Bell Measurements and Teleportation
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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 bji 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 Rout
Rin Lout
Lin At Lout
Rin Rout
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
a1b1 a2b2 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?