Continuous variable quantum entanglement and its applications

Continuous Variable Quantum
Entanglement and Its applications
Quantum Optics Group
Department of Physics
The Australian National University
Canberra, ACT 0200
Australian Centre for
Quantum-Atom Optics
The Australian National University
Canberra, ACT 0200
Ping Koy Lam

• Entanglement in General
• Continuous variable optical entanglement
• Entanglement measures
• Other types of entanglement
• Applications of entanglement
• Quantum teleportation
Outline

• Two objects are said to be entangled when their total wave-
function is not factorizable into wave-functions of the individual
objects.
• Not entangled
• Entangled
• Note: Entanglement is different to superposition.
What is entanglement?
!2 =
1
2
H2 + V2( )
! =
1
2
H1H2 + V1V2( )
!
" # $1 % &2( )

• P1 measures HV and get H
• P1 measures HV and get V
• P1 measures HV and get V
• P1 measures DA and get D
Why is it weird?
!
" =
1
2
H1H2 + V1V2( )
!
" =
1
2
D1D2 + A1A2( )
• P2 measuring HV MUST get H
• P2 measuring HV MUST get V
• P2 measuring DA can get D or A
• P2 measuring DA MUST get D
!
" =
1
2
H1H2 + V1V2( )
!
" =
1
2
H1H2 + V1V2( )
!
" =
1
2
H1H2 + V1V2( ) =
1
2
D1D2 + A1A2( ) =
1
2
L1L2 + R1R2( )
• Wave-function of the system collapses in a way that is completely determined
by the measurement outcome of P1.

How to create entanglement?
• Use conservation laws. Start with one system that can break up into sub-
systems.
• Eg. Nuclear fission with conservation of energy and momentum
• Eg. Parametric down conversion. Split one photon into two photons.
• Look at two non-commuting observables and “prove” via inference that
Heisenberg Uncertainty Principle (HUP) can appear to be violated.
• We get !Xinf•!P2 < HUP Limit?
• Resolution: Inference does not count!
• After particle 1 has been measured, the wave-function of particle 2 (or even
the system) is changed. This new wave-function still obeys the HUP.
Measure position !X1 Position inferred !Xinf
Measure momentum !P2
!
"X2
"P2
=
h
2
!
X,P[ ] = ih

A brief history of entanglement
1935: Einstein-Podolsky-Rosen’s proposal to prove quantum mechanics is incomplete
1935: Schrödinger coined the word “entanglement” - Verschränkung
1950: Gamma-ray pairs from positron & electrons produced by Wu & Shaknov.
1964: J. S. Bell proposed a theorem to exclude hidden variable theories.
1976: Entanglement between protons observed by Lamehi-Rachti & Mittig.
1980s: Low-energy photons from radiative atomic cascade by Aspect et al. Close a lot of
loopholes in a series of experiments.
1988: Light entanglement from crystals by Shih & Alley.
1989: Greenberger-Horne-Zeilinger entanglement.
1992: Entanglement from continuous-wave squeezers by Ou & Kimble et al.
1999: Entanglement from optical fibre by Silberhorn & Lam et al.
2001: Entanglement of atomic ensembles by Julsgaard & Polzik et al.
2002: Entanglement by a New Zealander, Bowen et al.
Future: Inter-species entanglement?
- Entanglement of light beams of different wavelengths
- Atom-light entanglement.
Future: Entanglement of Bose-Einstein Condensates?
Future: Macroscopic entanglement?
Future: Long lived entanglement?

• Subtract the intensities
(amplitudes) of the two
beams gives a very quiet
measurement: Intensity
difference squeezing.
• Sum the phases of the
two beams gives a very
quiet measurement as
well.
• What is the limit for
saying that there is
optical entanglement?
Continuous variable optical entanglement
• We want to look at the amplitude and the phase quadrature only.
!
X+
,X"
[ ]= 2i
!
V(X+
)V(X+
) =1

Parametric down conversion
pump light
EPR 1
EPR 2
crystal
• Pair productions => 2 photons production for each pump photon
=> Amplitude correlation
• Conserv. of energy => Anti-correlated k-vector
=> Phase anti-correlation
• One beam is vertically polarized and the other is horizontally polarized in
Type II Optical parametric oscillator/amplifiers.
• These two beams are entangled.

