### What is teleportation?
** In the Classical world: **
In o...
![teleportation scheme](http://i.imgur.com/yUuNyuj.png?1 "Teleporta...
An **EPR pair** is a pair of particles that are entangled with each...
The **singlet state** is an entangled state of system of two or mor...
Before discussing the teleportation phenomenon let us first underst...
#### A simple example of teleportation:
Consider two scientists Al...
It is interesting to notice that for the teleportation to be comple...
The teleportation procedure described above uses a system of 2 enta...
**Remarks about human teleportation:**
A human person is composed ...
#### Experimental results about Quantum Teleportation:
In Octobe...
VOLUME
70
29 MARCH l993
NUMBER 13
Teleporting
an Unknown
Quantum
State
via
Dual
Classical and
Einstein-Podolsky-Rosen
Channels
Charles
H.
Bennett,
~
)
Gilles
Brassard,
( )
Claude
Crepeau,
( )
(
)
Richard
Jozsa,
(
)
Asher
Peres,
~4)
and William
K.
Wootters(
)
'
IBM
Research
Division,
T.J.
watson Research
Center,
Yorktomn
Heights,
¹mYork 10598
(
lDepartement
IIto,
Universite
de
Montreal, C.
P
OI28,
Su.
ccursale
"A",
Montreal,
Quebec,
Canada HBC
817
(
lLaboratoire
d'Informatique
de
1'Ecole
Normale
Superieure,
g5
rue
d'Ulm,
7M80 Paris CEDEX
05,
France~
i
l
lDepartment
of
Physics,
Technion Israel
In—
stitute
of
Technology,
MOOO
Haifa,
Israel
l
lDepartment
of
Physics,
Williams
College,
Williamstoivn,
Massachusetts
OIP67
(Received
2 December
1992)
An
unknown
quantum
state
]P)
can be disassembled
into,
then
later reconstructed
from,
purely
classical information
and
purely
nonclassical
Einstein-Podolsky-Rosen
(EPR)
correlations.
To
do
so
the
sender,
"Alice,
"
and the
receiver,
"Bob,
"
must
prearrange
the
sharing
of an
EPR-correlated
pair
of
particles.
Alice
makes a
joint
measurement on her EPR
particle
and the unknown
quantum
system,
and sends Bob the classical result of this measurement.
Knowing
this,
Bob can convert the
state
of
his EPR
particle
into an exact
replica
of
the unknown state
]P)
which
Alice
destroyed.
PACS numbers: 03.65.
Bz,
42.50.
Dv,
89.
70.
+c
The existence
of
long
range
correlations between
Einstein-Podolsky-Rosen
(EP
R)
[1]
pairs
of
particles
raises the
question
of
their
use
for information
transfer.
Einstein
himself
used
the
word
"telepathically"
in
this
contempt
[2].
It
is
known that
instantaneous
information
transfer is definitely
impossible
[3].
Here,
we
show that
EPR
correlations can
nevertheless assist
in
the
"telepor-
tation"
of an intact
quantum
state
from one
place
to
another,
by
a sender who
knows
neither
the state
to be
teleported
nor
the
location of
the intended
receiver.
Suppose
one
observer,
whom we
shall call
"Alice,
"
has
been
given
a
quantum system
such as
a
photon
or
spin-&
particle, prepared
in
a
state
]P)
unknown
to
her,
and she
wishes to
communicate
to
another
observer,
"Bob,
"
suf-
ficient information about the
quantum
system
for him to
make an accurate
copy
of it.
Knowing
the
state vector
]P)
itself would
be
sufficient information,
but in
general
there is
no
way
to
learn it.
Only
if Alice knows
before-
hand that
~qb)
belongs
to a
given
orthonormal set can
she
make
a
measurement whose result
will allow her to make
an
accurate
copy
of
[P).
Conversely,
if
the possibilities
for
~P)
include two or
more
nonorthogonal states,
then
no
measurement
will
yield
sufhcient
information
to
prepare
a
perfectly
accurate
copy.
A
trivial
way
for
Alice
to
provide
Bob
with all the
in-
formation
in
[P)
would be to
send
the
particle
itself. If
she
wants to
avoid transferring
the
original
particle,
she can
make
it.
interact unitarily
with another
system,
or
"an-
cilla,
"
initially
in a known state
~ap),
in such
a
way
that
after
the interaction the
original particle
is left in
a
stan-
dard
state
~Pp)
and the ancilla is in an
unknown
state
]a)
containing complete
information about
~P).
