing in classical physics that
prevents the image from being made with arbitrarily weak
light. In other words, the light can have an arbitrarily small
energy content.
In quantum mechanics, the situation is fundamentally
difffferent. Any interaction that is strong enough to measure
some aspect of a system is necessarily strong enough to dis
rupt some other aspect of the same system. Thus, you can
learn nothing about a quantum system without changing
something else.
This should be evident in the examples involving A and
σ. Suppose we begin with σ = +1 along the z axis. If we
measure σ again with A oriented along z, we will confifirm the
previous value. We can do this over and over without chang
ing the result. But consider this possibility: Between suc
cessive measurements along the z axis, we turn A through
90 degrees, make an intermediate measurement, and turn it
back to its original direction. Will a subsequent measure
ment along the z axis confifirm the original measurement?
The answer is no. The intermediate measurement along the
x axis will leave the spin in a completely random confifigura
tion as far as the next measurement is concerned. There is
no way to make the intermediate determination of the spin
without completely disrupting the fifinal measurement. One
might say that measuring one component of the spin destroys
the information about another component. In fact, one sim
ply cannot simultaneously know the components of the spin
along two difffferent axes, not in a reproducible way in any
case. There is something fundamentally difffferent about the
state of a quantum system and the state of a classical system.
1.5 Propositions
The space of states of a classical system is a mathematical
set. If the system is a coin, the space of states is a set of
two elements, H and T. Using set notation, we would write
{H, T}. If the system is a six-sided die, the space of states
has six elements labeled {1, 2, 3, 4, 5, 6}. The logic of set the
ory is called Boolean logic. Boolean logic is just a formalized
version of the familiar classical logic of propositions.
A fundamental idea in Boolean logic is the notion of a
truth-value. The truth-value of a proposition is either true
or f alse. Nothing in between is allowed. The related set
theory concept is a subset. Roughly speaking, a proposition
is true for all the elements in its corresponding subset and
false for all the elements not in this subset. For example,
if the set represents the possible states of a die, one can
consider the proposition
A: The die shows an odd-numbered face.
The corresponding subset contains the three elements
{1, 3, 5}.
Another proposition states
B: The die shows a number less than 4.
The corresponding subset contains the states {1, 2, 3}.
Every proposition has its opposite (also called its negation).
For example,
not A: The die does not show an odd-numbered face.
The subset for this negated proposition is {2, 4, 6}.
There are rules for combining propositions into more com
plex propositions, the most important being or, and, and
not. We just saw an example of not, which gets applied to
a single subset or proposition. And is straightforward, and
applies to a pair of propositions.3 It says they are both true.
Applied to two subsets, and gives the elements common to
both, that is, the intersection of the two subsets. In the die
example, the intersection of subsets A and B is the subset
of elements that are both odd and less than 4. Fig. 1.5 uses
a Venn diagram to show how this works.
The or rule is similar to
and, but has one additional
subtlety. In everyday speech, the word or is generally used
in the exclusive sense—the exclusive version is true if one or
the other of two propositions is true, but not both. However,
Boolean logic uses the inclusive version of or, which is true if
either or both of the propositions are true. Thus, according
to the inclusive or, the proposition
Albert Einstein discovered relativity or Isaac Newton was
English
is true. So is
Albert Einstein discovered relativity or Isaac Newton was
Russian.
The inclusive or is only wrong if both propositions are false.
For example,
Albert Einstein discovered America4 or Isaac Newton was
Russian.
The inclusive or has a set theoretic interpretation as the
union of two sets: it denotes the subset containing anything
in either or both of the component subsets. In the die ex
ample, (A or B) denotes the subset {1, 2, 3, 5}.
1.6 Testing Classical Propositions
Let’s return to the simple quantum system consisting of a
single spin, and the various propositions whose truth we
could test using the apparatus A. Consider the following
two propositions:
A: The z component of the spin is +1.
B: The x component of the spin is +1.
Each of these is meaningful and can be tested by orienting
A along the appropriate axis. The negation of each is also
meaningful. For example, the negation of the fifirst proposi
tion is
not A: The z component of the spin is −1.
But now consider the composite propositions
(A or B): The z component of the spin is +1 or the x
component of the spin is +1.
(A and B): The z component of the spin is +1 and the
x component of the spin is +1.
Consider how we would test the proposition (A or B).
If spins behaved classically (and of course they don’t), we
would proceed as follows:5
•
Gently measure σz and record the value. If it is +1,
we are fifinished: the proposition (A or B) is true. If σz
is −1, continue to the next step.
•
Gently measure σx. If it is +1, then the proposition
(A or
B) is true. If not, this means that neither σz
nor σx
was equal to +1, and (A or B) is false.
There is an alternative procedure, which is to interchange the
order of the two measurements. To emphasize this reversal
of ordering, we’ll call the new procedure (B or A):
•
Gently measure σx and record the value. If it is +1 we
are fifinished: The proposition (B or A) is true. If σx is
−1 continue to the next step.
•
Gently measure σz. If it is +1, then (B
or A) is true.
If not, it means that neither σx nor σz
was equal to
+1, and (B or A) is false.
In classical physics, the two orders of operation give the same
answer. The reason for this is that measurements can be
arbitrarily gentle—so gentle that they do not affffect the re
sults of subsequent measurements. Therefore, the propo
sition (A or B) has the same meaning as the proposition
(B or A).
