The full speci\ufb01cation of a superposition state |\u03c8i = \u03b1|0i+ \u03b2|1i is given by<\/p>\n\n\n\n
the complex numbers \u03b1 and \u03b2. The meaning of these numbers is physically<\/p>\n\n\n\n
derived by making measurements on this state, in the computational basis.<\/p>\n\n\n\n
This process would randomly \u201ccollapse\u201d the state to either |0i or |1i. The<\/p>\n\n\n\n
probability of obtaining |0i is |\u03b1|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
and of obtaining |1i is |\u03b2|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. This is true in<\/p>\n\n\n\n
a statistical sense: make the same measurements on a statistically large set<\/p>\n\n\n\n
of identically prepared qubits: an ensemble. A measurement on a single qubit<\/p>\n\n\n\n
state that is unknown projects it on to a basis state and the original state is<\/p>\n\n\n\n
destroyed.<\/p>\n\n\n\n
So if we are given a single quantum system in the state |\u03c8i then can<\/p>\n\n\n\n
we make clones (that is, exact copies) of the state so that we can gather<\/p>\n\n\n\n
the requisite measurement data? The answer given by quantum mechanics is<\/p>\n\n\n\n
\u201cNO\u201d.<\/p>\n\n\n\n
There exists no quantum mechanical way (i.e., a unitary operator) to take<\/p>\n\n\n\n
one unknown state and make multiple identical copies of it.<\/p>\n\n\n\n
This is the no cloning theorem \ufb01rst formulated in 1982 [76, 27], which<\/p>\n\n\n\n
states that an arbitrary quantum system cannot be cloned by a universal<\/p>\n\n\n\n
unitary transformation. If<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
is a unitary cloning machine, then its action<\/p>\n\n\n\n
would be de\ufb01ned as taking as input the state |\u03c8i to be cloned along with a<\/p>\n\n\n\n
\u201cblank\u201d state, say |0i, and produce as output the original state and its clone:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
|\u03c8i|0i = |\u03c8i|\u03c8i. (4.3)<\/p>\n\n\n\n
<\/p>\n\n\n\n
66 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
Theorem: A unitary transformation cannot make identical copies of an<\/p>\n\n\n\n
arbitrary quantum state.<\/p>\n\n\n\n
Proof. Suppose there does exist a cloning machine as de\ufb01ned by Equation 4.3.<\/p>\n\n\n\n
Consider its action on two arbitrary quantum states |\u03c8i and |\u03c6i:<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
|\u03c8i|0i = |\u03c8i|\u03c8i, (4.4a)<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
|\u03c6i|0i = |\u03c6i|\u03c6i. (4.4b)<\/p>\n\n\n\n
Take the inner product of (4.4a) with (4.4b),<\/p>\n\n\n\n
LHS = h\u03c6|h0|<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
|\u03c8i|0i<\/p>\n\n\n\n
= h\u03c6|\u03c8i,<\/p>\n\n\n\n
RHS = h\u03c6|h\u03c6|\u03c8i|\u03c8i<\/p>\n\n\n\n
= h\u03c6|\u03c8i<\/p>\n\n\n\n
2<\/p>\n\n\n\n
.<\/p>\n\n\n\n
The only way LHS = RHS is if h\u03c6|\u03c8i = 0 (they are orthogonal) or if<\/p>\n\n\n\n
h\u03c6|\u03c8i = 1 (they are identical). Thus a more rigorous statement of the no-<\/p>\n\n\n\n
cloning theorem would be that non-orthogonal states cannot be cloned by the<\/p>\n\n\n\n
same unitary operator.<\/p>\n\n\n\n
Another proof is as follows:<\/p>\n\n\n\n
Proof. Since<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
is linear, its operation on a linear combination of states will<\/p>\n\n\n\n
be<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
(|\u03c8i + |\u03c6i)|0i = |\u03c8i|\u03c8i + |\u03c6i|\u03c6i.<\/p>\n\n\n\n
However, a cloner of the state |\u03c8i + |\u03c6i must produce<\/p>\n\n\n\n
(|\u03c8i + |\u03c6i)(|\u03c8i + |\u03c6i) = |\u03c8i|\u03c8i + 2|\u03c8i|\u03c6i + |\u03c6i|\u03c6i,<\/p>\n\n\n\n
which is NOT what<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
produced! In fact, the output of the cloner is actually<\/p>\n\n\n\n
an ENTANGLED state (Section 4.4) while what we require is a product state.<\/p>\n\n\n\n
Due to this inconsistency,<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U<\/p>\n\n\n\n
cl<\/p>\n\n\n\n
does not exist.<\/p>\n\n\n\n
You will see an illustration of this using CNOT operations in Chapter 7.<\/p>\n\n\n\n
The converse of this theorem is also true. Sometimes referred to as the no<\/p>\n\n\n\n
deletion theorem [52], this states that given multiple copies of an unknown<\/p>\n\n\n\n
quantum state, no unitary transformation can delete one of the copies to<\/p>\n\n\n\n
give a blank (|0i). This theorem thus protects the information content in a<\/p>\n\n\n\n
qubit. Both these theorems are of great importance in the theory of quantum<\/p>\n\n\n\n
information<\/p>\n","protected":false},"excerpt":{"rendered":"
The full speci\ufb01cation of a superposition state |\u03c8i = \u03b1|0i+ \u03b2|1i is given by the complex numbers \u03b1 and \u03b2. The meaning of these numbers is physically derived by making measurements on this state, in the computational basis. This process would randomly \u201ccollapse\u201d the state to either |0i or |1i. The probability of obtaining |0i […]<\/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-4005","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\/4005","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=4005"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4005\/revisions"}],"predecessor-version":[{"id":4561,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4005\/revisions\/4561"}],"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=4005"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4005"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}