state process and detector tomography

State, Process and Detector Tomography

Lior Cohen

Quantum optics course29 March - 5 April 2019

Motivation

• N00N states

( )NN 0021

+=Y

( )NNNN 000021

+=r

YY=r

Outline

• Continuous variable QST (Quantum state tomography)• Discrete variable QST– 1 qubit– Higher dimension

• Process tomography• Detector tomography – State reconstruction

• Detector calibration – State reconstruction

Wigner function - reminder

( ) ò¥

¥-+-= e ipyyqyqdypqW !

!/2|ˆ|1, r

p

• Quasi-probabilistic distribution

• Can be negative (only quantum state)

• Contains all the information about the state (including photon statistics)

Wigner function - definition

( ) ò¥


!/2|ˆ|1, r

p

( ) }ˆ)(ˆ{ rxx DTrW =

( ) ( ) ( )xxa

xaax 2dWW eò¥

¥-

- **

=

• Wigner function

• Characteristic function

• Wigner function ??


( ) }ˆ)(ˆ{ rxx DTrW = ( ) ( ) ( )xxa

xaax 2dWW eò¥

¥-

- **

=


( ) ò¥


!/2|ˆ|1, r

p

( ) }ˆ)(ˆ{ rxx DTrW =

( ) ( ) ( )ir

iirir dWW e riir xxxxaa

xaxa

ò¥

¥-

--=

2,,

• Wigner function

• Characteristic function

• Wigner function

Homodyne detection – measuring the Rotated quadratures

( )abbaiddccnn dc ˆˆˆˆˆˆˆˆˆˆ ++++ -=-=-

( ) 2/ˆˆˆ pq

qqbb +

-+ =-= Xaai ee ii

aaX ee ii ˆˆˆ qqq

-+ +=

• Rotated quadratures

( ) ( ) ( ) ( ) ( )( )ò¥

¥-+-= dppxpxWxW qqqq

pq cossin,sincos1,

!

Rotated quadratures

aaX ee ii ˆˆˆ qqq

-+ +=

( ) ( ) ( ) ( ) ( )( )ò¥

¥-+-= dppxpxWxW qqqq

pq cossin,sincos1,

!

( ) ( ) ( )qhqhrhrhqh qq cos,sin}ˆˆ{)}ˆˆ{exp(, -=== WeiDTrXiTrW i

( ) ( ) hqhqhdWxW e ix

ò¥

¥-

-= ,,

Rotated quadratures

reconstruction of Wigner function

( ) ( )qhqhqh cos,sin, -=WW

( ) ( ) ( )ir

iirir dWW e riir xxxxaa

xaxa

ò¥

¥-

--=

2,,

( ) ( ) ( )qhhqhaa

qhaqha ddWW e iriir ò ò

+-=

sincos2,,

( ) ( ) ( ) dxddxWW e xiir

ir qhhqaaqaqah

ò ò ò-+-

=sincos,,

Reconstruction of Wigner function –squeezed vacuum state

G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387, 471 (1997).

Reconstruction of Wigner function –displaced single-photon

Zavatta A, Viciani S, Bellini M. Phys. Rev. A. 23 (2005).

Reconstruction of Wigner function –Cat states

S Deléglise et al. Nature 455, 510-514 (2008) doi:10.1038/nature07288

|lñjq

>y|

|hñ

|rñ

|pñ

|vñ

|mñ

S3

S1

S2

Logical qubit Polarization qubit

Ψ = # 0 + & 1 Ψ = # ℎ + & )

* = 12 ℎ + )

, = 12 ℎ − )

. = 12 ℎ + / )

0 = 12 / ℎ + )

