![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg65.png\")
<\/p>\n\n\n\nBell\u2019s original work, and many subsequent variants show how quantum<\/p>\n\n\n\n
correlations in an entangled state are essentially di\ufb00erent from classical ones.<\/p>\n\n\n\n
One of the inequalities of Bell applies to a physical system consisting of two<\/p>\n\n\n\n
subsystems, obeying the principle of local realism. He shows that the quantum<\/p>\n\n\n\n
statistics for such a system involving entangled subsystems will necessarily vi-<\/p>\n\n\n\n
olate this inequality, a statement generically known as \u201cBell\u2019s theorem\u201d [64].<\/p>\n\n\n\n
Subsequently many similar inequalities were discovered by various authors.<\/p>\n\n\n\n
(These are reviewed in [18].) We will discuss one of them (not Bell\u2019s original<\/p>\n\n\n\n
one!) to show how quantum correlations are intrinsically di\ufb00erent from classi-<\/p>\n\n\n\n
cal (local realist) ones. This follows original work by Clauser, Horne, Shimony<\/p>\n\n\n\n
and Holt [17](CHSH).<\/p>\n\n\n\n
We consider spin as an example but the derivation holds true for any<\/p>\n\n\n\n
dichotomic variable, i.e., one with measurements outcomes described by two<\/p>\n\n\n\n
values, \u00b11. Let\u2019s revisit the experiment of Figure 4.2.<\/p>\n\n\n\n
Consider a source emitting a very large number N of entangled spin-half<\/p>\n\n\n\n
pairs, and four arbitrary directions \u02c6a,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
for SG machines chosen by Alice<\/p>\n\n\n\n
and Bob for measuring. Suppose that before measurement, the spin of the i<\/p>\n\n\n\n
th<\/p>\n\n\n\n
pair has hidden, \ufb01xed values r<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(a) and r<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
) for particle (1), s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b) and s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)<\/p>\n\n\n\n
for particle (2) along the respective axes. The correlation between particles<\/p>\n\n\n\n
(1) and (2) can be measured by the average value of the product of spin<\/p>\n\n\n\n
measurements:<\/p>\n\n\n\n
C(a, b) =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
N<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i<\/p>\n\n\n\n
r<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(a)s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b). (4.14)<\/p>\n\n\n\n
We will have similar expressions for C(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, b), C(a, b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
), and C(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
), if the ex-<\/p>\n\n\n\n
periments used those pairs of axes for measurement. These expressions for the<\/p>\n\n\n\n
average are the same as for classical statistical averages.<\/p>\n\n\n\n
CHSH in their worked aimed to calculate the quantity<\/p>\n\n\n\n
C(a, b) + C(a, b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
) + C(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, b) \u2212 C(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
). (4.15)<\/p>\n\n\n\n
We\u2019ll \ufb01rst see what the \u201cclassical\u201d value is, assuming hidden variable descrip-<\/p>\n\n\n\n
tion and then compare it with the predictions of quantum mechanics. First<\/p>\n\n\n\n
look at the possible combinations of spin values (in units of ~\/2) for the i<\/p>\n\n\n\n
th<\/p>\n\n\n\n
pair. We introduce the notation<\/p>\n\n\n\n
T<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= r<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(a)[s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b) + s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)], T<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= r<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)[s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b) \u2212 s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)].<\/p>\n\n\n\n
<\/p>\n\n\n\n
74 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
Observe that T<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ T<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= \u00b12 always. For instance, when r<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(a) = +1, r<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
) =<\/p>\n\n\n\n
\u22121, s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b) = \u22121, s<\/p>\n\n\n\n
