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

,452

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.

1

21

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

22112211221122112211 babababababababababa

22112211221122112211 babababababababababa

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?