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Moments Coherent-State Quantum Process Tomography (McsQPT)

Joint work with: S. Rahimi-Keshari, A. T. Rezakhani, T. C. Ralph

Masoud GhalaiiNOV. 2013

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Basic concepts—Phase space, Wigner function, HD, …

Harmonic oscillator

Annihilation and creation operators

Phase space

22 2 ˆ1 pˆ x̂

2 2H m

m ˆˆ[x, p] i

†

ˆˆ x̂+ipˆˆ x̂-ip

a

a

†ˆ ˆ[ , ] 1a a

† 1ˆ ˆ ˆ( )2

H a a

�̂�=∑𝑘

❑( �̂�𝑘† �̂�𝑘+ 12 )

𝛼

+i𝑥

𝑝

𝑊 �̂� (𝑥 ,𝑝 ) := 12𝜋 ℏ∫−∞

+∞

𝑑𝜂𝑒−𝑖𝜂𝑝 /ℏ⟨ 𝑥+𝜂 /2∨�̂�∨𝑥−𝜂 /2⟩ Leonhardth, Measuring the Quantum State of Light (1997) Garrison and Chiao, Quantum Optics (2008)

�̂�|𝛼 ⟩ :=𝛼∨𝛼 ⟩

Wigner function

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Some foremost quantum states—Alex Lvovsky laboratory

Smithy, et al. –PRL (1992)

S(ξ) :=e¿ ¿

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Basic concepts—Homodyne detection, state tomography, …

[Balanced] homodyne detection

N̂− /¿ ¿

Characteristic function; Lvovsky and Raymer –RMP (2009)

Cahil and Glauber –Phys.Rev. (1969)

𝑥𝑝

𝑊 �̂� (𝑥 ,𝑝)

𝑥𝜃𝑁−

𝛼𝐿𝑂

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Quantum [optical] process tomography

Quantum process tomography Two-mode squeezed vacuum Coherent-state Quantum Process Tomography

Mohseni, Rezakhani, and Lidar –PRA (2008)

D’Ariano and Lo Presti –PRL (2001)

Lobino, et al. –Science (2008)

- Readily available in lab- Could be produce in many amplitudes and phases- Characterization of outputs by using HD

A general dynamical map

A set of fixed Hermitian operators

ℰ (𝜚 )=∑𝑖

❑𝐴𝑖𝜚 𝐴𝑖†

{𝐸 𝑖}𝑖=0𝑑2−1

Tr (𝐸 𝑖†𝐸 𝑗 )=𝑑𝛿𝑖𝑗

ℰ (𝜚)= ∑𝑚 ,𝑛=0

𝑑2−1

❑𝜒𝑚𝑛𝐸𝑚 𝜚 𝐸𝑛†

𝜒𝑚𝑛=∑𝑖 , 𝑗

❑𝑎𝑚𝑖𝑎𝑛𝑗∗

𝐴𝑖=∑𝑚=0

𝑑2− 1

❑𝑎𝑚𝑖𝐸𝑚

𝑆 (𝜉 ):=𝑒¿ ¿

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Coherent-state quantum process tomography (csQPT)

Glauber-Sudarshan Representation

Sudarshan –PRL (1963) || Glauber –PRL (1963)

Lobino, et al. –Science (2008)

Introductory Quantum Optics, Gerry and Knight (2005)

ϱ̂=∫ d2α P ϱ̂(α)∨α ⟩⟨ α∨¿

ℰ jkmn=∫d2α PL ,mn(α ) ϱ̂ jk (α )

ℰ ( ϱ̂)=∫ d2α P ϱ̂(α)ℰ ¿

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csQPT in Fock basis

Rahimi-Keshari, et al. –NJP (2011)

ℰ jkmn=

1√m!n!