i<\/p>\n\n\n\n
(b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
) = +1, then T<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= \u22122 and T<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 0. You can see similar<\/p>\n\n\n\n
results for all other combinations of values for these two spin measurements.<\/p>\n\n\n\n
To evaluate the sum 4.15, we just sum T<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ T<\/p>\n\n\n\n
2<\/p>\n\n\n\n
over all i and divide by N:<\/p>\n\n\n\n
|C(a, b) + C(a, b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
) + C(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, b) \u2212 C(a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)| \u2264 2.. (4.16)<\/p>\n\n\n\n
This is the CHSH inequality.<\/p>\n\n\n\n
What does quantum mechanics predict for the sum (4.15)? Remember that<\/p>\n\n\n\n
the spins are not to have \ufb01xed values before measurement. The correlation<\/p>\n\n\n\n
between spins are now the quantum mechanical expectation values of spin<\/p>\n\n\n\n
operator products in the state of Equation 4.13:<\/p>\n\n\n\n
C(a, b) = h<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
a<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
b<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
. (4.17)<\/p>\n\n\n\n
Note that the operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
a<\/p>\n\n\n\n
, spin along direction \u02c6a is just ~\u03c3 \u00b7\u02c6a (in units of ~\/2).<\/p>\n\n\n\n
You would have shown in Problem 3.12 (b) of Chapter 3, that the eigenvectors<\/p>\n\n\n\n
of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
a<\/p>\n\n\n\n
are given by<\/p>\n\n\n\n
|\u02c6a\u00b1i = e<\/p>\n\n\n\n
\u2212i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
k\u00b7~\u03c3<\/p>\n\n\n\n
|Z\u00b1i.<\/p>\n\n\n\n
Here<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
k is a direction perpendicular to both \u02c6z and \u02c6a, i.e., parallel to \u02c6z \u00d7 \u02c6a.<\/p>\n\n\n\n
Example 4.4.1. Let\u2019s \ufb01nd the expectation value of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
a<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
b<\/p>\n\n\n\n
in the Bell state<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
a<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i = ~\u03c3 \u00b7 \u02c6a(|01i \u2212 |10i)<\/p>\n\n\n\n
= (a<\/p>\n\n\n\n
x<\/p>\n\n\n\n
X<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ a<\/p>\n\n\n\n
y<\/p>\n\n\n\n
Y<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ a<\/p>\n\n\n\n
z<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
1<\/p>\n\n\n\n
)(|01i \u2212 |10i)<\/p>\n\n\n\n
= a<\/p>\n\n\n\n
x<\/p>\n\n\n\n
(|11i \u2212 |00i) \u2212 ia<\/p>\n\n\n\n
y<\/p>\n\n\n\n
(|01i + |10i) + a<\/p>\n\n\n\n
z<\/p>\n\n\n\n
(|01i + |10i)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
a<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
b<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i = \u2212a<\/p>\n\n\n\n
x<\/p>\n\n\n\n
b<\/p>\n\n\n\n
x<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i \u2212 ia<\/p>\n\n\n\n
x<\/p>\n\n\n\n
b<\/p>\n\n\n\n
y<\/p>\n\n\n\n
(|10i + |01i) + a<\/p>\n\n\n\n
x<\/p>\n\n\n\n
b<\/p>\n\n\n\n
z<\/p>\n\n\n\n
(|10i + |11i)<\/p>\n\n\n\n
\u2212ia<\/p>\n\n\n\n
y<\/p>\n\n\n\n
b<\/p>\n\n\n\n
x<\/p>\n\n\n\n
(|00i + |11i) \u2212 a<\/p>\n\n\n\n
y<\/p>\n\n\n\n
b<\/p>\n\n\n\n
y<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i + ia<\/p>\n\n\n\n
y<\/p>\n\n\n\n
b<\/p>\n\n\n\n
z<\/p>\n\n\n\n
(|01i \u2212 |10i)<\/p>\n\n\n\n
+a<\/p>\n\n\n\n
z<\/p>\n\n\n\n
b<\/p>\n\n\n\n
x<\/p>\n\n\n\n
(|11i + |00i) + ia<\/p>\n\n\n\n
z<\/p>\n\n\n\n
b<\/p>\n\n\n\n
y<\/p>\n\n\n\n
(|01i \u2212 |10i) \u2212 a<\/p>\n\n\n\n
z<\/p>\n\n\n\n
b<\/p>\n\n\n\n
z<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i,<\/p>\n\n\n\n
h\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
a<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
S<\/p>\n\n\n\n
b<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
11<\/p>\n\n\n\n
i = \u2212a<\/p>\n\n\n\n
x<\/p>\n\n\n\n
b<\/p>\n\n\n\n
x<\/p>\n\n\n\n
\u2212 a<\/p>\n\n\n\n
y<\/p>\n\n\n\n
b<\/p>\n\n\n\n
y<\/p>\n\n\n\n
\u2212 a<\/p>\n\n\n\n
z<\/p>\n\n\n\n
b<\/p>\n\n\n\n
z<\/p>\n\n\n\n
= \u2212\u02c6a \u00b7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b.<\/p>\n\n\n\n
Then the left-hand side of Equation 4.16 is<\/p>\n\n\n\n
