Once you have measured Sz, you can
measure it again and you will get the
same answer as you got the first time
around. Sx and Sy work the same way.
So far, you might have thought that
each component of the electron’s spin
stores one bit of information.
But if you try to measure more than
one of the components, again something strange happens. Take an electron, measure Sz, and suppose that
the outcome is + 1. Now measure Sx
(obtaining either + 1 or − 1) and then
Figure 1. the no-cloning theorem.
Imagine that someone prepares a single
qubit in the state |ψñ = a|0ñ + b|1ñ and gives it
to you without telling you what a and b are.
your goal is to copy that qubit. We will call
whatever algorithm you use (the supposed
“quantum copy machine”) C. you feed C the
(unknown) qubit |ψñ and an output qubit
that starts in the state |0ñ and your machine
needs to output the original qubit and transform |0ñ into a copy of |ψñ.
you do not know in advance what a and
b are, so your copy machine has to work
for any values. In particular, your machine
needs to work if a = 1 and b = 0, which means
C (|0ñ|0ñ) = |0ñ|0ñ.
similarly, your copy machine needs to work
if a = 0 and b = 1, which means
C (|1ñ|0ñ) = |1ñ|1ñ.
but quantum mechanics is linear, so any
copy machine you could possibly build has
to be linear as well. this means that the
operator C is linear, so we can do some linear algebra:
C (|ψñ|0ñ) = C ((a|0ñ + b |1ñ) |0ñ)
= aC (|0ñ|0ñ) + bC (|1ñ|0ñ)
= a|0ñ|0ñ + b |1ñ|1ñ. ( 1)
your copy machine was supposed to copy
any state, so the output should have been
(a|0ñ + b |1ñ) (a |0ñ + b|1ñ)
If both a and b are nonzero, then the output
( 1) of your machine is not correct. this
means that your copy machine C cannot
measure Sz again. You would expect to
get Sz = + 1 as before, but if you do this
experiment you will get + 1 half the time
and − 1 half the time. Measuring the
electron’s spin therefore changes the
spin state of the electron. Physicists
have come to realize that this is not
a limitation of their experiments but
rather that the universe fundamentally operates this way.
No matter what encoding you use
or how perfect an apparatus you can
build, you can only ever reliably encode
one bit worth of recoverable classical
information in the spin of an electron. 20
Nonetheless, an electron behaves very
differently than a classical bit. If we use
electron spins instead of classical bits
to store information, we can perform
tasks that are completely impossible
with ordinary computers.
An electron’s spin is an example of a
mathematical object called a qubit.
A classical bit can take either of the two
values 0 or 1. But a qubit is described
mathematically by a normalized state
in a two-dimensional complex vec-
tor space. We will use notation from
physics to denote vectors that repre-
sent quantum states, writing a vector
named v as |vñ. We can write any one-
qubit state as
|qñ = a|0ñ + b|1ñ
where the states |0ñ and |1ñ form a
basis for the 2D vector space and
where a and b are complex numbers
that satisfy |a| 2 + |b| 2 = 1. If neither a
nor b is zero, then we call the state |qñ a
superposition of |0ñ and |1ñ because the
qubit |qñ is, in a sense, in both states
Just as one qubit can be in the state
|0ñ or |1ñ or some superposition (linear
combination) of both, n qubits can be
in any superposition of the states
|0 . . . 00ñ, |0 . . . 01ñ, |0 . . . 10ñ,
|0 . . . 11ñ, . . ., | 1 . . . 11ñ
So, an n qubit state is a vector in a
The simplest kind of measurement
one can perform on a single qubit is
one that answers this question: is the
qubit in a given state |rñ= a|0ñ+ b|1ñ?
Let us say our qubit is prepared in the
state |qñ as above and we make this
measurement. Then there are two
possible outcomes. We might get the
answer yes, in which case the state
of the system would change instanta-
neously from |qñ to |rñ. The probability
that this happens is given by the com-
plex inner product squared of the two
states in question
Pr [yes] = |a*a + b*b| 2.
If on the other hand we obtain the mea-
surement outcome no, then the state
of the system would instantaneously
change from |qñ to the state |r ⊥ñ = b*|0ñ
− a*|1ñ that is perpendicular to |rñ. This
happens with probability
Pr [no] = |a*b – b*a| 2 = 1 − Pr [yes].
We can use this mathematical frame-
work to explain the measurement statis-
tics of electron spin. We define the states
|Sz = +1ñ = |0ñ
|Sz = –1ñ = |1ñ;
these two states form a basis for a one-qubit vector space. Then
=+ 〉= 〉+ 〉
= − 〉= 〉− 〉
| 1 (|0 | 1 )
| 1 (|0 | 1 ).
=+ 〉= 〉+ 〉
=− 〉= 〉− 〉
| 1 (|0 | 1 )
| 1 (|0 | 1 ).
Measuring the spin component Sx is the
same as measuring whether the state
being tested is |Sx =+1ñ; the outcome
yes means Sx =+ 1 and the outcome no
means Sx = − 1. If the spin started in the
|Sz =+1ñ state then, upon measuring
Sx, we will obtain + 1 or − 1 with equal
probability and the state after the measurement would be either |Sx =+1ñ or
|Sx = −1ñ. If we then measure Sz again, we
obtain + 1 or − 1 with equal probability.
Physicists are trying to build
devices that can manipulate electrons
or other qubits in a manner analogous to the way ordinary computers
manipulate bits in their memories.
Such a device, if it worked reliably
and could store many qubits, would
be a functioning quantum computer.