If
Al-
ice now sends
Bob the
ancilla
(perhaps
technically
easier
than
sending
the
original
particle),
Bob can reverse her
actions to
prepare
a
replica
of her original
state
~P).
This
"spin-exchange
measurement"
[4]
illustrates
an
essential
feature
of
quantum
information: it
can be
swapped
from
one
system
to another,
but
it
cannot be duplicated
or
"cloned"
[5].
In this
regard
it
is
quite
unlike
classical
information,
which can be duplicated
at
will. The most
tangible
manifestation
of the nonclassicality
of
quantum
information is
the
violation
of
Bell
s
inequalities
[6)
ob-
served
[7]
in experiments
on
EPR
states.
Other
rnanifes-
tations
include
the
possibility
of
quantum
cryptography
[8),
quantum
parallel
computation
[9],
and
the
superior-
ity
of
interactive
measurements
for
extracting
informa-
1993 The American
Physical
Society
1895
VOLUME
70,
NUMBER 13 PHYSICAL REVIEW
LETTERS
29
MARcH
1993
tion
from
a
pair
of
identically
prepared
particles
[10].
The
spin-exchange
method of
sending
full information
to Bob still
lumps
classical and nonclassical information
together
in
a
single
transmission.
Below,
we show how
Alice can
divide
the
full
information
encoded
in
I&/)
into
two
parts,
one
purely
classical and the other
purely
non-
classical,
and
send
them
to Bob
through
two diferent
channels.
Having
received these two transmissions,
Bob
can
construct
an
accurate
replica
of
IP).
Of course
Alice's
original
IP)
is
destroyed
in
the
process,
as
it must
be
to
obey
the no-cloning theorem. We
call the
process
we are
about to describe
teleportation,
a
term
from science
fic-
tion
meaning
to make a
person
or
object disappear
while
an
exact
replica
appears
somewhere else. It must
be
em-
phasized
that our
teleportation,
unlike some science
fic-
tion
versions,
defies no
physical
laws. In
particular,
it
cannot take
place
instantaneously
or over
a
spacelike
in-
terval,
because it
requires, among
other
things,
sending
a classical
message
from Alice
to
Bob. The net result
of teleportation
is
completely
prosaic:
the removal of
IP)
from
Alice's
hands and its
appearance
in
Bob's
hands
a
suitable time
later.
The
only
remarkable feature is
that,
in the
interim,
the
information
in
IP)
has been
cleanly
separated
into classical and nonclassical
parts.
First we
shall show how to
teleport
the
quantum
state
IP&
of
a
spin-2
particle.
Later we discuss
teleportation
of
more
complicated
states.
The nonclassical
part
is transmitted first.
To do
so,
two
spin-&
particles
are
prepared
in an EPR
singlet
state
lc'i~
)
=
(+)
(I
Ti&
l~)
+
I
lx)
I
T~&)
(I
Ti)
I
T~)
+
I
li)
I
l~))
(2&
Note
that these four
states are
a
complete
orthonormal
basis for
particles
1 and 2.
It
is convenient
to write
the unknown
state of the first
particle
as
lei)
=
~ITi&+
l
lli),
the other
(particle
3)
is
given
to Bob.
Although
this
establishes
the
possibility
of
nonclassical
correlations
be-
tween Alice
and
Bob,
the EPR
pair
at this
stage
contains
no information
about
IP).
Indeed
the entire
system,
com-
prising
Alice's
unknown
particle
1
and the
EPR
pair,
is
in
a
pure
product state,
I/i)
I@&3
),
involving neither
classical
correlation nor
quantum
entanglement between
the
unknown
particle
and the EPR
pair.
Therefore no
measurement
on
either
member
of the EPR
pair,
or
both
together,
can
yield
any
information
about
I
P&.
An
entan-
glement
between
these two
subsystems
is
brought
about
in
the next
step.
To
couple
the
first
particle
with the EPR
pair,
Alice
performs
a
complete
measurement of
the
von Neumann
type
on the
joint system
consisting
of
particle
1
and parti-
cle 2
(her
EPR
particle).
This measurement
is
performed
in the Bell
operator
basis
[ll]
consisting
of
I@i&
)
and
1@~3
)
=
(
—
)
2 3 2 3
with
Ial
+
Ibl
=
1.
