Qu it - SNUkye/MAQIT_2016/slide/15g_kim.pdf · Summary • Optical Coherent State: Even...

QuditJaewan Kim ( [email protected])

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



Bell State



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)

Qu it - SNUkye/MAQIT_2016/slide/15g_kim.pdf · Summary • Optical Coherent State: Even...

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