Download Quantum Teleportation

Quantum Teleportation
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Teleportation is commonly understood as a
fictional method for transferring an object
between two locations by a process of
dislocation, information transmission, and
reconstruction
The destruction of the original object and the
creation of an exact replica
The actual object does not traverse the
intervening distance
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Teleportation should be impossible
because of the Heisenberg Uncertainty
 It is impossible to measure all the
attributes of a quantum state exactly
 It is impossible to measure both position
and momentum
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But it is possible to exploit another
aspect of the quantum theory, the
notation of the entangeled states and
nonlocal influences to create an exact
replica of an arbitrary quantum state, but
only if the original state was destroyed
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Only “quantum states” are teleported, not the
object itself
We can not use this scheme to teleport an
electron from one place to another
But we can teleport the spin orientation of one
electron at a particular location to another
location
The effect is nearly the same
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The state of the particle is destroyed and
reincarnated on another particle at the destination
without the original particle traversing the
intervening distance
If we can teleport a quantum state
between two locations, and we use
quantum states to encode qubits, then
we can teleport a qubit between two
locations
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Quantum teleportation relies upon quantum
phenomenon known as the EPR effect
EPR named after Albert Einstein, Boris
Podolsky, Nathan Rosen
EPR effect describes the enduring
interconnection between pairs of quantum
systems that had in some earlier time
interacted with one another
Difference between local and nonlocal
interactions
A local interaction is one that involves direct
contact or employees an intermediary that is
direct contact
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friction, gravity
Photons travel at the speed of light which is a
finite speed
Influence cannot propagate faster than the
speed of light
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If two events occur in the regions of
space time such that no signal could ever
reach one region from another, these
events ought to be completely
independent of one another
 Events are called to be spacelike
separated
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Local interactions:
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They are mediated by another entity such as a
particle or field
They propagate no faster than the speed of light
Their strength drops of with distance
Locality predicts that events in spacelike
separated regions ought to be independent of
one another
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All known forces in the Universe are local
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What is left to be nonlocal?
• Collapse of the state vector
• Nothing explains in quantum theory the mechanism of the
collapse
• Collapse of the state vector into a sharp state (eigenstate)
involves no forces
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Nonlocal interaction is not limited by speed of light,
not mediated by anything, and does not drop off in
strength with distance
• Conflict with Einstein’s Theory of Special Relativity
• Nothing can travel faster than ligth
• Problems with time
Reality is non local because of the
collapse of the state vector
 Measurement of a pairs of entangled
states whose components particles are
spacelike saparated
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A state z ∈ H4 of a two-qubit system is
decomposable if z can be written as a
product of states in H2
z=x"y
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A state that is not decomposable is
entangled
!
The state 12 ( 00 + 01 + 10 + 11 )
 Is decomposable because
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!
1
00 + 01 + 10 + 11 ) =
(
2
1
1
1
= (0 0 + 0 1 + 1 0 + 1 1)=
0 + 1 )"
(
(0 + 1)
2
2
2
!
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1
The state 2 ( 00 + 11 )
 Is entangled, to prove it we assume the
contrary 1 ( 00 + 11 ) = (a0 0 + a1 1 )(b0 0 + b1 1 ) =
2
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!
= a0b0 00 + a0b1 01 + a1b0 10 + a1b1 11 "
a0b0 =
1
2
a0b1 = 0
a1b0 = 0
1
a1b1 =
2
contradiction
!
1
( 00 + 11 )
If two qubits are entangled state
2
then observing one of them will give 0 or 1,
both with probability 1/2
 It is not possible to observe
! different values on
the qubits
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Experiments have shown that this correlation can
remain even if the qubits are separated more than
10 km
1
 If
( 00 + 11 ) is an input to a Hadamard
2
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operator, the output is the same again
!
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States can become entangled during
unitary evolution of two separated states
into entanglement states
 Consider two separate quantum systems
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" =
1
( "a + "b
2
$0 = " % # 0 =
$0 = " , # 0 =
)
#0
1
( "a + "b ) % #0
2
1
( " a + " b ), #0
2
!
"(t) = U(T) "(0)
1
U ( #a , $0 + #b , $0
2
1
"(t) =
( #a , $a + #b , $b )
2
"(t) =
!
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)
The structure of the state vector can no longer
be written as a simple product
We can only entangle them by applying the
inverse unitary operator
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The entangled bits or qubits of a state are
called an ebit
An ebit is a shared resource
An ebit is allways disrtributed between two
particles (qubits) 1
2
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( 00
+ 11 )
An ebit provides a channel for communication
Once either particle comprising the ebit is
measured,!the states of both particles become
definite
However the channel is restricted in the
sense that one cannot send an
intentional message (deliberate
sequence of 0’s and 1’s)
 The measuring process is random
 However a way was found...
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Action at Distance
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Suppose a quantum system such as an atom
emits a pair of photons
Polarization of states are entangled
Neither photon has a definite value for its
polarization until its polarization is actually
measured
When are the polarized states actually
determined?
Second photon is determined at the instant that
the polarization of the first photon is measured
• No of the 4 forces in the Universe are involved
• Nonlocal influence
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What would happen if you measured the
polarization of both photons simultaneously,
that no signal could travel from one photon to
another
No influence can propagate faster than light
Correlation in the measured polarization of the
photons hinted the existence of “hidden
variables” that determined the polarization
states of the photons from the outset
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Bell’s Theorem
Alice and Bop each have a polarizer
oriented angle θ1 and θ2
 Each angle can have a (v) vertical or (h)
horizontal value
 The angle between their polarizers is
θ12= θ1 - θ2
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Suppose we add a third measuring polarizer θ3
from the same common reference as the two
first
From classical optics we arrive at
Bell’s inequality states in correspondence to
the “triangle inequality”
1 2
1
1
sin (" 2 # "1 ) + sin 2 (" 3 # " 2 ) $ sin 2 (" 3 # "1 )
2
2
2
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If reality is local, the inequality should be
always true
!