The
complete
state of
the
three
particles
before
Alice's
measurement
is thus
The
subscripts
2
and 3
label the
particles
in
this
EPR
pair.
Alice's
original
particle,
whose
unknown
state
IP)
she seeks
to
teleport
to
Bob,
will be
designated
by
a
subscript
1 when
necessary. These three
particles
may
be
of diferent
kinds,
e.
g.
,
one
or more
may
be
photons,
the
polarization
degree
of freedom
having
the same
algebra
as a
0
(I
Ti) T~)
I
ls)
—
Ti)
I
l~)
I
T3)
&
+
(I
li)
I
T2)
I
L3)
I
ll)
l2&
I
T3))
spin.
In
this
equation,
each
direct
product
I
i)l 3)
can
be
ex-
ne
EPR
particle
(particle
2)
is
given
to
Alice,
while
pressed
in
terms
of the
Bell
operator
basis
vectors
IC&z
)
(+)
and
I@i~
),
and we
obtain
I+i~3)
=
g
[l@xa'&
(
—
~l
T3)
—
l
lls&)
+
l@gg')
(
—
~l
T3)
+
l
l13))
+
IC'ig'&
(&I
ls)
+
l
I
T3))
+ IC'i~')
(aIL3&
—
l
I
T3&)]
(,0)
IA),
(, D)
lda).
(6)
It follows
that,
regardless of
the unknown
state
I/i),
the
four
measurement
outcomes
are
equally likely,
each
oc-
curring
with
probability
1/4.
Furthermore, after
Alice
s
measurement,
Bob's
particle
3 will
have been
projected
into
one of the
four
pure
states
superposed
in
Eq.
(5),
according to the
measurement
outcome.
These
are,
re-
spectively,
—
lds)
—
=
—
(t,
),
(
~,
)
its),
Each of
these
possible resultant
states for
Bob's
EPR
particle
is related
in a
simple
way
to the
original state
IP)
which
Alice
sought
to
teleport. In the
case of the first
(singlet)
outcome,
Bob's
state is
the same
except
for an
irrelevant
phase
factor, so Bob
need
do
nothing
further
to
produce
a
replica
of
Alice's
spin.
In the
three
other
cases,
Bob
must
apply
one
of
the
unitary
operators in
Eq.
(6),
corresponding,
respectively,
to
180'
rotations
around
the
z, x,
and
y
axes,
in
order to
convert
his EPR
particle
into
a
replica
of
Alice's
original
state
IP).
(If
IP)
represents
a
photon
polarization
state,
a suitable
combination
of
half-
1896
VOLUME
70,
NUMBER 13
PHYSICAL
REVIEW
LETTERS
29 MARCH
1993
wave
plates
will
perform
these
unitary
operations.
)
Thus
an
accurate
teleportation
can
be
achieved
in all
cases
by
having
Alice tell
Bob
the classical outcome of her
mea-
surement,
after which Bob
applies
the
required
rotation
to transform the state
of
his
particle
into a
replica
of
IP).
Alice,
on the other
hand,
is left with
particles
1 and 2 in
one of the
states
4~2
)
or
IC~~
),
without
any
trace
of
(+)
'
(+)
the
original state
I
P)
.
Unlike the
quantum
correlation of
Bob's
EPR
particle
3 to
Alice's
particle
2,
the result
of
Alice's
measurement
is
purely
classical
information,
which can be
transmit-
ted, copied,
and stored at
will
in
any
suitable
physical
medium. In
particular,
this information need not
be
de-
stroyed
or canceled
to
bring
the
teleportation
process
to
a
successful conclusion:
The
teleportation
of
lg)
from
Alice
to
Bob has the side effect of
producing
two bits of
random classical information, uncorrelated to
lg),
which
are left behind
at
the
end of the
process.
Since
teleportation
is
a
linear
operation
applied
to
the
quantum
state
IP),
it
will
work
not
only
with
pure
states,
but also
with mixed or
entangled
states. For
example,
let
Alice's
original particle
1
be
itself
part
of an
EPR
singlet
with another
particle,
labeled
0,
which
may
be
far
away
from
both
Alice
and Bob.
Then,
after
teleportation,
particles
0
and 3
would
be
left in a
singlet state,
even
though
they
had
originally
belonged
to
separate
EPR
pairs.
