possible to construct higher-dimensional states by taking direct products<\/p>\n\n\n\n
of lower-dimensional states. However not all higher-dimensional states can be<\/p>\n\n\n\n
constructed this way. There will always exist states that cannot be expressed as<\/p>\n\n\n\n
a direct product. Such states are called entangled states. This nomenclature<\/p>\n\n\n\n
is due to Erwin Schr\u00a8odinger who \ufb01rst discovered the implication of such states<\/p>\n\n\n\n
in 1935 [61].<\/p>\n\n\n\n
For example, consider two generic qubits<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i = \u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|0i + \u03b2<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|1i, |\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i = \u03b1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|0i + \u03b2<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|1i. (4.6)<\/p>\n\n\n\n
If you form the direct product, you get<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
1<\/p>\n\n\n\n
i \u2297 |\u03c8<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
“<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
2<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u03b2<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
. (4.7)<\/p>\n\n\n\n
This is called a product state. Now the most general 2-qubit state is a<\/p>\n\n\n\n
superposition of the form<\/p>\n\n\n\n
|\u03c6i<\/p>\n\n\n\n
2<\/p>\n\n\n\n
= c<\/p>\n\n\n\n
0<\/p>\n\n\n\n
|0i + c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|1i + c<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|2i + c<\/p>\n\n\n\n
3<\/p>\n\n\n\n
|3i. (4.8)<\/p>\n\n\n\n
Equation 4.7 is of a special form:<\/p>\n\n\n\n
c<\/p>\n\n\n\n
0<\/p>\n\n\n\n
c<\/p>\n\n\n\n
3<\/p>\n\n\n\n
= c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
c<\/p>\n\n\n\n
2<\/p>\n\n\n\n
. (4.9)<\/p>\n\n\n\n
Not all states satisfy this property. Those states which do NOT are called<\/p>\n\n\n\n
entangled states. Equation 4.9 is the criterion for a 2-qubit state to be a<\/p>\n\n\n\n
product state.<\/p>\n\n\n\n
For example, the state<\/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) is entangled while<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|00i + |01i)<\/p>\n\n\n\n
is not. A state like |00i + |10i + |11i is partially entangled.<\/p>\n\n\n\n
Box 4.1: Bell States<\/p>\n\n\n\n
The classic examples of entangled states are the Bell states, so named<\/p>\n\n\n\n
in honor of John Bell [5] whose famous arguments resolved the Einstein\u2013<\/p>\n\n\n\n
Podolsky\u2013Rosen paradox [31] involving entangled states. They are also re-<\/p>\n\n\n\n
ferred to as EPR states for this reason. These states exhibit maximum corre-<\/p>\n\n\n\n
lation or anticorrelation between their components:<\/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) ; (4.10a)<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
01<\/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 \u2212 |11i) ; (4.10b)<\/p>\n\n\n\n
|\u03b2<\/p>\n\n\n\n
10<\/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
(|01i + |10i) ; (4.10c)<\/p>\n\n\n\n
<\/p>\n\n\n\n
Properties of Qubits 69<\/p>\n\n\n\n
|\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
(|01i \u2212 |10i) (4.10d)<\/p>\n\n\n\n
In the states |\u03b2<\/p>\n\n\n\n
00<\/p>\n\n\n\n
i and |\u03b2<\/p>\n\n\n\n
01<\/p>\n\n\n\n
i, the spin values of each component are always<\/p>\n\n\n\n
the same (correlated), while they are always opposite (anticorrelated) for the<\/p>\n\n\n\n
other two states.<\/p>\n\n\n\n
Verify that these states are mutually orthogonal. They can thus be used<\/p>\n\n\n\n
as a basis for the 2-qubit Hilbert space H<\/p>\n\n\n\n
2<\/p>\n\n\n\n
.<\/p>\n\n\n\n
When you have more than two qubits, you can have entanglement between<\/p>\n\n\n\n
all or some of the component qubits. In a 3-qubit system, for example, you<\/p>\n\n\n\n
could have entanglement between all three:<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
3<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
3<\/p>\n\n\n\n
(|010i + |101i) , (4.11)<\/p>\n\n\n\n
which is one of the so-called GHZ states (after Greenberger, Horne and<\/p>\n\n\n\n
Zeilinger [39]). Note for this particular state that each of the component qubits<\/p>\n\n\n\n
are anticorrelated, with the \ufb01rst and third having the opposite anticorrelation<\/p>\n\n\n\n
as the second.<\/p>\n\n\n\n
You could have entanglement between two qubits alone, for example:<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
