![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bgd0.png\")
<\/p>\n\n\n\nAn interesting aspect of quantum information transfer is how one can<\/p>\n\n\n\n
actually transfer two classical bits of information while physically transmitting<\/p>\n\n\n\n
only one qubit. This process seems to involve compressing two bits into one<\/p>\n\n\n\n
qubit and is accordingly called dense coding. The key to the process is the<\/p>\n\n\n\n
use of entanglement. This protocol preceded and inspired the teleportation<\/p>\n\n\n\n
protocol discussed above [10]. So our friends Alice and Bob enter the picture<\/p>\n\n\n\n
with their shared Bell state, which they are going to use as a resource to<\/p>\n\n\n\n
communicate two bits of information between them.<\/p>\n\n\n\n
The trick is fairly simple. Suppose Alice and Bob share the Bell state |\u03b2<\/p>\n\n\n\n
00<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
Alice performs a local operation on her piece of the entangled pair depending<\/p>\n\n\n\n
on the two-bit number she wishes to communicate, and then transfers the<\/p>\n\n\n\n
qubit over an appropriate quantum channel to Bob. Bob then measures both<\/p>\n\n\n\n
qubits in the Bell basis to obtain the two-bit number. The local operation<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
that Alice performs is according to Table 9.2.<\/p>\n\n\n\n
TABLE 9.2: Operations for super-dense coding.<\/p>\n\n\n\n
Number Operation<\/p>\n\n\n\n
00<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A =<\/p>\n\n\n\n
01<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
X<\/p>\n\n\n\n
10<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
11<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
X<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
Let\u2019s check how this works on an example: suppose Alice wishes to com-<\/p>\n\n\n\n
municate the number 2 or 10 in binary. The sequence of operations undergone<\/p>\n\n\n\n
by the Bell pair is then as follows:<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
00<\/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
[|00i + |11i]<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
Z<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
[|00i \u2212 |11i]<\/p>\n\n\n\n
Bell basis change<\/p>\n\n\n\n
\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2192 |10i. (9.2)<\/p>\n\n\n\n
You can verify the last step by performing the operations for the Bell mea-<\/p>\n\n\n\n
surement explicitly as a CNOT and then an H on the \ufb01rst qubit.<\/p>\n\n\n\n
Exercise 9.1. Show how the above dense coding protocol works if the entangled<\/p>\n\n\n\n
state shared by Alice and Bob was |\u03b2<\/p>\n\n\n\n
11<\/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
[|10i \u2212 |01i].<\/p>\n\n\n\n
<\/p>\n\n\n\n
Information and Communication<\/p>\n","protected":false},"excerpt":{"rendered":"
An interesting aspect of quantum information transfer is how one can actually transfer two classical bits of information while physically transmitting only one qubit. This process seems to involve compressing two bits into one qubit and is accordingly called dense coding. The key to the process is the use of entanglement. This protocol preceded and […]<\/p>\n","protected":false},"author":1,"featured_media":4043,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[498],"tags":[],"class_list":["post-4109","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-information-and-communication"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/informative.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4109","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=4109"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4109\/revisions"}],"predecessor-version":[{"id":4110,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4109\/revisions\/4110"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4043"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4109"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4109"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4109"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}