All
of
what we have
said above can
be generalized
to
systems
having
N
&
2
orthogonal
states. In
place
of an EPR
spin pair
in the
singlet state,
Alice would
use a
pair
of
N-state
particles
in
a
completely
entangled
state. For
definiteness let us
write this
entangled
state
as
P
.
I j) Ij
)
/~N,
where
j
=
0,
1,
. . .
,
N
—
1
labels the
N
elements
of an orthonormal
basis for each of
the
N-state
systems.
As
before,
Alice
performs
a
joint
measurement
on
particles
1 and
2.
One
such measurement
that has
the
desired effect is
the one whose
eigenstates
are
lg„),
defined
by
lg„~)
=
)
e
"'~"
j)
S
I(j
+
m)
mod
N)
/~¹
(lu2)
I»)
+
I») lqs))
(9)
where
(Iu), lv))
and
(Ip),
Iq))
are
any
two
pairs
of
or-
thonormal
states.
These are
maximally
entangled
states
[ll],
having
maximally
random
marginal
statistics
for
measurements
on
either
particle separately.
States
which
are
less
entangled
reduce
the
fidelity
of
teleportation,
and/or
the
range
of
states
lg)
that can
be accurately
tele-
ported.
The
states
in
Eq.
(9)
are also
precisely
those
ob-
tainable
from
the EPR
singlet
by
a
local
one-particle
uni-
tary
operation
[12].
Their
use
for the
nonclassical
channel
is
entirely
equivalent
to that of
the
singlet
(1).
Maximal
entanglement
is
necessary
and
suKcient for
faithful
tele-
P)
will
be
reconstructed
(in
the
spin-2 case)
as a
ran-
dom mixture
of
the
four
states
of
Eq.
(6).
For
any
lg),
this
is a maximally
mixed
state,
giving
no information
about the
input
state
IP).
It
could
not
be
otherwise,
be-
cause
any
correlation between the
input
and the
guessed
output
could be used to
send
a superluminal
signal.
One
may
still
inquire
whether accurate
teleportation
of a
two-state
particle requires
a
full
two
bits of classical
information. Could
it
be
done,
for
example,
using
only
two or three
distinct
classical
messages
instead
of
four,
or
four
messages
of
unequal probability?
Later we
show
that a
full two
bits
of classical channel
capacity
are
neces-
sary.
Accurate teleportation
using
a
classical
channel of
any
lesser
capacity
would allow Bob to
send
superlumi-
nal
messages
through
the
teleported
particle,
by
guessing
the classical
message
before it
arrived
(cf.
Fig.
2).
Conversely
one
may
inquire
whether other states
be-
sides an EPR
singlet
can
be used
as
the nonclassical
chan-
nel of the teleportation process.
Clearly
any
direct
prod-
uct
state
of
particles
2
and
3
is
useless,
because
for such
states manipulation
of
particle
2 has
no effect
on what
can
be predicted
about particle
3.
Consider now a
non-
factorable state
IT2s)
. It can
readily
be
seen that
after
Alice's
measurement,
Bob's
particle
3
will be
related to
IP&)
by
four
fixed
unitary
operations
if
and
only
if
IT23)
has
the
form
Once Bob learns from Alice that
she
has obtained the
re-
sult
nm,
he
performs
on his
previously
entangled particle
(particle
3)
the
unitary
transformation
Two bits
Two
bits
U„=
)
e
'""~
A;)
((k+
m)
modNI.
k
EPR
pair
Two bits
EPR
pair
This transformation
brings
Bob
s
particle
to the origi-
nal
state of
Alice's
particle
1,
and the
teleportation
is
complete.
The classical
message
plays
an essential
role in telepor-
tation. To
see
why,
suppose
that
Bob
is
impatient,
and
tries
to
complete
the
teleportation
by
guessing
Alice's
classical
message
before it arrives.
Then
Alice's
expected
FIG.
1. Spacetime diagrams
for
(a)
quantum
teleporta-
tion,
and
(b)
4-way
coding
[12].
As
usual,
time increases
from bottom
to
top.
The
solid lines represent
a
classical
pair
of
bits,
the
dashed
lines an EPR
pair
of
particles (which
may
be
of
different
types),
and the
wavy
line a
quantum
parti-
cle
in
an
unknown
state
IP).
