![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bgcd.png\")
<\/p>\n\n\n\nAs an illustration of the power of entanglement as a resource, we examine<\/p>\n\n\n\n
the rather dramatically titled protocol of quantum state teleportation. This<\/p>\n\n\n\n
idea was \ufb01rst introduced by Bennett in 1993 [6]. The teleportation problem is<\/p>\n\n\n\n
the following: Alice needs to transfer to Bob (at a distant location) an unknown<\/p>\n\n\n\n
qubit |\u03c8i generically denoted by \u03b1|0i+ \u03b2|1i. The key point here is that Alice<\/p>\n\n\n\n
does not know what \u03b1 and \u03b2 are. Quantum channels are not available for<\/p>\n\n\n\n
use, so she cannot simply transmit the qubit to Bob. The unknown state of<\/p>\n\n\n\n
the qubit cannot be determined since a measurement would destroy the state.<\/p>\n\n\n\n
Multiple measurements need to be performed on identical copies of the state<\/p>\n\n\n\n
in order to estimate \u03b1 and \u03b2, but Alice has only one copy, and the no-cloning<\/p>\n\n\n\n
theorem forbids her from making more copies.<\/p>\n\n\n\n
Prior to the process, we assume that Alice and Bob share an entangled<\/p>\n\n\n\n
pair of qubits in the state |\u03b2<\/p>\n\n\n\n
00<\/p>\n\n\n\n
i. The protocol, illustrated in Figure 9.2, works<\/p>\n\n\n\n
as follows: Alice \ufb01rst makes a Bell measurement on the two qubits in her<\/p>\n\n\n\n
possession (one unknown qubit and the other entangled with Bob\u2019s qubit).<\/p>\n\n\n\n
Refer to Figure 7.12 of Chapter 7 for the circuit equivalent to this process.<\/p>\n\n\n\n
The results of her measurements are two classical bits of information, which<\/p>\n\n\n\n
Alice now transmits to Bob, through a standard classical channel. Then Bob<\/p>\n\n\n\n
can basically retrieve the quantum state |\u03c8i by performing certain predeter-<\/p>\n\n\n\n
mined operations<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B on his qubit, that depend on the result of Alice\u2019s mea-<\/p>\n\n\n\n
surements. We can see how this works by representing the process as a circuit<\/p>\n\n\n\n
(Example 7.2) and working through it. Bell measurement involves transform-<\/p>\n\n\n\n
ing the two qubits into the Bell basis and then measuring them. The state of<\/p>\n\n\n\n
the three particles just before Alice measures her two qubits is<\/p>\n\n\n\n
|\u03c6i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
|00i<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
\u03b1|0i + \u03b2|1i<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
+<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
|01i<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
\u03b1|1i + \u03b2|0i<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
+<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
|10i<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
\u03b1|0i \u2212 \u03b2|1i<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
+<\/p>\n\n\n\n
1<\/p>\n\n\n\n
4<\/p>\n\n\n\n
|11i<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
\u03b1|1i \u2212 \u03b2|0i<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
(9.1)<\/p>\n\n\n\n
<\/p>\n\n\n\n
180 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
Upon Alice\u2019s measurement, all three qubits collapse to one of the states in<\/p>\n\n\n\n
Table 7.1. Thus to retrieve |\u03c8i, Bob must perform one of the set of conditional<\/p>\n\n\n\n
operations in Table 9.1.<\/p>\n\n\n\n
TABLE 9.1: Bob\u2019s conditional operations in the teleportation protocol.<\/p>\n\n\n\n
Alice transmits Bob performs<\/p>\n\n\n\n
00<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B = (Identity)<\/p>\n\n\n\n
01<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B = X<\/p>\n\n\n\n
10<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B = Z<\/p>\n\n\n\n
11<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B = ZX<\/p>\n\n\n\n
The entire protocol can be represented by the circuit in Figure 9.3.<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
Bell Measurement<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
Classical<\/p>\n\n\n\n
Alice<\/p>\n\n\n\n
\u2022<\/p>\n\n\n\n
Communication<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
00<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\ue01a<\/p>\n\n\n\n
Bob<\/p>\n\n\n\n
X Z<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
FIGURE 9.3: Circuit for teleportation.<\/p>\n\n\n\n
We\u2019ve worked through this circuit in Example 7.2, and you should have no<\/p>\n\n\n\n
doubts that the state |\u03c8i, which was initially with Alice, is \ufb01nally in Bob\u2019s line.<\/p>\n\n\n\n
This process uses up the entangled pair, which is why we regard entanglement<\/p>\n\n\n\n
as a resource.<\/p>\n\n\n\n
9.1.2 How teleportation does not imply faster-than-light<\/p>\n\n\n\n
communication<\/p>\n\n\n\n
A niggling question (which certainly worried Einstein as recorded in the<\/p>\n\n\n\n
EPR paper [31]) would be how the information contained in |\u03c8i was \u201cinstan-<\/p>\n\n\n\n
taneously\u201d transferred from Alice to Bob when Alice measured her qubits.<\/p>\n\n\n\n