𝜕αm𝜕αn ¿¿

[ℰ( ρ)] jk= ∑m,n=0

∞

ℰ jkmn ρmn

ρ= ∑m, n=0

∞

ρmn∨m⟩⟨ n∨¿

𝑥

𝑝

𝐸𝑅𝑅𝑂𝑅   →  ∼ 1𝑁

𝐸𝑅𝑅𝑂𝑅   →  ∼ 1√𝑁

¿𝛼 ⟩ ℰ ¿ℰ

{¿ n⟩}n=0∞

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Normally-ordered moments—Definition

ϱ̂=∫ d2α P ϱ̂(α)∨α ⟩⟨ α∨¿M jk=∫d2α P ϱ̂ (α )α k α j

P ϱ̂ (α )= ∑j ,k=0

∞

❑(−1 )k + j

j ! k !M

jkδ ( j ,k ) (α , α )

𝑀 𝑗𝑘≔Tr [ �̂�  𝑎† 𝑘𝑎 𝑗 ] 𝑀 𝑗1𝑘1 𝑗 2𝑘2≔Tr [ �̂� 𝑎†𝑘1𝑎 𝑗1𝑏† 𝑘2𝑏 𝑗2]

New method:

M-CS-QPTδ ( j , k )(α ,α)=𝜕αj 𝜕αk  δ2(α ,α )

δ 2(α ,α )=∫ d2ξ   eξ α −ξ α C. T. Lee –PRA (1992)

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csQPT by using moments—motivations

Prediction of non-classicality of process Characterization of Gaussian processes Direct measurement of moments

Shchukin and Vogel –PRA (2005)

Rahimi-Keshari, et al. –PRL (2013)

Definition: A quantum process is nonclassical if it transforms an input coherent state to a nonclassical state.

Some Gaussian processes

AmplificationAttenuation

Squeezing Displacement

ThermalizationIdentity

. . .

csQPT by using moments—Formalism

ϱ̂=∫ d2α P ϱ̂(α)∨α ⟩⟨ α∨¿ ℰ ( ϱ̂)=∫ d2α P ϱ̂(α )ℰ ¿

Pout ( β)=∫ d2α Pin(α )Pℰ (β∨α)

M jk(α )=∑m,n𝜕αm𝜕αnM jk (α )¿α=0

αmα n

m! n!

M jkout=∑

m ,nℰ jk

mn Mmnin

ℰ jkmn=

1m! n! 𝜕α

m𝜕αnM jk (α )¿α=0

ℰ

ℰ ( ϱ̂)=∫ d2 β Pout (β )∨β ⟩⟨ β∨¿

ℰ ¿

×β k β j

M jkout=∫ d2α Pin (α)M jk(α )

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M jkout=∫ d2β Pout (β )βk β j

M jkout (α )=∫ d2β Pℰ (β∨α ) βk β j

Scratch paper!

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McsQPT—Formalism

ℰ jkmn=

1m! n! 𝜕α

m𝜕αnM jk (α )¿α=0

χ [ ϱ̂ ] (ξ ):=Tr [ ϱ̂ eξ a†− ξ a]

ℰ jkmn=

(−1 ) j

m! n!𝜕αm𝜕α

n𝜕ξk𝜕ξ

j χ ℰ (ξ∨α )¿α , ξ=0

¿

ℰ 𝐣𝐤𝐦𝐧=∏

s=1

M

❑ 1ms !ns!

𝜕α s

ms𝜕α s

n s ¿

|𝛂 ⟩ :=¿α 1 , α2 ,…,α M ⟩

Some multi-mode processes

- Beam splitter- Squeezers - Noises and loses- Entanglement-based protocols for Quantum Key Distribution (QKD) and Quantum Secret Sharing- Evolution of highly entangled multimode state, known as a cluster state

McsQPT—Formalism—Generalization (M-mode case)

¿¿¿

Adesso and Illuminati –arxiv (2005)

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McsQPT—Other examples …

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McsQPT—Gaussian processes—Gaussian states

Gaussian states

Mean value vector Covariance matrix

Single-mode Gaussian state

Ferarro, et al. –arXiv (2005) || Adesso and Illuminati –J.Phys.A. (2007)

𝓦 [ϱG ] (𝐑 ) := 1(2π )n√¿¿¿

[𝐕 ] jk :=12⟨ R̂ j R̂ k+ R̂k R̂ j ⟩−⟨ R̂ j ⟩⟨ R̂k ⟩

𝒲[𝜚𝐺 ](𝑥 ,𝑝)=1

2𝜋 √¿¿ ¿V=(Δx x Δxp

Δxp Δ p p)

¿R :=(R1  R2   ...   R2 N )T

: (x p)TR

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McsQPT—Gaussian processes—Moments vs. R & V