Alice
(A)
performs
a
quantum
measurement,
and
Bob
(B)
a
unitary
operation.
1897
Vo~UME
70,
NUMBER 13
PH
YSICAL
REVIEW
LETTERS
29
M&RcH
1993
FIG. 2.
Spacetime diagram
of
a
more
complex
4-way
cod-
ing
scheme in which
the
modulated
EPR
particle
(wavy
line)
is
teleported
rather
than
being
transmitted
directly.
This
dia-
gram
can be used to
prove
that a
classical
channel of
two bits
of
capacity
is
necessary
for teleportation.
To do
so,
assume
on
the
contrary
that the teleportation
from
A'
to
B'
uses
an
internal classical
channel of
capacity
C
(
2
bits,
but
is still
able
to
transmit
the
wavy
particle's
state
accurately
from
A'
to
B',
and
therefore
still transmit the
external
two-bit
mes-
sage
accurately
from
B
to
A.
The
assumed
lower
capacity
(
2
of
the internal
channel
means that if
B'
were to
guess
the internal
classical
message
superluminally
instead
of
wait-
ing
for
it
to
arrive,
his
probability
2
of
guessing
correctly
would exceed
1/4,
resulting
in
a
probability
greater
than
1/4
for
successful
superluminal
transmission
of the external
two-
bit
message
from
B
to
A. This in turn
entails the existence
of
two
distinct external
two-bit
messages,
r
and
s,
such that
P(r]s),
the
probability
of
superluminally
receiving r
if
s was
sent,
is less than
1/4,
while
P(r]r),
the
probability
of
super-
luminally receiving
r
if
r was
sent,
is
greater
than
1/4.
By
redundant
coding,
even this statistical
difkrence
between
7-
and 8
could be used
to send
reliable superluminal
messages;
therefore
reliable teleportation
of
a
two-state
particle
cannot
be
achieved
with
a
classical
channel
of less than two
bits of
capacity.
By
the
same
argument,
reliable
teleportation
of an
¹tate
particle
requires
a
classical
channel
of
21og~(AI')
bits
capacity.
Bob the
original
quantum particle,
or a spin-exchanged
version
of
it,
if
she does
not
know where he
is;
but she
can
still
teleport
the
quantum
state to
him,
by
broadcasting
the classical
information to all
places
where he
might
be.
Teleportation
resembles
another recent
scheme for
us-
ing
EPR
correlations
to
help
transmit
useful
information.
In
"4-way
coding"
[12]
modulation
of
one
member of
an
EPR
pair
serves
to
reliably
encode a
2-bit
message
in
the
joint
state
of the
complete pair.
Teleportation
and
4-way
coding
can
be
seen
as
variations
on the same
un-
derlying
process,
illustrated
by
the
spacetime diagrams
in
Fig.
1. Note
that
closed
loops
are involved
for both
pro-
cesses.
Trying
to
draw
similar
"Feynman diagrams" with
tree structure,
rather
than
loops,
would
lead to
physically
impossible
processes.
On
the other
hand,
more
complicated
closed-loop
di-
agrams
are
possible,
such as
Fig.
2,
obtained
by
substi-
tuting
Fig.
1(a)
into
the
wavy
line
of
Fig.
1(b).
This
represents
a
4-way
coding
scheme in
which
the
modu-
lated
EPR
particle
is teleported
instead of
being
trans-
mitted directly.
Two
incoming
classical bits on
the lower
left are reproduced reliably
on
the
upper
right,
with
the
assistance
of two
shared EPR
pairs
and two
other
clas-
sical
bits,
uncorrelated
with the
external
bits,
in an
in-
ternal
channel
from
A'
to
O'.
This
diagram
is of interest
because
it can
be
used
to show
that
a
full
two bits
of
classical
channel
capacity
are
necessary
for accurate
tele-
portation
of a
two-state
particle
(cf.
caption).
Work
by
G.
B.is
supported
by
NSERC's
E.W. R.
Stea-
cie Memorial
Fellowship
and
Quebec's FCAR. A.
P.
was
supported
by
the Gerard
Swope
Fund
and the
Fund for
Encouragement
of Research.
Laboratoire
d'Informatique
de
1'Ecole
Normale
Superieure
is
associee au
CNRS
URA
1327.
portation.
Although
it is
currently
unfeasible to
store
separated
EPR
particles
for more than
a
brief
time,
if
it becomes
feasible
to
do
so,
quantum
teleportation
could be
quite
useful.