The key point here is that no such signaling that is faster than light (thereby<\/p>\n\n\n\n
violating the special theory of relativity) is in fact occurring. Until Bob actu-<\/p>\n\n\n\n
ally knows what the outcome of Alice\u2019s measurements were, he does not know<\/p>\n\n\n\n
that he is in possession of the qubit |\u03c8i. Thus, information is transferred only<\/p>\n\n\n\n
when Alice conveys to him her measurement outcomes, and in this scheme,<\/p>\n\n\n\n
she does not signal faster than light, but is in fact using conventional (classi-<\/p>\n\n\n\n
cal) methods of communication. In fact, the processes adopted in this typical<\/p>\n\n\n\n
protocol are an example of \u201clocal operations and classical communication\u201d or<\/p>\n\n\n\n
LOCC, which is one of the key phrases in quantum information theory.<\/p>\n\n\n\n
![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bgce.png\")
<\/p>\n\n\n\n
Information and Communication 181<\/p>\n\n\n\n
Box 9.1: No Signaling Theorem<\/p>\n\n\n\n
The fact that quantum mechanics does not allow distant parties to ex-<\/p>\n\n\n\n
change information instantaneously using the non-local correlations of entan-<\/p>\n\n\n\n
glement, can be proved neatly using the density operator formalism. Suppose<\/p>\n\n\n\n
Alice and Bob share a state<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i,j<\/p>\n\n\n\n
p<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
|ii<\/p>\n\n\n\n
A<\/p>\n\n\n\n
|ji<\/p>\n\n\n\n
B<\/p>\n\n\n\n
that may be entangled. Suppose Alice performs a measurement on her system,<\/p>\n\n\n\n
characterized by generalized measurement operators M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
. How does this a\ufb00ect<\/p>\n\n\n\n
the state of Bob\u2019s system? Bob\u2019s new density matrix is<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
0B<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
“<\/p>\n\n\n\n
X<\/p>\n\n\n\n
m<\/p>\n\n\n\n
(M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u2297 )\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
(M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u2297 )<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
m<\/p>\n\n\n\n
Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
(M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u2297 )\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
(M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u2297 )<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
m<\/p>\n\n\n\n
Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\ue002<\/p>\n\n\n\n
(M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u2297 )\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
\ue003<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
“<\/p>\n\n\n\n
X<\/p>\n\n\n\n
m<\/p>\n\n\n\n
(M<\/p>\n\n\n\n
\u2020<\/p>\n\n\n\n
m<\/p>\n\n\n\n
M<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u2297 )\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
#<\/p>\n\n\n\n
= Tr<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u03c1<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
= \u03c1<\/p>\n\n\n\n
B<\/p>\n\n\n\n
.<\/p>\n\n\n\n
Thus it is not possible to a\ufb00ect Bob\u2019s state by any local operation performed by<\/p>\n\n\n\n
Alice: Bob\u2019s knowledge cannot be changed \u2014 information cannot be conveyed<\/p>\n\n\n\n
\u2014 by Alice through the non-local correlations of entangled states.<\/p>\n\n\n\n
9.1.3 How teleportation does not imply cloning<\/p>\n\n\n\n
Another common misconception for a beginner in quantum mechanics is<\/p>\n\n\n\n
that teleportation looks as if the state |\u03c8i is copied out from Alice\u2019s location to<\/p>\n\n\n\n
Bob\u2019s. A little consideration will show that in fact this is not happening. The<\/p>\n\n\n\n
moment Alice measures her qubits, the state |\u03c8i ceases to exist on her end.<\/p>\n\n\n\n
She only has two classical bits with her. The unknown state with its implicit<\/p>\n\n\n\n
\u03b1 and \u03b2 coe\ufb03cients is completely transferred to Bob. The state |\u03c8i exists only<\/p>\n\n\n\n
in one location: either at Alice\u2019s end or at Bob\u2019s, and is NOT cloned at any<\/p>\n\n\n\n
point.<\/p>\n","protected":false},"excerpt":{"rendered":"
As an illustration of the power of entanglement as a resource, we examine the rather dramatically titled protocol of quantum state teleportation. This idea was \ufb01rst introduced by Bennett in 1993 [6]. The teleportation problem is the following: Alice needs to transfer to Bob (at a distant location) an unknown qubit |\u03c8i generically denoted by […]<\/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-4107","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\/4107","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=4107"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4107\/revisions"}],"predecessor-version":[{"id":4108,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4107\/revisions\/4108"}],"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=4107"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4107"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}