Bell’s original work, and many subsequent variants show how quantum
correlations in an entangled state are essentially different from classical ones.
One of the inequalities of Bell applies to a physical system consisting of two
subsystems, obeying the principle of local realism. He shows that the quantum
statistics for such a system involving entangled subsystems will necessarily vi-
olate this inequality, a statement generically known as “Bell’s theorem” [64].
Subsequently many similar inequalities were discovered by various authors.
(These are reviewed in [18].) We will discuss one of them (not Bell’s original
one!) to show how quantum correlations are intrinsically different from classi-
cal (local realist) ones. This follows original work by Clauser, Horne, Shimony
and Holt [17](CHSH).
We consider spin as an example but the derivation holds true for any
dichotomic variable, i.e., one with measurements outcomes described by two
values, ±1. Let’s revisit the experiment of Figure 4.2.
Consider a source emitting a very large number N of entangled spin-half
pairs, and four arbitrary directions ˆa,
ˆ
a
0
,
ˆ
b,
ˆ
b
0
for SG machines chosen by Alice
and Bob for measuring. Suppose that before measurement, the spin of the i
th
pair has hidden, fixed values r
i
(a) and r
i
(a
0
) for particle (1), s
i
(b) and s
i
(b
0
)
for particle (2) along the respective axes. The correlation between particles
(1) and (2) can be measured by the average value of the product of spin
measurements:
C(a, b) =
1
N
X
i
r
i
(a)s
i
(b). (4.14)
We will have similar expressions for C(a
0
, b), C(a, b
0
), and C(a
0
, b
0
), if the ex-
periments used those pairs of axes for measurement. These expressions for the
average are the same as for classical statistical averages.
CHSH in their worked aimed to calculate the quantity
C(a, b) + C(a, b
0
) + C(a
0
, b) − C(a
0
, b
0
). (4.15)
We’ll first see what the “classical” value is, assuming hidden variable descrip-
tion and then compare it with the predictions of quantum mechanics. First
look at the possible combinations of spin values (in units of ~/2) for the i
th
pair. We introduce the notation
T
1
= r
i
(a)[s
i
(b) + s
i
(b
0
)], T
2
= r
i
(a
0
)[s
i
(b) − s
i
(b
0
)].
74 Introduction to Quantum Physics and Information Processing
Observe that T
1
+ T
2
= ±2 always. For instance, when r
i
(a) = +1, r
i
(a
0
) =
−1, s
i
(b) = −1, s
i
(b
0
) = +1, then T
1
= −2 and T
2
= 0. You can see similar
results for all other combinations of values for these two spin measurements.
To evaluate the sum 4.15, we just sum T
1
+ T
2
over all i and divide by N:
|C(a, b) + C(a, b
0
) + C(a
0
, b) − C(a
0
, b
0
)| ≤ 2.. (4.16)
This is the CHSH inequality.
What does quantum mechanics predict for the sum (4.15)? Remember that
the spins are not to have fixed values before measurement. The correlation
between spins are now the quantum mechanical expectation values of spin
operator products in the state of Equation 4.13:
C(a, b) = h
ˆ
S
a
ˆ
S
b
i
β
11
. (4.17)
Note that the operator
ˆ
S
a
, spin along direction ˆa is just ~σ ·ˆa (in units of ~/2).
You would have shown in Problem 3.12 (b) of Chapter 3, that the eigenvectors
of
ˆ
S
a
are given by
|ˆa±i = e
−i
ˆ
k·~σ
|Z±i.
Here
ˆ
k is a direction perpendicular to both ˆz and ˆa, i.e., parallel to ˆz × ˆa.
Example 4.4.1. Let’s find the expectation value of
ˆ
S
a
ˆ
S
b
in the Bell state
|β
11
i.
ˆ
S
a
|β
11
i = ~σ · ˆa(|01i − |10i)
= (a
x
X
1
+ a
y
Y
1
+ a
z
Z
1
)(|01i − |10i)
= a
x
(|11i − |00i) − ia
y
(|01i + |10i) + a
z
(|01i + |10i)
ˆ
S
a
ˆ
S
b
|β
11
i = −a
x
b
x
|β
11
i − ia
x
b
y
(|10i + |01i) + a
x
b
z
(|10i + |11i)
−ia
y
b
x
(|00i + |11i) − a
y
b
y
|β
11
i + ia
y
b
z
(|01i − |10i)
+a
z
b
x
(|11i + |00i) + ia
z
b
y
(|01i − |10i) − a
z
b
z
|β
11
i,
hβ
11
|
ˆ
S
a
ˆ
S
b
|β
11
i = −a
x
b
x
− a
y
b
y
− a
z
b
z
= −ˆa ·
ˆ
b.