Squeezing with OPO/A
• For degenerate Type I OPO/A, the signal and idler beams have the same
polarization.
• The single output of the OPO/A is squeezed.
*

Squeezing and entanglement
• Can we use squeezed light to generate entanglement?
• Squeezing:
• One beam only
• Sub-quantum noise stability (quantum
correlations) exists in one quadrature at
the expense of making the orthogonal
quadrature very noisy
• Completely un-interested in the other
quadrature => Do not really care
whether state is minimum uncertainty
limited. Do not care about state purity.
• Entanglement
• Must be between 2 beams
• Must have quantum correlations
established on both non-commuting
quadratures
• Does worry about all quadratures!
Purity matters.
?

Generating quadrature entanglement
x
y
1
2
Ou et al., Phys. Rev. Lett. 68, 3663 (1992)
• Need to mix two squeezed beams with a 90 degree phase difference on a
50/50 beam splitter
!
X1,2
+
<1< X1,2
"
!
Xx
+
=
1
2
X1
+
+ X2
"
( )
!
Xx
"
=
1
2
X1
"
+ X2
+
( )
!
Xy
+
=
1
2
X1
+
" X2
"
( )
!
Xy
"
=
1
2
X1
"
" X2
+
( )

Entanglement generation experiment
SQZ
SQZ
CV
Entanglement
Pump
Seed
Seed

Looking within the uncertainty circle
EPR
Output X
EPR
Output Y
1a
1b
2a
2b
Individually, each beam is very noise in every quadrature
Combined, they are correlated in phase, and anti-correlated in
amplitude beyond the quantum limit.

Looking within the uncertainty circle
Output X
EPR
Output Y
1a
2a
1b
2b
Seems to demonstrate that EPR’s idea is right?
Is Heisenberg Uncertainty Principle being violated?
EPR
Resolving the paradox:

Cross correlations between beams

The sum and difference variances
Amplitude
Anti-correlations
Phase
Correlations
Xy
"
Xx
+
Xy
+
Xx
"
!
V Xx
"
" Xy
"
( ) 2 = V Xx
+
( )
!
V Xx
+
+ Xy
+
( ) 2 = V Xx
+
( )!
Xx
+
=
1
2
X1
+
+ X2
"
( )
!
Xx
"
=
1
2
X1
"
+ X2
+
( )
!
Xy
+
=
1
2
X1
+
" X2
"
( )
!
Xy
"
=
1
2
X1
"
" X2
+
( )
!
X1,2
+
<1< X1,2
"

Inseparability Criterion
• In the spirit of the Schrödinger Picture
• Measures the degree of inseparability of two entangled
wavefunctions
• Looks at the quadrature amplitudes’ quantum correlations
• The sum/difference correlations of the amplitude/phase between
the two sub-systems must both be less than the HUP
• Insensitive to the purity of states.
Duan et al., Phys. Rev. Lett. 84, 4002 (2001)
!
V Xx
"
" Xy
"
( ) 2 = V X2
+
( )
!
V Xx
+
+ Xy
+
( ) 2 = V X1
+
( )
!
V Xx
+
+ Xy
+
( )V Xx
"
" Xy
"
( ) 2 <1

State purity
• Minimum uncertainty states are pure
• Mixed states of squeezed light !
"
!
1
"
!
1
"
+ m
!
"

The conditional variances
Amplitude
Anti-correlations
Phase
Correlations
Xy
"
Xx
+
Xy
+
Xx
"
!
Vx|y
+
= V Xx
+
( )"
#Xx
+
#Xy
+
V Xy
+
( )
2
!
Vx|y
"
= V Xx
"
( )"
#Xx
"
#Xy
"
V Xy
"
( )
2
!
" = X1,2
+
<1< X1,2
#
=
1
"

EPR criterion
• More in the spirit of the Heisenberg Picture
• Measures how well we can demonstrate the EPR paradox
• Looks at conditional variances of the quadrature amplitudes
• The product of the amplitude and phase quadratures conditional
variances must be less than the Heisenberg Uncertainty Limit
• Takes into account the purity of the entanglement
Reid and Drummond, Phys. Rev. Lett. 60, 2731 (1988)
!
Vx|y
+
Vx|y
"
<1
!
Vx|y
+
= V Xx
+
( )"
#Xx
+
#Xy
+
V Xy
+
( )
2
!
Vx|y
"
= V Xx
"
( )"
#Xx
"
#Xy
"
V Xy
"
( )
2