12<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
3<\/p>\n\n\n\n
(|000i + |110i) (4.12)<\/p>\n\n\n\n
One can imagine more possible combinations of partial entanglement. Thus<\/p>\n\n\n\n
for larger dimensional systems, entanglement becomes more complicated.<\/p>\n\n\n\n
Entangled states are just some among the possible states of higher di-<\/p>\n\n\n\n
mensional quantum systems. Why do we single them out for a special name<\/p>\n\n\n\n
and status? What does it mean for a state to be entangled? We have already<\/p>\n\n\n\n
pointed out that entangled states have properties that make them correlated<\/p>\n\n\n\n
to each other. When two (or more) systems are in an entangled state, each<\/p>\n\n\n\n
component system does not have a de\ufb01nite state. This is what it means to<\/p>\n\n\n\n
say that the superposition cannot be written as a product of states of the<\/p>\n\n\n\n
component systems.<\/p>\n\n\n\n
Let us examine the meaning of correlations in the context of a two-qubit<\/p>\n\n\n\n
system in the entangled spin state<\/p>\n\n\n\n
|\u03c8i = |\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
Assume we have a beam of atom pairs in this state, and that we separate<\/p>\n\n\n\n
each pair carefully without changing the state and send one atom each to<\/p>\n\n\n\n
Alice and Bob, who proceed to measure the S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
value on their atom. Each<\/p>\n\n\n\n
has equal probability of having a value \u00b11\/2. Suppose Alice measures a value<\/p>\n\n\n\n
+1\/2 on her atom. This means its state has collapsed to |0i. But this is<\/p>\n\n\n\n
<\/p>\n\n\n\n
70 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
possible only if the combined state collapses to |00i, so that Bob\u2019s atom also<\/p>\n\n\n\n
collapses to |0i. This happens even without Bob making a measurement on<\/p>\n\n\n\n
his atom. If Bob now measures S<\/p>\n\n\n\n
z<\/p>\n\n\n\n
, he will get a value +1\/2. Similarly, had<\/p>\n\n\n\n
Alice obtained \u22121\/2, Bob would also measure the same value. There is perfect<\/p>\n\n\n\n
correlation between the spins of the two particles. Alice and Bob can verify<\/p>\n\n\n\n
this by making measurements on a large number of qubit pairs in the same<\/p>\n\n\n\n
state and comparing the values. As another example, if the state were the<\/p>\n\n\n\n
so-called singlet state<\/p>\n\n\n\n
|\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
(|01i \u2212 |10i,<\/p>\n\n\n\n
and the same experiment is performed, then there is perfect anticorrelation<\/p>\n\n\n\n
between the spins of the two qubits.<\/p>\n\n\n\n
In contrast, suppose that the spins were in the state<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
2<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|00i + |10i).<\/p>\n\n\n\n
It\u2019s easy to see that this state can be expressed as<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
2<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(|0i + |1i)|0i,<\/p>\n\n\n\n
decomposed into a product of states of each spin. In this un-entangled state,<\/p>\n\n\n\n
each spin does possess a de\ufb01nite state. The superposition in the state of the<\/p>\n\n\n\n
\ufb01rst spin is merely a basis state in another basis: the S<\/p>\n\n\n\n
x<\/p>\n\n\n\n
basis. Here there is<\/p>\n\n\n\n
no correlation between spin measurements made by Alice and those obtained<\/p>\n\n\n\n
by Bob<\/p>\n","protected":false},"excerpt":{"rendered":"
possible to construct higher-dimensional states by taking direct products of lower-dimensional states. However not all higher-dimensional states can be constructed this way. There will always exist states that cannot be expressed as a direct product. Such states are called entangled states. This nomenclature is due to Erwin Schr\u00a8odinger who \ufb01rst discovered the implication of such […]<\/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-4011","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\/4011","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=4011"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4011\/revisions"}],"predecessor-version":[{"id":4562,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4011\/revisions\/4562"}],"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=4011"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4011"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4011"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}