𝑀 𝑗1𝑘1 𝑗2 𝑘2=⟨ 𝑎† 𝑘1𝑎 𝑗1𝑏† 𝑘2𝑏 𝑗 2 ⟩

𝑀 1000❑ =⟨ �̂� ⟩=𝑥1

❑+𝑖𝑝1❑

𝑀 0010❑ =⟨ �̂�⟩=𝑥2

❑+ 𝑖𝑝2❑

𝑀 1100❑ =⟨ �̂�† �̂� ⟩=(𝑣11

❑+𝑣22❑ −1)+¿𝑀 1000

❑ ¿2

𝑀 0011❑ =⟨ �̂�† �̂�⟩=(𝑣33

❑ +𝑣44❑ −1)+¿𝑀 0010

❑ ¿2

𝑀 2000❑ =⟨ �̂�2 ⟩=(𝑣11

❑−𝑣22❑ +2𝑖𝑣12

❑)+𝑀10002

𝑀 0020❑ =⟨ �̂�2 ⟩=(𝑣33

❑ −𝑣44❑ +2 𝑖𝑣34

❑ )+𝑀 00102

𝑀1010❑ = ⟨ �̂��̂� ⟩=(𝑣13

❑ −𝑣24❑ +𝑖𝑣14

❑ +𝑖𝑣23❑ )+( ⟨ �̂�1 ⟩⟨ �̂�2 ⟩−⟨ �̂�1 ⟩⟨ �̂�2⟩+ 𝑖⟨ �̂�1 ⟩⟨ �̂�2 ⟩+𝑖⟨ �̂�1 ⟩⟨ �̂�2 ⟩ )

𝑀 1001❑ = ⟨ �̂��̂�† ⟩=(𝑣13

❑ +𝑣24❑ −𝑖𝑣14

❑ +𝑖𝑣23❑ )+( ⟨ �̂�1⟩⟨ �̂�2 ⟩+⟨ �̂�1 ⟩⟨ �̂�2 ⟩−𝑖⟨ �̂�1⟩⟨ �̂�2 ⟩+𝑖⟨ �̂�1 ⟩⟨ �̂�2 ⟩ )

# of moments which completely characterize any M-mode Gaussian state is “M(M+2).”

𝑀 10❑ =⟨ �̂� ⟩=𝑥1

❑+𝑖𝑝1❑

𝑀 20❑ =⟨ �̂�2 ⟩=(𝑣11

❑−𝑣22❑ +2 𝑖𝑣12

❑ )+𝑀102

𝑀 11❑=⟨ �̂�† �̂� ⟩=(𝑣11

❑+𝑣22❑ −1)+¿𝑀 10

❑ ¿2

𝑀 𝑗1𝑘1=⟨ 𝑎† 𝑘1𝑎 𝑗 1 ⟩

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McsQPT—Gaussian processes—Number of resources

𝐑 ′=𝐓𝐑+𝐝𝐕 ′=𝐓𝐕 𝐓𝑇+𝐍

𝑅′𝑖= t𝑖𝑗𝑅 𝑗+𝑑𝑖𝑖 , 𝑗=1 ,... ,2𝑁

𝑅′𝑖= t𝑖𝑗𝑅 𝑗 𝑖=1, ... ,2𝑁 , 𝑗=1 , ... ,2𝑁+1

𝐑 ′=~𝐓~𝐑

𝑑𝑖≡ t𝑖 ,2𝑁+1 𝑅2𝑁+1=1

Holevo and Werner –PRA (2001) || Weedbrook, et al. –RMP (2012)

, 𝑅 ′ 𝑖′

(𝑘 )= t𝑖 ′ 𝑗𝑅 𝑗(𝑘) 𝑗 ,𝑘=1 , ...,2𝑁+1

{𝑅 ′𝑖 ′

(1)=t𝑖′ 𝑗𝑅 𝑗(1 ) 𝑗=1 ,... ,2𝑁+1 ,

𝑅 ′𝑖′( 2)=t𝑖′ 𝑗𝑅 𝑗

(2 ) 𝑗=1 , ... ,2𝑁+1 ,...