Alice
and Bob would
only
need
a
stockpile
of
EPR
pairs
(whose
reliability
can
be tested
by
violations
of
Bell's
inequality
[7])
and
a
channel
capable
of
carry-
ing
robust classical
messages.
Alice could
then
teleport
quantum
states
to
Bob over
arbitrarily
great
distances,
without
worrying
about the eEects of attenuation and
noise
on,
say,
a
single photon
sent
through
a
long
op-
tical Aber. As
an
application
of
teleportation,
consider
the
problem
investigated
by
Peres and
Wootters
[10],
in
which
Bob
already
has
another
copy
of
~P).
If
he
acquires
Alice's
copy,
he can measure
both
together,
thereby
de-
termining the
state
[P)
more
accurately
than can be done
by
making
a
separate
measurement on each one.
Finally,
teleportation
has the
advantage
of still
being
possible
in
situations
where
Alice
and
Bob,
after
sharing
their EPR
pairs,
have wandered
about
independently
and
no
longer
know
each
others'
locations. Alice cannot,
reliably
send
Permanent
address.
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The teleportation procedure described above uses a system of 2 entangled particles in the singlet state. The same procedure can be generalised to systems with N>2 orthogonal states.
An **EPR pair** is a pair of particles that are entangled with each other. Entangled particles are such that each particle **cannot be described independently from the other**. For instance if you measure the spin of the first particle you will immediately know the spin of the second particle independently of the distance that separates the two particles.
---
Read more about the Einstein-Podolski-Rosen paradox here: [original EPR paper](http://fermatslibrary.com/s/on-the-epr-paradox)
Funny, didactic video about Quantum Entanglement: [Veritasium: Quantum Entanglement](https://www.youtube.com/watch?v=ZuvK-od647c)
Read more about Quantum Entanglement here: [Wikipedia: Quantum Entanglement](https://en.wikipedia.org/wiki/Quantum_entanglement)
It is interesting to notice that for the teleportation to be complete Alice needs to send a message to Bob. This message cannot travel faster than the speed of light which in turn **forbids faster than the speed of light teleportation.**
Once Bob receives Alice's message he will need to apply some simple unitary transformations to his particle in order to recover the state that Alice wanted to send him. Alice in turn is left with two spins in a different state while Bob now has the $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ spin that Alice wanted to send him.
#### Experimental results about Quantum Teleportation:
In October of 2015 physicists broke distance record in quantum teleportation. Researchers at the National Institute of Standards and Technology (NIST) have teleported quantum information carried in light particles over 100 kilometers of optical fiber. Read their original paper here: [Quantum teleportation over 100 km of fiber using highly-efficient superconducting
nanowire single photon detectors](http://arxiv.org/pdf/1510.00476v1.pdf)
In 1997 physicists at the Institut für Experimentalphysik at the Universität Innsbruck in Austria were the first to realise quantum teleportation with photons. If you want to learn more about it read their paper: http://www.pdx.edu/nanogroup/sites/www.pdx.edu.nanogroup/files/(1998)%20Zeilinger%20Experimental%20Quantum%20Teleportation.pdf
Before discussing the teleportation phenomenon let us first understand the operations that we can perform on a quantum system. We are interested in **unitary operations**, a class of transformations that reveal nothing about the quantum state thus do not destroy any of the hidden “quantum information”.
There are 3 types of operations of interest in the case of teleportation:
**X-operation:** this operation rotates a spin by 180 degrees around the x-axis:
\begin{eqnarray*}
\newcommand{\ket}[1]{|{#1}\rangle}
X(\, a\ket{\uparrow} + b\ket{\downarrow} \,) &=& a\ket{\downarrow} + b\ket{\uparrow}
\end{eqnarray*}
This operation changes a spin up into a spin down and vice-versa.
