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School of Physics
something
and Astronomy
FACULTY OF MATHEMATICAL
OTHER
AND PHYSICAL SCIENCES
Putting entanglement to work:
Super-dense coding, teleportation and metrology
Jacob Dunningham
Paraty, August 2007
Overview
What can we do with entanglement?
• Superdense coding
• Teleportation
• Quantum cryptography
• Quantum computing
• Quantum-limited measurements
Other lectures
Everyday
World
Quantum
Information
Multi-particle
WILD
Bunnies
PEDIGREE
Entanglements
Cats
Bats
Superdense coding
Suppose Alice and Bob want to communicate
Classically, if Alice sends Bob one bit - only one “piece” of information is
shared.
Is it possible to do better using quantum physics?
Superdense coding
Suppose Alice and Bob want to communicate
Classically, if Alice sends Bob one bit - only one “piece” of information is
shared.
Is it possible to do better using quantum physics?
Problem:
Can Alice send two classical bits of information
to Bob if she is only allowed to physically send
one qubit to him?
Answer:
Yes - by making use of entangled states
Superdense coding
Third party
Alice
00 : I 01: Z
10 : X 11:iY
1 qubit
1 qubit
Bob
Alice makes one of four measurements and then sends her qubit to Bob
Superdense coding
Procedure:
- If Alice wants to send…
00
She does nothing
01
She applies the phase flip Z to her qubit
10
She applies the quantum NOT gate X
11
She applies the iY gate
Superdense coding
Bell States
Superdense coding
Bell States
Finally, Alice sends her (single) qubit to Bob
Bob now has one of the Bell states. These are orthonormal and can be
distinguished by measurements.
Teleportation
It is also possible to transfer an unknown quantum state to a distant party
using entanglement - teleportation
Suppose Alice has a state of the form:
Unknown coefficients
Teleportation
It is also possible to transfer an unknown quantum state to a distant party
using entanglement - teleportation
Suppose Alice has a state of the form:
Unknown coefficients
And that Alice and Bob share a Bell state
Overall:
Teleportation
Alice has the first two qubits (on the left)
This can be rewritten in terms of the Bell states as:
Teleportation
Alice has the first two qubits (on the left)
This can be rewritten in terms of the Bell states as:
If Alice detects in the Bell-state basis,
Bob’s state is projected onto one of these
Teleportation
If Alice measures
Bob gets
Successful teleportation
The other outcomes require state rotations by Bob
Teleportation
If Alice measures
Bob gets
Successful teleportation
The other outcomes require state rotations by Bob
Alice measures
Bob gets
Operation
Teleportation
If Alice measures
Bob gets
Successful teleportation
The other outcomes require state rotations by Bob
Alice measures
Bob gets
Operation
Alice must communicate her measurement outcome to Bob by classical
means. This prevents information being transmitted faster than light.
Precision measurements
Cold is good
Colder is better:
Atomic fountains are
today’s best clocks,
realizing the SI second
to better than 10-15.
-wave
Cavity
~1 cm hole
For earthbound clocks, laser
cooling is sufficient
~ meter high fountains give ~ 1 second return times, so
~1 cm/s velocity spread, i.e. T~1 K Cs is sufficient.
NIST – U. Colorado – JPL
Primary Atomic Reference Clock in Space (PARCS)
Space-borne clocks, with
much longer interaction times
than possible on earth, will
benefit from colder atoms. A
BEC can provide the really
cold temperatures needed.
BEC in space
A 100 m diameter
trapped 87Rb
condensate…
…adiabatically
expanded to 1 cm…
…would, upon release, expand with less than
1 m/sec.
In principle, observation times longer than
1000 seconds would be possible
We’re missing a trick…
Interferometer
Interferometer
Interferometer
The phase shift can be found from:
Then
What is a beam splitter?
What is a beam splitter?
A state rotation….
What is a beam splitter?
“Any physical process that
transforms states in the same
way as a beam splitter”
What is a beam splitter?
“Any physical process that
transforms states in the same
way as a beam splitter”
For atoms, this is equivalent to
tunnelling between two potential wells
What is a beam splitter?
What is a beam splitter?
Using the identity:
What is a beam splitter?
Using the identity:
Shot noise
For a coherent state
at 1 and a vacuum
at 2:
Shot noise
For a coherent state
at 1 and a vacuum
at 2:
The uncertainty in the phase can be found from
‘Shot noise’
Due to discreteness of outcomes - same noise as coin toss
Squeezing
Not squeezed
Squeezed
Squeezing
If we use a squeezed state
as the input to the beam splitter:
Since:
i.e. there is no advantage to squeezing the input state
Squeezed vacuum
Bizarrely, by squeezing the vacuum we can do better
For modest squeezing:
This reduces to:
Squeezed vacuum
Bizarrely, by squeezing the vacuum we can do better
For modest squeezing:
This reduces to:
There is an optimum degree of squeezing. This gives:
But we can still do better…..
Heisenberg limit
The ultimate quantum limit is the Heisenberg limit:
Heisenberg limit
The ultimate quantum limit is the Heisenberg limit:
For a system with a fixed total number of particles, N,
The maximum number uncertainty in any part of the system is N
This mean that the uncertainty in the phase is:
Heisenberg limit
The ultimate quantum limit is the Heisenberg limit:
For a system with a fixed total number of particles, N,
The maximum number uncertainty in any part of the system is N
This mean that the uncertainty in the phase is:
Lower bound (equality)
is the Heisenberg limit
c.f. Shot noise
Heisenberg limit
The ultimate quantum limit is the Heisenberg limit:
For a system with a fixed total number of particles, N,
The maximum number uncertainty in any part of the system is N
How do we reach this limit?
This mean that the uncertainty in the phase is:
Lower bound (equality)
is the Heisenberg limit
c.f. Shot noise
The cat gets the cream
Stream of single particles:
i
1, 0  e
0,1
The cat gets the cream
Stream of single particles:
i
1, 0  e
0,1
Cat state
iN
N ,0  e
0, N
N-fold enhancement
of phase shift
Bucky-ball
Quic kTime™ and a
TIFF (Unc ompres sed) dec ompres sor
are needed to see this pic ture.
Bucky-ball
Quic kTime™ and a
TIFF (Unc ompres sed) dec ompres sor
are needed to see this pic ture.
C60 molecules
(1999)
Bucky-ball
Quic kTime™ and a
TIFF (Unc ompres sed) dec ompres sor
are needed to see this pic ture.
C60 molecules
(1999)
PROBLEM: Cats are fragile – any advantage is lost
Let’s try something else…..
Number correlated state
U
Bose-Hubbard model
For simplicity, we can consider a two-site system
Adiabatically turn down coupling, J
Initially, system in binomial state
1/ 2
N
 ~  
k  k 
Then as U/J:
0
k
N k
the state evolves to:
  N /2 N /2
Entangling lots of atoms
Measurement scheme
?
What input state should we use to measure, , with
Heisenberg limited accuracy?
Hong-Ou-Mandel effect
Input state:
Hong-Ou-Mandel effect
Input state:
The effect of the beam splitter is
found by transforming the operators
Hong-Ou-Mandel effect
Input state:
The effect of the beam splitter is
found by transforming the operators
The beam splitter ‘creates’
entanglement
The same approach can be taken for:
  N /2 N /2
  N /2 N /2
Beam splitter
N=20
Independent
 
