Post on 28-Jan-2020
QuditJaewan Kim ( jaewan@kias.re.kr)
Korea Institute for Advanced Study JK, J. Lee, S.-W. Ji, H. Nha, P. Anisimov, J. P. Dowling, Opt Comm 337, 79 (2015)
MOLDING a Quantum State
t
HX M
MM|Ψ⟩
|0⟩|0⟩|0⟩
…X1H2CX13|Ψ⟩
Molding
E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46 (2001).
M. A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004).
M. A. Nielsen and C. M. Dawson, Phys. Rev. A 71, 042323 (2005).
SCULPTURING a Quantum State- Cluster State [One-way] Quantum Computing -
|+⟩ |+⟩ |+⟩ |+⟩ |+⟩
|+⟩ |+⟩ |+⟩|+⟩ |+⟩ |+⟩
|+⟩
|+⟩ |+⟩ |+⟩ |+⟩ |+⟩ |+⟩
|+⟩ |+⟩ |+⟩ |+⟩ |+⟩ |+⟩
1. Initialize each qubit in |+⟩=(|0⟩+ |1⟩)/√¯ state.
2. Contolled-Phase(CZ) between the neighboring qubits.
3. Single qubit manipulations and single qubit measurements only [Sculpturing].No two qubit operations!
R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003).
2
Cont-Z and |+⟩- Commuting with each other- Symmetric w.r.t. control and target
Even superposition of computational basis states
B. C. Sanders and G. J. Milburn, Phys. Rev. A 45, 1919 (1992).M. Paternostra et al., Phys. Rev. A 67, 023811 (2003).Wang W.-F. et al., Chin. Phys. Lett. 25, 839 (2008)Nguyen B. A. and J. Kim, Phys. Rev. A 80, 042316 (2009).
1 1 2 212 121 2
1 2 2 1 2 212 12
1 2 2 1 2 2
1 2 1 2
2 212 23 21 23 21 231 2 3 1 3 1 3
1 2 3 1 2 3
0 1 0 12 2
0 0 1 1 0 12 2 2 2
0 0 1 1 0 12 2 2 2
1 10 12 2
0 12 2
1 10 1
Bell State
GHZ sta2
t2
e
Z Z
Z Z
Z Z Z Z Z Z
+ +⎛ ⎞⎛ ⎞+ + = ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠+ +⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠+ −⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
= + + −
+ + + = + + + + +
= + + + − −
An Interesting Observation of Exponential Function
( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )
2
22 1
1
22
2 11 2
3 1
1
0
2 1
0
0 1
terms
1!
2
2!
2 1
1
2
1 !
! !1!
2 1 !
3 1
1 !
!
2 !!
for 0,1, , 1!
d d d
d
dd d
d
d
d
kk
k
d
md
km
d
x
x
xd
x
ddx
dx
x xd
x
dx
d d
d
x
f x
xf x
f xf f
k dk
x
md
f x x
xe
+
−
+
−
−
+ +
−
=+
−
∞
=
+ + + +
+
+ + + +
+
+
−
+
=
+ +++ −
−
++
+
= + + + +
=
= = −+
∑
∑
L
L
L
14444444424444
L
44443
L
L
2
22
( )x
dO eπ
−
2 ideπ
ω =
i
1 dω=
1 dω=
Conjugate Relations
( )
( ) ( )
( ) ( )
2
1
01
0
with .
1
si
x ds
dks
k ss
dks
s kk
e x e e
f x e xd
e x f x
πω ω
ω
ω
−−
=
−
=
≡ =
⎧=⎪⎪
⎨⎪ =⎪⎩
∑
∑
2 ideπ
ω =
i
1 dω=
An Interesting Observation of a Coherent State
22
22
( )dO eπ α−
( )( )
( ) ( ) ( )( )
2 2 2 2
2 2 2 2
1 2 2 1
2 2 2 2
2
0 1 2 12 2 2 2
2
0
1 2 2 1! 1 ! 2 ! 2 1 !
0 1 2 10! 1! 2! 1 !
!
d d d d
d
n
n
e d e d e d e dd d d d
e e e e dd
e nn
α α α α
α α α α
α
α α α α
α α α α
αα
+ + −− − − −
−− − − −
∞−
=
+ + + + + + + −+ + −
+ + + + −−
=
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
∑
L
L
( ) ( ) ( ) ( )( )
( )
( )
( )( )
2 2 2 22 2 1 2 2 3 1
2 2 2 2
2
2
2
1
0
2
0
2
as0
2 2 1 2 2 3 12 ! 2 1 ! 2 2 ! 3 1 !