**Z-operation:** this operation rotates a spin 180 degrees around the z-axis:
\begin{eqnarray*}
\newcommand{\ket}[1]{|{#1}\rangle}
Z(\, a\ket{\uparrow} + b\ket{\downarrow} \,) &=& a\ket{\downarrow} - b\ket{\uparrow}
\end{eqnarray*}
This operation exchanges the left and right spin states
**Controlled-X (C-X) operation:** this operation acts on two spins at the time and products the following outcome:
\begin{eqnarray*}
\newcommand{\ket}[1]{|{#1}\rangle}
C-X(\, a\ket{\uparrow,\uparrow} + b\ket{\uparrow,\downarrow} + c\ket{\downarrow,\uparrow} + d\ket{\downarrow,\downarrow} \,)\\ \\=
a\ket{\uparrow,\uparrow} + b\ket{\uparrow,\downarrow} + c\ket{\downarrow,\downarrow} + d\ket{\downarrow,\uparrow}
\end{eqnarray*}
This operations flips the second spin if the first spin is down but if the first spin is up then it leaves the second spin invariant.
Alice wants to teleport $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ to Bob. They are in their respective labs arbitrarily far apart.
1. Alice and Bob share an entangled pair of particles A and B, let's call this system $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\Psi_{AB}}$
2. Alice receives the particle $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ that she wants to teleport to Bob
3. Alice performs transformations on the particles A $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ so that $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\Psi_{AB}}$ and $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ become entangled. Alice then performs a measurements on particles A $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ and communicates her results to Bob.
4. Once Bob receives the results he does the necessary transformations and the outcome is the particle $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ that Alice wanted to transfer him. Teleportation done! Alice is left with two different entangled particles, because performing measurements on quantum particles changes them.
**Remarks about human teleportation:**
A human person is composed of more than $10^{29}$ particles, each of which has a given position, momentum and spin. In addition to teleporting spins, you also need to teleport photons, gluons and other energy particles that compose a human being. Teleporting all of this quantum information is going to be significantly harder than a few spins. It is probably a good guess that teleportation of humans will never be possible.
The **singlet state** is an entangled state of system of two or more particles that have total spin 0. This state has some important properties:
- If we measure the spin along any axis, the outcome will always yield $+\frac{1}{2}$ with 50% probability and $−\frac{1}{2}$ with 50% probability. This means that we cannot retrieve any information about the outcome of a single spin.
- If we measure one spin along any axis and then measure the other spin along the same axis the results will
always be anti-correlated. This means that as soon as we measure one spin, we know the state of the second spin.
In Dirac notation the singlet state of two particles (A and B) can be written as:
\begin{eqnarray*}
\newcommand{\ket}[1]{|{#1}\rangle}
\ket{\Psi_{AB}} &=& \frac{1}{\sqrt{2}}\ket{\uparrow_A,\downarrow_B} - \frac{1}{\sqrt{2}}\ket{\downarrow_A,\uparrow_B}
\end{eqnarray*}
In the context of the paper the authors use 2 and 3 to designate the two entangled particles in the singlet state that will be used to teleport the state $\newcommand{\ket}[1]{|#1\rangle}\newcommand{\bra}[1]{\langle #1|}
\ket{\phi} $.
#### A simple example of teleportation:
Consider two scientists Alice and Bob that share two particles, A (for Alice) and B (for Bob), that are in the singlet state. Alice and Bob's labs are arbitrarily far apart and Alice wants to teleport an spin state $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ to Bob. With the notions of singlet state and unitary operaitons let us compute the teleportation of a spin $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ from Alice to Bob.
The singlet state that Alice and Bob share can be written:
\begin{eqnarray}
\newcommand{\ket}[1]{|{#1}\rangle}
\ket{\Psi_{AB}} &=& \frac{1}{\sqrt{2}}\ket{\uparrow_A,\downarrow_B} - \frac{1}{\sqrt{2}}\ket{\downarrow_A,\uparrow_B}
\end{eqnarray}
And the state that Alice wants to send to Bob can be written as:
\begin{eqnarray}
\newcommand{\ket}[1]{|{#1}\rangle}
\ket{\phi} &=& a\ket{\uparrow_{\phi}} + b\ket{\downarrow_{\phi}}
\end{eqnarray}
The first step that Alice performs is an entanglement of her particle with the spin she wants to teleport. The state fo all three spins then becomes:
\begin{eqnarray}
\newcommand{\ket}[1]{|{#1}\rangle}
\ket{\Psi_{\phi AB}} = \frac{a}{\sqrt{2}}\ket{\uparrow_{\phi},\uparrow_A,\downarrow_B} \nonumber \\
- \frac{a}{\sqrt{2}}\ket{\uparrow_{\phi},\downarrow_A,\uparrow_B} \nonumber \\
+ \frac{b}{\sqrt{2}}\ket{\downarrow_{\phi},\uparrow_A,\downarrow_B} \nonumber \\
- \frac{b}{\sqrt{2}}\ket{\downarrow_{\phi},\downarrow_A,\uparrow_B} \\
\end{eqnarray}
Our goal is to teleport the spin state $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ from Alice to Bob. Alice stars by applying the unitary **C-X operation** to state given by equation (3).