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However if we fix θ1=0
1 2
1
1
sin (" 2 ) + sin 2 (" 3 # " 2 ) $ sin 2 (" 3 )
2
2
2
Is not true any more
 Hidden variable theories are wrong
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!
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More than theoretical result, has been
demonstrated experimentally using detection
of photons
Teleport of a qubit
How to teleport a qubit?
 Let Alice be the sender and Bob the
receiver
 A qubit is in a superposition between 0
and 1
 Alice does not know the details about the
qubit she wants to transfer....
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Prior to Alice attempting to teleport a quantum
state to Bob, Alice and Bob must conspire to
establish a kind of "quantum communication
channel"
This is done by Alice creating a pair of
entangled particles, shipping one member of
the pair off to Bob and keeping the other
member of the pair for herself
The entangled particles serve as the two ends
of an ebit
Their states, although individually ambiguous,
are highly correlated with one another in the
sense that measuring one end of the ebit
instantaneously determines the value of a
similar experiment performed at the other end
of the ebit
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This "influence" however does not propagate
through the space separating the ends of the ebit in
the conventional sense
Rather there is a lingering correlation between the
states of the particles at either end of the ebit that
becomes manifest when either particle is measured.
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Below we describe a simulation of the operation of a
teleportation "circuit" devised by Canadian computer
scientist Gilles Brassard
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This is not a circuit in the usual sense because some of the
lines passing through the circuit are actually severed
However, this state does not pass
through the circuit in the conventional
sense
 the information in the unknown state is
disassembled into a classical part and a
quantum part, the two parts are
transmitted through separate channels
and re-combined by the receiver, Bob, to
re-incarnate the original state
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The circuit is composed of five different
gates, L, R, S, T, and a XOR gate
L, R, S and T are all gates that act on single
qubits
 L and R perform rotations, and S and T
perform phase shifts
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Rotation and Phase shift
1 #1 "1&
1
1
0 + 1 );L 1 =
(
(" 0 + 1 )
%
(;L 0 =
2 $1 1 '
2
2
1 # 1 1&
1
1
L=
( 0 " 1 );L 1 = 2 ( 0 + 1 )
%
(;R 0 =
2 $"1 1'
2
# i 0&
S =%
(;S 0 = i 0 ;S 1 = 1
$ 0 1'
#"1 0 &
T =%
(;T 0 = " 0 ;T 1 = "i1
$ 0 "i'
L=
!
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XOR operator
"1
$
0
XOR = $
$0
$
#0
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!
0 0 0%
'
1 0 0'
0 0 1'
'
0 1 0&
We embed these gates in a larger circuit so
that we can create versions of these quantum
operations that act on selected lines in a
quantum circuit
Preparation (Alice's job)
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For Alice to teleport an arbitrary, unknown, quantum
state to Bob, she begins by creating a pair of particles
whose quantum states are highly correlated with one
another
Alice keeps one of these particles and sends the other
to Bob (via the bottom line of the teleportation "circuit")
Although the two particles become physically remote
from one another, the correlation between their states
persists so long as neither particle is measured nor
interacts with its environment in any way
Such correlated particles are referred to as "ebits" in
the jargon of quantum computing
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To create the ebits, Alice pushes two
standard states (two particles each in
state |0>) through the following circuit.
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The action of the circuit is a state
1
( 00 + 11 )
2
!
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Entanglement (also Alice's job)
Next, suppose Alice wants to teleport a
particular quantum state to Bob
 Alice need not know what this state is in
order to teleport it successfully, so
without loss of generality we can say that
the state is "unknown”
 To teleport the state, Alice entangles it
with one of the ebits she created in Step 1
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Measurement (Alice's last job)
Next Alice measures the bits on lines 1
and 2 (the top and middle line)
 Top and middle output
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Tells what bits she measured by a
conventional channel (2-bit channel)
Bob's job
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Bob receive's Alice's 2-bit classical
message and immediately converts
those bits to corresponding kets for input
into the quantum circuit shown below
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..in the upper and middle lines
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That even although Alice does not measure the
output state at the bottom line of her circuit, it is
affected by her measurements
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This is because it is entangled with the state on the
middle line
The state entering Bob’s portion of circuit
contain the complete information about the
unknown state "
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Portioned between a classical and quantum channel
!
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Bob’s circuit can be thought of as
applying a different rotation to the state
on the bottom input to his circuit
depending on the answer Alice send him
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A possible design of teleportation
Computer Simulation
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unknownState = .5 ket[0] + Sqrt[1-.5^2] ket[1]
0.5 ket[0] + 0.866025 ket[1]
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The joint state of the 3 qubits after Alice has
pushed the 3 inputs through her circuit:
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0.25 ket[0, 0, 0] + 0.433013 ket[0, 0, 1] + 0.433013 ket[0, 1, 0] +
0.25 ket[0, 1, 1] - 0.25 ket[1, 0, 0] + 0.433013 ket[1, 0,
1] +
0.433013 ket[1, 1, 0] - 0.25 ket[1, 1, 1]
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Next Alice measures the bits on lines 1
and 2
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{{1, 1}, 0.866025 ket[1, 1, 0] - 0.5 ket[1,
1, 1]}
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Bit of line 1 and 2 are both one
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All possible values could be:
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Bob receive's Alice's 2-bit classical message
{1, 1} and immediately converts those bits to
corresponding kets for input into the
quantum circuit
• 0.5 ket[1, 1, 0] + 0.866025 ket[1, 1, 1]
•
0.5 ket[0] + 0.866025 ket[1] 0
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