10 1 2
1
with!
!
d d d d
d d d d
d
dk
k md
dm
k md
d d
m
e d e d e d e dd d d d
d
d d d d
kd
k d e k mdk md
k ke
d k md
α α α α
α
α
α
α α α α
α
α
+ + −− − − −
−
=
+∞−
=
+∞
−
→∞=
+ + + + + + + −+ + −
+ +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
+
−= + +
=
= ⋅ ++
= ⎯+
∑
∑
∑
L
L
L
2
1 .
As , 1 and .d d d d kl
d
k k k lα δ
⎯⎯⎯→
∴ →∞ = =practically dα ≥
Pseudo-number State
( )
2
2
0 !
k md
dm
k d e k mdk md
α α +∞−
=
= ⋅ ++
∑
d“Photon number comb”
0
0
d
dα
th
th
Conju(Coherent States gate Basis; States)Phase Pseudo- States .
is the 0 P state ( ) and an Even Superposition of
=
Comp
is
seudo-numbe
the 0
utational
stat
r
Pseudo
basis
Co-pha n u(se je
144424443
%
Computational( Sta Ba tesis s)NumPseudo- States . ber
) and an Even
gate basSuperposition of
is
14444244443
Pseudo-Number Basisand Pseudo-Phase Basis
1X
2X
= 0dα %
ldl ω α=%
1
01
0
Define .
1
1
ld
dkl
d dld
lkd d
l
l
k ld
l kd
ω α
ω
ω
−−
=−
=
=
⎧ =⎪⎪⎨⎪ =⎪⎩
∑
∑
%
%
%
Orthogonal & Normalized
1
01
0
1
1
dkl
d dld
lkd d
l
k ld
l kd
ω
ω
−−
=−
=
⎧ =⎪⎪⎨⎪ =⎪⎩
∑
∑
%
%
Exact Conjugate Relation
Pseudo-number states: Orthogonal, but not normalized.Pseudo-phase states: Normalized, but not orthogonal.As |α| gets bigger, they become orthonormalized.
22
22
( )dO eπ α−
practically dα ≥
( ) }{
( ) }{2
1
01
0
0 , 1 , , 1
Computational Basis: pseudo-number basis
0 , 1 , , 1
Conjugate Basis: pseudo-phase basis
1
1
d d d
d d d
id
dkl
d dsd
lkd d
k
d
d
e
k ld
l kd
π
α ωα
ω
ω
ω
−−
=
−
=
= = −
−
=
⎧=⎪⎪
⎨⎪ =⎪⎩
∑
∑
% % L
L
%
%
Qubits and Qudits
}{
}{
( )
( )
Computational Basis
0 , 1
Conjugate Basis
,
1 0 12
1 0 12
+ −
+ = +
− = −
Qubit Operators and Qudit Operators
Cf. D. L. Zhou et al., Phys. Rev. A 68, 062303 (2003).
12 1 2 1 21 1
12 1 2 1 21 1
0 1
1 0
1 0
0 1
1 00 1
1 00 1
0 11 0
1 00 1
CNOT 0 0 1 1
C-Z 0 0 1 1
1 11 0 11 12
0
0
0
0
X
Z
X I X
Z I Z
H
−
⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤
= ⎢ ⎥−⎣ ⎦= = ⊗ + ⊗
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦= = ⊗ + ⊗
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤= = + + −⎢ ⎥−⎣ ⎦
2 3
2 4
2
1 2 1 2 1 21 1 1
2
2
1
1 21 2
1
0
1 1
0 0
1 0 0 0
0
0 0
0
0
1 1 1 1
1
1
12
0 1 0
0 1
0
1 0
0 1
1 0
0 0 1 1 2 2
1 2ˆ ˆ
1
ˆ
C-Z
d
d
dk
d dk
d d d d d d
d d d d
d dkl
k l
I Z Z
k k l ln n
k k
d
n
X
Z
Z
H
ω
ω
ω
ω
ω ω ω
ω ω
ω
ω ω
ω−
−
−
=
− −
= =
⊗ + ⊗ + ⊗ +
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
= =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
=
= =
=
=
∑
∑∑
L
O
O O M
O
L L
L
O
O O
O
L
( )
6
3 6
211
1
0d
d
d
d
kk k
ω
ω ω
ω −
−
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥
=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
∑M O
M M O
%
2 ideπ
ω =
Generalized Z OperatorPseudo-Number Operator~ Pegg-Barnett Number Operator
Phase shifterGeneralized Hadamard Operator
One-step teleportation
Generalized X OperatorPseudo-Phase Operator~ Pegg-Barnett Phase Operator
Generalized Cont-Z OperatorCross Kerr Interaction
D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989).