Discrete QST - Single qubit

|lñjq

>y|

|hñ

|rñ

|pñ

|vñ

|mñ

S3

S1

S2

State reconstruction

å=

=3

0

ˆˆi

iiS sr

mp IIS -=1

rl IIS -=2vh IIS +=0

vh IIS -=3

SPD

PBS

!/#

!/$

N%&'( =D2

Discrete QST - Single qubit

1 0 0 0 1/ 2 1/ 2 1/ 2 / 2ˆ ˆ ˆ ˆ, , , .0 0 0 1 1/ 2 1/ 2 / 2 1/ 2H V P R

ii

r r r r-æ ö æ ö æ ö æ ö

= = = =ç ÷ ç ÷ ç ÷ ç ÷è ø è ø è ø è ø

1/ 2 0ˆ .0 1/ 2TotalyMixedr

æ ö= ç ÷è ø

|Hñ |Vñ

|lñjq

>y|

|hñ

|rñ

|pñ

|vñ

|mñ

S3

S1

S2


SPD

PBS

!/#

!/$

N%&'( =D2

Discrete QST - two qubits

1 2 1 21 2

3,, 0

1ˆ ˆ ˆ4 i i i ii i

S=

r = s Ä så

Discrete QST - two qubits

Discrete QST – four qubits

HHHHVHHHHVHHVVHH

HHVHVHVHHVVHVVVH

HHHVVHHVHVHVVVHV

HHVVVHVVHVVVVVVV

HHHHVHHH

HVHHVVHH

HHVHVHVH

HVVHVVVHHHHV

VHHVHVHV

VVHVHHVV

VHVVHVVV

VVVV

0

0.1

0.2

0.3

0.4

0.5

twelve qubits

“12-photon entanglement and scalable scattershot boson sampling with optimal entangled-photon pairsfrom parametric down-conversion”Han-Sen Zhong, Yuan Li, Wei Li, Li-Chao Peng, Zu-En Su, Yi Hu, Yu-Ming He, Xing Ding, W.-J. Zhang, Hao Li,L. Zhang, Z. Wang, L.-X. You, Xi-Lin Wang, Xiao Jiang, Li Li, Yu-Ao Chen, Nai-Le Liu, Chao-Yang Lu, Jian-Wei PanarXiv:1810.04823 (Submitted on 11 Oct 2018)

“…and detected by 24 superconducting nanowire single-photon detectors…”

“…the 12-photon coincidence isabout one per hour.”

Quantum process

S3

S1

S2

• Mapping of any arbitrary to : .ˆ 'rr̂ ( )ˆ ˆ'r e r=

S3

S1

S2

e

Quantum process

• Isotropic depolarization

S3

1

S1

1

1S2

•Amplitude damping (T1)

•Dephasing (T2)

( )ˆ ˆ ˆ ˆ ˆ( ) (1 )3 x x y y z zppe r r s rs s rs s rs= - + + +

† †0 0 1 1ˆ ˆ ˆ( ) E E E Ee r r r= +

0 1

1 0 0,0 1 0 0

E E gg

æ ö æ ö= =ç ÷ ç ÷ç ÷ç ÷- è øè ø

ˆ ˆ ˆ( ) 1 z ze r a r a s rs= × + - ×S3

1

S11

1S2

S3

1

S1

1

1S2

Quantum process

S3

S1

S2

ˆ 'rr̂ ( )ˆ ˆ'r e r=

S3

S1

S2

e

• The c(4x4) matrix uniquely describes a qubit process:

.• Quantum process tomography of a channel requires several

quantum state tomography measurements of different states.

( ) †ˆ ˆˆ ˆij i jijE Ee r c r=å

• Mapping of any arbitrary to : .

A. Shaham and H. S. Eisenberg, Phys. Rev. A 83, 022303 (2011).

Detector tomography (QDT)

• k photons hit PNRD, what is the measured statistics?

• QST -> known measurements, finding state• QDT-> known states, finding measurement

}ˆ{)( MTrmp!

r=

åå ==P=k

kknk

k

nkn kkTrnp rqrqr )()( ˆ}ˆˆ{)(kk

k

nkn å=P )(ˆ q

QDT – On-off detector

J. S. Lundeen et. al. “Tomography of quantum detectors,” Nat. Phys. 5, 27–30 (2008).

åå -==k

knk

kkk

nk e

knp

2

!)(

2)()( aa

qrq



åå -==k

knk

kkk

nk e

knp

2

!)(),(

2)()( aa

qarqa



å=k

kknknp )(),( )( arqa

11 ´´´ = FFNNp rqMFFNMNp ´´´ = rq

QDT – time-multiplexed detector




MFFNMNp ´´´ = rq


11 ´´´ = FFNNp rq 111

´´-´ = FNNF p rq

J. Renema, et. al. "Tomography and state reconstruction with superconducting single-photon detectors." Phys. Rev. A 86 062113 (2012).

QDT and state reconstruction

ConsProsKnown states requiredNo assumptions on

detector:Inversion problems1. Uniformity

2. Unknown non-lineareffects

Detector calibration

realmodeldet rq=p

Array detector - model

11 ´´´ = FFNNp rq

111

´´-´ = FNNF p rq

detection efficiency,

crosstalk,

X

X

dark counts,

finite size,

100 elements

x~

h

d

N

÷÷÷÷÷÷

ø

ö

çççççç

è

æ

=

!

!"

nP

PP

P1

0

real

÷÷÷÷÷÷÷÷

ø

ö

çççççççç

è

æ

=

N

s

P

P

PP

P

!