|\u02c6a \u00b7 (<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b +<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
) +<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u00b7 (<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u2212<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
)| \u2264 |\u02c6a||<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b +<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
| + |<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
||<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u2212<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
| (4.18)<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2(<\/p>\n\n\n\n
p<\/p>\n\n\n\n
1 + cos \u03c6 +<\/p>\n\n\n\n
p<\/p>\n\n\n\n
1 \u2212 cos \u03c6)(4.19)<\/p>\n\n\n\n
where cos \u03c6 =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u00b7<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
. (4.20)<\/p>\n\n\n\n
Now the minimum value this can take is obviously when cos \u03c6 = 0, and that<\/p>\n\n\n\n
value is 2<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2, greater than the CHSH bound. Thus there exist con\ufb01gurations<\/p>\n\n\n\n
<\/p>\n\n\n\n
Properties of Qubits 75<\/p>\n\n\n\n
of detectors that can violate the CHSH inequality. See for instance Figure<\/p>\n\n\n\n
4.3. This leads us to conclude that quantum mechanics is NOT compatible<\/p>\n\n\n\n
with a local realistic description, that is, the assumption that the spins have<\/p>\n\n\n\n
values before they are measured must be wrong. The entangled state vector<\/p>\n\n\n\n
describes the pair as a single whole, with no room for describing the states<\/p>\n\n\n\n
of the individual constituents. They have no well-de\ufb01ned spin in such a state.<\/p>\n\n\n\n
There is therefore no way of setting about deriving the CHSH inequality for<\/p>\n\n\n\n
such a system: the spin values of particles (1) and (2) do not exist before they<\/p>\n\n\n\n
are measured.<\/p>\n\n\n\n
Example 4.4.2. Let\u2019s examine the directions for which the CHSH inequality is<\/p>\n\n\n\n
maximally violated. If cos \u03c6 = 0, then we have<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u22a5<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
. The RHS of inequality<\/p>\n\n\n\n
4.18 also shows that \u02c6a and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b +<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
must be parallel or antiparallel, and so<\/p>\n\n\n\n
also<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b \u2212<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
must be parallel or antiparallel. One way of picking such<\/p>\n\n\n\n
directions is for Alice to choose \u02c6z and \u02c6x while Bob chooses the \u00b145<\/p>\n\n\n\n
\u25e6<\/p>\n\n\n\n
directions<\/p>\n\n\n\n
(<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
\u02c6x + \u02c6z<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
and<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
\u02c6z \u2212 \u02c6x<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
), as in Figure 4.3. Other sets of combinations are<\/p>\n\n\n\n
also possible that satisfy the above criterion (\ufb01nd them!). In the language<\/p>\n\n\n\n
of quantum mechanics, we must speak of the operators corresponding to the<\/p>\n\n\n\n
measurement axes of A and B: in other words, we talk of then measuring<\/p>\n\n\n\n
the operator \u03c3<\/p>\n\n\n\n
a<\/p>\n\n\n\n
or \u03c3<\/p>\n\n\n\n
b<\/p>\n\n\n\n
. Thus we speak of correlations between certain pairs of<\/p>\n\n\n\n
observables that violate the CHSH bound for classical correlations.<\/p>\n\n\n\n
FIGURE 4.3: Directions for SG detectors a, a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, b and b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
and the corresponding<\/p>\n\n\n\n
observables measured by Alice and Bob, that maximally violate the CHSH<\/p>\n\n\n\n
inequality.<\/p>\n\n\n\n
The beauty of Bell\u2019s inequalities was that for the \ufb01rst time they provided a<\/p>\n\n\n\n
way to test quantum mechanics experimentally. The \ufb01rst experimental realiza-<\/p>\n\n\n\n
tion of this was performed by the group led by Alain Aspect in 1981 [2]. Since<\/p>\n\n\n\n
then, many experiments have been performed that con\ufb01rm the violation of<\/p>\n\n\n\n
the inequalities, and the corresponding interpretation of quantum mechanics<\/p>\n\n\n\n
