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LINEAR OPTICAL QUANTUM INFORMATION
PROCESSING, IMAGING, AND SENSING:
WHAT’S NEW WITH N00N STATES?
Jonathan P. Dowling
Hearne Institute for Theoretical Physics
Louisiana State University
Baton Rouge, Louisiana
quantum.phys.lsu.edu
14 JUNE 2007 ICQI-07, Rochester
Hearne Institute for Theoretical Physics
Quantum Science & Technologies Group
H.Cable, C.Wildfeuer, H.Lee, S.Huver, W.Plick, G.Deng, R.Glasser, S.Vinjanampathy,
K.Jacobs, D.Uskov, JP.Dowling, P.Lougovski, N.VanMeter, M.Wilde, G.Selvaraj, A.DaSilva
Not Shown: M.A. Can, A.Chiruvelli, GA.Durkin, M.Erickson, L. Florescu,
M.Florescu, M.Han, KT.Kapale, SJ. Olsen, S.Thanvanthri, Z.Wu, J.Zuo
Outline
1. Quantum Computing & Projective
Measurements
2. Quantum Imaging, Metrology, & Sensing
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
CNOT with Optical Nonlinearity
The Controlled-NOT can be implemented using a Kerr medium:
|0= |H Polarization
|1= |V Qubits
(3)
PBS
R is a /2 polarization rotation,
followed by a polarization dependent
phase shift .
Rpol
z
Unfortunately, the interaction (3) is extremely weak*:
10-22 at the single photon level — This is not practical!
*R.W. Boyd, J. Mod. Opt. 46, 367 (1999).
Two Roads to Optical CNOT
I. Enhance
Nonlinearity with
Cavity, EIT — Kimble,
Walther, Haroche,
Lukin, Zubairy, et al.
II. Exploit
Nonlinearity of
Measurement —
Knill, LaFlamme,
Milburn, Franson, et
al.
Cavity QED
Linear Optical Quantum Computing
Linear Optics can be Used to Construct
CNOT and a Scaleable Quantum Computer:
 0   1  2
Knill E, Laflamme R, Milburn GJ
NATURE 409 (6816): 46-52 JAN 4 2001
 0   1  2
Milburn
Franson JD, Donegan MM, Fitch MJ, et al.
PRL 89 (13): Art. No. 137901 SEP 23 2002
Road to
EntangledParticle
Interferometry:
An Early
Example of
Entanglement
Generation by
Erasure of
Which-Path
Information
Followed by
WHY IS A KERR NONLINEARITY LIKE
A PROJECTIVE MEASUREMENT?
LOQC
KLM
Photon-Photon
XOR Gate
Cavity QED
EIT
Photon-Photon
Nonlinearity
Projective
Measurement
Kerr Material
Projective Measurement
Yields Effective “Kerr”!
GG Lapaire, P Kok,
JPD, JE Sipe, PRA 68
(2003) 042314
A Revolution in Nonlinear Optics at the Few Photon Level:
No Longer Limited by the Nonlinearities We Find in Nature!
NON-Unitary Gates 
KLM CSIGN Hamiltonian
Effective Unitary Gates
Franson CNOT Hamiltonian
Nonlinear Single-Photon
Quantum Non-Demolition
You want to know if there is a single photon in mode b,
without destroying it.
Cross-Kerr Hamiltonian: HKerr =
|in b
|1 a
 a †a b †b
Kerr medium
|1
D2
D1
“1”
Again, with  = 10–22, this is impossible.
*N Imoto, HA Haus, and Y Yamamoto, Phys. Rev. A. 32, 2287 (1985).
Linear Single-Photon
Quantum Non-Demolition
The success probability is less
than 1 (namely 1/8).
D0
|1
The input state is constrained
to be a superposition of 0, 1,
and 2 photons only.
Conditioned on a detector
coincidence in D1 and D2.
D1
D2
 /2
|1
 /2
Effective  = 1/8
 21 Orders of
Magnitude
Improvement!
|0
|in =
2
cn |n

n=0
|1
P Kok, H Lee, and JPD, PRA 66 (2003) 063814
Outline
1. Quantum Computing & Projective
Measurements
2. Quantum Imaging, Metrology, & Sensing
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
Quantum Metrology with N00N States
H Lee, P Kok, JPD,
J Mod Opt 49,
(2002) 2325.
Shotnoise to
Heisenberg Limit
Supersensitivity!
AN Boto, DS
Abrams, CP
Williams, JPD, PRL
85 (2000) 2733
a† N a N
Superresolution!
Quantum Lithography Experiment
|20>+|02
>
|10>+|01
>
Canonical Metrology
Suppose we have an ensemble of N states | = (|0 + ei |1)/2,
and we measure the following observable: A = |0 1| + |1 0|


The expectation value is given by: |A| = N cos 
and the variance (A)2 is given by: N(1cos2)