\begin{eqnarray}
\newcommand{\ket}[1]{|{#1}\rangle}
CX(\ket{\Psi_{\phi AB}}) = \frac{a}{\sqrt{2}}\ket{\uparrow_{\phi},\uparrow_A,\downarrow_B} \nonumber \\
- \frac{a}{\sqrt{2}}\ket{\uparrow_{\phi},\downarrow_A,\uparrow_B} \nonumber \\
+ \frac{b}{\sqrt{2}}\ket{\downarrow_{\phi},\downarrow_A,\downarrow_B} \nonumber \\
- \frac{b}{\sqrt{2}}\ket{\downarrow_{\phi},\uparrow_A,\uparrow_B} \\
\end{eqnarray}
Now Alice will rotate $\newcommand{\ket}[1]{|{#1}\rangle}\ket{\phi}$ around the z-axis and measure the spin of her particle B in the z-direction. Let us assune that Alice's result was a spin up. In this case we can rewrite equation (4):
\begin{eqnarray}
\newcommand{\ket}[1]{|{#1}\rangle}
\ket{\Psi_{\phi AB}} = a\ket{\uparrow_{\phi},\uparrow_A,\downarrow_B} + b\ket{\downarrow_{\phi},\uparrow_A,\uparrow_B} \\
\end{eqnarray}
there is a $50\%$ probability of obtaining spin up, thus $\frac{a^2+b^2}{2} = \frac{1}{2}$.
We can now write the first spin in terms of $\ket{\leftarrow}$ and $\ket{\rightarrow}$. We can rewrite (5):
\begin{eqnarray}
\ket{\Psi_{\phi AB}} = \frac{a}{\sqrt{2}} (\ket{\rightarrow_{\phi},\uparrow_A,\downarrow_B} + \ket{\leftarrow_{\phi},\uparrow_A,\downarrow_B}) \nonumber \\
+ \frac{b}{\sqrt{2}} (\ket{\rightarrow_{\phi},\uparrow_A,\uparrow_B} + \ket{\leftarrow_{\phi},\uparrow_A,\uparrow_B})
\end{eqnarray}
Alice now performs a measurement along the x-direction on spin $\ket{\phi}$. Let us assume that she measures the spin right outcome. Then we can wrote (6):
\begin{eqnarray}
\ket{\Psi_{\phi AB}} = a \, \ket{\rightarrow_{\phi},\uparrow_A,\downarrow_B} + b \, \ket{\rightarrow_{\phi},\uparrow_A,\uparrow_B}
\end{eqnarray}
Now Alice can communicate her result to Bob. Once Bob receives Alice's message he will apply an X operation to his particle and he will be left with the original spin that Alice wanted to teleport.
\begin{eqnarray}
X(\ket{\Psi_{\phi AB}}) = a \, \ket{\rightarrow_{\phi},\uparrow_A,\uparrow_B} + b \, \ket{\rightarrow_{\phi},\uparrow_A,\downarrow_B}
\end{eqnarray}
The particle in Bob's Lab is:
\begin{eqnarray}
\ket{\phi} = a \, \ket{\uparrow_B} + b \, \ket{\downarrow_B}
\end{eqnarray}
### What is teleportation?
** In the Classical world: **
In order to teleport information in the classical world you need to:
1. Fully measure the state of the input
2. Transmit the results to a distant location
3. Reconstruct the original from the received description
** In the Quantum world:**
In the Quantum world everything is different due to Heisenberg's uncertainty principle. The uncertainty principle forbids any measurement from extracting all the information of an object, which means that it is impossible to fully measure the state of an input. If one cannot extract enough information from an object to make a perfect copy, it becomes impossible to create a perfect copy. This paper was the first to introduce the concept of quantum teleportation. The authors found that it is possible to scan part of the information from an object A (the one we want to teleport) while causing the remaining part of the information to pass, via the Einstein-Podolsky-Rosen effect, into another object C which has never been in contact with A and is arbitrarily far away.