( ) ( ) ( )1
0
1
01
0
1 1
ct c c c c0 0
1 with 1 1
1 2ˆ ˆ
ˆ
d
dd d dl
dl
d dld
d dl
d dlm
d d d dl m
X l l d
Z l l
H l l
CZ Z l l m mn n
nω
ω ω
ω
−
=
−
=
−
=
− −
= =
= − − ≡ −
= =
=
= = ⊗ =
∑
∑
∑
∑∑
%
Maximal Entanglement ofPseudo-Number State and Pseudo-Phase State
Generalized Controlled-Z Operator
1 2ˆ ˆH n nχ= − Cross Kerr Interaction
2 ,L ddπχ α= <~
1 21 2 1 2
2 ˆ ˆˆ ˆ ˆ ˆ12
i n ni Ln n n ndU e e Zπ
χ ω= = = =(d ≅ 10~1000 ?!)
1 2
1 1ˆ ˆ
12 1 20 0
1 1
0 01 1
0 0
1
1
1 1
d dn n
d dk l
d dkl
d dk l
d d
d d d dl k
Z k ld
k ld
l l k kd d
α α ω
ω
− −
= =− −
= =− −
= =
=
=
= =
∑ ∑
∑∑
∑ ∑% %
“Deterministic” Generation of a Qudit Cluster State
TayloringScissors:measurements in pseudo-number basis (Z)
Stitches:measurements in pseudo-phase basis (X)
p q
rp,q r lattice
n nω α∈
∏ ∏) )
One-step d-dim Teleportation
1 2
1 12
12 1 2 10 0
21
1 2
12
Projective Measurement
2into
2 2
2 2
2 2 2
2 2 2
1
d ddn n
dl m
dlml d
l m
l d dl
dlkl d
l k
lkl d
l
lkl d
l
lkl d
l
kl d
lk
dk
mZ l
dm
a ld
a l l
ka l
d
a l
a H l
H a l
H Z a l
H Z
φ α ω
ω
ω
ω
ω
ω
φ
− −
= =
−
−
−
−
−
−
=
=
=
⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭
⎯⎯⎯⎯⎯⎯⎯→
=
=
=
=
∑ ∑
∑∑
∑
∑ ∑
∑
∑
∑
∑
) )
%
%
%%
%
1dk%
d-dim Teleportation
d-dim Bell state
1dk%
2ds%
Xs Z-k R
d-dim Teleportation
1dk%
2ds%
Pseudo-Phase Measurementby Homodyne Detection
2X
1X1X
2Xo
l dl
s ds
c l
c s
ψ =
′=
∑
∑ %
0
1X
2X
Homodyne
Homodyne
dk%
12 1 2 1 2
l d s dl s
l d dl
c l c s
Z c l l
ψ
ψ α
′= =
=
∑ ∑
∑
%
%
1ψ
2α
2dk%1dk
Pseudo-Number Measurement1dk
Postselection of high Number state
( )
2
2
0 !
k md
dm
k d e k mdk md
α α +∞−
=
= ⋅ ++
∑
d “Photon number comb”
d ′
l md k nd N′+ = + =
0.4
0.3
0.2
0.1
39000 40000 41000 Photon Number
Am
plitu
de
Quantum Repeater
Quantum Repeater
Quantum Repeater
Quantum Repeater
TayloringScissors:measurements in pseudo-number basis (Z)
Stitches:measurements in pseudo-phase basis (X)
Bell State
TayloringScissors:measurements in pseudo-number basis (Z)
Stitches:measurements in pseudo-phase basis (X)
GHZ State
Coherent State
Spin Coherent State Qudit
≈
Modulo-d spin state & Spin coherent state
Ising Interaction CZ
Summary• Optical Coherent State: Even superposition of d-dim
pseudo-number computational basis states• Generalized Cont-Z can be implemented
by Cross-Kerr interaction (d ≅ 10~1000 ?!)Max Entanglement Qudit Cluster State
• d-dim teleportation • Pseudo-Phase Measurement by Homodyne detection• Pseudo-Number Measurement• Network for Quantum Communication• Spin coherent state qudit• Qudit Cluster Quantum Computation …
# Decoherence# Single qudit operation with non-integer power
JK, J. Lee, S.-W. Ji, H. Nha, P. Anisimov, J. P. Dowling, Opt Comm 337, 79 (2015)