The formalism for quantum systems developed so far applies to what are<\/p>\n\n\n\n
called pure states. A system in a pure state is completely speci\ufb01ed by the<\/p>\n\n\n\n
state vector. A complete set of experimental tests will determine the system<\/p>\n\n\n\n
state fully: we have maximal knowledge of the system. For example, for a spin<\/p>\n\n\n\n
system, we can \ufb01nd a particular orientation of an SG machine such that the<\/p>\n\n\n\n
state is in its + or \u2212 port. This also means that the state is an eigenvector of<\/p>\n\n\n\n
some operator, or is always a linear combination of the computational basis<\/p>\n\n\n\n
states.<\/p>\n\n\n\n
As opposed to this, as in most practical cases, we only have incomplete<\/p>\n\n\n\n
knowledge of the state. This means that the state is in practice not an eigen-<\/p>\n\n\n\n
state of an observable, but consists of a mixture of eigenstates with classical<\/p>\n\n\n\n
probabilities of being in each state. Such a state is called a mixed state<\/p>\n\n\n\n
and CANNOT be represented by a state vector. The most convenient way of<\/p>\n\n\n\n
representing and dealing with such systems, is through the density operator<\/p>\n\n\n\n
formulation, as proposed \ufb01rst by von Neumann [70] in 1927.<\/p>\n\n\n\n
1<\/p>\n\n\n\n
For instance, how do we describe the state of an unpolarized beam of spins,<\/p>\n\n\n\n
such as those emitted from the oven in the original Stern\u2013Gerlach experiment?<\/p>\n\n\n\n
We will \ufb01nd that on analyzing such a beam using an SG machine in any<\/p>\n\n\n\n
orientation, it is split into up-spin and down-spin beams of equal intensities.<\/p>\n\n\n\n
The state of this beam can be regarded as a 50-50 mixture of basis states of<\/p>\n\n\n\n
any representation. This is an example of a mixed state. We cannot represent<\/p>\n\n\n\n
it as a superposition of any basis states. However, the output of an SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
\u2191<\/p>\n\n\n\n
\ufb01lter, which splits into up-spin and down-spin beams of equal intensities when<\/p>\n\n\n\n
passed through SG<\/p>\n\n\n\n
x<\/p>\n\n\n\n
or SG<\/p>\n\n\n\n
y<\/p>\n\n\n\n
machines is a pure state that can be represented<\/p>\n\n\n\n
by the state vectors<\/p>\n\n\n\n
|0i \u2261 |\u2191<\/p>\n\n\n\n
z<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
[|\u2191<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i + |\u2193<\/p>\n\n\n\n
x<\/p>\n\n\n\n
i] =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
[|\u2191<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i + i|\u2193<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i…<\/p>\n\n\n\n
Another point to keep in mind is that we have so far been describing closed<\/p>\n\n\n\n
quantum systems, that are isolated from the environment or not a\ufb00ected by it.<\/p>\n\n\n\n
More realistic systems are open to the environment, the e\ufb00ect of which must<\/p>\n\n\n\n
be taken into account in some fashion, though one may not have complete<\/p>\n\n\n\n
1<\/p>\n\n\n\n
The density operator was also independently proposed by Lev Landau [45] and by Felix<\/p>\n\n\n\n
Bloch.<\/p>\n\n\n\n
77<\/p>\n\n\n\n
78 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
information as to how the environment a\ufb00ects the system. One way of dealing<\/p>\n\n\n\n
with such situations is to regard the system along with the environment as a<\/p>\n\n\n\n
big super-system that is closed. So when we concentrate only on the system,<\/p>\n\n\n\n
we have to average out the e\ufb00ect of the environment. The resulting system<\/p>\n\n\n\n
state typically is mixed, and one needs the density operator formulation to<\/p>\n\n\n\n
describe it. The material in this chapter is of a slightly advanced character<\/p>\n\n\n\n
and may be skipped at \ufb01rst reading<\/p>\n","protected":false},"excerpt":{"rendered":"
The formalism for quantum systems developed so far applies to what are called pure states. A system in a pure state is completely speci\ufb01ed by the state vector. A complete set of experimental tests will determine the system state fully: we have maximal knowledge of the system. For example, for a spin system, we can […]<\/p>\n","protected":false},"author":1,"featured_media":4020,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[491],"tags":[],"class_list":["post-4021","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-5-grand-unification"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4021","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=4021"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4021\/revisions"}],"predecessor-version":[{"id":4022,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4021\/revisions\/4022"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4020"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4021"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4021"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}