We would in general consider not just a single quantum system, represent-<\/p>\n\n\n\n
ing one qubit, but a multiple qubit system that will consist of distinct and<\/p>\n\n\n\n
non-interacting component single-qubit systems. The quantum states of the<\/p>\n\n\n\n
composite system are elements of a larger Hilbert space composed of the single<\/p>\n\n\n\n
qubit Hilbert spaces. For this we take what is called a direct product of the<\/p>\n\n\n\n
single-qubit basis states, to form basis states of the larger Hilbert space. This<\/p>\n\n\n\n
direct product is also called a tensor product, represented by the symbol<\/p>\n\n\n\n
\u2297. The elements of the tensor product basis consist of ordered sequences of<\/p>\n\n\n\n
elements from the bases of each of the component Hilbert spaces.<\/p>\n\n\n\n
Postulate 5. The Hilbert space of a composite system S is the direct product<\/p>\n\n\n\n
of Hilbert spaces of the components A, B, C…:<\/p>\n\n\n\n
H<\/p>\n\n\n\n
S<\/p>\n\n\n\n
= H<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 H<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\u2297 H<\/p>\n\n\n\n
C<\/p>\n\n\n\n
… (3.27)<\/p>\n\n\n\n
If the subsystems have basis states {|e<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i}, {|e<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i}…, then each basis state of<\/p>\n\n\n\n
the full system is a tensor product of the form<\/p>\n\n\n\n
|e<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i = |e<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2297 |e<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2297 …<\/p>\n\n\n\n
A general state of the composite system can be expressed as a linear combina-<\/p>\n\n\n\n
tion of basis states of the composite Hilbert space.<\/p>\n\n\n\n
For example, a 2-qubit system would consist of two non-interacting single<\/p>\n\n\n\n
qubits (say the individual z-spins of two isolated electrons), each with a 2-d<\/p>\n\n\n\n
Hilbert space H<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. The Hilbert space of the 2-qubit system is then<\/p>\n\n\n\n
H<\/p>\n\n\n\n
4<\/p>\n\n\n\n
= H<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u2297 H<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. (3.28)<\/p>\n\n\n\n
If we label the bases of the H<\/p>\n\n\n\n
2<\/p>\n\n\n\n
s as {|0i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
, |1i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
} and {|0i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
, |1i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
}, we get the<\/p>\n\n\n\n
basis for H<\/p>\n\n\n\n
4<\/p>\n\n\n\n
as the ordered pairs<\/p>\n\n\n\n
\ue008<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
, |1i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\ue009<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
\ue008<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
, |1i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\ue009<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\ue008<\/p>\n\n\n\n
|0i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 |0i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
, |0i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 |1i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
, |1i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 |0i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
, |1i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 |1i<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\ue009<\/p>\n\n\n\n
. (3.29)<\/p>\n\n\n\n
The notation |ai \u2297 |bi is shortened to |abi and we write the basis for H<\/p>\n\n\n\n
4<\/p>\n\n\n\n
as<\/p>\n\n\n\n
{|00i, |01i, |10i, |11i}. (3.30)<\/p>\n\n\n\n
We see binary representations of the numbers 0 to 3 emerging in this 2-qubit<\/p>\n\n\n\n
system.<\/p>\n\n\n\n
In matrix notation, these basis vectors are generated by direct products. To<\/p>\n\n\n\n
write the direct product of two matrices, we should realize that every element<\/p>\n\n\n\n
of one matrix is associated with every element of the other. This is done in<\/p>\n\n\n\n
the following manner:<\/p>\n","protected":false},"excerpt":{"rendered":"
We would in general consider not just a single quantum system, represent- ing one qubit, but a multiple qubit system that will consist of distinct and non-interacting component single-qubit systems. The quantum states of the composite system are elements of a larger Hilbert space composed of the single qubit Hilbert spaces. For this we take […]<\/p>\n","protected":false},"author":1,"featured_media":3974,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[489],"tags":[],"class_list":["post-3997","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-4-quantum-mechanics"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-computer-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3997","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=3997"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3997\/revisions"}],"predecessor-version":[{"id":3998,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3997\/revisions\/3998"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3974"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3997"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3997"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3997"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}