Classically, the outcomes of decision processes are always distinguishable:<\/p>\n\n\n\n
it is taken for granted that a tossed coin will land either on heads or on tails<\/p>\n\n\n\n
and upon looking at it, we can distinguish the di\ufb00erent outcomes with cer-<\/p>\n\n\n\n
tainty. In applications to quantum information processing too, we will usually<\/p>\n\n\n\n
measure the output state after a process. If this state is to give us answers to<\/p>\n\n\n\n
the problem we are trying to solve, it is important to be able to distinguish<\/p>\n\n\n\n
alternate outcomes. In quantum, basis states can get transformed to superpo-<\/p>\n\n\n\n
sitions. Alternate outcomes may be possible that must be distinguishable. It<\/p>\n\n\n\n
is easy to see that this is possible if the states are orthogonal.<\/p>\n\n\n\n
Suppose the possible \ufb01nal states are |\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i and |\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i that are not orthogonal,<\/p>\n\n\n\n
h\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i 6= 0. This means that one can write the second state in terms of the<\/p>\n\n\n\n
\ufb01rst and its orthogonal complement |\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u22a5<\/p>\n\n\n\n
:<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i = a|\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i + b|\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u22a5<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Thus on measuring the output, there is a probability |a|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
that we get |\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i<\/p>\n\n\n\n
even if the output state being measured was |\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i. There is a probability |a|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
of getting the wrong outcome when measuring \u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. The two output states as-<\/p>\n\n\n\n
sumed here cannot therefore be distinguished reliably. This fact can be proved<\/p>\n\n\n\n
rigorously by showing that one cannot invent any measurement operator that<\/p>\n\n\n\n
gives distinct outcomes with certainty on measuring a set of states that are<\/p>\n\n\n\n
not mutually orthogonal. This property is exploited in secure quantum key<\/p>\n\n\n\n
distribution to make the communication safe.<\/p>\n\n\n\n
Other means of distinguishing non-orthogonal states have been invented in<\/p>\n\n\n\n
which the space of states is extended, and the notion of measurement is gen-<\/p>\n\n\n\n
eralized. These so-called unambiguous state discrimination techniques allow<\/p>\n\n\n\n
for the possibility of getting inconclusive results after measurement. However<\/p>\n\n\n\n
if positive results are obtained then they do tell the two states apart.<\/p>\n","protected":false},"excerpt":{"rendered":"
Classically, the outcomes of decision processes are always distinguishable: it is taken for granted that a tossed coin will land either on heads or on tails and upon looking at it, we can distinguish the di\ufb00erent outcomes with cer- tainty. In applications to quantum information processing too, we will usually measure the output state after […]<\/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-4007","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\/4007","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=4007"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4007\/revisions"}],"predecessor-version":[{"id":4008,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4007\/revisions\/4008"}],"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=4007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}