as theory that intrinsically does not obey \u201clocal realism\u201d.<\/p>\n\n\n\n
However, some researchers have tried to come up with non-local theories<\/p>\n\n\n\n
that still are consistent with relativity, notably the GRW [37] theory of Ghi-<\/p>\n\n\n\n
<\/p>\n\n\n\n
76 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
rardi, Rimini, and Weber, and Bohmian mechanics [22]. The debate still con-<\/p>\n\n\n\n
tinues as people come up with plausible non-local realistic theories to replace<\/p>\n\n\n\n
quantum mechanics!<\/p>\n\n\n\n
This section ought to have convinced you that quantum entanglement is<\/p>\n\n\n\n
something new and more than classical correlations: leading to its exploitation<\/p>\n\n\n\n
as a resource in information processing.<\/p>\n\n\n\n
Many of the original papers cited in this chapter are reprinted in an in-<\/p>\n\n\n\n
valuable volume by Wheeler and Zurek [72]. A wonderful discussion of many<\/p>\n\n\n\n
of the properties of quantum systems discussed here is given in the book by<\/p>\n\n\n\n
Aharonov and Rohrlich [1].<\/p>\n\n\n\n
Problems<\/p>\n\n\n\n
4.1. Find out what the action of each of the \u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
operators is on the Bloch sphere<\/p>\n\n\n\n
by checking their e\ufb00ects on the eigenvectors |Z\u00b1i, |X\u00b1i and |Y \u00b1i.<\/p>\n\n\n\n
4.2. Prove that the Bell states are mutually orthogonal and that they form<\/p>\n\n\n\n
a basis for H<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. You must be able to express an arbitrary 2-qubit state<\/p>\n\n\n\n
|\u03c8i = a|00i + b|01i + c|10i + d|11i as a linear superposition of the Bell<\/p>\n\n\n\n
states. Find the coe\ufb03cients in this superposition in terms of a, b, c, and d.<\/p>\n\n\n\n
4.3. Entanglement and basis change: suppose |s<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i and |s<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i, linear combinations<\/p>\n\n\n\n
of the basis states |0i and |1i form an orthonormal basis for a spin Hilbert<\/p>\n\n\n\n
space. Show that the two-spin entangled \u201csinglet\u201d state<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|s<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i \u2297 |s<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i \u2212 |s<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i \u2297 |s<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i)<\/p>\n\n\n\n
is equivalent to<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|01i \u2212 |10i).<\/p>\n\n\n\n
Check that this preservation of the form of entanglement does not hold for<\/p>\n\n\n\n
the other three Bell states in the transformed basis.<\/p>\n\n\n\n
4.4. We found the directions \u02c6a,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
a<\/p>\n\n\n\n
0<\/p>\n\n\n\n
,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b, and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
b<\/p>\n\n\n\n
0<\/p>\n\n\n\n
of Stern\u2013Gerlach machines for<\/p>\n\n\n\n
which the CHSH inequality is maximally violated for spin half particles.<\/p>\n\n\n\n
Translate this experiment to photon polarization measurements and \ufb01nd<\/p>\n\n\n\n
the corresponding directions for the axes of polarizers used by Alice and<\/p>\n\n\n\n
Bob that would maximally violate the CHSH inequality.<\/p>\n","protected":false},"excerpt":{"rendered":"
Bell\u2019s original work, and many subsequent variants show how quantum correlations in an entangled state are essentially di\ufb00erent 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 […]<\/p>\n","protected":false},"author":1,"featured_media":4002,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[490],"tags":[],"class_list":["post-4017","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-5-interpreting-quantum-physics"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/atom-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4017","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4017"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4017\/revisions"}],"predecessor-version":[{"id":4563,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4017\/revisions\/4563"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4002"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4017"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4017"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}