Then the left-hand side of Equation 4.16 is
|ˆa · (
ˆ
b +
ˆ
b
0
) +
ˆ
a
0
· (
ˆ
b −
ˆ
b
0
)| ≤ |ˆa||
ˆ
b +
ˆ
b
0
| + |
ˆ
a
0
||
ˆ
b −
ˆ
b
0
| (4.18)
=
√
2(
p
1 + cos φ +
p
1 − cos φ)(4.19)
where cos φ =
ˆ
b ·
ˆ
b
0
. (4.20)
Now the minimum value this can take is obviously when cos φ = 0, and that
value is 2
√
2, greater than the CHSH bound. Thus there exist configurations
Properties of Qubits 75
of detectors that can violate the CHSH inequality. See for instance Figure
4.3. This leads us to conclude that quantum mechanics is NOT compatible
with a local realistic description, that is, the assumption that the spins have
values before they are measured must be wrong. The entangled state vector
describes the pair as a single whole, with no room for describing the states
of the individual constituents. They have no well-defined spin in such a state.
There is therefore no way of setting about deriving the CHSH inequality for
such a system: the spin values of particles (1) and (2) do not exist before they
are measured.
Example 4.4.2. Let’s examine the directions for which the CHSH inequality is
maximally violated. If cos φ = 0, then we have
ˆ
b ⊥
ˆ
b
0
. The RHS of inequality
4.18 also shows that ˆa and
ˆ
b +
ˆ
b
0
must be parallel or antiparallel, and so
also
ˆ
a
0
and
ˆ
b −
ˆ
b
0
must be parallel or antiparallel. One way of picking such
directions is for Alice to choose ˆz and ˆx while Bob chooses the ±45
◦
directions
(
1
√
2
ˆx + ˆz
and
1
√
2
ˆz − ˆx
), as in Figure 4.3. Other sets of combinations are
also possible that satisfy the above criterion (find them!). In the language
of quantum mechanics, we must speak of the operators corresponding to the
measurement axes of A and B: in other words, we talk of then measuring
the operator σ
a
or σ
b
. Thus we speak of correlations between certain pairs of
observables that violate the CHSH bound for classical correlations.
FIGURE 4.3: Directions for SG detectors a, a
0
, b and b
0
and the corresponding
observables measured by Alice and Bob, that maximally violate the CHSH
inequality.
The beauty of Bell’s inequalities was that for the first time they provided a
way to test quantum mechanics experimentally. The first experimental realiza-
tion of this was performed by the group led by Alain Aspect in 1981 [2]. Since
then, many experiments have been performed that confirm the violation of
the inequalities, and the corresponding interpretation of quantum mechanics
as theory that intrinsically does not obey “local realism”.
However, some researchers have tried to come up with non-local theories
that still are consistent with relativity, notably the GRW [37] theory of Ghi-
76 Introduction to Quantum Physics and Information Processing
rardi, Rimini, and Weber, and Bohmian mechanics [22]. The debate still con-
tinues as people come up with plausible non-local realistic theories to replace
quantum mechanics!
This section ought to have convinced you that quantum entanglement is
something new and more than classical correlations: leading to its exploitation
as a resource in information processing.
Many of the original papers cited in this chapter are reprinted in an in-
valuable volume by Wheeler and Zurek [72]. A wonderful discussion of many
of the properties of quantum systems discussed here is given in the book by
Aharonov and Rohrlich [1].
Problems
4.1. Find out what the action of each of the σ
i
operators is on the Bloch sphere
by checking their effects on the eigenvectors |Z±i, |X±i and |Y ±i.
4.2. Prove that the Bell states are mutually orthogonal and that they form
a basis for H
2
. You must be able to express an arbitrary 2-qubit state
|ψi = a|00i + b|01i + c|10i + d|11i as a linear superposition of the Bell
states. Find the coefficients in this superposition in terms of a, b, c, and d.
4.3. Entanglement and basis change: suppose |s
1
i and |s
2
i, linear combinations
of the basis states |0i and |1i form an orthonormal basis for a spin Hilbert
space. Show that the two-spin entangled “singlet” state
1
√
2
(|s
1
i ⊗ |s
2
i − |s
2
i ⊗ |s
1
i)
is equivalent to
1
√
2
(|01i − |10i).
Check that this preservation of the form of entanglement does not hold for
the other three Bell states in the transformed basis.
4.4. We found the directions ˆa,
ˆ
a
0
,
ˆ
b, and
ˆ
b
0
of Stern–Gerlach machines for
which the CHSH inequality is maximally violated for spin half particles.
Translate this experiment to photon polarization measurements and find
the corresponding directions for the axes of polarizers used by Alice and
Bob that would maximally violate the CHSH inequality.
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