𝑅 ′ 𝑖′(2𝑁+1)= t𝑖′ 𝑗𝑅 𝑗

(2𝑁+ 1) 𝑗=1 , ... ,2𝑁 +1

𝑅′𝑖′(𝑘)=𝑅 𝑗

(𝑘) t𝑖′ 𝑗 ℛ ′𝑖 ′=ℛ𝒯𝑖′𝒯𝑖 ′=ℛ−1ℛ ′

𝑖′ , 𝑖′=1, ... ,2𝑁 +1

, 𝐑 ,𝐕

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ℛ=(𝑅1(1 ) 𝑅2

(1) ... 𝑅2𝑁(1 ) 1

𝑅1(2 ) 𝑅2

( 2 ) ... 𝑅2𝑁(2 ) 1

. ¿ ¿ . ¿¿¿ .¿¿¿¿¿𝑅1

( 2𝑁+1 )¿𝑅2(2𝑁+1 )¿...¿𝑅2𝑁

(2𝑁 +1)¿1¿¿ ) ℛ ′𝑖 ′=(

𝑅 ′𝑖 ′( 1)

𝑅 ′𝑖′( 2 )

.

.

.𝑅′𝑖′( 2𝑁+1 )

)𝒯𝑖 ′=(t𝑖′ 1t𝑖′ 2

.

.

.t𝑖′ , 2𝑁+1

)𝑖 ′=1 ,... ,2𝑁 = 𝒯𝑖 ′=ℛ−1ℛ ′𝑖′

Proposition: Any trace-preserving M-mode Gaussian process could be completely characterized using only “2M + 1” input M-mode coherent state, which scales with number of modes, linearly.

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McsQPT—Examples—Process of decoherence

Vacuum state Squeezed vacuum Thermal squeezed

ThermalBath 1

4

¿0 ⟩𝑆 (𝜉 )∨0 ⟩

)

¿𝑆 (𝜉 ):=𝑒¿ ¿

Richter and Vogel –PRA (2007)

¿𝑏(𝑡)=N(1−𝜈2(𝑡)) ,  𝜈(𝑡 )=𝑒−𝛾𝑡

⟨ (𝛥 �̂� )2 ⟩=⟨ �̂�2⟩− ⟨ �̂� ⟩2

⟨ (𝛥 �̂� )2 ⟩= 14 (1+2𝑀 11

❑ +𝑀 20❑ +𝑀 20

❑ )

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McsQPT—Examples—Process of decoherence

ℰ 𝑗𝑘𝑚𝑛 (𝑡)=𝑗 !𝑘 !𝑚 !𝑛 !

N 𝑗−𝑚

( 𝑗−𝑚¿ !𝜈𝑚+𝑛(𝑡)[1−𝜈2(𝑡)] 𝑗−𝑚𝛿 𝑗 −𝑚 ,𝑘−𝑛

𝜒 ¿

𝑀 𝑗𝑘out (𝑡)= 𝑗 !𝑘!∑

𝑚=0

N 𝑗−𝑚

( 𝑗−𝑚¿!𝜈2𝑚− 𝑗+𝑘(𝑡)[1−𝜈2(𝑡 )] 𝑗−𝑚𝑀𝑚 ,𝑚− 𝑗+𝑘

in

ℰ 𝑗𝑘𝑚𝑛=

(−1 ) 𝑗

𝑚 !𝑛! 𝜕𝛼𝑚𝜕𝛼𝑛 𝜕𝜉𝑘𝜕𝜉𝑗 𝜒ℰ (𝜉∨𝛼)¿𝛼 , 𝜉=0

𝑀 𝑗𝑘out=∑

𝑚 ,𝑛ℰ 𝑗𝑘𝑚𝑛𝑀𝑚𝑛

in

Reminder!

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Thanks to you

and thanks to

Dr. RezakhaniMohammad Mehbodi

Maryam Afsary

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Breitenbach, et al. –Nature (1997)

Backup slide—1

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Backup slide—2

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ℛ𝑁=1=(x1(1 ) p1

( 1) 1x1

(2 ) p1( 2) 1

x1(3 ) p1

( 3 ) 1) ,ℛ𝑁=2=(x1

( 1) p1( 1) x2

(1 ) p2(1 ) 1

x1( 2) p1

( 2) x2(2 ) p2

(2 ) 1x1

( 3) p1( 3) x2

(3 ) p2(3 ) 1

x1( 4 ) p1

( 4 ) x2( 4 ) p2

( 4 ) 1x1

( 5) p1( 5) x2

(5 ) p2(5 ) 1

)

Backup slide—3

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¿¿

Backup slide—4