Other forms of quadrature entanglement
• Can we have entanglement that has cross quadrature correlations
between beams?
• Can we have entanglement that has same sign correlations for
both quadratures?
!
Vx+|y"
+
= V Xx
+
( )"
#Xx
+
#Xy
"
V Xy
"
( )
2
<1
!
Vx"|y+
"
= V Xx
"
( )"
#Xx
"
#Xy
+
V Xy
+
( )
2
<1
!
V Xx
"
" Xy
+
( ) 2 <1
!
V Xx
+
+ Xy
"
( ) 2 <1
!
Vx|y
+
= V Xx
+
( )"
#Xx
+
#Xy
+
V Xy
+
( )
2
<1
!
Vx|y
"
= V Xx
"
( )"
#Xx
"
#Xy
"
V Xy
"
( )
2
<1
V Xx
"
" Xy
"
( ) 2 <1V Xx
+
" Xy
+
( ) 2 <1

No cloning theorem
Let U be the cloning operator such that
U |#> = | # > $ | # > and
U |%> = | % > $ | % >
For a state in superposition |&> = 1/!2 ( |%> + |#> ), we have
U |&> = 1/!2 (U | % > + U | # >) = U 1/!2 (| % > + | # >)
Assuming QM is linear
Should the answer be:
U |&> = 1/!2 (| % > $ | % > + | # > $ | # >)
or
U |&> = 1/!2 [(| % > + | # >) $ (| % > + | # >)] (q.e.d.)

Polarization entanglement
(D, D)
(H, V)
(L,R)
!
ˆS1, ˆS2[ ]= 2iˆS3
ˆS2, ˆS3[ ]= 2iˆS1
ˆS3, ˆS1[ ]= 2iˆS2
CommutationCommutation rrelationelationss
ooff StokesStokes operatorsoperators
.ˆˆˆˆˆ
ˆˆˆˆˆ
,ˆˆˆˆˆ
,ˆˆˆˆˆ
††
3
††
2
††
1
††
0
!!
!!
i
VH
i
HV
i
HV
i
VH
VVHH
VVHH
eaaieaaiS
eaaeaaS
aaaaS
aaaaS
"=
+=
"=
+=
"
"
S1!
S3!
S2!
S3!
S2!S1!
2 1
2
1
1 2
2
1
1 22
1
a)
2
1
2
1
1 2
2
1
1 22
1
S1!
S3!
S2!
S3!
S2!S1!
c)
2 1
2
1
1 2
2
1
1 22 1
S1!
S3!
S2!
S3!
S2!S1!
b)
2
1
2
1
1 2
2
1
1 22 1
S1!
S3!
S2!
S3!
S2!S1!

Spatial entanglement
(a)
(b)
(c)
PP
PP
BS
O PA
O PA
! 0
! 0
+
+
SD
Optical
cavity
SD
Optical
cavity
(d)
(e)
(a)
(b)
BS
OPA
OPA
HD
HD
LO
(c) (d)
(e)
TEM00
TEM00
TEM10
TEM10
TEM10
LOTEM10
!
!
+
+
• Near field-Far field entanglement
– Squeeze 2 TEM10 modes and interfere on a beam splitter
• Position-momentum entanglement
– equivalent to near field-far field entanglement
• Split detector entanglement
– Squeeze 2 flipped modes and interfere on a beam splitter
!
x, px[ ] =1
!
y, py[ ]=1

Virtual entanglement
Real entanglement
Virtual entanglement
No entanglement

Applications of entanglement
• Quantum information processing
– C-not gates in quantum computation
– Grover’s algorithm
– Shor’s algorithm
– Quantum games
• Quantum communication and cryptography
– Quantum key distribution
– Super-dense coding
– Secret sharing network
• Quantum metrology
– Ultra-sensitive interferometric measurements
– Sub-diffraction limited imaging resolution
– Time keeping, lithography, etc.

Continuous variable quantum entanglement and its applications

and the machine in
the movie The Fly.
Oxford English Dictionary:
Old definition: The conveyance of persons (esp. of
oneself) or things by psychic power.
New definition: In futuristic description, apparently
instantaneous transportation of persons, etc., across
space by advanced technological means.

