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 of their state spaces.
• Schmidt decomposition: • If and the state is said
to be separable. If more than one then is said to be entangled.
• The state of one system cannot be specified without the other.
i i
,
' 'ij i j i i ii j i
a b 0j ib 0ib
0ib
Bell States
• For two two-state systems denoted each by the Bell states form a basis for the whole system and are maximally entangled:
where is anti-symmetric and are symmetric with respect to particle interchanging.
1
21
21
21
2
,
, ,
Bell Measurements
out in out intA L R R L
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out in out inU V H H V
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.
out in out inBS rU A R R L L
out in out in out in out inPBS r R R L L t L R R LU A H H H H A V V V V Polarization
beam splitter
Half wave plate at 45 degrees
Examples for photons: Beam splitter:
Example: distinguishing anti-symmetric and symmetric states - Hong–Ou–Mandel effect
1
2
1
21
21
2
2 2
1 2 2 1
1
0
cos( ) 1 2 sin( ) 2 1
BS r t
r t
r t t r
BS
U A d L d R A d L d R
A A
A A A A
U d L d R i d L d 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
• By measuring the Bell operator we have created entanglement!
1
2
Beam splitter operator representation for a single photon:
OR ?
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.
0 , , 0
0 , , 0
0 , , 0
0 , , 0
ij ij ij ij
ij ij ij ij
ij ij ij ij
ij ij ij ij
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
• 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.
1 0
0 1z
0 1
1 0x
0 1
1 0yi
Experimentally• Alice takes two photons (2,3) from a PDC in an anti-symmetric
entangled state and sends photon3 to Bob.• Alice creates photon1 at 45 degrees, measures only
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.
1
2
Teleportation with complete BSM
1 2 4
1 2 4
VV H
H H V
1 2 4
1 2 4
HV H
V H V
Teleportation with complete BSM
4
4
145 135
21
45 1352
V
H
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 2log 3 1.58.
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21
21
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:
• Now, performing a Bell measurement on photons 2+3 results in entanglement of 1+4 into the same state as 2+3.
OR
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
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Take only “four mode” cases
Nature 423, 417-422 (22 May 2003)
VVVHorHHHV
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:
• Create an EPR pair of fidelity between every two adjacent nodes.
1F
1 2 1, ... .NC C C nN L
2C 6C 8C1C 7C3C 4C 5C
1,EPR F2M 1,EPR F 1,EPR F 1,EPR F 1,EPR F 1,EPR F 1,EPR F 1,EPR F 1,EPR F
nN L 23 9N Example:
• At every Node perform a Bell measurement of one photon on both sides.
2C 6C 8C1C 7C3C 4C 5C
, LEPR F 2M , LEPR F , LEPR F
i kLC
• Purify the entanglement between using M copies to achieve higher fidelity.
i kLC
M 1,EPR F F 1,EPR F F 1,EPR F F
2C 6C 8C1C 7C3C 4C 5C
, LEPR F F
2C 6C 8C1C 7C3C 4C 5CM
1,EPR F F 1
Resources (number of EPR pairs): log ( ) 1L Mn n nR M N M L N Polynomial in resources, logarithmic (n) in time!
2C 6C 8C1C 7C3C 4C 5C
• Repeat the process for the new state until A and B share an entangled pair.
Why ask questions when you can go home?