!"1

0

det

FN´q


)()1()1(1)1(

1~11~),,,~,(

00 00

0

2

nPmn

jjk

kN

ddkpkN

Npx

Npx

psp

nNxdP

nn

mnm

m

k

j

jkmm

N

k

pNkp

N

p

spps

s

N åå åå

å¥

=

-¥

= =

-

=

--

=

--

-÷÷ø

öççè

æ-÷÷

ø

öççè

æ÷÷ø

öççè

æ-÷÷

ø

öççè

æ--

÷÷ø

öççè

æ÷øö

çèæ --÷÷

ø

öççè

æ÷øö

çèæ -÷÷

ø

öççè

æ-

=

hh

h

LC, Y. Pilnyak, D. Istrati, N. M. Studer, J. P. Dowling, and H. S. Eisenberg, ”Absolute self-calibration of single-photon and multiplexed photon-number-resolving detectors,” PhysRevA.98.013811 (2018).

x~

h

dN

https://doi.org/10.1103/PhysRevA.98.013811


)()1()1(1)1(

1~11~),,,~,(

00 00

0

2

nPmn

jjk

kN

ddkpkN

Npx

Npx

psp

nNxdP

nn

mnm

m

k

j

jkmm

N

k

pNkp

N

p

spps

s

N åå åå

å¥

=

-¥

= =

-

=

--

=

--

-÷÷ø

öççè

æ-÷÷

ø

öççè

æ÷÷ø

öççè

æ-÷÷

ø

öççè

æ--

÷÷ø

öççè

æ÷øö

çèæ --÷÷

ø

öççè

æ÷øö

çèæ -÷÷

ø

öççè

æ-

=

hh

h x~

h

dN

SPD calibration

hd

)()1()1(1);,(

)()1()1();,(

01

00

nPdndp

nPdndp

nn

n

nn

n

å

å¥

=

¥

=

---=

--=

hh

hh

SPD calibration

01

2210

11

1

ppnn

dp

-=

-+-

=hhd

tntndppO

--+

==1

2221

0

1det

hh

BBOType IISPD

PBSH

TMSV

V

NDF

Variable transmission t?=h

hd

SPD calibration - results

0.0 0.2 0.4 0.6 0.8 1.00

0.5

1

1.5

2x10-3

Ode

t

Transmission

SPD1 SMSV SPD2 SMSV SPD1 TMSV SPD2 TMSV

x10-3

LC, Y. Pilnyak, D. Istrati, N. M. Studer, J. P. Dowling, and H. S. Eisenberg, ”Absolute self-calibration of single-photon and multiplexed photon-number-resolving detectors,” PhysRevA.98.013811 (2018).

dtntnd

ppO

--+

==1

2221

0

1det

hh

SPD# SV light Our method Klyshko’s method1 SMSV 11.3±1.1% 11.8±0.9%2 SMSV 7.4±0.9% 8.1±0.9% 1 TMSV 17.4±1.0% 17.3±0.8%2 TMSV 12.7±0.9% 11.7±0.8%

https://doi.org/10.1103/PhysRevA.98.013811

Detector calibration - PNRD

Oscilloscope signal

11

time

Signal height histogramvoltage


å÷÷÷

ø

ö

ççç

è

æ-

-÷÷ø

öççè

æ÷øö

çèæ -÷÷

ø

öççè

æ÷øö

çèæ --÷÷

ø

öççè

æ-÷÷

ø

öççè

æ-=

=

---

s

p

p

Nn

psspnN

s de

Npx

Npx

psp

pN

edP0

2

11

1~1~1)1(

h

h

0 0.15 0.3 0.45 0.6 0.75 0.90

5

10

15

20

Cou

nts

(kH

z)

Voltage (Volt)0 0.15 0.3 0.45 0.6 0.75 0.9

0.1

1

10

100

1000

10000

Cou

nts

(Hz)

Voltage (Volt)

state reconstruction - PNRD

111

´´-´ = FNNF p rq

1.06.7003.0048.0~

104.1

11

1~1~1)1(

5

0

2

±=±=

´»

÷÷÷

ø

ö

ççç

è

æ-

-÷÷ø

öççè

æ÷øö

çèæ -÷÷

ø

öççè

æ÷øö

çèæ --÷÷

ø

öççè

æ-÷÷

ø

öççè

æ-=

-

=

--- å

nxd

de

Npx

Npx

psp

pN

edPs

p

p

Nn

psspnN

s

h

h

h


019.0

15.07.72

111

=×

±== ´´

-´

np FNNF rq

QDT vs. detector modelingdetector modelingQDT

No assumptions on detector:1. Uniformity

2. Unknown non-lineareffectsKnown states required

4 parameters from experiment

Many parameters from experiment

Inversion problemInversion problems

reconstructionreconstruction

Summary- tools of quantum optics

Photon number detection

Quantum state tomography

Photon number detectors

Quantum light sources

Squeezed light 1-Fock state

state process and detector tomography

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