Teleportation definition
• Teleportation is the disembodied transportation of an object that
involves
– Thorough measurements of an input
– Transmission of the measured results
– Perfect reconstruction of the input at a different location
‘Alice’
‘Bob’
(2)
Transmission
Alice and Bob are the names
given to:
“A” the sender and
“B” the receiver.
(1)
Measurement
(3)
Reconstruction

Teleportation objective
• To prove that we can reconstruct the quantum state of light at a
distance without paying any “quantum duty” of measurements.
• To teleport a laser beam that carries information.
• We encode small signals on the sideband frequencies of the light
beam both on the phase as well as the amplitude quadratures.
• Equivalent to AM and FM simulcast.
• Need to show that information on both quadrature can in
principle be perfectly reconstructed at a distance.
Phase
Amplitude

How big is the quantum noise
• If the intensity “stick” is the width of Australia, then how big is
the quantum noise?
Assuming our experimental parameters (5kHz linewidth, 10mW
@ 1064nm) then the quantum noise is a 1m gym ball.

The classical teleporter
• Simultaneous measurements of the conjugate observables will
introduce vacuum noise.

The quantum teleporter
• Plug the vacuum noise input with entanglement!

Teleportation fidelity
• Results can be analysed by comparing an ensemble of the input
and the output states.
• We can use fidelity = <#in|'out|#in>
• F = 1 is perfect teleportation
• F = 0.67 is the no-cloning limit
• F = 0.5 is the classical limit
• Problem: Cannot tell whether
there are quantum correlations
between two objects via fidelity.
0
0.1
0.2
0.3
0.4
0.5
0.6
(a) (b)
0 0.5 1
Gain of teleportation
0 0.5 1 1.5
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.7
|(!+ + !- ) - (!+ + !- )|in out outin
X+
0
0 5 10
10

Teleportation information
• Results can also be analysed by measuring the signal-to-noise of
both amplitude and phase quadratures.
• Need to encode signal on both quadratures and measure the ratio of
signal and noise power
• SNRout = 100% (0 dB) of SNRin
is perfect teleportation
• SNRout = 50% (3 dB) of SNRin
is the no-cloning limit
• SNRout = 33% (4.8 dB) of SNRin
is the classical limit
-2
0
2
4
6
8
10
12
14
16
8.36 8.38 8.4 8.42 8.44
X+ X-
Input
Input
X+
8.36 8.38 8.4 8.42 8.44
Frequency (MHz)
Output
Time (minutes)
X-
(c)
(b)
(d)
Output

T-V diagram (Ralph-Lam criteria)
• Analyse teleportation in terms of signal transfer coefficients and
quantum correlations.
• Horizontal Axis: T = SNRampli + SNRphase information axis
• Vertical Axis: V = Vx+|y+ + Vx-|y- correlation axis
Ralph and Lam, Phys. Rev. Lett. 81, 5668 (1998).
Grangier et. al, Nature 396, 537 (1998).

The Copenhagen explanation
No Entanglement Entanglement
Alice
Bob
Alice
Bob
Wavefunctions collapse instantaneously.

Cramer’s transactional interpretation
Quantum events can
be describe by the
interferences of
advanced and
retarded waves.
How can two bits
produce one qubits?
In a quantum teleporter, information has to travel backward in
time from Alice to the source of the EPR and then forward in time
from the EPR to Bob.
Qubits ( 2 bits + E-bit

Looking at the Wigner functions
Classical Teleportation
Quantum Teleportation

The Heisenberg Picture
Quantum
information is
contained in the
classical channels.
They are buried in
the EPR noise.
We can think of the EPR source as being twin plasterers. One
‘packed’ the quantum information and the other ‘unpacked’ the
quantum information.
Alice measured (signal + NOISE)
Bob reconstruct with: (signal + NOISE) - NOISE = signal

Photonic description of entanglement
ntotal = nx + ny =
1
4
!2
Xx
+
+ !2
Xx
"
+ !2
Xy
+
+ !2
Xy
"
( )"1
!
nmin = sinh2
r1 + sinh2
r2
nexcess = ntotal ! nmin ! nbias
nbias =
1
2
!2
Xx± y
+
4
+
1
!2
Xx± y
+ +
!2
Xx± y
"
4
+
1
!2
Xx± y
"
#
$
%
%
&
'
(
( " nmin"1

Continuous variable quantum entanglement and its applications

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