2ln2
Ne2 t
2/3 of atoms lost
N=20
Independent
 
2ln2
Ne2 t
Cat State
1
cos( N )  1  e 2 N t 
2
1
t
2 N
2/3 of atoms lost
N=20
Independent
 
2ln2
Ne2 t
Cat State
1
cos( N )  1  e 2 N t 
2
1
t
2 N
2/3 of atoms lost
Not
Entangled
Mott
Detectors
 Efficiency
Classical Limit
Reference: T. Kim et al. PRA 60, 708 (1999).
Classical Limit
To reach the Heisenberg
limit we need:
>11/N
1 


1
N
Reference: T. Kim et al. PRA 60, 708 (1999).
How do we resolve this?
Entangled
Not
Entangled
Mott
?
Pass back through Mott transition
Reversible process – adiabatically lower potential barrier
Disentangle particles
Independent
particles
Entangling and
disentangling
N=20 and J = p/30
Both cases are robust to
loss and insensitive to
detector inefficiencies
Entangled
Not
Entangled
Mott
Allows sub-SQL
resolution but still
robust to loss
Mott
Not
Entangled
Allows sub-SQL
resolution but not
destroyed by
imperfect detectors
N
Collapse time ~1/N
Atoms
Position
Atoms
Position
A phase shift of N=1, i.e. =1/N, is amplified into a
dramatic observable: Interference fringes appear
and disappear.
Summary
What can we do with entanglement?
• Superdense coding
• Teleportation
• Quantum cryptography
• Quantum computing
• Quantum-limited measurements