The unknown phase can be estimated with accuracy:
A
1
 =
=
| d A/d | N

This is the standard shot-noise limit.
note the
square-root
P Kok, SL Braunstein, and JP Dowling, Journal of Optics B 6, (2004) S811
Quantum
Lithography & Metrology
 N   N,0  0,N
Now we consider the state
AN  0,N N,0  N,0 0,N
Quantum Lithography*:

Quantum Metrology:
N |AN|N = cos N
High-Frequency
Lithography
Effect
AN
1
H =
=
| d AN/d |
N



and we measure

Heisenberg Limit:
No Square Root!
P. Kok, H. Lee, and J.P. Dowling, Phys. Rev. A 65, 052104 (2002).
Outline
1. Quantum Computing & Projective
Measurements
2. Quantum Imaging, Metrology, & Sensing
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
Showdown at High-N00N!
How do we make High-N00N!?
|N,0 + |0,N
With a large cross-Kerr
nonlinearity!* H =  a†a b†b
|1
|0
|N
|0
This is not practical! —
need  =  but  = 10–22 !
N00N States
In Chapter 11
*C Gerry, and RA Campos, Phys. Rev. A 64, 063814 (2001).
Solution: Replace the Kerr with
Projective Measurements!
single photon detection
at each detector
a’
a
OPO
3a3
b
b’
b
6
4
2
0
a
a
a
a
0
2
4
6
b
b
3a1b
b
1a3b
4
a'
0
b'
0
a'
4
b'
b
Probability of success:
3
64
Best we found:
3 Cascading
16 Not
Efficient!
H Lee, P Kok, NJ Cerf, and JP Dowling, Phys. Rev. A 65, R030101 (2002).
These Ideas Implemented in
Recent Experiments!
|10::01>
|10::01>
|20::02>
|20::02>
|30::03>
|40::04>
|30::03>
Local and Global Distinguishability
in Quantum Interferometry
GA Durkin & JPD, quant-ph/0607088
A statistical distinguishability based on relative entropy characterizes the
fitness of quantum states for phase estimation. This criterion is used to
interpolate between two regimes, of local and global phase
distinguishability.
The analysis demonstrates that, in a passive MZI, the Heisenberg limit
is the true upper limit for local phase sensitivity — and Only N00N
States Reach It!
N00N
NOON-States Violate Bell’s Inequalities
CF Wildfeuer, AP Lund and JP Dowling, quant-ph/0610180
Probabilities of correlated clicks and independent clicks
Pab (,),Pa (),Pb ()
Building a Clauser-Horne Bell inequality from the expectation
values Pab (,),Pa (),Pb ()

1 Pab (,)  Pab (,)  Pab (,)  Pab (,)  Pa ()  Pb ()  0


Shared Local Oscillator Acts
As Common Reference
Frame!
Bell
Violation!
Outline
1. Quantum Computing & Projective
Measurements
2. Quantum Imaging, Metrology, & Sensing
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes
Efficient Schemes for
Generating N00N States!
|N>|0>
Constrained
|N0::0N>
Desired
|1,1,1>
Number
Resolving
Detectors
Question: Do there exist operators “U” that produce “N00N” States Efficiently?
Answer: YES!
H Cable, R Glasser, & JPD, quant-ph/0704.0678. Linear!
N VanMeter, P Lougovski, D Uskov, JPD, quant-ph/0612154. Linear!
KT Kapale & JPD, quant-ph/0612196. (Nonlinear.)
Quantum P00Per Scooper!
H Cable, R Glasser, & JPD, quant-ph/0704.0678.
2-mode
squeezing process
Old Scheme
linear
optical
processing
χ
OPO
beam
splitter
New Scheme
How to eliminate
the “POOP”?
U(50:50)|4>|4>
0.3
|amplitude|^2
0.25
0.2
0.15
0.1
0.05
0
|0>|8>
|2>|6>
|4>|4>
Fock basis state
|6>|2>
|8>|0>
quant-ph/0608170
G. S. Agarwal, K. W. Chan,
R. W. Boyd, H. Cable
and JPD
Quantum P00Per Scoopers!
H Cable, R Glasser, & JPD, quant-ph/0704.0678.
“Pizza
Pie”
Phase
Shifter
Spinning glass wheel. Each segment a
different thickness.
N00N is in Decoherence-Free Subspace!
Feed-Forward-Based Circuit
Generates and manipulates special
cat states for conversion to N00N
states.
First theoretical scheme scalable to
many particle experiments!
Linear-Optical Quantum-State
Generation: A N00N-State Example
N VanMeter, D Uskov, P Lougovski, K Kieling, J Eisert, JPD, quant-ph/0612154




2
2
2
0
0.03
( 50  05 )
2
U
1
0



This counter example disproves
the N00N Conjecture: That N
Modes Required for N00N.
The upper bound on the resources scales quadratically!
Upper bound theorem:
The maximal size of a N00N
state generated in m modes via
single photon detection in m–2
modes is O(m2).
Conclusions
1. Quantum Computing & Projective
Measurements
2. Quantum Imaging & Metrology
3. Showdown at High N00N!
4. Efficient N00N-State Generating Schemes