![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg59.png\")
<\/p>\n\n\n\nA generic qubit could have a non-de\ufb01nite state expressed as a superposition<\/p>\n\n\n\n
|\u03c8i = \u03b1|0i + \u03b2|1i, |\u03b1|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ |\u03b2|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1.<\/p>\n\n\n\n
How do we picture a qubit? As a vector in Hilbert space, the description<\/p>\n\n\n\n
is abstract. The 2-d Hilbert space is a space with 4 dimensions. To get a<\/p>\n\n\n\n
better feel for the sort of vector a quantum state is, we look at a geometrical<\/p>\n\n\n\n
visualization of a qubit.<\/p>\n\n\n\n
The space of all possible single qubits is spanned by all values of the four<\/p>\n\n\n\n
real numbers de\ufb01ned by \u03b1 and \u03b2 but subject to the constraint of normalization:<\/p>\n\n\n\n
|\u03b1|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
+ |\u03b2|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1. We have an additional constraint in the form of equivalence<\/p>\n\n\n\n
of all states di\ufb00ering by an overall phase. The four parameters thus reduce to<\/p>\n\n\n\n
two, which determine the surface of a unit sphere in the space of parameters.<\/p>\n\n\n\n
Let\u2019s see how.<\/p>\n\n\n\n
Recall the representation of |\u03c8i in polar form (Equation 3.5):<\/p>\n\n\n\n
|\u03c8i = r<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|0i + r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
|1i,<\/p>\n\n\n\n
where we\u2019ve written \u03c6 = \u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2212\u03b8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
, the relative phase between the basis vectors.<\/p>\n\n\n\n
We can further parametrize r<\/p>\n\n\n\n
1<\/p>\n\n\n\n
and r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
in terms of a single angle \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
1<\/p>\n\n\n\n
+ r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= 1 =\u21d2 r<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= cos \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, r<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= sin \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
.<\/p>\n\n\n\n
63<\/p>\n\n\n\n
<\/p>\n\n\n\n
64 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
We now have<\/p>\n\n\n\n
|\u03c8i = cos \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|0i + sin \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
|1i,<\/p>\n\n\n\n
which is the standard representation of a point on the unit sphere by spherical<\/p>\n\n\n\n
polar coordinates \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u2208 [0, \u03c0] and \u03c6 \u2208 [0, 2\u03c0].<\/p>\n\n\n\n
But we still have one further condition, which is often not intuitively ob-<\/p>\n\n\n\n
vious. For a given state at (\u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
, \u03c6), consider the point on this sphere that is<\/p>\n\n\n\n
diametrically opposite: i.e., at (\u03c0 \u2212 \u03b8, \u03c0 + \u03c6) :<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
antipode<\/p>\n\n\n\n
= \u2212cos \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|0i \u2212 sin \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
|1i = \u2212|\u03c8i,<\/p>\n\n\n\n
which is physically indistinguishable from |\u03c8i. Thus the upper hemisphere of<\/p>\n\n\n\n
the sphere is su\ufb03cient to represent the states of a qubit, i.e., \u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u2208 [0, \u03c0\/2].<\/p>\n\n\n\n
It is useful to regard this space as still a sphere by replacing the parameter<\/p>\n\n\n\n
\u03b8<\/p>\n\n\n\n
0<\/p>\n\n\n\n
by \u03b8\/2, \u03b8 \u2208 [0, \u03c0]. Geometrically this is visualized as \u201cfolding\u201d the lower<\/p>\n\n\n\n
hemisphere on the upper, to obtain the Bloch sphere. The usual sphere is<\/p>\n\n\n\n
a \u201cdouble cover\u201d of the Bloch sphere. We \ufb01nally have a representation of the<\/p>\n\n\n\n
qubit as a unique point on this sphere (Figure 4.1):<\/p>\n\n\n\n
|\u03c8i = cos<\/p>\n\n\n\n
\u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|0i + e<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
sin<\/p>\n\n\n\n
\u03b8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1i; (4.1)<\/p>\n\n\n\n
0 \u2264 \u03b8 \u2264 \u03c0, 0 \u2264 \u03c6 \u2264 2\u03c0.<\/p>\n\n\n\n
The vector<\/p>\n\n\n\n
~p \u2261 (cos \u03c6 sin \u03b8, sin \u03c6 sin \u03b8, cos \u03b8) (4.2)<\/p>\n\n\n\n
is called the Bloch vector, after a notation invented by Felix Bloch in 1943<\/p>\n\n\n\n
to depict the polarization states of light. Note that this sphere is not to be<\/p>\n\n\n\n
regarded as one in 3-d coordinate space.<\/p>\n\n\n\n
FIGURE 4.1: The Bloch sphere.<\/p>\n\n\n\n
On this sphere, the north pole represents |0i and the south pole, |1i. In<\/p>\n\n\n\n
general, antipodal points on the Bloch sphere represent orthogonal state.<\/p>\n\n\n\n
<\/p>\n\n\n\n
Properties of Qubits 65<\/p>\n\n\n\n
This picture is useful for visualizing the e\ufb00ects of single qubit transforma-<\/p>\n\n\n\n
tions, which would take a point on this sphere to another.<\/p>\n\n\n\n
There is no known simple generalization of this idea for multiple qubits,<\/p>\n\n\n\n
but it is useful for testing out ideas on gates and transformations for single<\/p>\n\n\n\n
qubits.<\/p>\n\n\n\n
Exercise 4.1. Using the polar representation for complex numbers \u03b1 and \u03b2, ob-<\/p>\n\n\n\n
tain the relationship between the angles \u03b8 and \u03c6 and the magnitude and<\/p>\n\n\n\n
phase of \u03b1 and \u03b2.<\/p>\n\n\n\n
Exercise 4.2. Figure out the location on the Bloch sphere of the states<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i + |1i) and<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i \u2212 |1i).<\/p>\n\n\n\n
Exercise 4.3. Show that antipodal states on the Bloch sphere (i.e., those at<\/p>\n\n\n\n
(\u03b8, \u03c6) and at (\u03c0 \u2212 \u03b8, \u03c0 + \u03c6) are orthogonal<\/p>\n","protected":false},"excerpt":{"rendered":"
A generic qubit could have a non-de\ufb01nite state expressed as a superposition |\u03c8i = \u03b1|0i + \u03b2|1i, |\u03b1| 2 + |\u03b2| 2 = 1. How do we picture a qubit? As a vector in Hilbert space, the description is abstract. The 2-d Hilbert space is a space with 4 dimensions. To get a better feel […]<\/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-4003","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\/4003","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=4003"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4003\/revisions"}],"predecessor-version":[{"id":4560,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4003\/revisions\/4560"}],"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=4003"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4003"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4003"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}