1.7 Testing Quantum Propositions
Now we come to the quantum world that I described earlier.
Let us imagine a situation in which someone (or something)
unknown to us has secretly prepared a spin in the σz = +1
state. Our job is to use the apparatus A to determine
whether the proposition (A or B) is true or false. We will
try using the procedures outlined above.
We begin by measuring σz. Since the unknown agent has
set things up, we will discover that σz = +1
. It is unnecessary
to go on: (A or B) is true. Nevertheless, we could test σx
just to see what happens. The answer is unpredictable. We
randomly fifind that σx = +1 or σx = −1. But neither of these
outcomes affffects the truth of proposition (A or B).
But now let’s reverse the order of measurement. As be
fore, we’ll call the reversed procedure (B or A), and this time
we’ll measure σx fifirst. Because the unknown agent set the
spin to +1 along the z axis, the measurement of σx is ran
dom. If it turns out that σx = +1, we are fifinished: (B or A)
is true. But suppose we fifind the opposite result, σx = −1.
The spin is oriented along the −x direction. Let’s pause here
brieflfly, to make sure we understand what just happened. As
a result of our fifirst measurement, the spin is no longer in its
original state σz = +1
. It is in a new state, which is either
σx = +1 or σx = −1. Please take a moment to let this idea
sink in. We cannot overstate its importance.
Now we’re ready to test the second half of proposition
(B or A). Rotate the apparatus A to the z axis and mea
sure σz. According to quantum mechanics, the result will be
randomly ±1. This means that there is a 25 percent probabil
ity that the experiment produces σx = −1 and σz = −1. In
other words, with a probability of 1
4 , we fifind that (B or A)
is false; this occurs despite the fact that the hidden agent
had originally made sure that σz = +1
.
Evidently, in this example, the inclusive or is not sym
metric. The truth of (A or B) may depend on the order in
which we confifirm the two propositions. This is not a small
thing; it means not only that the laws of quantum physics
are difffferent from their classical counterparts, but that the
very foundations of logic are difffferent in quantum physics as
well.
What about (A and B)? Suppose our fifirst measure
ment yields σz = +1 and the second, σx = +1
. This is of
course a possible outcome. We would be inclined to say that
(A and B) is true. But in science, especially in physics,
the truth of a proposition implies that the proposition can
be verifified by subsequent observation. In classical physics,
the gentleness of observations implies that subsequent exper
iments are unaffffected and will confifirm an earlier experiment.
A coin that turns up Heads will not be flflipped to Tails by
the act of observing it—at least not classically. Quantum
mechanically, the second measurement (σx = +1) ruins the
possibility of verifying the fifirst. Once σx
has been prepared
along the x axis, another mesurement of
σz will give a ran
dom answer. Thus (A and B) is not confifirmable: the second
piece of the experiment interferes with the possibility of con-
fifirming the fifirst piece.
If you know a bit about quantum mechanics, you proba
bly recognize that we are talking about the uncertainty prin
ciple. The uncertainty principle doesn’t apply only to posi
tion and momentum (or velocity); it applies to many pairs
of measurable quantities. In the case of the spin, it applies
to propositions involving two difffferent components of σ. In
the case of position and momentum, the two propositions we
might consider are:
A certain particle has position x.
That same particle has momentum p.
From these, we can form the two composite propositions
The particle has position x and the particle has
momentum p.
The particle has position x or the particle has
momentum p.
Awkward as they are, both of these propositions have mean
ing in the English language, and in classical physics as well.
However, in quantum physics, the fifirst of these propositions
is completely meaningless (not even wrong), and the second
one means something quite difffferent from what you might
think. It all comes down to a deep logical difffference between
the classical and quantum concepts of the state of a system.
Explaining the quantum concept of state will require some
abstract mathematics, so let’s pause for a brief interlude on
complex numbers and vector spaces. The need for complex
quantities will become clear later on, when we study the
mathematical representation of spin states.
1.8 Mathematical Interlude:
Complex Numbers
Everyone who has gotten this far in the Theoretical Mini
mum series knows about complex numbers. Nevertheless, I
will spend a few lines reminding you of the essentials. Fig.
1.6 shows some of their basic elements.
A complex number z is the sum of a real number and an
imaginary number. We can write it as
z = x + iy,
where x and y are real and i2 = −1. Complex numbers can
be added, multiplied, and divided by the standard rules of
arithmetic. They can be visualized as points on the complex
plane with coordinates x, y. They can also be represented in
polar coordinates:
z = reiθ = r(cos θ + isin θ).
Adding complex numbers is easy in component form: just
add the components. Similarly, multiplying them is easy
in their polar form: Simply multiply the radii and add the
angles:
r1eiθ1 r2eiθ2 = (r1r2) ei(θ1+θ2).
Every complex number z has a complex conjugate z∗ that is
obtained by simply reversing the sign of the imaginary part.
If
z = x + iy = reiθ,
then
z∗ = x − iy = re−iθ.
Multiplying a complex number and its conjugate always gives
a positive real result:
z∗z = r2.
It is of course true that every complex conjugate is itself a
complex number, but it’s often helpful to think of z and z∗
as belonging to separate “dual” number systems. Dual here
means that for every z there is a unique z∗ and vice versa.
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