Transcript Bell Measurements and Teleportation

Bell Measurements and
Teleportation
Overview
• Entanglement
• Bell states and Bell measurements
• Limitations on Bell measurements using linear
devices
• Teleportation
• Dense coding
• Entanglement swapping
• Entanglement purification
• Quantum repeaters
Entanglement
• Two systems described by two separable
Hilbert spaces.
• States of the two systems can be
described by the tensor product
i    i 
of their state spaces.
• Schmidt decomposition:    aij i   j   bi  'i   'i 
i, j
i
• If bi  0 and bji  0 the state is said to
be separable. If more than one bi  0
then   is said to be entangled.
• The state of one system cannot be
specified without the other.

 

Bell States
• For two two-state systems denoted each by
 ,  the Bell states form a basis for the whole
system and are maximally entangled:
1
2
1

2
1

2
1

2
 








 
 
 
  
 
 
 
where  is anti-symmetric and  ,  ,  are
symmetric with respect to particle interchanging.




Bell Measurements
Distinguishing Bell states using linear elements such as beam
splitters, phase shifters, photo-detectors etc.
All elements can be described by unitary transformations. In linear ones
particle number is conserved.
Examples for photons:
Beam splitter:  U BS  Ar Rout

Rin  Lout




Lin  At Lout

 
Rin  Rout
Polarization
 U PBS  Ar H Rout H Rin  H Lout H Lin  At VLout VRin  VRout VLin
beam splitter
Half wave plate
 U   Vout Hin  Hout Vin
at 45 degrees
,45
2

Lin

Example: distinguishing anti-symmetric and
symmetric states - Hong–Ou–Mandel effect



1
   
2
1

   
2
1

   
2
 
 

1
   
2

OR






?
Beam splitter operator representation for a single photon:



U BS  Ar d 1 L  d 2 R  At d 2 L  d 1 R

Ar  At  1
2
2
Ar At  At Ar  0



U BS  cos( ) d 1 L  d 2 R  i sin( ) d 2 L  d 1 R

• Double transmission obtains a minus sign relative to double reflection.
•symmetric states have zero amplitude for d1-d2 coincidence.
• d1 + d2 simultaneous “click”  the state has collapsed to
 

1
   
2
• By measuring the Bell operator we have created entanglement!

Distinguishing Bell States
• The goal: To create a set of unitary
operators that would make a different set
of detectors “click” for each Bell state.
ij  0  ij ,  ij ,  ij  0
ij  0  ij ,  ij ,  ij  0
 ij  0  ij , ij ,  ij  0
 ij  0  ij ,  ij , ij  0
Distinguishing Bell states – cont.
A scheme to measure 2+ Bell states.
•Turns out this is the best we can do with linear elements.
•Non-linear devices can achieve a complete measurement but
with low efficiency.
Teleportation
• Alice wants to send a quantum bit  to Bob.
• She cannot measure the state and send the
results.
• If she sends the qubit
itself it might deteriorate on
the way or take too much
time to get there if it is a
state of a massive object.
Teleportation – cont.
• Alice has a photon-qubit
that she
wants to teleport.
• Alice creates two entangled photons, 2 and 3,
and sends photon3 to Bob.
• She performs a Bell measurement on photon1
and photon2 and sends Bob the result.
• Bob performs a transformation of his photon3
according to Alice’s Bell measurement result and
photon3 becomes a replica of photon1.
How does it work?
• Before Alice’s Bell measurement the complete
state is:


which can be expressed as
 1 0 
 z  

 0 1
0 1
x  

1 0
 0 1
i y  

1 0 
• By performing a Bell measurement on photons 1
and 2 they make photon3 collapse into one of the
above states.
• By sending the result Alice instructs Bob which
transformation to perform – Pauli matrices.
Experimentally
• Alice takes two photons (2,3) from a PDC in an anti-symmetric
entangled state and sends photon3 to Bob.
1
• Alice creates photon1 at 45 degrees, measures only          
2
on photons 1 and 2 and indicates to Bob about it.
• In this configuration, Bob’s photon is immediately a replica of photon1.
• Photon1 is destroyed in accordance with the no-cloning theorem.
Teleportation with complete BSM
VV
1 2  H4
H1V2  H 4
H1H 2  V4
V1H 2  V4
Teleportation with complete BSM
H4

1
45  135
2
1

45  135
2
V4 



Very low efficiency…
Dense Coding


• By manipulating one photon


entangled in a Bell state we can


convert it to another Bell state.


• Manipulation of one photon = four Bell states
= two bits!
• We can measure 2+“1” out of four Bell states.
• A “trit”: enhancement of the channel capacity
by a factor of log2 3  1.58.
1
2
1

2
1

2
1

2
 
   

   


   
   
Dense Coding Experiment
Phys. Rev. Lett. 76, 4656–4659
Entanglement Swapping
• Making photons that have never interacted
entangle using mediators.
• We want to entangle photons 1 and 4.
• We entangle photons 1 with 2 and 3 with 4.
The complete state is:
OR
• Now, performing a Bell measurement on
photons 2+3 results in entanglement of 1+4
into the same state as 2+3.
Entanglement Swapping
Experiment
Entanglement Purification Motivation
• Distribution of entangled states between
distant locations is essential for quantum
communication over large distances.
• The quality of entangled states generally
decreases exponentially with the channel
length.
• Error correction in quantum computation.
Entanglement purification
Take only “four mode” cases
HHHV or VVVH
 a1b1  a2b2   a1 b1  a2 b2   a1 b1  a2 b2   a1 b1  a2 b2   a1 b1  a2 b2
a1b1 a2b2   a1 b1  a2 b2   a1 b1  a2 b2   a1 b1  a2 b2   a1 b1  a2 b2
)May 2003 22( 417-422 ,423 Nature
Quantum Repeaters
• Classical repeaters: divide the channel into N
segments and enhance the signal at each
node.
• Qubits cannot be cloned at each node and
re-sent.
• Quantum repeaters: A teleportation scheme
involving entanglement swapping and
purification.
• Works in logarithmic time and polynomial in
resources with respect to the channel length.
The Scheme
• Divide the channel between A and B into N
segments by N-1 nodes: C1, C2 ...CN 1. N  Ln
• Create an EPR pair of fidelity F1 between every
two adjacent nodes.
N  Ln  Example:
M2
EPR, F1
EPR, F1
C1
EPR, F1
C2
EPR, F1
C3
N  32  9
EPR, F1
C4
EPR, F1
C5
EPR, F1
C6
EPR, F1
C7
EPR, F1
C8
• At every Node C perform a Bell measurement
of one photon on both sides.
i kL
M2
 EPR, FL
C1
C2
  EPR, FL
C3
C4
C5
  EPR, FL
C6
C7
C8

• Purify the entanglement between Ci kL using M
copies to achieve higher fidelity.
M
 EPR, F  F1   EPR, F  F1   EPR, F  F1 
C1
C3
C2
C4
C5
C6
C7
C8
• Repeat the process for the new state until A and
B share an entangled pair.
M

C1
1
EPR, F  FL
C3
C2

C1
C4
C5
C6
C7

C8
EPR, F  F
1
C2
Resources (number of EPR pairs):
C3
C4
C5
C6
C7

C8
R  M n N  M n Ln  N logL ( M )1
Polynomial in resources, logarithmic (n) in time!
Why ask questions when you can go home?